diff --git "a/FdAzT4oBgHgl3EQfUfww/content/tmp_files/2301.01266v1.pdf.txt" "b/FdAzT4oBgHgl3EQfUfww/content/tmp_files/2301.01266v1.pdf.txt" new file mode 100644--- /dev/null +++ "b/FdAzT4oBgHgl3EQfUfww/content/tmp_files/2301.01266v1.pdf.txt" @@ -0,0 +1,5838 @@ +HIGGS-COULOMB CORRESPONDENCE AND WALL-CROSSING IN ABELIAN GLSMS +KONSTANTIN ALESHKIN AND CHIU-CHU MELISSA LIU +Dedicated to the memory of Professor Bumsig Kim +Abstract. We compute I-functions and central charges for abelian GLSMs using virtual matrix factorizations of +Favero and Kim. In the Calabi-Yau case we provide analytic continuation for the central charges by explicit integral +formulas. The integrals in question are called hemisphere partition functions and we call the integral representation +Higgs-Coulomb correspondence. We then use it to prove GIT stability wall-crossing for central charges. +Contents +1. +Introduction +1 +2. +Geometry of gauged linear sigma models and A-model state spaces +2 +3. +Categories of B-branes and K-theories +10 +4. +The Higgs Branch +10 +5. +The Coulomb Branch +32 +Appendix A. +Convergence of multivariate hypergeometric functions +41 +References +46 +1. Introduction +2d gauged linear sigma models (GLSMs) were introduced by Witten in 1993. Following [FJR], the input data +of a GLSM is a 5-tuple (V, G, C∗ +R, W, ζ), where V is a finite dimensional complex vector space, G ⊂ GL(V ) is +a reductive linear group known as the gauged group, C∗ +R ∼= C∗ acts linearly on V and the action commutes +with the G-action, W : V → C is a G-invariant polynomial which is quasi-homogeneous with respect to the C∗ +R- +action, and ζ is a G-character with the property V ss +G (ζ) = V s +G(ζ), i.e., every ζ-semistable point is ζ-stable, so +that the GIT quotient stack Xζ = [V//ζG] = [V ss +G (ζ)/G] is an orbifold (i.e. smooth DM stack with trivial generic +stabilizer). The space of stability conditions is ˆG ⊗Z R ∼= RdimC Z(G), where ˆG = Hom(G, C∗) is the group of G +characters and Z(G) is the center of G; it is decomposed into chambers called phases. The G-invariant polynomial +W descends to wζ : Xζ → C; the pair (Xζ, wζ) is a Landau-Ginzburg (LG) model, where wζ is known as the +superpotential. GLSM invariants of (V, G, C∗ +R, W, ζ) are, roughly speaking, virtual counts of curves in the critical +locus Zζ := Crit(wζ) = [(Crit(W) ∩ V ss +G (ζ)) /G] which is often assumed to be compact/proper but can be singular. +Different phases of a GLSM (that is GLSMs which differ only by the choice of a stability parameter) are closely +related to each other. For example, let G = C∗ act on V = C6 by weights (1, 1, 1, 1, 1, −5) with W = pW5 where +W5 = x5 +1 + · · · + x5 +5 is the Fermat quintic polynomial in 5 variables. In the CY/geometric phase ζ > 0, Xζ = KP4, +Zζ = X5 := {W5 = 0} ⊂ P4 is the Fermat quintic threefold, and GLSM invariants are (up to sign) Gromov-Witten +(GW) invariants of X5. +In the LG phase ζ < 0, Xζ ∼= [C5/µ5], where µ5 is the group of 5-th roots of unity +acts diagonally on C5, Zζ is supported at the origin, and GLSM invariants are Fan-Jarvis-Ruan-Witten (FJRW) +invariants of the affine LG model ([C5/µ5], W5). Chiodo-Ruan [CR] proved genus-zero LG/CY correspondence +for quintic threefolds relating GW invariants of X5 and FJRW invariants of ([C5/µ5], W5). Their proof can be +summarized into two steps. +(1) (ϵ-wall-crossing) In the CY (resp. LG) phase, the Givental-style mirror theorem says the J-function which +governs the genus-zero GW (resp. FJRW) is related to the I-function, which can be expressed in terms of +explicit hypergeometric series, by explicit change of variables known as the mirror map. +(2) (ζ-wall-crossing) The I-function admits a Mellin-Barnes integral representation. I-functions in the two +phases are related by analytic continuation given by deforming the contour of integration in C. +The interpretation of Step (1) as ϵ-wall-crossing appeared in later work. For each ϵ ∈ Q>0, Ciocan-Fontanine–Kim– +Maulik [CKM] introduced ϵ-stable quasimaps to certain GIT quotient W//G. Ciocan-Fontanine and Kim [CK] +1 +arXiv:2301.01266v1 [math.AG] 3 Jan 2023 + +introduced Jϵ which is a generating function of invariants defined by genus-zero ϵ-stable quasimaps; Jϵ specializes +to the I-function and the J-function as ϵ → 0+ and ϵ → +∞, respectively. In the presence of a good torus action, +they proved ϵ-wall-crossing which relates Jϵ to the J-function J = J∞ for any ϵ ∈ Q>0, by change of variables. +They computed the I-function I = J0+ explicitly. In particular, they recover the mirror theorem in the geometric +phase first proved by Givental [Gi96] and Lian-Liu-Yau [LLY]. The mirror theorem in the LG phase was first proved +by Chiodo-Ruan [CR] and later reproved by Ross-Ruan [RR] via ϵ-wall-crossing. +For a general GLSM, GLSM invariants are defined by integrating against virtual cycles on moduli of ϵ-stable LG +quasimaps. The virtual cycle is constructed for narrow sectors by Fan-Jarvis-Ruan [FJR] via cosection localization, +and for both narrow and broad sectors by Favero-Kim [FK] via matrix factorization; Favero-Kim’s construction +generalizes previous constructions for affine LG models [PV16] and for convex hybrid models [CFGKS]. When +ϵ > 0, the definition relies on a good lift ˜ζ which is a character of the group Γ ⊂ GL(V ) generated by G and C∗ +R, +such that V ss +Γ (˜ζ) = V ss +G (ζ). Such a good lift does not always exist. At ϵ = 0+, a good lift is not needed. In this +paper we focus on the ϵ → 0+ stability condition and study genus-zero GLSM invariants and ζ-wall-crossing for +abelian GLSMs where G = (C∗)κ. In this case, Xζ is a smooth toric DM stack. Let �T-be the diagonal subgroup +of GL(V ). Using the work of Favero-Kim [FK], we define and compute K-theoretic GLSM I-function IK +w which +takes values in the K-theory of category of matrix factorizations on the inertial stack IXζ, and the (cohomological) +GLSM I-function Iw which takes values in the GLSM state space Hw. We also define and compute K-theorectic +�T-equivariant I-function IK +� +T of (V, G, C∗ +R, 0, ζ) which takes values in K � +T (IXζ), and �T-equivariant I-function I � +T of +(V, G, C∗ +R, 0, ζ) which takes values in H � +T (IXζ). +In Gromov-Witten theory I-functions fail to capture integral structure on cohomology [Ir09]. Hosono defined an +object called a central charge that sees the Gamma integral structure. Integral structures are crucial for integral +representations. Central charges are power series which are constructed from both J-function and objects of the +derived category of coherent sheaves of the target manifold. Hori and Romo [HR] constructed explicit analytic +functions called hemisphere partition functions and conjectured that their power series expansions are equal to +the central charges in appropriate cases. +We define the GLSM central charge of a matrix factorization B of +(Xζ, wζ) as Zw(B) = ⟨Iw, Γwchw([B])⟩ where Γw is an appropriate version of Iritani’s Γ-class and [B] is the K- +theory class of B. We also define the �T-equivariant central charge of a �T-equivariant perfect complex B on Xζ as +Z � +T (B) = ⟨I � +T , Γ � +T ch � +T ([B])⟩. We show that our central charges indeed have integral representations of the hemisphere +partition function form (Theorem 5.6). +We call this representation Higgs-Coulomb correspondence because in +physics GLSM central charges can be obtained by a version of the Higgs branch localization and hemisphere +partition functions are computed by the Coulomb branch localization (c.f. [BC]). +The integral representations +we obtain depend continuously on the complexified stability parameter θ = ζ + 2π√−1B and do not have any +restrictions on ζ. +In this philosophy ζ-wall-crossing follows immediately by analytic continuation in ζ (Theorem 5.21). Remarkably, +matrix factorizations of the central charges in question are related by the Fourier-Mukai transform [BP10]. Let ζ± +represent stability conditions in two adjacent chambers and B+ be a matrix factorizations in the phase corresponding +to ζ+. Then, the analytic continuation of the central charge of B+ is a central charge of B− = FM(B+). Particular +Fourier-Mukai kernel is choosen by B = Im(θ)/2π and convergence of the integral representation is equivalent to +the so-called Grade Restriction Rule [HL,BFK,CIJS]. +(1.1) +B +B+ +B− +π+ +π− +F M +Acknowledgements. We wish to thank Daniel Halpern-Leistner, Kentaro Hori, Hiroshi Iritani, Andrei Okounkov, +Tudor P˘adurariu, Renata Picciotto, Alexander Polishchuk, Che Shen, Yefeng Shen, Mark Shoemaker, and Yang +Zhou for helpful communications. We thank the hospitality and support of the Simons Center for Geometry and +Physics (SCGP) during the program Integrability, Enumerative Geometry and Quantization (August 22-September +23, 2022) where part of the paper was completed. The authors are partially supported by NSF grant DMS-1564497. +2. Geometry of gauged linear sigma models and A-model state spaces +In this paper, all the schemes and algebraic stacks are defined over Spec C, where C is the field of complex +numbers. +2 + +2.1. Gauged linear sigma models. We start with the setup of a general gauged linear sigma model (GLSM) +following [FJR,FK]. Part of our formulation in Section 4 is closer to that in the more general setting in [CJR]. +The input data of a GLSM is a 5-tuple (V, G, C∗ +R, W, ζ), where +(1) (linear space) V = SpecC[x1, . . . , xn+κ] ≃ Cn+κ is a finite dimensional complex vector space, where κ = +dim G. +(2) (gauge group) G ⊂ GL(V ) is a reductive algebraic group. +(3) (vector R-symmetry) C∗ +R ∼= C∗ acts linearly and faithfully on V , so we may view C∗ +R as a subgroup of +GL(V ). We assume that +(a) the intersection G ∩ C∗ +R is finite, and +(b) the C∗ +R-action commutes with the G-action. +The finite group G ∩ C∗ +R must be cyclic, generated by an element J of finite order r ∈ Z>0, given explicitly +in Equation (2.2) below. The surjective group homomorphism C∗ +R → C∗ +ω := C∗/⟨J⟩ is a degree r covering +map. Let Γ := GC∗ +R ⊂ GL(V ) be the subgroup generated by G and C∗ +R. Then we have a short exact +sequence of groups: +(2.1) +1 → G → Γ +χ→ C∗ +ω → 1. +By (b), the Γ-action on V induces a C∗ +ω-action on the smooth Artin stack [V/G] in the sense of [Ro]. The +R-charges are +(q1, . . . , qn+κ) = +�2c1 +r , . . . , 2cn+κ +r +� +, +where c1, . . . , cn+κ ∈ Z are the weights of the C∗ +R-action on V ≃ Cn+κ. (Note that gcd(c1, . . . , cn+κ) = 1 +since C∗ +R acts faithfully on V .) Let +(2.2) +J = (e2π√−1c1/r, . . . , e2π√−1cn+κ/r). +(4) (superpotential) W : V → C is a G-invariant regular function which is a quasi-homogeneous polynomial of +degree r with respect to the C∗ +R-action on V ; in other words, W ∈ C[x1, . . . , xn+κ]G and +W(tc1x1, . . . , tcn+κxn+κ) = trW(x1, . . . , xn+κ), +t ∈ C∗ +R, +(x1, . . . , xn+κ) ∈ V. +It descends to a function w : [V/G] → C of degree 1 with respect to the C∗ +ω-action on [V/G]. +(5) (stability condition) ζ ∈ Hom(G, C∗) = Hom(Gab, C∗), where Gab = G/[G, G] is the abelianization of G. +We view Hom(G, C∗) as an additive group and let χζ : G → C∗ denote the associated G-character. Let +V ss +G (ζ) (respectively V s +G(ζ)) be the set of semistable (respectively stable) points in V determined by the +G-linearization on the trivial line bundle V × C → V given by g · (v, t) = (g · p, χζ(g)t). We assume that +V s +G(ζ) = V ss +G (ζ). Then the quotient stack Xζ := [V ss +G (ζ)/G] is an orbifold (i.e. a smooth Deligne-Mumford +stack with trivial generic stabilizer) of dimension n. The GIT quotient Xζ := V �ζ G = V ss +G (ζ)/G is the +coarse moduli space of Xζ. +Let Zζ := [(Crit(W) ∩ V ss +G (ζ))/G] be the critical locus of wζ := w|Xζ : Xζ → C. We say ζ is in a geometric phase +if Crit(W) ∩ V ss(ζ) is non-singular, which implies Zζ is an orbifold. +The central charge of the GLSM is (cf. [FJR, Definition 3.2.3]) +(2.3) +ˆc := dim V − dim G − 2ˆq = n − 2ˆq, +where ˆq = 1 +2 +n+κ +� +j=1 +qj = 1 +r +n+κ +� +j=1 +cj. +Remark 2.1. By the condition (4) above, the superpotential W is semi-invariant with respect to Γ in the sense +of [PV11, Section 2], i.e., +(2.4) +W(γ · x) = χ(γ)W(x) +for any γ ∈ Γ and x ∈ V. +Remark 2.2. In this paper the stability condition ζ is an element in Hom(G, C∗) and corresponds to the symbol θ +in [FJR,FK]. The 1-dimensional torus C∗ +ω in this paper corresponds to C∗ +ω in [CJR]. +Remark 2.3. At this point, we do not assume the critical locus Zζ is proper. This allows us to include the case +without superpotential as a special case where the superpotential and the R-charges are zero, i.e. W = 0 and qj = 0 +for j = 1, . . . , n + κ; note that in this special case the C∗ +R-action on V is trivial, and in particular, not faithful. +3 + +2.2. Abelian GLSMs. In the rest of this paper, we consider abelian GLSMs where the gauge group G = (C∗)κ is +a complex algebraic torus. In this case [G, G] = {1} and Gab = G. +Up to an inner automorphism of GL(V ), we may assume the image of ρV : G → GL(V ) is contained in the +diagonal torus �T ≃ (C∗)n+κ ⊂ GL(V ) ∼= GLn+κ(C). Then Xζ is an n-dimensional toric orbifold, and its coarse +moduli Xζ is a semi-projective simplicial toric variety which contains T := �T/G ∼= (C∗)n as a Zariski dense open +subset. We have a short exact sequence of abelian groups +(2.5) +1 → G +ρV +−→ �T −→ T → 1. +Remark 2.4. The notation in this subsection is similar to but slightly different from that in [CIJ]: G, �T, and T +in this paper correspond to K, T, and Q in [CIJ, Section 4.3], respectively; κ and n in this paper correspond to r +and m − r in [CIJ, Section 4.1], respectively. +• Applying Hom(C∗, −) to (4.38), we obtain the following short exact sequence of cocharater lattices +(2.6) +0 → L := Hom(C∗, G) −→ � +N := Hom(C∗, �T) −→ N := Hom(C∗, T) → 0, +where L ∼= Zκ, � +N ∼= Zn+κ, and N ∼= Zn. +• Applying Hom(−, C∗) to (4.38), or equivalently dualizing (2.6), we obtain the following short exact sequence +of character lattices: +(2.7) +0 → M := Hom(T, C∗) −→ � +M := Hom( �T, C∗) −→ L∨ := Hom(G, C∗) → 0 +The map L → � +N ∼= Zn+κ is given by (D1, . . . , Dn+κ) where Di ∈ L∨. The stability condition ζ is an element +in Hom(G, C∗) = L∨. If we choose a Z-basis {ξ1, . . . , ξκ} of L (which is equivalent to a choice of an isomorphism +G ≃ (C∗)κ) and let {ξ∗ +1, . . . , ξ∗ +κ} be the dual Z-basis of L∨, then +Di = +κ +� +a=1 +Qa +i ξ∗ +a +for some Qa +i ∈ Z. Given any t = �κ +a=1 taξ∗ +a ∈ L∨, where t1, . . . , tκ ∈ Z, let χt : G → C∗ be the corresponding G +character given by +(2.8) +χt(s1, . . . , sκ) = +κ +� +a=1 +sta +a . +Then the map G ≃ (C∗)κ → �T ∼= (C∗)n+κ is given by +(2.9) +s = (s1, . . . , sκ) ∈ G �→ +� +χD1(s), . . . , χDn+κ(s) +� +∈ �T, +where χDi(s) = +κ +� +a=1 +sQa +i +a . +Given a lattice Λ ∼= Zr and a field F, we define ΛF := Λ ⊗Z F ∼= Fr; in this paper, F = Q, R, or C. +Remark 2.5. The map G → �T is injective iff D1, . . . , Dn+κ generate the lattice L∨ over Z. In Section 5 (The +Coulomb Branch), we work with the weaker assumption that D1, . . . , Dn+κ span the vector space L∨ +Q over Q, or +equivalently, the kernel K of the group homomorphism G → �T is finite. Then Xζ is a smooth toric DM stack with +a generic stabilizer K; it is a toric orbifold iff K is trivial. It is also possible to work in this generality in Section +4 (The Higgs branch). We assume K is trivial in Section 4 mainly because [FJR] and [FK] assume so, but K can +be non-trivial in orbifold quasimap theory [CCK] which can be viewed as a mathematical theory of GLSM without +superpotential. +Let GR ≃ U(1)κ be the maximal compact subgroup of G ≃ (C∗)κ. Then LR is canonically isomorphic to the Lie +algebra gR of GR. The G-action on V = SpecC[x1, . . . , xn+κ] restricts to a Hamiltonian GR-action on the K¨ahler +manifold (V, +√−1 +2 +�n+κ +i=1 dxi ∧ d¯xi) with a moment map +µ : V −→ g∨ +R = L∨ +R, +(x1, . . . , xn+κ) = 1 +2 +κ +� +a=1 +Qa +i |xi|2ξ∗ +a. +Then +Xζ = [V ss +G (ζ)/G] = [µ−1(ζ)/GR]. +From this perspective, the stability condition ζ is a regular value of the moment map µ, and can be an element in +L∨ +R ∼= Rκ. +4 + +2.3. Anticones and the extended stacky fan. The triple (V, G, ζ), which is part of the input data +(V, G, C∗ +R, W, ζ) of the given abelian GLSM, determines a set Aζ of anticones and an extended stacky fan +Σζ = (N, Σζ, β, Sζ). We describe them in this subsection, and describe V ss +G (ζ) ⊂ V in terms of anticones. +We fix an isomorphism V = SpecC[x1, . . . , xn+κ] ∼= Cn+κ, which determines an ordered Z-basis (�e1, . . . , �en+κ) +of � +N. In particular, +� +N = +n+κ +� +i=1 +Z�ei. +Let vi ∈ N be the image of ei under � +N → N. Define β = (v1, . . . , vn+κ). +Given a subset I of {1, . . . , n + κ}, let I′ = {1, . . . , n + κ} \ I be its complement, and define +∠I = { +� +i∈I +aiDi : ai ∈ R, ai > 0} ⊂ L∨ +R, +σI = { +� +i∈I +aivi : ai ∈ R, ai ≥ 0} ⊂ NR. +If I = ∅ is the empty set, define σ∅ = {0}. +For a fixed G-action on V , a stability condition ζ ∈ L∨ +R determines the following three sets. +Aζ += +{I ⊂ {1, . . . , n + κ} : ζ ∈ ∠I}, +Σζ += +{σI : I′ ∈ Aζ}, +Sζ += +{i ∈ {1, . . . , n + κ} : σ{i} /∈ Σζ} = {i ∈ {1, . . . , n + κ} : {i}′ /∈ Aζ}. +Note that Sζ ⊂ I for any I ∈ Aζ. +Assumption 2.6. We choose the stability condition ζ ∈ L∨ +R such that the following three equivalent conditions are +satisfied. +(i) For any I ∈ Aζ, {Di : i ∈ I} spans L∨ +Q as a vector space over Q. +(ii) For any I ∈ Aζ, {vi : i ∈ I′} is a set of linearly independent vectors in NQ, or equivalently, σI′ is a +simplicial cone in NR. +(iii) Σζ is a simplicial fan in NR. +Elements in Σζ are called cones, while elements in Aζ are called anticones. By (i), the cardinality |I| of any +anticone I ∈ Aζ is greater or equal to κ. Let Σζ(d) be the set of d-dimensional cones in Σζ. Then σ ∈ Σζ(d) iff +σ = σI where |I| = d and I′ ∈ Aζ. Let +Amin +ζ += {I ∈ Aζ : |I| = κ} = {I ∈ Aζ : σI′ ∈ Σζ(n)} +be the set of minimal anti-cones. +The irrelevant ideal of ζ is the ideal Bζ in C[x] := C[x1, . . . , xn+κ] generated by {xI := � +i∈I xi : I ∈ Aζ}. Let +Zζ = Z(Bζ) be the closed subvariety of V = SpecC[x] defined by the irrelevant ideal Bζ ⊂ C[x], and let Uζ = V \Zζ. +If ζ ∈ L∨ is a G character, then Uζ = V ss +G (ζ), and Zζ = V un +G (ζ) is the set of unstable points defined by ζ. +For any I ∈ Aζ, define +(2.10) +VI = V \ Z(xI) = {(x1, . . . , xn+κ) ∈ V : xi ̸= 0 if i ∈ I} = (C∗)I × CI′. +Then +Uζ = +� +I∈Aζ +VI. +Note that if I, J ∈ Aζ and I ⊂ J then VJ ⊂ VI. Therefore, +Uζ = +� +I∈Amin +ζ +VI. +We now give an alternative description of Zζ. If I ⊂ {1, . . . , n + κ}, and I /∈ Aζ, or equivalently σI′ /∈ Σζ, define +ZI = CI × {0}I′ = {(x1, . . . , xn+κ) ∈ V : xi = 0 if i ∈ I′}. +Then +Zζ = +� +I /∈Aζ +ZI. +Define +Cζ := +� +I∈Aζ +∠I = +� +I∈Amin +ζ +∠I ⊂ L∨ +R. +The open cone Cζ is called the extended ample cone in [CIJ]. It is a chamber in the space of stability conditions: if +ζ′ ∈ Cζ then Σζ′ = Σζ and Uζ′ = Uζ. We recall the following facts. +5 + +(1) The quotient stack +Xζ = [Uζ/G] +is the smooth toric Deligne-Mumford (DM) stack defined by the stacky fan (N, Σζ, β). See Borisov-Chen- +Smith [BCS] for definition of toric Deligne-Mumford stacks in terms of stacky fans. +(2) The coarse moduli space of Xζ is the categorical (and geometric) quotient +Xζ = Uζ/G +which is the toric variety defined by the simplicial fan Σζ ⊂ NR. See [Fu93, CLS] for an introduction of +toric varieties, and in particular the definition of general normal toric varieties in terms of fans. +(3) If ζ ∈ L∨ is a G-character then +Xζ = [V ss +G (ζ)/G] +is the GIT quotient stack, and +Xζ = V ss +G (ζ)/G = V �ζ G +is the GIT quotient. +(4) The triple (V, G, ζ) determines a particular presentation of Xζ as a quotient stack [Uζ/G] and an extended +stacky fan +Σζ = (N, Σζ, β, Sζ), +a notion introduced by Jiang [Ji08]. +2.4. Closed toric substacks and their generic stabilizers. The �T-divisor �Dj = {xj = 0} ⊂ V = SpecC[x] +restricts to a �T-divisor �Dj∩Uζ ⊂ Uζ which descends to a T-divisor Dj = [( �Dj∩Uζ)/G] in the toric stack Xζ = [Uζ/G] +and a T-divisor Dj in the toric variety Xζ. Note that Dj and Dj are empty if j ∈ Sζ. +Given any σ ∈ Σζ(d), where 0 ≤ d ≤ n, we have σ = σI′ for some I ∈ Aζ with |I| = κ + n − d. Let +V(σ) = +� +i∈I′ +Di ⊂ Xζ, +V (σ) = +� +i∈I′ +Di ⊂ Xζ. +Then V(σ) (resp. V (σ)) is an (n − d)-dimensional closed toric substack (resp. subvariety) of Xζ (resp. Xζ). The +generic stabilizer of the toric stack V(σ) is the finite group +Gσ = +� +i∈I +Ker(χDi) ⊂ G +where χDi : G → C∗ is defined as in (2.9). If τ, σ ∈ Σζ and τ ⊂ σ then V(τ) ⊃ V(σ), so Gτ ⊂ Gσ. In particular, +V({0}) = Xζ and G{0} is trivial. If I ∈ Amin +ζ +, then σI′ ∈ Σζ(n) and pI := V(σI′) ≃ [•/GσI′ ] = BGσI′ is the unique +T-fixed point in +XI := [VI/G] ≃ +�� +(C∗)I × CI′� +/G +� +≃ [CI′/GσI′ ] ≃ [Cn/GσI′ ]. +Here • = SpecC is a point, BGσI′ is the classifying space of GσI′ , and VI is defined by Equation (2.10). +2.5. Line bundles. Let +U T +j := OXζ(−Dj) ∈ PicT (Xζ), +uT +j := −(c1)T (U T +j ) ∈ H2 +T (Xζ; Q). +Then uT +j is the T-equivariant Poincar´e dual of Dj. Note that uT +j = 0 if j ∈ Sζ. The T-equivariant Chern character +of U T +j is chT (U T +j ) = e−uT +j . +The T-equivariant line bundles U T +j generate KT (Xζ) as an algebra over Z. Any group homomorphism A → T +induces a map [Xζ/A] → [Xζ/T] = [V ss +G (ζ)/ �T] and ring homomorphisms +KT (Xζ) → KA(Xζ), +φ∗ : H∗ +T (Xζ; Q) → H∗ +A(Xζ; Q). +Let Uj ∈ Pic(Xζ) (resp. uj ∈ H2(Xζ; Q)) be the image of U T +j +∈ PicT (Xζ) (resp. uT +j ∈ H2 +T (Xζ; Q)) under +the surjective group homomorphism PicT (Xζ) → Pic(Xζ) (resp. H2 +T (Xζ; Q) → H2(Xζ; Q)) induced by the group +homomorphism {1} → T. Then c1(Uj) = −uj and ch(Uj) = e−uj. +Let U � +T +j ∈ Pic � +T (Xζ) (resp. u � +T +j ∈ H2 +� +T (Xζ; Q)) be the image of U T +j ∈ PicT (Xζ) (resp. uT +j ∈ H2 +T (Xζ; Q)) under the +group homomorphism PicT (Xζ) → Pic � +T (Xζ) (resp. H2 +T (Xζ; Q) → H2 +� +T (Xζ; Q)) induced by the group homomorphism +�T → T = �T/G. Then (c1) � +T (U � +T +j ) = −u � +T +j and ch � +T (U � +T +j ) = e−u +� +T +j . +For i = 1, . . . , n + κ, let χDi : G → C∗ be defined as in (2.9). For a = 1, . . . , κ, let χξ∗ +a : G → C∗ be the character +associated to ξ∗ +a ∈ L∨, i.e., χξ∗ +a(s1, . . . , sκ) = sa. Let G act on Uζ × C by +s · (x1, . . . , xn+κ, y) = (χD1(s)x1, . . . , χDn+κxκ, χξ∗ +a(s−1)y), +6 + +This defines a G-equivariant line bundle on Uζ, or equivalently a line bundle Pa on Xζ = [Uζ/G]. Let pa = −c1(Pa) ∈ +H2(Xζ; Q). Then ch(Pa) = e−pa. For j = 1, . . . , n + κ, we have +(2.11) +Uj = +κ +� +a=1 +P +Qa +j +a +∈ Pic(Xζ), +uj = +κ +� +a=1 +Qa +j pa ∈ H2(Xζ; Q). +Let Λj ∈ Pic � +T (•) = Pic(B �T) be the �T-equivariant line bundle over a point • defined by the �T-character �tj, and +let λj = −(c1) � +T (Λj) ∈ H2 +� +T (•; Z) = H2(B �T; Z). Then +K � +T (•) = Z[Λ±1 +1 , . . . , Λ±1 +n+κ], +H∗(B �T; Z) = Z[λ1, . . . , λn+κ]. +For later convenience, we introduce the following definition. +Definition 2.7. Let I ∈ Amin +ζ +be a minimal anticone. Then {Di : i ∈ I} is a Q-basis of L∨ +Q. Let {D∗,I +i +: i ∈ I} be +the dual Q-basis of LQ, i.e., for any i, j ∈ I, +⟨Di, D∗,I +j ⟩ = δij +where ⟨ , ⟩ : L∨ +Q × LQ → Q is the natural pairing between dual vector spaces over Q. +Given any I ∈ Amin +ζ +, the inclusion ιI : pI �→ Xζ of the torus fixed point pI induces a ring homomorphism +ι∗ +I : H∗ +� +T (Xζ; Q) −→ H∗ +� +T (pI; Q) ≃ H∗(B �T; Q) = Q[λ1, . . . , λn+κ]. +For i = 1, . . . , n + κ, +(2.12) +ι∗ +Iu +� +T +i = +� +j∈I +⟨Di, D∗,I +j ⟩λj − λi +∀I ∈ Amin +ζ +. +In particular, if i ∈ I then ι∗ +Iu � +T +i = 0, which is consistent with the fact that the T-fixed point pI is not contained in +the divisor Di. +Let �T × G act on Uζ × C by +(�t, s) · (x1, . . . , xn+κ, y) = (�t1χD1(s)x1, . . . , �tn+κχDn+κxκ, χξ∗ +a(s−1)y), +where �t = (�t1, . . . , �tn+κ) ∈ �T and s ∈ G. This defines a �T × G-equivariant line bundle on Uζ, or equivalently a +�T-equivariant line bundle P � +T +a on Xζ = [Uζ/G]. Let p � +T +a = −c1(P � +T +a ) ∈ H2 +� +T (Xζ; Q). Then ch � +T (P � +T +a ) = e−p +� +T +a . For +a = 1 . . . , κ, +(2.13) +ι∗ +Ip +� +T +a = +� +j∈I +⟨ξ∗ +a, D∗,I +j ⟩λj +∀I ∈ Amin +ζ +. +We have +(2.14) +U +� +T +j = +κ +� +a=1 +(P +� +T +a )Qa +j · Λ−1 +j +∈ Pic � +T (Xζ), +u +� +T +j = +κ +� +a=1 +Qa +j p +� +T +a − λj ∈ H2 +� +T (Xζ; Q). +Under K � +T (Xζ) −→ K(Xζ), U � +T +j , P � +T +a , and Λj are mapped to Uj, Pa, and 1, respectively; under H2 +� +T (Xζ; Q) → +H2(Xζ; Q), u � +T +j , p � +T +a , and λj are mapped to uj, pa, and 0, respectively. Our definitions of u � +T +j , U � +T +j , p � +T +a , P � +T +a , λj, Λj +are consistent with the convention in Givental’s papers [GiV,GiVI] on permutation-equivariant quantum K-theory +of toric manifolds. +2.6. The inertia stack. +2.6.1. The inertia stack of a general algebraic stack. Given a general algebraic stack X, the inertia stack IX of X +is the fiber product +IX +� +� +X +∆ +� +X +∆ � +∆ � X × X +where ∆ : X → X ×X is the diagonal morphism. IX is an algebraic stack, and in particular a groupoid. An object +in the groupoid IX is a pair (x, g) where x is an object in the groupoid X and k is an element in the automorphism +group AutX (x) = HomX (x, x) of x. Morphisms between two objects in IX are +HomIX ((x1, g1), (x2, g2)) = {h ∈ HomX (x1, x2) : h ◦ g1 = g2 ◦ h}. +7 + +The map (x, g) �→ (x, g−1), where (x, g) is an object in IX, defines an involution +inv : IX → IX. +2.6.2. The inertia stack of a quotient stack. If X = [U/G] is a quotient stack, where U is a scheme and G is an +algebraic group, then the inertia stack is also a quotient stack: +IX = [IU/G] +where IU := {(x, g) ∈ U × G | g · x = x} is a closed subscheme of U × G, and the G-action on IU is given by +h · (x, g) = (h · x, hgh−1), +where h ∈ G and (x, g) ∈ IU. +In particular, if G is abelian then the action is given by h · (x, g) = (h · x, g). +The involution inv : IX → IX is induced by the G-equivariant involution IU → IU given by (x, g) �→ (x, g−1). +2.6.3. The inertial stack of the toric orbifold Xζ. Let Xζ = [Uζ/G] be as in Section 2.3. To describe its inertia stack +IXζ more explicitly, we introduce some definitions. Given σ = σI ∈ Σζ, where I ⊂ {1, . . . , n + κ}, define +(2.15) +Box(σ) := +� +v ∈ N : v = +� +i∈I +civi, 0 ≤ ci < 1 +� +and +(2.16) +Box′(σ) := +� +v ∈ N : v = +� +i∈I +civi, 0 < ci < 1 +� +. +Define +(2.17) +Box(Σζ) := +� +σ∈Σζ +Box(σ) = +� +σ∈Σζ(n) +Box(σ). +which is a finite set. For any v ∈ Box(Σζ) there exists a unique σ ∈ Σζ such that v ∈ Box′(σ). Therefore, +(2.18) +Box(Σζ) = +� +σ∈Σζ +Box′(σ), +where the right hand side of (2.18) is a disjoint union. Given any σ = σI ∈ Σζ, where I′ ∈ Aζ, define +(2.19) +age(v) = +� +i∈I +ci ∈ Q +if v = +� +i∈I +civi ∈ Box′(σ). +Suppose that σ = σI ∈ Σζ where I′ ∈ Aζ. There is a bijection Box(σ) −→ Gσ given by +(2.20) +v = +� +i∈I +civi �→ g(v) = (a1, · · · , an+κ) ∈ Gσ ⊂ G ⊂ �T ≃ (C∗)n+κ where ai = +� +e2π√−1ci, +i ∈ I, +1, +i ∈ I′. +The map v �→ g(v) defines a bijection +Box(Σζ) = +� +σ∈Σζ +Box(σ) −→ +� +σ∈Σζ +Gσ = {g ∈ G : (Uζ)g is non-empty.}, +where (Uζ)g = {x ∈ Uζ : g · x = x}. We have +(2.21) +IUζ = {(x, g) ∈ Uζ × G : g · x = x} = +� +v∈Box(Σζ) +(Uζ)g(v) × {g(v)}. +The above union is a disjoint union of connected components. +IXζ = [IUζ/G] = +� +v∈Box(Σζ) +Xζ,v +where Xζ,v ≃ [(Uζ)g(v)/G]. +In particular, g(0) = 1 is the identity element of G and Xζ,0 ≃ Xζ. There is an involution inv : Box(Σζ) → Box(Σζ) +characterized by g(inv(v)) = g(v)−1 ∈ G. The involution inv : IXζ −→ IXζ maps Xζ,v isomorphically to Xζ,inv(v). +Observe that +age(v) + age(inv(v)) + dim Xζ,v = n = dim Xζ. +Let +(2.22) +η = (eπ√−1q1, . . . , eπ√−1qn+κ) ∈ C∗ +R. +8 + +Then η2 = J = (e2π√−1q1, . . . , e2π√−1qn+κ) and W(η · x) = −W(x). Let invR : IXζ → IXζ be the map induced by +the map IUζ → IUζ given by (x, g) �→ (η · x, g−1). Then invR maps Xζ,v isomorphically to Xζ,inv(v). Note that in +general invR ◦ invR is not the identity map, i.e., invR is not an involution. +2.7. A-model GLSM state spaces. Let M be a large positive number such that the real part of any critical +values of W is less than M, and define w∞ +ζ := (ReW)−1(M, ∞) ⊂ Xζ. As a graded vector space over Q, the rational +GLSM state space of (V, G, C∗ +R, W, ζ) is +(2.23) +Hw,Q = +� +v∈Box(Σζ) +H∗(Xζ,v, w∞ +ζ,v; Q)[2 (age(v) − ˆq)]. +where wζ,v = wζ +�� +Xζ,v. In particular, Hw,Q ∼= H∗(IXζ; Q) as a vector space over Q. Note that invR maps (Xζ,v, wζ,v) +diffeomorphically to (Xζ,inv(v), −wζ,inv(v)) and induces an isomorphism +(2.24) +inv∗ +R : H∗(Xζ,inv(v), −w∞ +ζ,inv(v); Q) +∼ += +−→ H∗(Xζ,v, w∞ +ζ,v; Q). +By [FK, Corollary 4.3], Crit(wζ,v) ⊂ Zζ = Crit(wζ). If Zζ is proper then there is a nondegenerate pairing +(2.25) +H∗(Xζ,v, w∞ +ζ,v; Q) × H∗(Xζ,v, −w∞ +ζ,v; Q) → Q. +for all v ∈ Box(Σζ). Combining (2.24) and (2.25), we obtain a nondegenerate pairing +H∗(Xζ,v, w∞ +ζ,v; Q) × H∗(Xζ,inv(v), w∞ +ζ,inv(v); Q) → Q +for all v ∈ Box(Σζ), thus a non-degenerate pairing ( , )w : Hw,Q × Hw,Q → Q. +As a graded vector space over C, the GLSM state space of (V, G, C∗ +R, W, ζ) is +Hw = +� +v∈Box(Σζ) +H∗ � +Xζ,v, (Ω• +Xζ,v, dwζ,v) +� +[2(age(v) − ˆq)] +where +H∗ � +Xζ,v, (Ω• +Xζ,v, dwζ,v) +� +≃ H∗(Xζ,v, w∞ +ζ,v; C). +Therefore, Hw ≃ Hw,Q ⊗Q C as a graded vector space over C. +We say v ∈ Box(Σζ) is narrow if Xζ,v is compact. If v is narrow then wζ,v is constant and w∞ +ζ,v is empty, so +H∗(Xζ,v, w∞ +ζ,v; C) = H∗(Xζ,v; C). +In this case, we call the above vector space a narrow sector. We say v is broad if v is not narrow, and call +H∗(Xζ,v, w∞ +ζ,v; C) +a broad sector. +We also consider a closely related GLSM (V, G, C∗ +R, 0, ζ) obtained by replacing the superpotental W by zero. As +a graded vector space over C, the GLSM state space of (V, G, C∗ +R, 0, ζ) is +(2.26) +H = +� +v∈Box(Σζ) +H∗(Xζ,v; C)[2 (age(v) − ˆq)]. +Note that inv and invR are homotopic as maps from Xζ,v to Xζ,inv(v), so inv∗ +R = inv∗ : H∗(Xζ,inv(v); C) → +H∗(Xζ,v; C). +The action of �T on V commutes with the action of its subgroup G and C∗ +R, and preserves the zero superpotential +0 and the semistable locus V ss +G (ζ). We define the �T-equivariant state space H � +T of (V, G, C∗ +R, 0, ζ) as follows. As a +graded vector space over C, +(2.27) +H � +T = +� +v∈Box(Σζ) +H∗ +� +T (Xζ,v; C)[2 (age(v) − ˆq)] +where each H∗ +� +T (Xζ,v; C) is a module over H∗ +� +T (•; C) = H∗(B �T; C) = C[λ1, . . . , λn+κ]. Let C(λ) := C(λ1, . . . , λ) be +the fractional field of C[λ] := C[λ1, . . . , λn+κ]. There is a non-degenerate pairing +H � +T ⊗C[λ] C(λ) × H � +T ⊗C[λ] C(λ) → C(λ). +9 + +3. Categories of B-branes and K-theories +Given an abelian GLSM, we consider several versions of the category of B-branes and the A-model state space. +dg category +category of B-branes +K-theory +A-model state space +MF(Xζ, wζ) +DMF(Xζ, wζ) ≃ DSg(Xζ,0) +K(DMF(Xζ, wζ)) +Hw = +� +v∈Box(Σζ) +H∗(Xζ,v, w∞ +ζ,v) +Perf � +T (Xζ) +Db +� +T (Xζ) +K � +T (Xζ) +H � +T = +� +v∈Box(Σζ) +H∗ +� +T (Xζ,v) +Perf(Xζ) +Db(Xζ) +K(Xζ) +H = +� +v∈Box(Σζ) +H∗(Xζ,v) +Db +c(Xζ) +Kc(Xζ) +Hc = +� +v∈Box(Σζ) +H∗ +c (Xζ,v) +In each version, the Chern character sends the K-theory class of a B-brane to an element in the A-model state +space. +Kw = K(DMF(Xζ, wζ)) +chw +−→ +Hw := +� +v∈Box(Σζ) +Hw,v, +Hw,v = H∗(Xζ,v, wζ,v; C). +K � +T = K � +T (Xζ) +ch � +T +−→ +H � +T = +� +v∈Box(Σζ) +H � +T ,v, +H � +T ,v = H∗ +� +T (Xζ,v; C). +K = K(Xζ) +ch +−→ +H = +� +v∈Box(Σζ) +Hv, +Hv := H∗(Xζ,v; C). +Kc = Kc(Xζ) +chc +−→ +Hc = +� +v∈Box(Σζ) +Hc,v, +Hc,v = H∗ +c (Xζ,v; C). +K � +T is a commutative ring with unity and an algebra over K � +T (•) = K(B �T) = Z[Λ± +1 , . . . , Λ± +n+κ], and K is a +commutative ring with unity and an algebra over K(•) = Z. There is a surjective ring homomorphism K � +T → K. +Kc and Kw are modules over the ring K. There is a map Kc → K which is a morphism of K-modules; the image +Kct is an ideal in the ring K. Taking Euler characteristic defines non-degenerate pairings: +K � +T × K � +T → Q(Λ1, . . . , Λn+κ), +K × Kc → Z, +Kw × Kw → Z. +Fix v ∈ Box(Σζ). +H � +T ,v is a commutative ring with unity and an algebra over H∗ +� +T (•) = H∗(B �T) = +C[λ1, . . . , λn+κ], and Hv is a commutative ring with unity and an algebra over H∗(•) = C. There is a surjec- +tive ring homomorphism H � +T ,v → Hv. Hc,v and Hw,v are modules over the ring Hv. There is a map Hc,v → Hv +which is a morphism of Hv-modules; the image Hct,v is an ideal in the ring Hv. We have. non-degenerate pairings: +H � +T ,v × H � +T ,inv(v) → C(λ1, . . . , λn+κ), +Hv × Hc,inv(v) → C, +Hw,v × Hw,inv(v) → C. +4. The Higgs Branch +4.1. A mathematical theory of GLSM: an overview. Let (V, G, C∗ +R, W, ζ) be the input date of a GLSM. The +first four components (V, G, C∗ +R, W) give rise to the following diagram: +V +W +� +� +[V/G] +w +� +� +□ +C +� +[V/Γ] +ˆw +� +� +[C/C∗ +ω] +� +BΓ = [•/Γ] +Bχ � BC∗ +ω = [•/C∗ +ω] +where the middle square is Cartesian, the upper triangle and the lower square are commutative, and the bottom +right arrow Bχ : BΓ → BC∗ +ω is induced by the group homomorphism χ : Γ → C∗ +ω. A Landau-Ginzburg (LG) +quasimap to (V, G, C∗ +R, W, ζ) is a birational map from an orbicurve C to [V ss +G (ζ)/Γ] which extends to a representable +10 + +morphism f : C → [V/Γ] of smooth Artin stacks and satisfies certain stability conditions such that the following +diagram commutes: +[V/Γ] +π +� +C +f +� +P +� +ωlog +C +� +BΓ +Bχ +� +BC∗ +ω +Recall that BΓ is the classifying space of principal Γ-bundles, and [V/Γ] is the classifying space of a principal +Γ-bundle P → C together with a section u : C → P ×Γ V , where P ×Γ V → C is the rank n + κ vector bundle +associated to the representation Γ → GL(V ). A section u : C → P ×Γ V is equivalent to a Γ-equivariant map +˜f : P → V . More explicitly, we have the following cartesian diagram +P +� +� +• +� +C = P/Γ +π◦f � BΓ = [•/Γ] +Let pr1 : P × V −→ P and pr2 : P × V → V be the projection to the first and second factors, respectively. We +have a commutative diagram +P × V +pr1 +� +pr2 +� V +� +P +� +(idP , ˜ +f) +� +˜ +f +� +• +where all the arrows are Γ-equivariant. Taking the quotient of the above diagram by the Γ-action, we obtain the +following commutative diagram +P ×Γ V +� +� [V/Γ] +π +� +C +π◦f � +u +� +f +� +BΓ = [•/Γ] +4.2. Twisted curves and their moduli. We follow the presentation of [AV02,AGV] on twisted curves. A genus- +g, ℓ-pointed twisted prestable curve is a connected proper one-dimensional DM stack C together with ℓ disjoint +zero-dimensional integral closed substacks z1, . . . , zℓ ⊂ C, such that +(i) C is ´etale locally a nodal curve; +(ii) formally locally near a node, C is isomorphic to the quotient stack +[Spec(C[x, y]/(xy))/µr], +where η · (x, y) = (ηx, η−1y), η ∈ µr; +(iii) each marking zi ⊂ C is contained in the smooth locus of C; +(iv) C is a scheme away from the markings and the singular points of C; the coarse moduli space C of C is a +nodal curve of arithmetic genus g. +Let π : C → C be the coarse moduli morphism; let zi = π(zi). The resulting (C, z1, . . . , zℓ) is a genus-g, ℓ-pointed +prestable curve. We say (C, z1, . . . , zℓ) is stable if (C, z1, . . . , zℓ) is stable. The moduli Mtw +g,ℓ of genus-g, ℓ-pointed +twisted prestable curves is a smooth Artin stack of dimension 3g − 3 + ℓ. For i = 1, . . . , ℓ, let Li be the line bundle +on Mtw +g,ℓ whose fiber over (C, z1, . . . , zℓ) is the T ∗ +zjC, the cotangent line to the coarse curve C at zi. +The space of infinitesimal automorphisms of (C, z1, . . . , zℓ) is +Ext0 +OC(ΩC(z1 + · · · + zℓ), OC), +while the space of infinitesimal deformations of (C, z1, . . . , zℓ) is +Ext1 +OC(ΩC(z1 + · · · + zℓ), OC). +(C, z1, . . . zℓ) is stable if and only if Ext0 +OC(ΩC(z1 + · · · + zℓ), OC) = 0. +11 + +4.3. Line bundles over a twisted curve. Let (C, z1, . . . , zℓ) be a genus-g, ℓ-pointed twisted prestable curve, +where zi = Bµri. A line bundle L on C defines a morphism C −→ BC∗ such that we have the following Cartesian +diagram +L +� +� +□ +[C/C∗] +� +C +� [•/C] = BC∗ +where [C/C∗] → BC∗ is the universal line bundle over the classifying space BC∗ of principal C∗-bundles. +Given any line bundle L over C, there exists a positive integer m such that L⊗m = π∗M for some line bundle M +over the coarse moduli space C. Define +deg L = 1 +m deg M ∈ Q +• If zi is a scheme point, we define agezi(L) = 0. +• If zi = Bµri, where ri > 1, then the restriction Lzi of L to zi is an element in Pic(Bµri) = Hom(µri, C∗) ∼= +Z/riZ. There is a unique ai ∈ {0, 1, . . . , ri − 1} such that +Lzi ∼= (TziC)⊗ai. +Define agezi(L) = ai +ri +∈ (0, 1) ∩ Q. +There is a unique line bundle L on the coarse moduli C such that +L ≃ π∗L ⊗ OC( +ℓ +� +i=1 +aizi). +where +deg +� +OC +� +ℓ +� +i=1 +aizi +�� += +ℓ +� +i=1 +agezi(L), +deg(π∗L) = deg L ∈ Z. +So +deg L − +ℓ +� +i=1 +agezi(L) ∈ Z. +For i = 0, 1, hi(C, L) = hi(C, L), so +(4.1) +χ(C, L) = χ(C, L) = deg L + 1 − g − +ℓ +� +j=1 +agezj(L) +which is a special case of Kawasaki’s orbifold version of Riemann-Roch theorem [Ka79]. +The space of infinitesimal automorphisms of L on a fixed twisted prestable curve C is +Ext0 +OC(L, L) ≃ H0(C, OC). +The space of infinitesimal deformations of L on a fixed twisted prestable curved C is +Ext1 +OC(L, L) ≃ H1(C, OC). +4.4. Universal moduli of principal Γ-bundles. Let Mg,ℓ(BΓ) denote the moduli of pairs ((C, z1, . . . , zℓ), P), +where +• (C, z1, . . . , zℓ) is a genus-g, ℓ-pointed twisted prestable curve; +• P is a principal Γ-bundle over C which corresponds to a representable morphism C −→ BΓ. +Then Mg,ℓ(BΓ) is a smooth Artin stack. Note that Mg,ℓ(BΓ) can be identified with the Hom-stack +HomM(CM, BΓ × M) +where M = Mtw +g,ℓ. Forgetting the principal bundle P defines a (non-representable) morphism of smooth Artin stacks +πD/M : D := Mg,ℓ(BΓ) −→ M +which is smooth of relative dimension dim Γ(g − 1) = (κ + 1)(g − 1). So Mg,ℓ(BΓ) is a smooth Artin stack of +dimension +3g − 3 + ℓ + (κ + 1)(g − 1) = (4 + κ)(g − 1) + ℓ = (dim BΓ − 3)(1 − g) + ℓ +12 + +where dim BΓ = − dim Γ = −κ − 1. Let πD : CD → D := Mg,ℓ(BΓ) be the universal curve, let fD : CD → BΓ +be the universal map, and let PD → CD be the universal principal Γ-bundle. We have the following cartesian +diagrams: +CD +� +πD +� +□ +CM +πM +� +D +πD/M � M +PD +� +� +□ +• +� +CD +fD +� BΓ +Let �L := Hom(C∗, Γ) ∼= Zκ+1 be the cocharacter lattice of Γ. Its dual lattice �L∨ = Hom(Γ, C∗) is the character +lattice of Γ. A principal Γ-bundle P −→ C determines a map C = P/Γ → BΓ = [•/Γ] whose degree is an element +βΓ ∈ H2(BΓ; Q) = �LQ +characterized by the following property. For any Γ-character λ ∈ Hom(Γ, C∗) = �L∨ = H2(BΓ; Z), let P ×λ C → C +be the line bundle associated to the representation λ : Γ → C∗ = GL(1). Then +� +βΓ +λ = deg(P ×λ C) = +� +[C] +c1(P ×λ C) ∈ Q, +where +• +� +βΓ +λ denotes the natural pairing between βΓ ∈ �LQ = H2(BΓ; Q) and λ ∈ �L∨ ⊂ �L∨ +Q = H2(BΓ; Q), and +• +� +[C] +c1(P ×λ C) denotes the natural pairing between [C] ∈ H2(C; Q) and c1(P ×λ C) ∈ H2(C; Q). +In other words, βΓ ∈ H2(BΓ; Q) is the image of [C] ∈ H2(C; Q) under P∗ : H2(C; Q) −→ H2(BΓ; Q). +The monodromy of P at zj = Bµrj is an element γj ∈ Γ of order rj. The subset +�LQ/�L ∼= (Q/Z)κ+1 ⊂ �LC/�L ∼= (C/Z)κ+1 = (C∗)κ+1 +can be identified with +{γ ∈ Γ : ord(γ) is finite.}. +The monodromies (γ1, . . . , γℓ) ∈ (�LQ/�L)ℓ and the degree βΓ ∈ �LQ satisfy the following compatibility condition: +ℓ +� +j=1 +γj = βΓ + �L ∈ �LQ/�L. +Let Mg,⃗γ(BΓ, βΓ) ⊂ Mg,ℓ(BΓ) be the open and closed substack of pairs ((C, z1, . . . , zℓ), P) with degree βΓ ∈ �LQ +and monodromies ⃗γ = (γ1, . . . , γℓ) ∈ (�LQ/�L)ℓ. Then +Mg,ℓ(BΓ) = +� +⃗γ=(γ1,...,γℓ)∈(�LQ/�L)ℓ +βΓ∈�LQ, �ℓ +i=1 γi=βΓ+�L +Mg,⃗γ(BΓ, βΓ). +4.5. Universal moduli of Γ-structures. Polishchuk-Vaintrob introduced Γ-structures [PV16] which is an alter- +native formulation for W-structures in [FJR11]. +Given an object (C, z1, . . . , zℓ) in Mg,ℓ(•), a Γ-structure on (C, z1, . . . , zℓ) is a pair (P, ρ) where ((C, z1, . . . , zℓ), P) +is an object in Mg,ℓ(BΓ) and +ρ : P ×χ C +∼ += +−→ ωlog +C +is an isomorphism of line bundles on C. +Let Bg,ℓ be the moduli space of triples ((C, z1, . . . , zℓ), P, ρ), where (C, z1, . . . , zℓ) is an object in Mtw +g,ℓ(•) and +(P, ρ) is a Γ-structure on (C, z1, . . . , zℓ). We have a commutative diagram +B = Bg,ℓ +πB/D� +πB/M +� +D := Mg,ℓ(BΓ) +πD/M +� +M = Mtw +g,ℓ +where πB/D : B −→ D is given by forgetting ρ. +The map πB/M : B → M is smooth of relative dimension +dim G(g − 1) = κ(g − 1), so Bg,ℓ is a smooth Artin stack of dimension +3g − 3 + ℓ + κ(g − 1) = (3 + κ)(g − 1) + ℓ = (dim BG − 3)(1 − g) + ℓ. +13 + +The map +(ρV , ρR) : G × C∗ +R ∼= (C∗)κ+1 −→ Γ ∼= (C∗)κ+1 +(h, t) �→ ht +is a surjective group homomorphism and a covering map of degree r = ord(J). It induces +(ρV , ρR)∗ : H2(BG; Z) × H2(BC∗ +R; Z) = L × +� +Z[P1] +� +−→ H2(BΓ; Z) = �L +which is an inclusion of lattices of finite index r. (We recall that BC∗ +R = P∞, and we let [P1] ∈ H2(P∞; Z) be the +class of P1 ⊂ P∞.) Therefore, we obtain an isomorphism +(ρV , ρR)∗ : H2(BG; Q) × H2(BC∗ +R; Q) = LQ × +� +Q[P1] +� +∼ += +−→ H2(BΓ; Q) = �LQ. +Let (βG, βR) = (ρV , ρR)−1 +∗ (βΓ). Then βR = 2g − 2 + ℓ +r +[P1] ∈ Q[P1] = H2(BC∗ +R; Q) = H2(P∞; Q). +Let Bg,ℓ(βG) ⊂ Bg,ℓ be the open and closed substack of triples ((C, z1, . . . , zℓ), P, ρ) with degree +βΓ = +� +βG, 2g − 2 + ℓ +r +[P1] +� +∈ LQ × +� +Q[P1] +� ∼= H2(BΓ; Q). +Then +Bg,ℓ = +� +βG∈LQ +Bg,ℓ(βG). +For each βG, let Bg,⃗γ(βG) ⊂ Bg,ℓ(βG) be the open and closed substack of triples ((C, z1, . . . , zℓ), P, ρ) with mon- +odromies ⃗γ = (γ1, . . . , γℓ) ∈ (LQ/L)ℓ. Then +Bg,ℓ(βG) = +� +⃗γ=(γ1,...,γℓ)∈(LQ/L)ℓ +�ℓ +i=1 γi=βG+L +Bg,⃗γ(βG). +4.6. Moduli of sections. In this subsection we fix g, ℓ, βG and ⃗γ = (γ1, . . . , γℓ) and let B = Bg,⃗γ(βG). +Following [BF97, Section 1], we introduce the following definition. Given a coherent sheaf F of OX-modules on +an algebraic stack X, let C(F) := SpecX(SymF∨) be the abelian cone associated to F. In particular, when F is +locally free, C(F) = tot(F) is the vector bundle associated to F. +Let PB → CB be the universal principal Γ-bundle over the universal curve πB : CB → B. Recall that Γ is a +subgroup of the diagonal torus �T ⊂ GL(V ), so +VB := PB ×Γ V = +n+κ +� +i=1 +Li,B +where each Li,B is a line bundle over CB. Consider the moduli of sections +Bg,⃗γ(V, βG) := C (πB∗VB) +which parametrizes 4-tuples ((C, z1, . . . , zℓ), P, ρ, u) where the triple ((C, z1, . . . , zℓ), P, ρ) is an object in B(•) and +u = (u1, . . . , un+κ) ∈ H0(C, P ×Γ V ) = H0(C, +n+κ +� +j=1 +Lj) +where uj ∈ H0(C, Lj). Let +Bg,ℓ(V, βG) := C +� +πBg,ℓ(βG)∗VBg,ℓ(βG) +� +. +Then +Bg,ℓ(V, βG) = +� +⃗γ=(γ1,...,γℓ)∈(LQ/L)ℓ +�ℓ +i=1 γi=βG+L +Bg,⃗γ(V, βG). +Given a fixed triple ((C, z1, . . . , zℓ), P, ρ), the space of infinitesimal deformations of the section uj of the line +bundle Lj is H0(C, Lj) and the space of obstructions to deforming uj is H1(C, Lj). We now compute +χ(C, Lj) = h0(C, Lj) − h1(C, Lj). +The line bundle Lj is of degree +⟨Dj, βG⟩ + qj +2 (2g − 2 + ℓ) +14 + +and has monodromy e2π√−1ageγi(Dj) around zi, where ageγi(Dj) ∈ Q ∩ [0, 1) is the unique representative in [0, 1) of +the pairing ⟨Dj, βG⟩ ∈ Q/Z. Note that ageγi(Dj) = ageziLj. We have +⟨Dj, βG⟩ + qj +2 (2g − 2 + ℓ) − +ℓ +� +i=1 +ageγi(Dj) ∈ Z. +By Kawasaki’s orbifold version of the Riemann-Roch theorem [Ka79], +χ(C, Lj) = ⟨Dj, βG⟩ + qj +2 (2g − 2 + ℓ) + (1 − g) − +ℓ +� +i=1 +ageγi(Dj). +Let +(4.2) +(c1)G(V ) = +n+κ +� +j=1 +Dj ∈ L∨, +ageγ(V ) = +n+κ +� +j=1 +ageγi(Dj) ∈ Q, +and recall that ˆq = 1 +2 +�n+κ +j=1 qj. Then +χ(C, P ×Γ V ) = +n+κ +� +j=1 +χ(C, Lj) = ⟨(c1)G(V ), βG⟩ + (n + κ − 2ˆq)(1 − g) − +ℓ +� +i=1 +� +ageγi(V ) − ˆq +� +. +There is a map πS/B : S = Bg,⃗γ(V, βG) −→ B = Bg,⃗γ(βG) given by ((C, z1, . . . , zℓ), P, ρ, u) �→ ((C, z1, . . . , zℓ), P, ρ), +i.e, forgetting the section u. The map πS/B is virtually smooth: there is a relative perfect obstruction theory ES/B, +where +E∨ +S/B = π∗ +S/BRπB∗VB. +The relative virtual dimension of πS/B is +(4.3) +dvir +S/B = ⟨(c1)G(V ), βG⟩ + (n + κ − 2ˆq)(1 − g) − +ℓ +� +i=1 +� +ageγi(V ) − ˆq +� +. +Therefore, the moduli of sections S is a possibly singular, but virtually smooth Artin stack; it is equipped with a +perfect obstruction theory ES of virtual dimension +dvir +S = dvir +S/B + dim B = ⟨(c1)G(V ), βG⟩ + (n + κ − 2ˆq)(1 − g) − +ℓ +� +i=1 +� +ageγi(V ) − ˆq +� ++ (κ + 3)(g − 1) + ℓ +which can be rewritten as +(4.4) +dvir +S = ⟨(c1)G(V ), βG⟩ + (ˆc − 3)(1 − g) + ℓ − +ℓ +� +i=1 +� +ageγi(V ) − ˆq +� +where ˆc = n − 2ˆq is the central charge of the GLSM. Note that if γ ∈ LQ/L corresponds to v ∈ Box(Σζ) then +ageγ(V ) = age(v). +The definition of the central charge ˆc and the degree shift 2 (age(v) − ˆq) in the definition of the A-model state spaces +are motivated by the formula (4.4) of the virtual dimension, which is consistent with [FJR, Lemma 6.1.7]. +Let PS → CS be the universal principal Γ-bundle over the universal curve πS : CS → S, and let uS : CS → +VS := PS ×Γ V be the universal section. We have the following commutative diagram. +VS +� +� +□ +VB +� +CS +� +πS +� +uS +� +□ +CB +� +πB +� +□ +CM +πM +� +S +πS/B � B +πB/M � M +15 + +4.7. Landau-Ginzburg quasimaps and their moduli. +Definition 4.1 (prestable LG quasimaps). A prestable genus-g, ℓ-pointed, degree βG Landau-Ginzburg (LG) +quasimap to the 5-tuple X = (V, G, C∗ +R, W, ζ) is a 4-tuple Q = ((C, z1, . . . , zℓ), P, ρ, u), which is an object in +Bg,ℓ(V, βG)(•) such that the base locus +B(Q) := u−1(P ×Γ V us +G (ζ)) ⊂ C +of Q is purely zero-dimensional and is disjoint from the marked points and the nodes in C. +Remark 4.2. Note that the �T-action preserves V ss +G (ζ) and V us +G (ζ). In particular, Γ acts on V us +G (ζ). +Let LGpre +g,ℓ (X, βG) be the moduli of prestable genus-g, ℓ-pointed, degree βG LG quasimaps to X. It is an open +substack of Bg,ℓ(V, βG). There are evaluation maps +evi : LGpre +g,ℓ (X, βG) → IXζ = +� +v∈Box(Σ) +Xζ,v, +i = 1, . . . , ℓ. +Given ⃗v = (v1, . . . , vℓ) ∈ Box(Σζ)ℓ, let +LGg,⃗v(X, βG) := +ℓ� +i=1 +ev−1 +i +(Xζ,vi) . +Then LGg,⃗v(X, βG) is an open substack of Bg,⃗γ=(γ1,...,γℓ)(V, βG), where γi ∈ LQ/L corresponds to vi ∈ Box(Σ). +Therefore, the virtual dimension of LGg,⃗v(X, βG) is +(4.5) +⟨(c1)G(V ), βG⟩ + (ˆc − 3)(1 − g) + ℓ − +ℓ +� +i=1 +(age(v) − ˆq) . +Definition 4.3 (good lift). ˜ζ ∈ �L∨ is a lift of ζ ∈ L∨ if ζ is the image of ˜ζ under +�L∨ = Hom(Γ, C∗) → L∨ = Hom(G, C∗). +˜ζ is a good lift of ζ if V ss +Γ (˜ζ) = V ss +G (ζ). +For any ˜ζ ∈ Hom(Γ, C∗) = �L∨, let χ˜ζ : Γ → C∗ denote the corresponding Γ-character, let L˜ζ ∈ PicΓ(V ) denote +the Γ-equivariant line bundle on V determined by χ˜ζ +χ +˜ζ1+˜ζ2 = χ +˜ζ1χ +˜ζ2, +L˜ζ1+˜ζ2 = L˜ζ1 ⊗ L˜ζ2. +A section s ∈ H0(V, L˜ζ)Γ defines a Γ-equivariant map V → C which induces a morphism +s : P ×Γ V → P ×χ˜ +ζ C. +Given any u ∈ H0(C, P ×Γ V ), let u∗s := s◦u ∈ H0(C, P ×χζ C). The following definition is [FJR, Definition 4.2.10] +(which is essentially [CKM, Definition 7.1.1]) in slightly different notation. +Definition 4.4 (length). Let Q = ((C, z1, . . . , zℓ), P, ρ, u) be a prestable LG quasimap to X = (V, G, C∗ +R, W, ζ), let +˜ζ ∈ �L∨ be a good lift of ζ ∈ L∨. The length of a point y in C with respect to Q and ˜ζ is defined to be +(4.6) +ℓy(Q, ˜ζ) := min +�ordy(u∗s) +m +��� s ∈ H0(V, Lm˜ζ = L⊗m +˜ζ +)Γ, m ∈ Z>0 +� +where ordy(u∗s) is the order of vanishing of the section u∗s ∈ H0(C, P ×χm˜ +ζ C) at y. +Definition 4.5 (ϵ-stable LG quasimaps). Let ˜ζ ∈ Hom(Γ, C∗) be a good lift of ζ ∈ Hom(G, C∗), and let ϵ be a +positive rational number. A prestable LG quasimap Q = ((C, z1, . . . , zℓ), P, ρ, u) is ϵ-stable with respect to ˜ζ if +(1) ωlog +C +⊗ (P טζ C)ϵ ∈ Pic(C) ⊗Z Q is an ample Q line bundle on C, and +(2) ϵℓy(Q, ˜ζ) ≤ 1 for every y ∈ C. +Remark 4.6. Let Cv (respectively Cv) be the connected component of the normalization of C (respectively C) +associated to a vertex v in the dual graph of the coarse moduli C of C. Let gv be the genus of Cv and let ℓv be +the number of points on Cv mapped to a marked point or a node under the normalization map, and let βΓ(v) ∈ +H2(BΓ; Q) = �LQ be the degree of Cv → C +P→ BΓ. Condition (1) in Definition 4.5 is equivalent to the following +condition: +2gv − 2 + ℓv + ϵ +� +βΓ(v) +˜ζ > 0 +for all vertex v in the dual graph of C. +16 + +Remark 4.7. Let m ∈ Z>0, ϵ ∈ Q>0, and ˜ζ ∈ �L∨. Then Q is ϵ-stable with respect to ˜ζ iff it is (ϵ/m)-stable with +respect to m˜ζ; see [CKM, Remark 7.1.4] for the analogous statement in quasimap theory. Given ν ∈ �L∨ +Q, choose +m ∈ Z>0 such that mν ∈ �L∨ and define Q to be ν-stable if it is (1/m)-stable with respect to mν; the definition is +independent of the choice of m and agrees with [FK, Definition 2.6]. +Let LGpre +g,ℓ (X, βG) (respectively LGϵ,˜ζ +g,ℓ(X, βG)) be the moduli of genus-g, ℓ-pointed, degree βG prestable (respec- +tively ϵ-stable with respect to ˜ζ) quasimaps to X := (V, G, C∗ +R, W, ζ). More generally, let Y be a Γ-invariant closed +subscheme of V such that Y ∩ V s +G(ζ) = Y ∩ V ss +G (ζ) is non-empty (e.g. Y = Crit(W)), and let Y = (Y, G, C∗ +R, W, ζ). +Let LGpre +g,ℓ (Y, βG) be the closed substack of LGpre +g,ℓ (X, βG) parametrizing Q = ((C, z1, . . . , zℓ), P, ρ, u) such that +u : C → [Y/Γ] ⊂ [V/Γ], and define LGϵ,˜ζ +g,ℓ(Y, βG) similarly. Fan-Jarvis-Ruan proved the following result: +Theorem 4.8 ( [FJR]). LGϵ,˜ζ +g,ℓ(Y, βG) is a separated Deligne-Mumford stack of finite type. It is proper over SpecC +if [(Y ∩ V ss +G (ζ))/G] is. +Given ⃗v = (v1, . . . , vℓ) ∈ Box(Σζ)ℓ, let +LGϵ,˜ζ +g,⃗v(X, βG) := LGϵ,˜ζ +g,ℓ(X, βG) ∩ LGg,⃗v(X, βG). +Then +LGϵ,˜ζ +g,ℓ(X, βG) = +� +⃗v∈Box(Σζ)ℓ +LGϵ,˜ζ +g,⃗v(X, βG). +In Section 4.8 and Section 4.9 below, we fix g,⃗v, ϵ, βG, and let +X = LGϵ,˜ζ +g,⃗v(X, βG), +Z = LGϵ,˜ζ +g,⃗v(Z, βG) +where Z = (Crit(W), G, C∗ +R, W, ζ). By Theorem 4.8, if +Zζ = [(Crit(W) ∩ V ss +G (ζ)) /G] = Crit(wζ) +is proper, then Z is proper. +4.8. Cosection localized virtual cycle and cosection localized virtual structure sheaf. Recall that v ∈ +Box(Σζ) is narrow if Xζ,v is compact. We assume v1, . . . , vℓ are narrow in this subsection. Under this assumption, +Fan-Jarvis-Ruan [FJR] constructed a cosection δ : ObX → OX whose zero locus is Z. Applying [KL13], they obtain +a cosection localized virtual cycle +[X]vir +loc ∈ A∗(Z; Q) +such that +ι∗[X]vir +loc = [X]vir ∈ A∗(X; Q) +where ι : Z → X is the inclusion, and [X]vir is the Behrend-Fantechi virtual fundamental class [BF97] defined by +the perfect obstruction theory described in previous subsections. When Z is proper, [X]vir +loc can be used to define +(cohomological) GLSM invariants in the narrow sector1. The construction of cosection localized virtual cycle in the +narrow sector in [FJR] can be viewed as generalization of Chang-Li-Li’s construction of Witten’s top Chern class +via cosection localization [CLL]. +We now consider the following particularly nice case, as in [BF97, Proposition 5.6]. +Situation 4.9. X is smooth of dimension r0 and ObX is locally free of rank r1. +In Situation 4.9, ObX := tot(ObX) is a vector bundle over X of rank r1, called the obstruction bundle, and the +virtual dimension is r = r0 − r1. Let δ∨ : OX → Ob∨ +X be the dual of the cosection, which is a section of Ob∨ +X. +[X]vir += +cr1(ObX) ∩ [X] = (−1)r1cr1(Ob∨ +X) ∩ [X] ∈ Ar(X; Q), +(4.7) +[X]vir +loc += +(−1)r1cr1(Ob∨ +X, δ∨) ∩ [X] ∈ Ar(Z; Q). +(4.8) +where +• [X] ∈ Ar0(X; Q) is the fundamental class of the smooth DM stack X, +• cr1(Ob∨ +X) ∩ − : Ak(X; Q) → Ak−r1(X; Q) is the top Chern class, and +• cr1(Ob∨ +X, δ∨) ∩ − : Ak(X; Q) → Ak−r1(Z; Q) is the localized top Chern class [Fu98, Chapter 14]. +1In [FJR, Section 6], GLSM correlators are defined for compact type insertions [FJR, Definition 4.1.4] which are more general than +narrow insertions. See [Sh] for subtleties of defining GLSM invariants involving compact type insertions which are not narrow, as well +as an alternative construction of genus-zero compact type GLSM invariants under additional assumptions. +17 + +Let K0(X) (resp. K0(X)) denote the Grothendieck group generated by coherent sheaves (resp. locally free +sheaves) on X with relations [F] = [F ′] + [F ′′] whenever there is a short exact sequence 0 → F ′ → F → F ′′ → 0. +Applying [KL18], one obtains a cosection localized virtual structure sheaf +Ovir +X,loc ∈ K0(Z) +such that +ι∗Ovir +X,loc = Ovir +X ∈ K0(X) +where Ovir +X is the virtual structure sheaf defined in [BF97] (see [BF97, Remark 5.4]) and [Lee]. When Z is proper, +Ovir +loc an be used to define K-theoretic GLSM invariants in the narrow sector. +In Situation 4.9, +Ovir +X = +r1 +� +i=0 +(−1)i ∧i Ob∨ +X = (−1)r1 det(Ob∨ +X) +r1 +� +i=0 +(−1)i ∧i ObX ∈ K0(X). +Note that +td(ObX)ch(Ovir +X ) = cr1(ObX), +so +[X]vir = td(ObX)ch(Ovir +X ) ∩ [X]. +The section δ∨ : OX → Ob∨ defines a Koszul complex +(4.9) +K(δ∨) := Symr1 � +ObX +iδ∨ +→ OX +� += +� +0 → ∧r1ObX +iδ∨ +→ ∧r1−1ObX → · · · → ∧1ObX +iδ∨ +→ OX → 0 +� +which is exact on X − Z. If e1, . . . , ek are local sections of ObX then +iδ∨(e1 ∧ · · · ∧ ek) = +k +� +i=1 +(−1)i−1⟨δ∨, ei⟩e1 ∧ · · · ∧ ei−1 ∧ ei+1 ∧ · · · ∧ ek. +Note that +td(Ob∨ +X)chX +Z (K(δ∨)) = cr1(Ob∨ +X, δ∨), +where chX +Z (K(δ∨))∩ : A∗(X; Q) → A∗(Z; Q) is the localized Chern character [Fu98, Chapter 18]. +4.9. Virtual factorization. Favero-Kim [FK] construct GLSM invariants for general choice of stability in both +narrow and broad sectors via matrix factorization, generalizing constructions in [PV16,CFGKS]. In this subsection +we briefly describe the construction (in slightly different notation). +4.9.1. The Artin stack C. Let ⃗γ = (γ1, . . . , γℓ), where γi ∈ LQ/L corresponds to vi ∈ Box(Σζ). Let B := Bg,⃗γ(βG) +be the universal moduli of Γ-structures defined in Section 4.5. Let PB → CB be the universal principal Γ-bundle +over the universal curve πB : CB → B, let VB = PB ×Γ V , and let C(πB∗VB) be the moduli of sections, which is +an abelian cone over B, as in Section 4.6. Then X is an open substack of C(πB∗VB), and the image of X under the +projection C(πB∗VB) → B (forgetting the section) is contained in a finite type open substack B◦ ⊂ B. Therefore, +X is an open substack of +C := C (πB◦∗VB◦) = B◦ ×B C (πB∗VB) . +For i = 1, . . . , ℓ, there are evaluations maps +(4.10) +evC +i : C −→ Xvi := [V g(vi)/G] +which restricts to +(4.11) +evi : X −→ Xζ,vi = +�� +V g(vi) ∩ V ss +G (ζ) +� +/G +� +⊂ Xvi. +18 + +4.9.2. The smooth Artin stack A and the smooth DM stack U. Over the finite type smooth Artin stack B◦, +RπB◦∗VB◦ admits a global resolution +RπB◦∗VB◦ = [A +dA +−→ B] +where A, B are locally free sheaves of OB◦-modules. +Let πA/B◦ : A := tot(A) → B◦ be the projection, let +tA ∈ Γ(A, π∗ +A/B◦A) be the tautological section, and let βA := +� +π∗ +A/B◦dA +� +◦ tA ∈ Γ(A, π∗ +A/B◦B). The zero locus of +tA is the zero section in A = tot(A), and the zero locus of βA is C. There exists an open substack U ⊂ A such +that U is a DM stack of finite type and the following diagram is a Cartesian square. +X = Z(βU) +ιX +� +jX +� +U +jU +� +C = Z(βA) +ιC +� A +In the above Cartesian diagram, the two vertical arrows are open embeddings, the two horizontal arrows are closed +embeddings, and βU = j∗ +UβA ∈ Γ(U, BU), where BU := j∗ +Uπ∗ +A/B◦B. The virtual dimension r of X is +r = dim B + rankA − rankB = dim U − rankB. +We have +[X]vir = crankB (BU, βU) ∩ [U], +where +• [U] ∈ Ar+rankB(U; Q) is the fundamental class of the smooth DM stack U, +• crankB (BU, βU) ∩ − : Ar+rankB(U) → Ar(X) is the localized top Chern class, and +• [X]vir ∈ Ar(X; Q) is the Behrend-Fantechi virtual fundamental class of X. +4.9.3. Evaluation maps. For i = 1, . . . , ℓ, the evaluation map evC +i +: C → Xvi extends to evA +i +: A → Xvi which +restricts to evU +i : U → Xζ,vi, so that we have the following commutative diagram +X +ιX � +� +U +evU +i � +evU +i � +� +Xζ,vi +� +C +ιC +� A +evA +i +� Xvi +where evU +i ◦ ιX = evi and evA +i ◦ ιC = evC +i , and all the vertical arrows are open embeddings. Let +⃗v = (v1, . . . , vℓ), +X⃗v := +ℓ +� +i=1 +Xvi, +Xζ,⃗v := +ℓ +� +i=1 +Xζ,vi. +A and evA +i are chosen such that +(4.12) +evA := +ℓ +� +i=1 +evA +i : A → X⃗v +is a surjective smooth map between smooth Artin stacks. More explicitly, given any object ξ = ((C, z1, . . . , zℓ), P, ρ) +in B◦(•), which corresponds to a morphism SpecC → B◦, let Aξ := SpecC ×B◦ A be the fiber of A = tot(A) over +ξ. We have the following linear maps between complex vector spaces: +(4.13) +H0(C, P ×Γ V ) +ιC,ξ +−→ Aξ +evU +ξ +−→ +ℓ +� +i=1 +H0(zi, (P ×Γ V )|zi) = +ℓ +� +i=1 +V g(vi) +where ιC,ξ is injective and evU +ξ is surjective. As a consequence, +(4.14) +evU := +ℓ +� +i=1 +evU +i : U → Xζ,⃗v +is a smooth map between smooth DM stacks. +19 + +4.9.4. Superpotentials and matrix factorizations. Let wvi : Xvi = [V g(vi)/G] → C denote the restriction of w : +[V/G] → C. Define a superpotential wA on A: +(4.15) +wA := +ℓ +� +i=1 +(evA +i )∗wvi ∈ Γ(A, OA) +which restricts to a superpotential on U: +(4.16) +wU := j∗ +UwA = +ℓ +� +i=1 +(evU +i )∗wζ,vi ∈ Γ(U, OU). +The sum of residues of a meromorphic 1-form on a curve is zero, so +ι∗ +CwA = 0, +ι∗ +XwU = 0. +When the GLSM X is a convex hypbrid model, it is shown in [CFGKS] that +(a) U can be chosen to be separated over SpecC. +(b) There exists a cosection α∨ +A : π∗ +A/B◦B → OA, or equivalently a section αA : OA → π∗ +A/B◦B∨, such that +⟨αA, βA⟩ = −wA. +(c) Let αU := j∗ +UαA ∈ Γ(U, B∨ +U). Then +Z = Z(αU) ∩ Z(βU) ⊂ X = Z(βU) ⊂ U. +In Section 4.14, we will provide explicit construction of U and α∨ +A satisfying (a)-(c) for the genus-zero one-pointed +moduli spaces used to define the GLSM I-functions for all abelian GLSMs. +When αA exists, one obtains a Koszul matrix factorization {αA, βA} of (A, −wA) defined by +{αA, βA} = +� � +i Λ2iπ∗ +A/B◦B∨ +∂ +� +� +i Λ2i+1π∗ +A/B◦B∨ +∂ +� +� +, +where ∂ = iβA + αA ∧ . +Then +KU := {αU, βU} = j∗ +U{αA, βA} +is a Koszul matrix factorization of (U, −wU). It is called the fundamental factorization in [PV16, CFGKS] and +called the virtual factorization in [FK]. +4.9.5. Localized Chern character and the virtual fundamental class. Let U be a smooth DM stack over C and let +w : U → C be a regular function. In [FK, Section B.4], Favero-Kim define the Atiyah class and the localized Chern +character of a matrix factorization for (U, w), following the construction of Kim-Polishchuk [KP22] when U is a +smooth scheme. Applying the definition to KU, one obtains a localized Chern character +chU +Z KU ∈ Heven +Z +(U, (Ω• +U, −dwU)) . +The virtual fundamental class of X is defined to be +[U]vir +w := +� +ℓ +� +i=1 +ri +� +tdchU +Z KU ∈ Heven +Z +(U, (Ω• +U, −dwU)) . +where ri = ord(g(vi)), or equivalently zi = Bµri. +4.10. Effective classes. Given X = (V, G, C∗ +R, W, ζ), and g, ℓ ∈ Z≥0, define +Keff(X)g,ℓ := {βG ∈ LQ : LGpre +g,ℓ (X, βG) is nonempty}. +We now give a more explicit description of Keff(X)g,ℓ when ℓ = 1. Let ((C, z), P, ρ, u) be an object in LGpre +g,1(X, βG). +Then +u(z) ∈ Xζ = +� +I∈Amin +ζ +XI. +If u(z) ∈ XI then ui(x) ̸= 0 for i ∈ I. We observe that +ui(z) ̸= 0 ⇒ deg Li ≥ 0 and agez(L) = 0 ⇔ deg Li ∈ Z≥0 +20 + +since deg Li − agez(L) ∈ Z. Therefore, +Keff(X)g,1 ⊂ +� +I∈Amin +ζ +Keff +I (X)g,1, +where +(4.17) +Keff +I (X)g,1 = {βG ∈ LQ : ⟨Di, βG⟩ + qi +2 (2g − 1) ∈ Z≥0 for all i ∈ I}. +Indeed, it is not hard to see that +(4.18) +Keff(X)g,1 = +� +I∈Amin +ζ +Keff +I (X)g,1. +Let {D∗I +i +: i ∈ I} be the Q-basis of LQ ∼= Qκ dual to the Q-basis {Di : i ∈ I} of L∨ +Q: for any i, j ∈ I, +⟨Di, D∗I +j ⟩ = δij +Then +Keff +I (Xζ, wζ)g,1 = +� � +i∈I +(mi − qi +2 (2g − 1))D∗I +i +: mi ∈ Z≥0 +� +4.11. Stacky loop spaces. In orbifold quasimap theory [CKK], the Jϵ-function is defined via C∗ localization on +genus-zero ϵ-stable quasimap graph spaces. When the gauge group G = (C∗)κ is an algebraic torus, the small +I-function I = J0+|t=0 can be computed completely explicitly by C∗ localization on stacky loop spaces. Similarly, +we may define the small Jϵ-function of a GLSM via C∗ localization on genus-zero ϵ-stable LG quasimap graph +spaces. When the gauge group G = (C∗)κ is an algebraic torus, the small I-function I = J0+|t=0 of the GLSM +can be computed by C∗ localization on the LG version of stacky loop spaces. In this paper, we define and compute +K-theoretic and cohomological I-functions via C∗ localization on stacky loop spaces defined in this subsection. +4.11.1. Classical version. Given any subset I of {1, . . . , n + κ}, define +VI = {(x1, . . . , xκ+n) ∈ V = Cn+κ | xi ̸= 0 if i ∈ I} = CI′ × (C∗)I +where I′ = {1, . . . , n + κ} \ I. Then +V ss +G (ζ) = +� +I∈Amin +ζ +VI, +and +Xζ = [V ss +G (ζ)/G] = +� +I∈Amin +ζ +XI +where XI = [VI/G] ⊂ Xζ is an open toric substack which is an affine toric orbifold defined by the n-dimensional +cone σI′ spanned by {vi : i ∈ I′}. XI contains a unique torus fixed (possibly stacky) point +pI = +�� +{0}I′ × (C∗)I� � +G +� += +�� +{0} +¯I × {1}I� � +GσI′ +� +∼= BGσI′ +where GσI′ ⊂ G is the stabilizer of the point {0}I′ × {1}I ∈ V (see Section 2.4). Let +NσI′ = +� +i∈I +Zvi. +Then GσI′ is a finite abelian group, and +GσI′ ∼= N/NσI′ , +Hom(GσI′ , C∗) ∼= L∨�� � +i∈I +ZDi +� +. +21 + +4.11.2. Quantum version. We introduce the following convention. Given any rational number m/a, where m ∈ Z, +a ∈ Z>0, and m, a are coprime, we define +Hi(P1, OP1(m +a )) := Hi(P[a, 1], OP[a,1](m)), +i = 0, 1. +Recall that the total space of OP1[a,1](m) is [ +� +(C2 − {0}) × C +� +/C∗], where C∗ acts by weights (a, 1, m). +Given any β ∈ Keff(Xζ, wζ)0,1, we let +(4.19) +Vβ = +n+κ +� +i=1 +Vβ,i, +where Vβ,i = H0 � +P1, OP1(⟨Di, β⟩ − qi +2 ) +� +, +and let +(4.20) +Wβ = +n+κ +� +i=1 +Wβ,i, +where Wβ,i = H1 � +P1, OP1(⟨Di, β⟩ − qi +2 ) +� +. +Let χDi : G → C∗, where 1 ≤ i ≤ n + κ, be defined as in Section 2. Let G act on Vβ,i and Wβ,i by g · u = χDi(g)u +where g ∈ G and u ∈ Vβ,i or Wβ,i. +Definition 4.10 (stacky loop space). We define the degree β stacky loop space by +(4.21) +Xζ,β := [V ss +β (ζ)/G]. +The stacky loop space in the above definition is the analogue of the stacky loop space in orbifold quasimap +theory [CCK, Section 4.2], which can be viewed as the orbifold version of Givental’s toric map space [Gi98, Section +5]. +We have +V ss +β (ζ) = +� +I∈Amin +ζ +Vβ,I +where +Vβ,I = {(u1, . . . , un+κ) ∈ Vβ | ui ̸= 0 if i ∈ I}. +Definition 4.11 (obstruction bundle and obstruction sheaf). We define the degree β obstruction bundle by +Obζ,β = [ +� +V ss +β (ζ) × Wβ +� +/G]. +The degree β obstruction sheaf Obζ,β is the locally free sheaf of OXζ,β-modules on Xζ,β associated to the vector +bundle Obζ,β. +The obstruction bundle Obζ,β is a toric vector bundle over the smooth toric DM stack Xζ,β, and +Obζ,β = Spec +� +SymOb∨ +ζ,β +� +. +Let TXζ,β be the tangent sheaf of Xζ,β. The two-term complex +(4.22) +� +TXζ,β +0 +−→ Obζ,β +� +is a perfect tangent-obstruction complex [LT98] on Xβ,ζ. Taking the dual of (4.22), we obtain +(4.23) +� +Ob∨ +ζ,β +0 +−→ ΩXζ,β +� +which is a perfect obstruction theory [BF97] on Xζ,β. In particular, the tangent-obstruction complex (4.22) and the +perfect obstruction theory are objects in D(Xζ,β), the derived category of coherent sheaves on Xζ,β. The virtual +tangent bundle is +T vir +Xζ,β = TXζ,β − ObXζ,β ∈ K(Xζ,β). +Let TXζ,β/BG be the relative tangent bundle of the smooth map Xζ,β → BG. We have the following short exact +sequence of vector bundles on Xζ,β: +(4.24) +0 → Xζ,β × LC → TXζ,β/BG = [ +� +V ss +β (ζ) × Vβ +� +/G] → TXζ,β → 0. +Given I ∈ Amin +ζ +, σI′ is an n-dimensional cone in Σ. Define +Keff +I (X, ζ)0,1 +−→ +Box(σI′) = +� � +i∈I′ +aivi : ai ∈ [0, 1) ∩ Q +� +β +�→ +v(β) := +� +i∈I′ +{−⟨Di, β⟩ + qi +2 }vi. +22 + +The big torus �T = (C∗)n+κ acts on Vβ by +(�t1, . . . , �tn+κ) · (u1, . . . , un+κ) = (�t1u1, . . . , �tn+κun+κ). +This induces an action of T = �T/G ∼= (C∗)n (the flavor torus) on Xζ,β. The tangent sheaf TXζ,β and the obstruction +sheaf Obζ,β are T-equivariant locally free sheaves Xζ,β, so the the perfect obstruction theory is T-equivariant and +is an object in DT (Xζ,β), the derived category of T-equivariant coherent sheaves on Xζ,β, and T vir +Xζ,β ∈ KT (Xζ,β). +Define +V ◦ +β,I = {(u1, . . . , un+κ) ∈ Vβ | ui(1, 0) ̸= 0 if i ∈ I}. +Then V ◦ +β,I is a Zariski open dense subset of Vβ,I, and +V ss +β (ζ)◦ := +� +I∈Amin +ζ +V ◦ +β,I +is a Zariski dense open subset of V ss +β (ζ). Define +X◦ +ζ,β := [V ss +β (ζ)◦/G] +which is the open substack of Xζ,β. (Our notation X◦ +ζ,β is motivated by Okounkov’s notation in [Ok20], in which +QM◦ denotes the open subtack of QM where ∞ = [1, 0] is not a base point.) +Given I ∈ Amin +ζ +, σI′ is a top-dimensional (i.e. n-dimensional) cone in Σ. Recall that g(v) ∈ GI is the image of +v = � +i∈I′ aivi ∈ Box(σI′) under the bijection Box(σI′) → GσI′ . The g(v)-fixed subspace of V is +V g(v) = {x = (x1, . . . , xn+κ) ∈ V | xi = 0 if i ∈ I′ and ai /∈ Z}. +The connected component Xζ,v of the inertia stack IXζ associated to v is Xζ,v = +�� +V g(v) ∩ V ss(ζ) +� +/G +� +which is an +open dense substack of the Artin stack Xv = [V g(v)/G]. There is an evaluation map +ev∞ : Xζ,β −→ Xv(β) +[u1, . . . , un+κ] �→ [u1(1, 0), . . . , un+κ(1, 0)]. +Then +ev−1 +∞ (Xζ,v(β)) = X◦ +ζ,β +4.12. Torus actions and C∗ +q fixed points. In orbifold quasimap theory, the I-function is obtained by torus +localization on the stacky loop space, using the C∗-action on P1, the coarse moduli space of P[a, 1] where a is a +positive integer. We denote this C∗ by C∗ +q since it corresponds to C× +q in [Ok20]. For �T-equivariant parameters, we +use notation similar to that in [CIJ] and [GiV]. +KC∗q(•) = K(BC∗ +q) = Z[q±1], +K � +T (•) = K(B �T) = Z[Λ±1 +1 , . . . , Λ±1 +n+κ], +Let +z = c1(q) ∈ H2 +C∗ +q(•; Z), +λj = −c1(Λj) ∈ H2 +� +T (•; Z). +Then +H∗ +C∗q(•; Z) = H∗(BC∗ +q; Z) = Z[z], +H∗ +� +T (•; Z) = H∗(B �T; Z) = Z[λ1, . . . , λn+κ], +� +M = +n+κ +� +j=1 +Zλj. +Let deg(x) = a, deg(y) = 1. Then +C[x, y] = +∞ +� +m=0 +C[x, y]m +where C[x, y]m denote the degree m part of the graded ring C[x, y]. If m ∈ Z≥0 then +H0(P[a, 1], OP1[a,1](m)) = C[x, y]m = +⌊ m +a ⌋ +� +k=0 +Cxkym−ka, +H1(P[a, 1], OP1[a,1](m)) = 0. +Let C∗ +q act on P[a, 1] by q · [x, y] = [qx, y] = [x, q−1/ay], and on C[x, y] by q · x = x, q · y = q−1/ay. Given any +number r ∈ Q, let ⌊r⌋ be the unique integer such that ⌊r⌋ ≤ r < ⌊r⌋ + 1, and let {r} := r − ⌊r⌋ ∈ [0, 1). As an +23 + +element in KC∗q(•), +H0(P[a, 1], OP1[a,1](m)) − H1(P[a, 1], OP1[a,1](m)) += +q−{ m +a } +1 − q−1 + q− m +a +1 − q = +∞ +� +k=0 +q−{ m +a }−k − +∞ +� +k=0 +q− m +a −1−k += +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +⌊ m +a ⌋ +� +k=0 +q−{ m +a }−k, +m ≥ 0 +0, +−a ≤ m ≤ −1 +− +−⌊ m +a ⌋−1 +� +k=1 +q−{ m +a }+k, +m ≤ −a − 1 +For j = 1, . . . , n + κ, we define +(4.25) +dj(β) := ⟨Dj, β⟩ − qj +2 ∈ Q +Then +(4.26) +V j +β = +� +� +� +� +� +� +� +⌊dj(β)⌋ +� +k=0 +Λjq−{dj(β)}−k, +dj(β) ≥ 0; +0, +dj(β) < 0 +, +W j +β = +� +� +� +� +� +� +� +0, +dj(β) ≥ −1, +−⌊dj(β)⌋−1 +� +k=1 +Λjq−{dj(β)}+k, +dj(β) < −1. +(V j +β )C∗ +q = +� +C, +dj(β) ∈ Z≥0, +0, +otherwise. +Write +V = +n+κ +� +j=1 +Vj = V + +β ⊕ V − +β ⊕ V ⊥ +β +where Vj = SpecC[xj], +(4.27) +V + +β := +� +j∈[1..n+κ] +dj(β)∈Z≥0 +Vj, +V − +β := +� +j∈[1..n+κ] +dj(β)∈Z<0 +Vj, +V ⊥ +β := +� +j∈[1..n+κ] +dj(β)/∈Z +Vj. +Then +V + +β = V +C∗ +q +β , +V + +β ⊕ V − +β = V g(v(β)). +We define +Fβ := +�� +V + +β ∩ V ss +G (ζ) +�� +G +� +. +Then Fβ is a closed substack of Xζ,v(β). +ev∞ : X◦ +ζ,β → Xζ,v(β) +restricts to an isomorphism +ev∞ : +� +X◦ +ζ,β +�C∗ +q → Fβ. +We identify +� +X◦ +ζ,β +�C∗ +q with Fβ under the above isomorphism. +If I ∈ Amin +ζ +and β ∈ Keff(Xζ, wζ) then the torus fixed point pI is contained in Fβ if and only of β ∈ Keff +I . +4.13. Virtual tangent and normal bundles. Since C∗ +q acts trivially on Fβ, it acts linearly on the fibers of any +C∗ +q-equivariant vector bundle on Fβ. If V is a C∗ +q-equivariant vector bundle over Fβ then +V = +� +d∈Z +Vd = V f ⊕ V m, +where Vd is the subbundle on which C∗ +q acts by weight d, and V m = +� +d̸=0 +Vd (resp. V f = V0) is the moving (resp. +fixed) part of V under the C∗ +q-action. Let +T 1 +β := TXζ,β +�� +Fβ , +T 2 +β := Obζ,β|Fβ . +Then T 1,f +β += TFβ is the tangent bundle of Fβ, and T 1,m +β += NFβ/X◦ +ζ,β is the normal bundle of Fβ in Xζ,β. +24 + +The virtual tangent bundle of Fβ is +T 1,f +β +− T 2,f +β += TFβ − 0 = TFβ. +Therefore, +[Fβ]vir = [Fβ]. +The virtual normal bundle of Fβ is defined to be +(4.28) +N vir +β +:= T 1,m +β +− T 2,m +β +∈ KC∗q× � +T (Fβ). +Let +ιβ→v(β) : Fβ → Xζ,v(β), +ιv : Xζ,v → Xζ, +ιβ = ιv(β) ◦ ιβ→v(β) : Fβ → Xζ +be inclusion maps. +Proposition 4.12. +(4.29) +N vir +β += ι∗ +β→v(β) � +N vir +β +where +(4.30) +� +N vir +β += +n+κ +� +j=1 +ι∗ +v(β)(U +� +T +j )−1� ∞ +� +k=0 +q−dj(β)+k − +∞ +� +k=0 +q{−dj(β)}+k� ++ +� +dj(β)∈Z<0 +ι∗ +v(β)(U +� +T +j )−1 +∈ KC∗q× � +T +� +Xζ,v(β) +� +. +Proof. +T 1,m +β += +� +dj(β)≥0 +ι∗ +β(U +� +T +j )−1� +� +k∈Z +0≤k 0. +Remark 5.3. The formula (5.4) is a multidimensional inverse Mellin transform (see e.g. [Ts]) of Γe2π√−1⟨t,σ⟩, +where the G-character t ∈ L∨ is identified with the line bundle Lt ∈ KG(V ) through the natural isomorphism. +Due to Proposition 5.8 +Remark 5.4. The right hand side of (5.4) converges in the domain +(5.5) +Ut := +� +B ∈ gR | |⟨B + t, ν⟩| < +� +i +|⟨Di, ν⟩|/4 for all ν ∈ gR\{0} +� +. +In particular, this holds if B = −t, so that Ut is non-empty for any character t. This condition is quite restrictive, +in particular none of the branes in K(MF([V/G], w)) in the the mirror quintic example satisfies this condition. +Expansion in phases. Now we turn to the description of the disk partition function component in the phases of +the GLSM. +Let C be a maximal cone of the secondary fan. The set of minimal anticones is defined as: +(5.6) +Amin +C +:= {I ⊂ [1..n + κ] | C ⊂ ∠I, |I| is minimal} +where ∠I is defined as in Section 2.3. The latter condition is equivalent to |I| = κ. To formulate our version of +Higgs-Coulomb Correspondence (Theorem 5.6), we recall some notations from Definition 2.7. For a fixed minimal +anticone I ∈ Amin +C +, the G-characters {Di : i ∈ I} form a Q-basis of L∨ +Q. We define {D∗,I +i +: i ∈ I} to be the dual +Q-basis of LQ. +Definition 5.5 (chamber hemisphere partition function). Let C be a maximal cone of the secondary fan. Define +the chamber hemisphere partition function +(5.7) +ZD2(Lt)C := (−1)κ +� +I∈Amin +C +���� +Λκ +a=1ξ∗ +a +Λκ +a=1Dia +���� +� +m∈(Z≥0)κ +� +i′∈I′ +Γ(⟨Di′, σm⟩ + αi′) +� +i∈I +(−1)mi +mi! +exp(⟨θ + 2π +√ +−1t, σm⟩), +where m = (mi1, . . . , miκ), I = {i1, . . . , iκ}, and σm = − � +i∈I(mi + αi)D∗,I +i +. +Theorem 5.6 (Higgs-Coulomb correspondence). Let UC := � +I∈Amin +C +UI ⊂ C, where UI is defined in Proposi- +tion A.3. UC is open and nonempty and if ζ ∈ UC and B ∈ Ut then we have the equality +(5.8) +ZD2(Lt) = ZD2(Lt)C +Proof. Under the assumptions in the theorem all the integrals and series in (5.8) converge due to Proposition 5.8 +and Proposition A.3. +Now we turn to the proof of the equality of the right hand side and left hand side. Our strategy is to deform the +integration contour in (5.4) while separating the contributions from different torus fixed points. +First we need to introduce some notations to work with contours. Let πi +mi denote a hyperplane ⟨Di, σ⟩+αi = −mi +and +π{i1,...,il} +{mi1,...,mil} := +l� +k=1 +{⟨Dik, σ⟩ + αik = −mik} . +Below we will use the following notations: Ik = {i1, . . . , ik} ⊂ {1, . . . , n + κ} such that Di1, . . . , Dik are linearly +independent if not stated otherwise. Ior +k stands for an ordering (i1, . . . , ik) of Ik. In addition mIk = (mi1, . . . , mik) ∈ +(Z≥0)k. +Given such Ik consider the following exact sequence: +1 → GIk → G +Di1,...,Dik +−→ +HIk ≃ (C∗)k → 1, +where HIk := G/GIk. +We also denote the corresponding Lie algebras by gIk (of dimension κ − k) and hIk +(of dimension k). +Canonically h∨ +Ik = ⟨Di1, . . . , Dik⟩. +Note that πIk +mIk is an affine space parallel to gIk so that +πg→hIk (πi +mi) is a hyperplane in hIk whenever i ∈ Ik. We will denote these hyperplanes by the same symbol. If +k = κ, then πIk +mIk is just a point and GIk = GσI′ +k is finite. +Given a point p ∈ πIk +mIk that is not contained in the other hyperplanes consider an analytic neighbourhood U +that does not intersect other hyperplanes and let [p] and U0 = πg→hIk (U) be their projections to the quotient hIk. +2The condition specifies a chamber in the hyperplane arrangement of pole hyperplanes of the integrand. The fact that such a chamber +is nonempty is not immediate. For the mirror quintic we can choose δ = (�100 +a=1 s1a)cξ101 − c �100 +a=1 ξa, where c is a small positive +number, 0 < 500c < 1. +33 + +The linear independence condition implies that �k +l=1 πilmil is a simple normal crossing divisor for any mIk +with the center at [p], so U0\ �k +l=1 πilmil is homotopically equivalent to (S1)k. +Let CIor +k +0 ([p]) denote a generator +in Hk(U0\ �k +l=1 πilmil ) oriented by the form (2π√−1)−kdDi1/Di1 ∧ . . . ∧ dDik/Dik, that is in the basis of (complex) +linear functions on hIk given by zl = Dil we have +(5.9) +1 +(2π√−1)k +� +C +Ior +k +0 +([p]) +Λk +l=1 +dzl +zl += 1. +Let also CIor +k +0 (p) denote a generator of +Hk(U\ +k� +l=1 +πilmil ) +projecting to CIor +k +0 ([p]). Let Ω ∈ Λκ−k(gIk)∨ +R be a volume form. Such a form defines an orientation of √−1(gIk)R. +This is canonically the Lie algebra of (GIk)comp, the maximal compact subgroup of GIk. +We define a cycle +CIor +k +Ω (p) ∈ Hκ(g\Polar, |ℑ(σ)| ≫ 0) +as a class of CIor +k +0 (p) + √−1(gIk)R oriented by the form (2π√−1)−kΛk +l=1dDil/Dil ∧ Ω. In the definition above Polar +stands for the polar divisor of the integrand, that is �n+κ +i=1 +� +mi≥0 πi +mi. If k = κ, then Ω ∈ R. If Ω = 1 we usually +omit it from the notation. +Remark 5.7. We will be considering integrals of the form +� +C +Ior +k +Ω +(p) dσ Γ · e⟨θ,σ⟩. Such integrals depend only on the +homology class of the cycle in the group +Hκ(g\ +n+κ +� +i=1 +� +m≥0 +πi +mi, |ℑ(σ)| ≫ 0). +In particular, two cycles are homologous if they are related by a smooth homotopy that leaves the set |ℑ(σ)| ≫ 0 +invariant and does not cross polar hyperplanes. +Lemma 5.8. Let CIor +k +Ω (p) be as above, k < κ and +(5.10) +|⟨B, ν⟩| < 1 +4 +n+κ +� +i=1 +⟨Di, ν⟩ for all ν ∈ gIk\{0}. +Then the integral +(5.11) +� +C +Ior +k +Ω +(p) +dσ Γ · e⟨θ,σ⟩ +is absolutely convergent. +In particular it is always convergent if B ∈ g⊥ +Ik. +We show that Γ · e⟨θ,σ⟩ uniformly +exponentially decays as |σ| → ∞ on the integration cycle. +Proof. We present the proof of this statement in Appendix A. +□ +Lemma 5.9 (One plane crossing). Let Ior +k = (Ior +k−1, i) such that gIk−1 ̸= gIk−1∪{i}, Ω be a volume form on (gIk−1)R +and p ∈ πIk +mIk be generic (does not intersect other polar hyperplanes). Pick f ∈ gIk−1\gIk. +(5.12) +C +Ior +k−1 +Ω +(p + εf) = CIor +k +ιf Ω(p) + C +Ior +k−1 +Ω +(p − εf), +where ιfΩ is a κ − k form on (gIk)R and ε small enough. +Proof. Let D(z, r) denote a disk with center at z ∈ C and radius r. Consider a linear coordinate system σ1, . . . , σκ +on g with the center at p such that CIor +k +Ω (r + εf) can be represented by the cycle +(∂D(0, α))k−1 × (ε + +√ +−1R) × ( +√ +−1Rκ−k) +for for some α ∈ R\{0} such that f = ∂/∂σk. Then the form Ω is equal to c dσk ∧ · · · ∧ dσκ, where c ∈ R\{0}. This +cycle is homotopic to the union +(∂D(0, α))k−1 × (−ε + +√ +−1R) × ( +√ +−1Rκ−k) +34 + +oriented by the same form and +(∂D(0, α))k × ( +√ +−1Rκ−k) +oriented by the form c dσk+1 ∧ . . . ∧ dσκ = ιfΩ. The homotopy is nontrivial only in the k-th component, where it +is shown in Figure 1. +ε +−ε +σk +Figure 1. Basic contour deformation +□ +Applying the lemma several times we get the following statement: +Lemma 5.10. Consider Ior +k−1, a volume form Ω on (gIk)R and p ∈ πIk−1 +mIk−1 generic (does not intersect other polar +hyperplanes) and f ∈ gIk−1\ � +i, gIk−1̸=gIk−1∪{i} gIk−1∪{i}. +(5.13) +C +Ior +k−1 +Ω +(p) = − +� +i/∈Ik−1 +� +mi≥0 +CIor +k +ιf Ω({p + [0, N]f} ∩ πi +mi) + C +Ior +k−1 +Ω +(p + Nf), +where ιfΩ is a κ − k form on (gIk)R and the cycle is empty if the intersection is. We note that only i such that +gIk−1 ̸= gIk−1∪{i} enter the right hand side. +Remark 5.11. Later we consider integrals of the form +� +C +dσ Γ · e⟨θ,σ⟩, +C ∈ Hκ(g\Polar, |ℑ(σ)| ≫ 0). Let {Cα}α∈A ⊂ Hκ(g\Polar, |ℑ(σ)| ≫ 0). If A is an infinite set, then by abuse of +notations when we write an expression of the form +(5.14) +� +α∈A +Cα = C +we actually mean +(5.15) +� +α∈A +� +Cα +dσ Γ · e⟨θ,σ⟩ = +� +C +dσ Γ · e⟨θ,σ⟩, +when the left hand side converges. We use this notation to simplify already bulky notations. +Corollary 5.12. Let p, Ω, N be as in Lemma 5.10 then there exist a constant const that does not depend on p and +f such that if ⟨ζ, f⟩ < const then +(5.16) +C +Ior +k−1 +Ω +(p) = − +� +i/∈Ik−1 +� +mi≥0 +CIor +k +ιf Ω({p + R≥0f} ∩ πi +mi). +Proof. Lemma 5.10 implies +(5.17) +� +C +Ior +k−1 +Ω +(p) +dσ Γ · e⟨θ,σ⟩ = − +� +i/∈Ik−1 +� +mi≥0 +� +C +Ior +k +ιf Ω({p+[0,N]f}∩πimi) +dσ Γ · e⟨θ,σ⟩ + +� +C +Ior +k−1 +Ω +(p+Nf) +dσ Γ · e⟨θ,σ⟩. +Using Proposition A.5 with p → p + Nf we can estimate the last term on the right hand side: +(5.18) +� +C +Ior +k−1 +Ω +(p+Nf) +dσ Γ · e⟨θ,σ⟩ ≤ e−const|p+Nf|, +35 + +In particular, the integral approaches 0 as N → ∞. Now the corollary follows from taking the limit N → ∞. +□ +Cdx∧dy(p) +C(1) +f ⊥(q) +C(2,1) +1 +((−4, −4)) +C(1,2) +1 +((−4, −1)) +gR +Figure 2. Contour deformations in projection to gR. +Figure 2 shows two consecutive applications of Corollary 5.12 in a simple example. The plane is a real part +gR of the two-dimensional Lie algebra g ≃ C2. +The picture shows only 2 series of polar divisors located at +x = −1, −2, −3, . . . and y = −1, −2, −3, . . .. The initial contour Cdx∧dy(p) is a purely imaginary contour depicted +by a green dot. +Its projection to the real Lie algebra is just a point. +Direction of deformation f is a dashed +green line. After first application of Corollary 5.12 the contour transforms into two series of copies of R × S1 +denoted by purple intervals. Indeed, projections of these contours to gR are intervals intersecting the corresponding +polar hyperplanes. Each of this contour is deformed parallel to the respective polar hyperplane applying Corol- +lary 5.12 again. The contours deform to a number of T 2 ≃ (S1)2 winding around each crossing of polar hyperplanes. +Let us introduce some more notations for cones of different dimensions in g. Given I such that {Di}i∈I form a +basis of g∨ consider the cone +(5.19) +∠∗,ζ +I += { +� +i∈I +sign(⟨ζ, D∗,I +i +⟩)ciD∗,I +i +| ci ≥ 0}, +and its subcones: +(5.20) +∠∗,ζ +J⊂I = ∠∗,ζ +I +∩ gJ. +The signs are chosen so that ⟨ζ, σ⟩ ≥ 0 for any σ ∈ ∠∗,ζ +J⊂I. We also define +signζ +J⊂I = (−1)#{i∈I\J | ⟨ζ,D∗,I +i +⟩<0}. +In particular, if I is a minimal anticone (which is the main case of interest) then ∠∗,ζ +I += ∠∗ +I, where ∠∗ +I ⊂ gR is the +dual cone of ∠I ⊂ g∨ +R, and signζ +J⊂I = 1 for all J ⊂ I. We also have ∠∗,ζ +∅⊂I = ∠∗,ζ +I +and ∠∗,ζ +I⊂I = 0. +Further recall that +Keff +I += +� � +i∈I +miD∗,I +i +��� mi ∈ Z≥0 +� +and define +Keff +I (α) := − +� +i∈I +αiD∗,I +i +− Keff +I += +� +− +� +i∈I +(αi + mi)D∗,I +i +��� mi ∈ Z≥0 +� +. +36 + +If αi = qi/2 then Keff +I (α) = −Keff +I (Xζ, wζ). If p ∈ πIk +mIk then +p−∠∗ζ +Ik⊂I = +� +− +� +i∈Ik +(αi+mi)D∗,I +i ++ +� +i∈I\Ik +(⟨Di, p⟩−ci)D∗,I +i +��� ci ∈ R, ci ≥ 0 if ⟨ζ, D∗,I +i +⟩ > 0 and ci ≤ 0 if ⟨��, D∗,I +i +⟩ < 0 +� +. +The following lemma is the main technical lemma of the proof. It relates integrals over (S1)k × Rκ−k to sums +of cycles over (S1)κ in some κ − k-dimensional cone by recursively applying Corollary 5.12 and checking that only +the contributions in the corresponding cone survive. +Lemma 5.13. Consider Ior +k , a volume form Ω on (gIk)R and p ∈ πIk +mIk generic (does not intersect other polar +hyperplanes). Then +(5.21) +CIor +k +Ω (p) = (−1)κ−k � +I,Ik⊂I +signζ +Ik⊂I +� +q∈(p−∠∗,ζ +Ik⊂I)∩Keff +I (α) +sign +�Λk +l=1Dil ∧ Ω +Λκ +l=1Dil +� +CIor(q). +We note that we can describe the set (p − ∠∗,ζ +Ik⊂I) ∩ Keff +I (α) explicitly as follows. Let −ni′ := ⌊⟨Di′, p⟩ + αi′⌋ ∈ Z for +i′ ∈ I\Ik. Then +(p − ∠∗,ζ +Ik⊂I) ∩ Keff +I (α) = +� +− +� +i∈I +(αi + mi)D∗,I +i +��� i ∈ I\Ik =⇒ mi ∈ Z and +� +ni ≤ mi +if ⟨ζ, D∗,I +i +⟩ > 0, +0 ≤ mi < ni +if ⟨ζ, D∗,I +i +⟩ < 0. +� +. +Proof. The right hand side is convergent due to Proposition A.3 since the series in the right hand side is a subseries +of the one considered in the proposition. +We prove the lemma by recursion in k. First, let k = κ − 1. We apply Corollary 5.12 where f ∈ (gIκ−1)R ≃ R +such that ⟨ζ, f⟩ < 0. Ω ∈ (gIκ−1)∨ +R ≃ g∨ +R/ +� +⊕i∈Iκ−1RDi +� +, so that for each i /∈ Iκ−1 one can write Ω = ciDi (where +right hand side is a representative). In particular, ιfΩ = ci⟨Di, f⟩, where +ci = Λκ−1 +l=1 Dil ∧ Ω +Λκ−1 +l=1 Dil ∧ Di +. +Furthermore, sign(⟨Di, f⟩) = sign(⟨ζ, D∗,I +i +⟩) = signζ +Iκ−1⊂I. So we have +(5.22) +C +Ior +κ−1 +Ω +(p) = − +� +i/∈Iκ−1 +� +mi≥0 +C(Ior,i) +ci⟨Di,f⟩({p + R≥0f} ∩ πi +mi). +Thus the theorem in this case follows. +We remark that the conditions in Corollary 5.12 are stricter than the convergence condition of the theorem, but +the statement is about equality of the analytic functions is some domain, so it extends to the maximal domain of +the mutual convergence. +Consider the general case 0 ≤ k < κ − 1. We choose a generic f ∈ (gIk)R ≃ Rκ−k such that ⟨ζ, f⟩ < 0 and apply +Corollary 5.12 +(5.23) +CIor +k +Ω (p) = − +� +i/∈Ik +� +mi≥0 +C(Ior +k ,i) +ιf Ω +({p + [0, N]f} ∩ πi +mi) + CIor +k +Ω (p + Nf). +The sum over mi ∈ Z restricts to a set +(5.24) +� +ni ≤ mi, ⟨Di, f⟩ > 0, +0 ≤ mi < ni, ⟨Di, f⟩ < 0, +where again −ni = ⌊⟨Di, p⟩ + αi⌋. In particular, if I is a minimal anticone then the second case does not appear. +Now we can use the recursion step for each term in the double sum on the right hand side. +(5.25) +C(Ior +k ,i) +ιf Ω +(˜p = {p + [0, N]f} ∩ πi +mi) = +� +I,Ik∪{i}⊂I +signζ +Ik∪{i}⊂I +� +q∈(˜p−∠∗,ζ +Ik∪{i}⊂I)∩Keff +I (α) +sign +�Λk +l=1Dil ∧ Di ∧ ιfΩ +Λκ +l=1Dil +� +CIor(q) +Let us analyze when q ∈ Keff +I (α) appears in the expansion of the formula (5.32) after applying (5.25). Such a +term appears when there exist i ∈ I′ +k, mi ∈ Z such that +(5.26) +q ∈ {p + [0, N]f} ∩ πi +mi − ∠∗,ζ +Ik∪{i}⊂I. +37 + +But � +i∈I′ +k ∠∗,ζ +Ik∪{i}⊂I = ∂∠∗,ζ +Ik⊂I. So we can rewrite equation (5.26) as +q ∈ p + cf − ∂∠∗,ζ +Ik⊂I +for some c > 0, or +(5.27) +q − cf ∈ p − ∂∠∗,ζ +Ik⊂I. +In other words, for each intersection of the ray q − R≥0f with the boundary of the cone p − ∠∗,ζ +Ik⊂I we have a +term with the cycle centered at q in the expansion of (5.32). Number of such intersections depends on whether +q ∈ p − ∠∗,ζ +Ik⊂I or not. +(1) q ∈ p − ∠∗,ζ +Ik⊂I. Scalar product of ζ with all the inward normals to ∂∠∗,ζ +Ik is by definition positive, f is a +linear combination of these normals, and ⟨ζ, f⟩ < 0. Then at least one coefficient of the linear combination +is negative, and q − R≥0f has one intersection point with p − ∂∠∗,ζ +Ik⊂I. The total contribution to the sum is +(5.28) +signζ +Ik∪{i}⊂Isign +�Λk +l=1Dil ∧ Di ∧ ιfΩ +Λκ +l=1Dil +� +CIor(q), +where +Ω = cDi ∧ Λκ +l=k+2Dil + · · · +and +ιfΩ = c⟨Di, f⟩Λκ +l=k+2Dil + · · · , +where dots denote the possible terms that vanish in the formula (5.28). Plugging this expression for the +sign, we get the equality +(5.29) +signζ +Ik∪{i}⊂Isign +�Λk +l=1Dil ∧ Di ∧ ιfΩ +Λκ +l=1Dil +� += signζ +Ik⊂Isign +�Λk +l=1Dil ∧ Ω +Λκ +l=1Dil +� +. +(2) q /∈ p − ∠∗,ζ +Ik⊂I. There are either 0 or two intersection points because the scalar product of f with at least +one inward normal to ∂∠∗,ζ +Ik⊂I is negative. The total contribution to the formula (5.32) is +(5.30) +� +signζ +Ik∪{i}⊂Isign +�Λk +l=1Dil ∧ Di ∧ ιfΩ +Λκ +l=1Dil +� ++ signζ +Ik∪{j}⊂Isign +�Λk +l=1Dil ∧ Dj ∧ ιfΩ +Λκ +l=1Dil +�� +CIor(q). +Here we can write +Ω = cDi ∧ Dj ∧ Λκ +l=k+3Dil + · · · , +where dots denote irrelevant terms again. In the first summand we have +ιfΩ = c⟨Di, f⟩Dj ∧ Λκ +l=k+3Dil + · · · , +and in the second one +ιfΩ = −c⟨Dj, f⟩Di ∧ Λκ +l=k+3Dil + · · · . +So the total coefficient is +(5.31) +signζ +Ik∪{i}sign(c⟨Dj, f⟩) + signζ +Ik∪{j}sign(c⟨Di, f⟩) = += signζ +Ik +� +sign(c⟨ζ, D∗I +j ⟩⟨Dj, f⟩) + sign(c⟨ζ, D∗,I +i +⟩⟨Di, f⟩) +� +. +In the last formula ⟨ζ, D∗,I +i +⟩Di and ⟨ζ, D∗,I +j ⟩Dj are inward normals of the cone ∠∗,ζ +Ik⊂I. However, we know that +the ray q − R≥0f intersects the cone p − ∠∗,ζ +Ik⊂I with inward pointing first intersection and outward pointing +second one, so the signs in the second line of the formula (5.31) are different and the total contribution +vanishes. +Collecting contributrions from all the points q ∈ Keff +I (α) for all I using the statement above we obtain the +statement of the lemma. The picture is shown in our toy example (we suppose that ⟨ζ, D∗{1,2} +1 +⟩, ⟨ζ, D∗{1,2} +2 +⟩ > 0) +in Figure 2. +□ +Using Lemma 5.13 in the case when CIor +Ω (p) = Cdσ(δ) (where we remind that our choice is dσ = Λκ +a=1ξ∗ +a) we write +(5.32) +� +Cdσ(δ) +dσ Γ · e⟨θ,σ⟩ = = (−1)κ +� +I∈Amin +C +� +q∈Keff +I (α) +sign +� Λκ +a=1ξ∗ +a +Λκ +a=1Dia +� � +CIor(q) +dσ Γ · e⟨θ,σ⟩. +Let us compute each term in the right hand side. +38 + +Lemma 5.14. +(5.33) +� +CIor(q) +dσ Γ · e⟨θ,σ⟩ = (2π +√ +−1)κ Λκ +a=1ξ∗ +a +Λκ +a=1Dia +� +i′∈I′ +Γ(⟨Di′, σm⟩ + αi′) +� +i∈I +(−1)mi +mi! +e⟨θ,σm⟩. +Proof. Each integral is a κ-dimensional torus around a simple normal crossing divisor and q = − � +i∈I(αi+mi)D∗,I +i +. +Let {σI +i }i∈I denote a set of coordinates on g corresponding to linear functions {Di}i∈I on g. Then we have +(5.34) +� +CIor(q) +dσ Γ · e⟨θ,σ⟩ = (2π +√ +−1)κ Res +CIor(q) dσ Γ · e⟨θ,σ⟩. +Next we use +(5.35) +dσ = Λκ +a=1ξ∗ +a +Λκ +a=1Dia +dσI +so that +(5.36) +Res +CIor(q) dσ Γ · e⟨θ,σ⟩ = Λκ +a=1ξ∗ +a +Λκ∗a +a=1Dia +κ +� +a=1 +Res +σI +ia→−αia−mia +dσI +ia Γ · e⟨θ,σ⟩ = += Λκ +a=1ξ∗ +a +Λκ∗a +a=1Dia +� +i′∈I′ +Γ(⟨Di′, σm⟩ + αi′) +� +i∈I +(−1)mi +mi! +e⟨θ,σm⟩, +where σm = q. +□ +Substituting this expression into (5.32) we prove the proposition. +□ +Corollary 5.15. Power series in (5.8) for different stability chambers are analytic continuations of each other in the +parameter ζ = ℜ(θ). The analytic continuation is through the integral representation of the disk partition function. +This analytic continuation can be understood as a particular case of the Crepant Transformation Conjecture [Ru99, +Ru02,Ru06] for toric Deligne-Mumford stacks proved in [CIJ]. +5.3. Disk partition functions and wall-crossing. We defined the chamber disk partition function components +corresponding to the phases of the GLSM (cones of the secondary fan). The wall-crossing is given by the analytic +continuation formula (5.4). +We would like to extend it to wall-crossing for arbitrary branes. +Let B = � +t ctLt ∈ K([V/G]). +Then for +different characters t1 and t2 the Higgs-Coulomb correspondence (5.8) hold for different domains B ∈ Ut1 and +B ∈ Ut2 correspondingly. It could happen that Ut1 ∩ Ut2 = ∅. Then we need another description for analytic +continuation. This description is given in Definition 5.18 for general one wall crossing. +First of all, we need to explain the wall-crossing setup that we use. We follow the one stated, for example, +in [BH2]. +Consider two maximal cones C+, C− of the secondary fan which are adjacent along a codimension one wall +h⊥ ⊂ g∨. Let h := (h⊥)⊥ ⊂ g and pick its integral generator h ∈ L, h = Ch. Also define hi := ⟨Di, h⟩. Then +�n+κ +i=1 hi = 0 since �n+κ +i=1 Di = 0 (Calabi-Yau condition). +In the terminology of the reference [BH2] h = (h1, . . . , hn+κ) defines the circuit that is associated to the wall- +crossing. We define +I± := {i | ± hi > 0}. +The circuit itself is {vi : i ∈ I+ ∪ I−}. +Let I0 ⊂ (I+ ∪ I−)′. If I = I0 ⊔ {i0} is a minimal anticone of C± where i0 ∈ I±, then I0 ⊔ {i±} is an anticone of +C± for each i± ∈ I±. Modification along the circuit is obtained by replacing all the cones of the form I0 ⊔ {i+} by +the anticones I0 ⊔ {i−}. We denote by A0 +C0 the set of all such I0. +Following [BH2] we call all the anticones of this form essential anticones. We have +Amin +C± = Aess +C± ⊔ Anoness +C± +. +Lemma 5.16. Nonessential anticones always contain at least one element from both I+ and I−. +Proof. Any minimal anticone must contain at least one number from I+ ∪ I− for dimensional reasons. If it does +not contain numbers from I−, then it is essential if it contains exactly one number from I+ and if it contains more +than 1, then the cones C± are adjacent along the wall of higher codimension. +□ +39 + +Remark 5.17. We use anticones instead of the cones because they are better adjusted to partition functions even +though they are completely interchangeable. Each minimal anticone corresponds to a torus fixed point of the corre- +sponding toric variety. +Definition 5.18 (Wall hemisphere partition function). Let C0 = C+ ∩ C− be a codimension one cone of the +secondary fan and +(5.37) +|⟨B + t, h⟩| < +� +i +|⟨Di, h⟩|/4. +For each Iκ−1 ∈ A0 +C0 and mIκ−1 ∈ (Z≥0)κ−1 we choose p(mIκ−1) ∈ πIκ−1 +mIκ−1 generic with the property that cardinality +of (p(m) ∓ R≥0h) ∩ Keff +Iκ−1∪{i±}(α) is bounded by c|m| for all i± ∈ I± (this property is satisfied if, for example, all +p(m) are on the same hyperplane) 3. +Define the wall hemisphere partition function corresponding to the wall C0: +(5.38) +ZD2(Lt)C0 := +� +Iκ−1∈A0 +C0 +Zess +Iκ−1(Lt) + +� +I∈Anoness +C± +ZI(Lt), +where +(5.39) +ZI(Lt) = +� +q∈Keff +I (α) +sign +� +dσ +Λκ +a=1Dia +� � +CIor(q) +dσ Γ · e⟨θ+2π√−1t,σ⟩, +I ∈ Anoness +C± +. +and +(5.40) +Zess +J (Lt) = +� +m∈(Z≥0)κ−1 +� +−sign +� +dσ +Λκ−1 +a=1 Dja ∧ Ω +� � +CJor +Ω +(p(m)) +dσ Γ · e⟨θ+2π√−1t,σ⟩+ ++ +� +i±∈I+∪I− +� +q∈(p(m)∓R≥0h)∩Keff +J∪{i±}(α) +sign +� +dσ +Λκ +a=1Dia +� � +CIor(q) +dσ Γ · e⟨θ+2π√−1t,σ⟩ +� +, +where Ω is any volume form in (gJ)R. Note that in (5.40) the sum in the second sum is finite. +Remark 5.19. There seems to be an infinite number of choices of points p(m) in the definition of the wall hemi- +sphere partition function. This is due to the fact that in the general case the cones in g corresponding to essential +anticones in the adjacent phases intersect and one cannot split the corresponding effective classes uniformly. As we +will see below the definition is independent of these choices. +Remark 5.20 (Grade restriction rule). The formula (5.37) is called the grade restriction rule since it puts a +restriction of the character t (grading of the brane). Disk partition function components can be analytically continued +directly along the walls of the secondary fan if the grade restriction rule is satisfied as is stated in the following +theorem. +Theorem 5.21 (Wall-crossing). In the setting of Definition 5.18 there exists a connected open set UC0 ⊂ gR such +that UC0 ∩ UC+ and UC0 ∩ UC− are nonempty and the wall hemisphere partition function converges for all ζ ∈ UC0, +B satisfies (5.37) and for all ζ ∈ UC0 ∩ UC± it is equal to the chamber hemisphere partition function +(5.41) +ZD2(Lt)C0 = ZD2(Lt)C± +In particular, the wall hemisphere partition function does not depend on the choices in the definition. +Proof. The convergence of the non-essential part of the right hand side is established in Proposition A.3. +Consider the essential part. Let ζ ∈ C+ ∪ C−. Then for each J we can write ζ = � +j∈J ζJ +j Dj, where ζJ +j > 0. +Note that all Dj are in the wall hyperplane (Rh)⊥, so ⟨Dj, h⟩ = 0 and for any c1 > 0 we can choose ζ far enough in +the interior of C+ ∪ C− such that ⟨ζ, p(m)/|p(m)|⟩ < −c1 for all m. Moreover, the same inequality holds for all q +in (5.40). Then we can use Proposition A.5 to estimate the integral summands and the residue summands in (5.40) +by e−c2|p(m)| and e−c2|q| respectively. The number of summands for each m is bounded by c3|m| because condition +q ∈ (p(m) ∓ R≥0h) ∩ Keff +J ⊔ {i±} is linear in m if p(m) are chosen as in the definition 5.18. Therefore, The essential +summand Zess +J (t) is bounded by � +mJ∈(Z≥0)κ−1 c3|mJ|e−c2dist(πJ +mJ ,0) < ∞. Moreover, the convergence is an open +condition on ζ, so it must hold in an open neighbourhood This finishes the proof of convergence. +3This property is rather technical and can be made even weaker. The meaning of this property is that p(m) must not go to infinity in +the direction of ±h too fast as |m| → ∞ +40 + +Consider an essential integral term in (5.40). If ζ ∈ C+ we can apply Corollary 5.12: +(5.42) +CJor +Ω (p(m)) = − +� +i±∈I± +� +mi±≥0 +C(Jor,i±) +ιhΩ +({p(m) + R≥0h} ∩ πi± +mi± ). +By slight abuse of notation we can write Ω = ci±Di± ∈ (gJ)∗ +R, so ιhΩ = ±ci±. So +(5.43) +sign +� +dσ +Λκ−1 +a=1 Dja ∧ Ω +� += sign +� +ci± +dσ +Λκ−1 +a=1 Dja ∧ Di± +� +, +and therefore +(5.44) +− sign +� +dσ +Λκ−1 +a=1 Dja ∧ Ω +� +CJor +Ω (p(m)) = ± +� +i± +� +q∈(p(m)+R≥0h)∩Keff +J∪{i±}(α) +sign +� +dσ +Λκ−1 +a=1 Dja ∧ Di± +� +C(Jor,i±)(q). +Combining this with the second term from (5.40) we obtain: +(5.45) +Zess +J (Lt) = +� +i+ +ZJ∪{i+}(Lt). +In the same way if ζ ∈ C− we compute +(5.46) +Zess +J (Lt) = +� +i− +ZJ∪{i−}(Lt). +The theorem is proved. 4 +□ +Appendix A. Convergence of multivariate hypergeometric functions +This is mostly known due to many people: [Ho] or, in the more systematic exposition [?], [?] and others. The +authors were not able to find some of the results, so we provide a short overview of the subject here. We use the +notations of the main part of the paper, particularly Section 5. +Let θ = ζ + 2π√−1B and σ = τ + √−1ν represent complex variables on g∨ and g correspondingly and Di ∈ +L∨, i ≤ n + κ be a collection of the vectors that spans g∨ over C such that �n+κ +i=1 Di = 0 (Calabi-Yau condition). +Let α ∈ Cn+κ be a generic vector and +(A.1) +Γ = Γ(σ) = +n+κ +� +i=1 +Γ(⟨Di, σ⟩ + αi). +We remind thatthe secondary fan Σ2 is defined as a fan with Σ2(1) = {Di}n+κ +i=1 and whose maximal cones are all +possible intersections of ∠I of dimension κ. +First of all we need some basic results about the gamma function. The Stirling approximation: +(A.2) +Γ(x + iy) = (2π)1/2zz−1/2e−z(1 + O( 1 +|z|)), +|Arg(z)| < π − α +where α > 0, and the Landau notation is +f(z) = O( 1 +|z|) is equivalent to lim sup +|z|→∞ +|z||f(z)| < ∞ +In particular, +(A.3) +|Γ(x + iy)| = (2π)1/2|z|x−1/2e−xe−yArg(z)(1 + O( 1 +|z|)), +|Arg(z)| < π − α. +Now let Sδ = R\ �∞ +n=0(−n − δ, −n + δ) be a domain in R separated from poles of the gamma function by a small +positive constant δ. +4We remark about the convergence again, the condition in Corollary 5.12 is stricter than in the theorem, but equality of analytic +functions extends to the maximal domain where both converge. +41 + +Lemma A.1. Let z = x + √−1y and x ∈ Sδ. Then +(A.4) +Γ(x) < const · |x|x− 1 +2 e−x. +(A.5) +|Γ(z)| < const · |z|x− 1 +2 e−min(x,0)e− π|y| +2 . +In addition, if x ∈ Sδ ∩ K, where K is a compact set, then +(A.6) +|Γ(z)| < const(|y| + 1)x−1/2e− π|y| +2 . +Proof. By Equation (A.3), there exist positive constants c1, c2 such that +(A.7) +c1|z|x− 1 +2 e−xe−yArg(z) < |Γ(z)| < c2|z|x− 1 +2 e−xe−yArg(z), +x > δ. +If x > 0, then x > δ since x ∈ Sδ. We can write Arg(z) = Arctan(y/x) = sign(y)π/2 − Arctan(x/y), so we have +−yArg(z) = −y(sign(y)π/2 − Arctan(x/y)) < π|y| +2 ++ x, +since Arctan(x/y) < x/y. Therefore, +(A.8) +|Γ(z)| < c2|z|x− 1 +2 e− π|y| +2 , +x > δ. +If x < 0, then | sin(πx)| > sin δ since x ∈ Sδ. We use the reflection formula and (A.7): +(A.9) +|Γ(z)| = +π +|Γ(1 − z) sin(πz)| < π +c1 +|1 − z|−(1−x)+1/2e1−xeyArg(1−z)| sin(πz)|−1, +x < 0. +If x < 0 then |1 − z| > |z|, |1 − z|x−1/2 < |z|x−1/2, and |Arg(1 − z)| < π +2 , so +(A.10) +|Γ(z)| < πe +c1 +|z|x−1/2e−xe +π|y| +2 | sin(πz)|−1. +The first formula (A.4) follows from (A.7) and (A.10) for y = 0. +We have sin(πz) = sin(π(x + √−1y)) = sin(πx) cosh(πy) + √−1 cos(πx) sinh(πy), +| sin(πz)| ≥ | sin(πx) cosh(πy)| > sin(πδ)eπy + e−πy +2 +> 1 +2 sin(πδ)eπ|y|. +Finally, we use this to simplify (A.10): +(A.11) +|Γ(z)| < const · |z|x−1/2e−xe +−π|y| +2 +. +Collecting the formulas (A.8) and (A.11) we get the inequality (A.5): +(A.12) +|Γ(z)| ≤ const · |z|x−1/2e− min(x,0)e−π|y|/2, +x ∈ Sδ, +where the constant depends on δ. +The third formula (A.6) follows from (A.12) and (|y| + const)x ∼ |y|x, y → ∞. +□ +We also recall the multivariate Cauchy-Hadamard theorem. +Theorem A.2 (Cauchy-Hadamard). Let +(A.13) +� +m∈(Z≥0)κ +cmzm. +The series (absolutely) converges in the polydisk with the multiradii r = (r1, . . . , rκ) if +(A.14) +lim +N→∞ supm,|m|=N(cmrm)1/|m| ≤ 1 +and such polydisk is maximal if the left hand side is equal to the right hand side. +Proposition A.3. +(1) The domain of convergence of +(A.15) +ZI = +� +m∈(Z≥0)κ +� +i′∈I′ +Γ(⟨Di′, σm⟩ + αi′) +� +i∈I +(−1)mi +mi! +exp(⟨θ, σm⟩), +is non-empty. Moreover, if the series converges at ζ0, then it also converges if ζ ∈ {ζ0} − ∠I. +42 + +(2) The domain of convergence of the series is UI + √−1Rκ, where UI ⊂ Rκ is defined by the constraints +(A.16) +⟨ζ + log Ψ(σ), σ⟩ = +� +i∈I +(⟨ζ, D∗,I +i +⟩ + log Ψi(σ))σi > 0, σ ∈ ∠∗ +I, +where Ψ(σ) = (Ψ1(σ), . . . , Ψκ(σ)) is the Horn vector defined below in the proof. +Proof. In this proof we work in the basis {D∗,I +i +}i∈I on g and the corresponding coordinate system on g and g∨. +That is if f ∈ g and f ∨ ∈ g∨, we write f = � +i fiD∗,I +i +, f ∗ = � +i f ∗ +i1Di, where fi := ⟨Di, f⟩ and f ∗ +i := ⟨f ∗, D∗I +i ⟩. +Let sI +i′i := ⟨Di′, D∗I +i ⟩. Consider the asymptotics of the series (A.15). We write the series as +(A.17) ZI = +� +m∈(Z≥0)κ +� +i′∈I′ +Γ(− +� +i +sI +i′i(mi + αi) + αi′) +� +i∈I +(−1)mi +mi! +exp(⟨θ, σm⟩) = exp(− +� +i +θiαi) +� +m∈(Z≥0)κ +cmzm, +where zm = �κ +i=1 zmi +i += � +i exp(−miθi). For generic α arguments of the gamma functions are uniformly separated +from Z≤0 by a number δ > 0 because sI +i′i are rational numbers. +Therefore we can apply Lemma A.1 and write the upper bound on the series expansion coefficients: +(A.18) +|cm| ≤ const · +� +i′∈I′(sI +i′m + ci′)(sI +i′m+ci′−1/2) +� +i∈I mmi−1/2 +i +, +where sI +i′m = +� +i∈I +sI +i′imi and ci′ = +� +i∈I +sI +i′iαi + αi′. +Thus we can write +(A.19) +lim +N→∞ +sup +m,|m|≥N +(cmrm)1/|m| ≤ +≤ lim +N→∞ +sup +m,|m|≥N +� +const +� +i′∈I′ +(sI +i′m + ci′)ci′−1/2 � +i +m−1/2 +i +· exp +� +−⟨ζ, m⟩ − +� +i′∈I′ +sI +i′m log(sI +i′m + ci′) − +� +i +mi log mi +��1/|m| +, +where |m| denotes any norm on Rn. +We define the Horn vector Ψ(σ) := � +i Ψi(σ)Di, where +(A.20) +Ψi(σ) := +σi +� +i′∈I′(sI +i′σ)sI +i′i . +In particular, log Ψ(σ) = � +i log Ψi(σ)Di, where log Ψi(σ) = log(σi) − � +i′∈I′ sI +i′i log(sI +i′σ). +We note that even +though log Ψi(σ) is not defined if any of σi, sI +i′σ = 0, the sum � +i σi log Ψi(σ) = ⟨log Ψ(σ), σ⟩ can be defined as a +limit. +Under the assumption of the proposition ζ = ℜ(θ) satisfies the equation +(A.21) +⟨ζ + log Ψ(σ), σ⟩ > 0 +for all σ ∈ Rκ\{0}, so we have +(A.22) +− ⟨ζ, m⟩ − +� +i′∈I′ +sI +i′m log(sI +i′m + ci′) − +� +i∈I +mi log mi ≤ +≤ −⟨ζ, m⟩ − +� +i′∈I′ +sI +i′m log(sI +i′m + ci′) − +� +i∈I +mi log mi + ⟨ζ + log Ψ(m), m⟩ = − +� +i′∈I′ +sI +i′m log(1 + ci′/(sI +i′m)), +where by slight abuse of notation we identify m = � +i∈I miD∗,I +i +. Consider the function x log(1 + c/x). As x → 0 we +have x log(1 + c/x) ∼ x log(c/x) → 0, and when x → ∞ then log(1 + c/x) = c/x + O(1/x2), so x log(1 + c/x) → c. +Therefore the last expression in (A.22) is bounded from above and below, so we can write +(A.23) +lim +N→∞ +sup +m,|m|≥N +(cmrm)1/|m| ≤ lim +N→∞ supm,|m|≥N exp(const +� +i′∈I′ +(sI +i′m + ci′)ci′−1/2 � +i∈I +m−1/2 +i +)1/|m| ≤ 1. +so the series (A.15) converges by the multivariate Cauchy-Hadamard theorem. If for some σ the inequality (A.21) +is not satisfied, then it is not satisfied for all m proportional to σ and the limit in (A.23) is greater than 1. This +proves the second claim of the proposition. +� +i Ψi(σ)σi is bounded and separated from 0 on the unit sphere, so � +i log Ψi(σ) is bounded on the same domain +below by a constant −N. First claim of the proposition is proved by choosing ζ = N � +i D∗,I +i +. +□ +43 + +Notation A.4. Below we work with estimates including many constants whose exact values are of no importance +and can be rather cumbersome. Notations const, consti denote various such constants. +Below we present the proof of Lemma 5.8 and Corollary 5.12. +Proof of Lemma 5.8. We use the formula (A.6) from Lemma A.1 to estimate the integrand: +(A.24) +| +n+κ +� +i=1 +Γ(⟨Di, σ⟩ + αi)e⟨θ,σ⟩| < const · +� +i/∈Ik +(|⟨Di, ℑ(σ)⟩| + 1)⟨Di,ℜ(σ)⟩+αi exp +� +�−2π⟨B, ℑ(σ)⟩ − π/2 +� +i/∈Ik +⟨Di, ℑ(σ)⟩ +� +� , +for σ on the integration contour. +We notice that condition (5.10) implies that +|⟨B, ν⟩| − 1 +4 +n+κ +� +i=1 +⟨Di, ν⟩ < −const · |ν|, +for some positive const since the expression is homogeneous of degree 1 in |ν|. +Therefore, expression in the exponential in (A.24) is bounded above by −c|ℑ(σ)| for some constant c > 0 (in +any norm on g) under the assumption of the lemma, so the absolute value of the integrand is bounded by an +exponentially decaying function. +□ +Proposition A.5. Let k < κ, Ior +k = (i1, . . . , ik) be such that {Di}i∈Ik are linearly independent. Consider a cycle +CIor +Ω (p) where p ∈ πIk +mIk and is separated from all other polar hyperplanes by positive number δ. Let B be such that +the integral +(A.25) +Int(p) := +� +C +Ior +Ik +Ω +(p) +dσ Γ · e⟨θ,σ⟩ +converges absolutely. There exist constants c1, c2 > 0 such that if ⟨ζ, p/|p|⟩ < −c1 then |Int(p)| < e−c2|p|, |p| ≫ 0. +Proof. Let us fix I and work in the basis given by {Di}i∈I. Using Fubini theorem we write +(A.26) +Int(p) = ± +� +Rκ−k +� +(S1)k dσ +n+κ +� +i=1 +Γ · e⟨θ,σ⟩ = += const +� +i∈Ik +(−1)mi +mi! +� +Rκ−k Ω +� +i/∈Ik +Γ(⟨Di, p + +√ +−1ℑ(σ)⟩ + αi)e⟨θ,σ⟩. +Let pi := ⟨Di, p⟩ and yi := ℑ(⟨Di, σ⟩), |y| := |ℑ(σ)|. Consider the main asymptotics of the Gamma functions +without argument shift by α: +(A.27) +n+κ +� +i=1 +|pi|−pi := exp +� +�− +� +i/∈Ik +pi log |pi| + +� +i∈Ik +(mi + αi) log(mi + αi) +� +� , +where the left hand side is defined as a limit if any pi = 0. Calabi-Yau condition �n+κ +i=1 Di = 0 implies that (A.27) +is a homogeneous function of degree 0 in p. We use it to simplify the asymptotics of the integrand: +(A.28) +� +i∈Ik +m−mi +i +� +i/∈Ik +|pi + αi + +√ +−1yi|pi � +i +|pi|−pi = += +� +i∈Ik +(mi + αi)αiexp +� +�� +i/∈Ik +pi log |1 + (αi + +√ +−1yi)/pi| + +� +i∈Ik +mi log(1 + αi/mk) +� +� ≤ +≤ econst·|p|exp +� +�� +i/∈Ik +pi log |1 + (αi + +√ +−1yi)/pi| +� +� , |p| → ∞. +In the last inequality we used the fact that (mi+αi)αi is bounded by a polynomial in |p| and mi log(1+αi/mi) < αi. +Analogously to the proof of Lemma 5.8 we have +(A.29) +e−2π⟨B,ℑ(σ)⟩−� +i/ +∈Ik π|yi|/2 ≤ e−const2|y|. +44 + +Now we apply Lemma A.1 to (A.25): +(A.30) +� +|pi|−pi|Int(p)| ≤ +� +i +|pi|−pi � +i∈Ik +m−mi +i +e− � +i∈Ik mie⟨ζ,p⟩× +× +� +Rκ−k Ω +� +i/∈Ik +|pi + αi + +√ +−1yi|pi+αi−1/2e− � +i/ +∈Ik min(pi+αi,0)e−2π⟨B,ℑ(σ)⟩−� +i/ +∈Ik π|yi|/2 ≤ +≤ econst4|p|+⟨ζ,p⟩e +� +i αi +� +Rκ−k Ω +� +i/∈Ik +|pi + αi + +√ +−1yi|αi−1/2exp +� +�� +i/∈Ik +pi log |1 + (αi + +√ +−1yi)/pi| +� +� e−const2|y|, +where in the last line we used (A.28) and (A.29). +We use the obvious inequality (a+b)x ≤ (2a)x+(2b)x for a, b > 0, x ∈ R\{0} to write |a+√−1b|x = (a2+b2)x/2 ≤ +2x/2(|a|x + |b|x). Furthermore, if pi < 0, +(A.31) +|1 + (αi + +√ +−1yi)/pi|pi < |1 + αi/pi|pi. +This estimate does not depend on y and is subexponential in p if pi is separated from −αi, so we can ignore these +terms in the estimate. Then +(A.32) +� +i/∈Ik,αi−1/2>0 +|pi + αi + +√ +−1yi|αi−1/2exp +� +� +� +i/∈Ik,pi>0 +pi log |1 + (αi + +√ +−1yi)/pi| +� +� < +< 2 +� +i/ +∈Ik (pi+αi−1/2)/2 +� +i/∈Ik,αi−1/2>0 +� +|pi + αi|αi−1/2 + |yi|αi−1/2� +� +i/∈Ik,pi>0 +((1 + |αi/pi|)pi + |yi|pi/|pi|pi) ≤ +≤ 2 +� +i/ +∈Ik (pi+αi−1/2)/2 +� +i/∈Ik,αi−1/2>0 +� +|pi + αi|αi−1/2 + |yi|αi−1/2� +� +i/∈Ik,pi>0 +(e|αi| + |yi|pi/|pi|pi), +where in the last inequality we use (1 + c/x)x = ex log(1+c/x) < ec, c, x > 0. The first product in the last line has +polynomial behaviour in pi so it is bounded by a polynomial in |p|. Also, factors 2pi are bounded by econst|p|. Let +us focus on the only non-trivial terms containing |yi|pi/ppi +i . We drop powers of |yi|αi−1/2 since they will produce +the same exponential asymptotics. In order to prove the claim of the proposition we prove that each integral +(A.33) +� +Rκ−k Ω +� +i∈J⊂I′ +k +|yi|pi/ppi +i e−const2|y| +is bounded above by econst|p|. Let ρ = |y|. Then in the polar coordinates on Rκ−k we write +(A.34) +� +Rκ−k Ω +� +i∈J⊂I′ +k +|y|pi/ppi +i e−const2|y| = const +� +i +p−pi +i +� ∞ +0 +dρ ρκ−k−1+� +i pie−const2ρ = += const · const +k−κ−� +i pi +2 +� +i +p−pi +i +Γ( +� +i +pi + κ − k). +If � +i pi is small, then so is the integral. Otherwise we can use the Stirling approximation. Γ(� +i pi+κ−k)/Γ(� +i pi) +is bounded by a polynomial in |p|, so we need to prove that +Γ( +� +i +pi) +� +i +p−pi +i +< econst|p|. +Stirling approximation implies: +(A.35) +Γ( +� +i +pi) < const( +� +i +pi) +� +i pi−1/2e− � +i pi. +Lemma A.6. Let a1, . . . , ak > 0 and a = �k +i=1 ai > 0. Then +(A.36) +aa � +i +a−ai +i +≤ ka +Proof. Equation (A.36) is equivalent to +(A.37) +k +� +i=1 +�ai +a +�− ai +a ≤ k +45 + +Define f : [0, ∞)k → R by +f(b1, . . . , bk) = +��k +i=1 b−bi +i +if b1, . . . , bk > 0, +0 +if bi = 0 for some i. +Then f is continuous, and is smooth on (0, ∞)k. +Let ∆k := {(b1, . . . , bk) ∈ R : bi ≥ 0, �k +i=1 bi = 1} be the +(k −1)-simplex. Then f is positive in the interior of ∆k and is zero on ∂∆k. Using Lagrange multiplier we compute +that +max +(b1,...,bk)∈∆k f(b1, . . . , bk) = f(1 +k , . . . , 1 +k ) = k. +Therefore, (A.37) holds whenever a1, . . . , ak > 0 and a = �k +i=1 ai. +□ +Applying Lemma A.6 we find that +(A.38) +Γ( +� +i +pi) +� +i +p−pi +i +≤ econst(� +i pi) ≤ econst|p|. +Returning to (A.30) we find +(A.39) +� +i +|pi|−pi|Int(p)| ≤ e⟨ζ,p⟩econst|p|, |p| → ∞, +or +(A.40) +|Int(p)| < e⟨ζ,p⟩+� +i pi log |pi|+const|p| = e(⟨ζ,p/|p|⟩+� +i pi/|p| log |pi|/|p|)+const)|p|, +where we used that � +i |pi|−pi is homogeneous of degree 0. 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