Sphere partition function and $t{t}^{\ast }$ geometry in abelian GLSM

1. Introduction

In this note, I recall the notions of integral and real structures in quantum cohomology introduced by Iritani^25,26,27 from the perspective of the quantum differential equation. We discuss the notion of $t{t}^{\ast }$-structure^9,14,23,27 in the Calabi–Yau case, where the structure can be constructed from the solution to the quantum differential equation and the real structure on the state space.

The authors in Refs. 4 and 13 provided an exact formula for the GLSM sphere partition function. It was conjectured in Refs. 5,6 and 29 that the sphere partition function computes the $t{t}^{\ast }$-geometry for the underlying model in the geometric Calabi–Yau phase of GLSM. There are various checks of this conjecture in the literature. In particular, the author did various computations of the sphere function and $t{t}^{\ast }$ geometry in the Landau–Ginzburg phase on the mirror side.^1,2 Erkinger and Knapp¹⁷ generalized these computations to hybrid model cases and conjectured the general formula.

The main result of this note is proof of the Erkinger–Knapp formula for general abelian Calabi–Yau GLSM, where the I-function has been defined in Ref. 3 using Favero and Kim¹⁹ formalism. The proof strategy is similar to the one used in Ref. 3 to prove the correspondence between the hemisphere partition function and certain components of the GLSM I-functions (quasi-central charges). Conceptually, we can turn off the superpotential of the GLSM and turn on equivariant variables on the underlying toric GIT quotient, where we can explicitly compute the $t{t}^{\ast }$-structure in Proposition 3.2. Then, we show that the nonequivariant limit exists and coincides with the GLSM expression with nonzero superpotential, see Theorem 4.1.

Mathematical GLSM invariants in the geometric phase are known to coincide with Gromov–Witten theory of a Calabi–Yau threefold complete intersection up to a sign¹² and an identification of the state spaces. In particular, the $t{t}^{\ast }$-structures of these theories are compatible. This correspondence is expected to hold more generally.

The picture turns out to be somewhat simpler for the sphere case compared to the hemisphere because one does not need to consider branes (the categories of matrix factorizations and equivariant coherent sheaves), however the convergence of the sphere partition function integral is more subtle than the hemisphere function.

2. Quantum Differential Equation, Integral and Real Structure

In this section, we review the relevant notions for the case of a smooth projective variety to simplify the notations. Later, in Sec. 3, we will use existing modifications for smooth DM toric stacks. In Sec. 4, we provide generalizations for GLSM.

2.1. QDE and $t{t}^{\ast }$ for a projective variety

Let X be a smooth projective variety over $ℂ$. Gromov–Witten theory of X endows $H^{\ast }(X)$ with a Frobenius manifold structure.¹⁵

Namely, let $\alpha ,\beta ,\gamma \in H^{\ast }(X)$.

The (big) quantum multiplication $\star :H^{\ast }(X;ℂ)\otimes H^{\ast }(X;ℂ)\to H^{\ast }(X;ℂ)\{t\}$ is defined by the following formula :

where the brackets on the right-hand side denote the Gromov–Witten invariants. We assume the convergence of the series for $t\in U\subset H^{\ast }(X)$ for the exposition, and $\{t\}$ denotes convergent power series in t. Furthermore, abusing the notations we write $H^{\ast }(X)$ instead of the convergence domain U. This multiplication is well known to be associative (WDVV equations) and commutative.

Frobenius manifold. The Frobenius manifold structure can be summarized using the Dubrovin connection. Consider the projection ${\pi }_1:H^{\ast }(X)\times {ℂ}_z^{\ast }\to H^{\ast }(X)$ and the trivial bundle :

$T{H}^{\ast }(X)$ is endowed with the trivial Levi-Civita connection ${\nabla }^{LC}$ coming from the constant Poincare pairing.

We can identify the fiber with $H^{\ast }(X)$ via $\partial /{\partial }_{t_i}={\partial }_{t_i}\to {\gamma }^i$. Dubrovin connection is the “A-model Gauss–Manin-type connection” on this trivial bundle given by

where

• the Euler vector field is

  $$E=c_1(X)+\sum _{i=1}^N\frac{2-deg({\gamma }^i)}{2}{t}_i{\partial }_{t_i}=\sum _{i,{\gamma }^i\in H^2(X)}{r}_i{\partial }_{t_i}+\sum _{i=1}^N\frac{2-deg({\gamma }^i)}{2}{t}_i{\partial }_{t_i},$$
  and $c_1(X)={\sum }_{i,{\gamma }^i\in H^2(X)}{r}_i{\gamma }^i$.

• The grading operator is

  $$\mathrm{\text{Gr}}=1-\frac{n}{2}\cdot \mathrm{\text{Id}}-{\nabla }^{LC}E=\frac{\mathrm{\text{deg}}}{2}-\frac{n}{2}.$$

The connection in z-direction is regular singular at $z=\infty$ and, in general, irregular singular at $z=0$. The constant shift in $\mathrm{\text{Gr}}$ is chosen such that the spectrum of $\mathrm{\text{Gr}}$ is symmetric.

The remarkable feature of the Dubrovin connection is that it is flat, and this flatness records most of the Frobenius manifold information.^a

Fundamental solution and J-function. The connection has a fundamental flat section of the form

where $L(t,z)$ is an End$(H^{\ast }(X))$-valued analytic function of $t,z$ and $L(t,z)=\mathrm{\text{Id}}+O(1/z)$.

The J-function is defined as

where the star means conjugation with respect to the Poincare pairing. The operators L and $L^{-1}$ conjugate the Dubrovin conection in the $T{H}^{\ast }(X)$ direction with the trivial Levi-Civita connection of the Poincare pairing :

Sesquilinear pairing. Let ${(-1)}^{\ast }$ denote the map sending $f(z)$ to $f(-z)$. For multivalued functions, it is defined by the analytic continuation along the positive direction. By the symmetry of the quantum multiplication, the z-sesquilinear pairing

is invariant with respect to $\nabla$, that is Following Ref. 21, we use uneven brackets to stress that this pairing is neither symmetric nor skew-symmetric in general. Since $L(t,z)$ is a fundamental solution to the tangent part of the quantum differential equation (1), the invariance of the pairing implies which is equivalent to the symplectic property $L^{-1}(t,z)=L^{\ast }(t,-z)$.

The pairing $[\circ ,\circ )$ is an A-model counterpart of the so-called higher residue pairing introduced by Saito.³²

On the other hand,

is skew-symmetric. It is ${\mathcal{𝒪}}_{H^{\ast }(X)}$-linear and is related to the Givental’s symplectic space $H^{\ast }(X;ℂ)((z^{-1}))$.²² This symplectic form plays a key role in the Lagrangian cone approach to Gromov–Witten theory.

Gamma structure. Iritani²⁶ introduced an integral structure on the cohomology and the space of flat sections of the Dubrovin connection called gamma integral structure. Let ${\{{\delta }_i\}}_{i=1}^n$ be the Chern roots of $TX$. Then the Gamma class is defined as

It is a regularized Euler class of the normal bundle to the space of positive loops to $\mathcal{𝒳}$. Let also the dual Gamma class be such that where the Todd class of X is

Let

be the modified Chern character.

Let $\mathcal{𝒮}(X)$ denote the space of multivalued flat sections of $\nabla$. The gamma integral structure²⁶ on $\mathcal{𝒮}(X)$ is an integral lattice in $\mathcal{𝒮}(X)$ induced by the following map :

This integral structure arises naturally from the integral structure on the K-theory in the following sense: the Euler pairing transforms to the sesquilinear pairing (2.1) under s : where we used

One can also define Gamma integral structure on the cohomology group $H^{\ast }(X)$. The fundamental solution to the QDE provides a $ℂ$-linear isomorphism of the cohomology of X with the space of flat sections :

This defines a nonstandard integral lattice and sesquilinear pairing on $H^{\ast }(X)$ by

Central charges. The quantum cohomology central charge of $\mathcal{ℱ}$ is a function on $H^{\ast }(X)\times {ℂ}_z$ :

Conjecturally, for fixed variables they are central charges of appropriate Bridgeland stability conditions on the category $D^b(X)$.

Central charges are given by specific linear combinations of the entries of the fundamental solution matrix

Central charges can be also written in terms of the J-function :

Real structure. Gamma integral structure on cohomology $H^{\ast }(X)$ is a lattice given by the image of $K(X)$ via

This integral structure induces a real subspace in $H^{\ast }(X;ℂ)$. A real structure is a complex anti-linear involution which leaves the real subspace invariant. By the HRR formula, the sesquilinear pairing is real with respect to the real structure, that is

Let $\alpha \in H^{\ast }(X;ℝ)$. Then

The right-hand side is The left-hand side is where I used $\overline{{(-1)}^{n/2}}=\overline{e^{\pi in/2}}={(-1)}^n{(-1)}^{n/2}$ for $n\in \mathbb{Z}$.

Therefore, we can write the real structure explicitly²⁷ as

Calabi–Yau case and $t{t}^{\ast }$-metric. Cecotti and Vafa⁹ introduced a hermitian metric on deformation spaces of 2d supersymmetric field theories. It has been studied by multiple people in different contexts, e.g. Refs. 14, 23 and 27.

In the Calabi–Yau case $c_1(X)=0$ and the second equation in (1) reduces to the homogeneity condition for the flat sections :

In this case it is convenient to set $z=1$. We then define the pairing $[\circ ,\circ )$ on $H^{\ast }(X)$ by

The pairing $[\circ ,\circ )$ is symmetric for even n and anti-symmetric for odd n.

In the case where X is a Calabi–Yau threefold, $t{t}^{\ast }$ geometry reduces to the so-called (projective or local) special Kahler geometry.²⁰ More generally, let X be a Calabi–Yau n-fold. Consider the restriction of the quantum connection (1) to $H^{\ast }(X)\times H^2(X)\to H^2(X)$. The A-model Hodge filtration is defined by $F_A^{n-p}{H}^{\ast }(X):=L^{-1}{\oplus }_{k=0}^p{H}^{2k}(X)$. The A-model filtration together with the integral structure (3) defines a pure Hodge structure of weight n around the large volume limit.²⁷ The sesquilinear pairing and a trivial connection induced from the trivial bundle structure turn it into a variation of polarized Hodge structures. In particular, $i^{p-q}[\kappa (L^{-1}\cdot \alpha ),L^{-1}\cdot \alpha )>0$ for any $\alpha \in H^{p,q}(X)\backslash \{0\}$ by Theorem 3.9 in Ref. 27.

The top filter $F_A^n{H}^{\ast }(X)=ℂ\cdot J$ is a line subbundle spanned by the J-function. This line bundle has a hermitian metric induced from the Hermitian pairing $[\kappa (\circ ),\circ )$ on $H^{\ast }(X)$. Zamolodchikov metric on the space $H^2(X)$ is defined as a Kahler metric with the Kahler potential given by

and does not depend on the choice of a section of the line bundle.

Let ${\{{\mathcal{ℱ}}_i\}}_i$ be a $\mathbb{Q}$-basis of $K(X)$ and ${\{{\mathcal{ℱ}}^i\}}_i$ be a dual basis, such that

The pairing can be written in terms of central charges :

Below we compute the Zamolodchikov metric for the case of abelian GLSMs or toric Landau–Ginzburg models.

3. Toric Case

3.1. Toric geometry

We briefly introduce the necessary notations from toric geometry mostly following the conventions of Ref. 3. Let $V\simeq {ℂ}^N$ be a complex vector space, $G\subset {({ℂ}^{\ast })}^N\subset GL(V)$ be a torus, $G\simeq {({ℂ}^{\ast })}^{\kappa }$. Let $\zeta \in \mathrm{\text{char}}{(G)}_{\mathbb{Q}}$ be such that $V^{\zeta -st}=V^{\zeta -ss}$. Then

is a smooth semi-projective toric DM stack (e.g. Ref. 28).

Let $\tilde{T}\simeq {ℂ}^N$ be the diagonal torus in $GL(V)$, and consider the exact sequence of groups :

We have short exact sequences associated to cocharacter and character lattices correspondingly : Let $v_i\in N$ be the image of the i-th basis vector of ${\mathbb{Z}}^N$ under $\beta$, and $D_i\in 𝕃$ be the image of the ith basis vector of ${({\mathbb{Z}}^N)}^{\vee }$ by $\psi$.

The GIT data above can be given in terms of the extended stacky fan²⁸

where $\mathrm{\Sigma }=\{\sigma \}$ is a rational polyhedral fan in $N_{ℝ}$ defined as

Let ${\mathcal{𝒜}}_{\zeta }$ be the set of all anticones, that is a set of all I such that $\zeta \in {\sum }_{i\in I}{ℝ}_{>0}{D}_i$.

Anticones are in one-to-one correspondence with cones. In particular, torus fixed points correspond to minimal anticones or maximal cones. We shall use the notations $I^{\prime }=[1\cdots N]\backslash I$. Furthermore, given an anticone $I\in {\mathcal{𝒜}}_{\zeta }$ let ${\sigma }_{I^{\prime }}$ be the associated cone and $p_I\in \mathcal{𝒳}$ be the corresponding torus fixed point.

Cohomology. If not specified, all the cohomology groups in this section have rational coefficients. We also use the notation $R_k:=R{\otimes }_{\mathbb{Z}}k$ for any field $k$ and a $\mathbb{Z}$-module R.

The second cohomology of $\mathcal{𝒳}$ can be read off the exact sequence

We identify the cohomology elements with (rational) characters of G. Choose a $\mathbb{Z}$-basis ${\{p_a\}}_{a=1}^{\kappa }$ of $H^2(\mathcal{𝒳};\mathbb{Z})\simeq {𝕃}^{\vee }$. Using the latter isomorphism we can write $D_i={\sum }_{a=1}^{\kappa }{D}_i^a{p}_a$ for some $D_i^a\in \mathbb{Z}$, $1\le i\le N$, $1\le a\le \kappa$. The matrix ${\{D_i^a\}}_{i,a}$ is called the charge matrix in the physical literature. Let ${\mathcal{𝒟}}_i$ be a T-equivariant divisor in ${\mathcal{𝒳}}_{\zeta }$ given by equation $\{x_i=0\}$ in homogeneous coordinates. We denote its T-equivariant Poincare dual class by $u_i^T\in H_T^2(\mathcal{𝒳})$.

The nonequivariant limit is a class $u_i\in H^2(\mathcal{𝒳})$ which can be written as

Let $H_{\tilde{T}}^{\ast }(pt)=\mathbb{Z}[{\lambda }_1,\dots ,{\lambda }_N]$. The group homomorphism $\tilde{T}\to T$ induces the equivariant cohomology map $H_T^{\ast }(\mathcal{𝒳})\to H_{\tilde{T}}^{\ast }(\mathcal{𝒳})$. Let $u_i^{\tilde{T}}$ be the image of $u_i^T$ under this map. Explicitly, we can write

There exist a Stanley–Reisner-type descriptions of the cohomology ring^7,28 :

where $m\in H_T^2(pt)\simeq M_{ℂ}$ acts by ${\sum }_i〈v_i$, $m〉u_i^T$. and

Let $f(\{p_a^{\tilde{T}}\},\{{\lambda }_i\})\in H_{\tilde{T}}^{\ast }(\mathcal{𝒳})$ be a class in the equivariant cohomology ring. We can choose a lift $\tilde{f}\in ℂ[{\{p_a^{\tilde{T}}\}}_a,{\{{\lambda }_i\}}_i]$ which is a regular function on $\mathrm{\text{Spec}}(H_{\tilde{T}\times G}^{\ast }(pt))\simeq (𝕃\oplus \text{Ñ})\otimes ℂ$. Let $G_I\subset G$ denote the stabilizer of the torus fixed point $p_I$ corresponding to the anticone I. It is well known that the equivariant integral of f can be computed by residues :

where the contour is the product of circles around the hyperplanes ${\{u_i^{\tilde{T}}=0\}}_{i\in I}\subset (𝕃\oplus \text{Ñ})\otimes ℂ$ in positive directions. The integral does not depend on the choice of the lift $\tilde{f}$, is $ℂ[{\{{\lambda }_i\}}_i]$-linear and can be extended to an analytic completion in the equivariant variables.

The equivariant Poincare pairing is defined by the equivariant integration :

This pairing does not have a nonequivariant limit $\lambda \to 0$ when $\mathcal{𝒳}$ is nonproper. The dualizing module for $H^{\ast }(\mathcal{𝒳})$ is the cohomology with compact support. The combinatorial definition in the toric case is due to Borisov and Horja.⁸ $H_{T,c}^{\ast }(\mathcal{𝒳})$ is an ideal in $H_T^{\ast }(\mathcal{𝒳})$ isomorphic to a module of $H_T^{\ast }(X)$ with generators and relations The equivariant Poincare pairing turns into a nondegenerate pairing : The nonequivariant limit is a perfect pairing :

Inertia stack and orbifold cohomology. The inertia stack of $\mathcal{𝒳}$ is a DM stack

where $\text{Box}(\mathrm{\Sigma })={\bigcup }_{I\in {\mathcal{𝒜}}_{\zeta }}{\text{Box}}_I$, and Each ${\mathcal{𝒳}}_v$ is called a twisted sector and is a toric DM substack of $\mathcal{𝒳}$ given by where $V_v=\{x\in V,x_i=0\mathrm{\text{if}}{c}_i\ne 0\}$ for $v={\sum }_{i\in I^{\prime }}{c}_i{v}_i$.

The degree shift function $\mathrm{\text{age}}:\text{Box}(\mathrm{\Sigma })\to \mathbb{Q}$ is defined as $\mathrm{\text{age}}({\sum }_i{c}_i{v}_i)={\sum }_i{c}_i$ for the minimal representation of v. The orbifold (Chen–Ruan) cohomology is

We denote the generator of the v-twisted sector by $1_v$. By definition it has degree $2\mathrm{\text{age}}(v)$. We let $deg$ be the true degree operator, where degree is shifted by $\mathrm{\text{age}}$ and ${deg}_0$ denotes the degree on $H^{\ast }(\mathcal{ℐ}\mathcal{𝒳})$. Let $v={\sum }_i{c}_i{v}_i$. We denote by $-v$ the Box element equal to ${\sum }_i\{-c_i\}{v}_i$, where the brackets stand for the fractional part. There is an involution $\mathrm{\text{inv}}$ on $\mathcal{ℐ}\mathcal{𝒳}$ sending ${\mathcal{𝒳}}_v\to {\mathcal{𝒳}}_{-v}$ via the identity map. This involution induces an involution $H_{\mathrm{\text{orb}}}^{\ast }(\mathcal{𝒳})\to H_{\mathrm{\text{orb}}}^{\ast }(\mathcal{𝒳})$ sending $1_v$ to $1_{-v}$. $H_{\mathrm{\text{orb}}}^{\ast }(\mathcal{𝒳})$ is endowed with the orbifold Poincare pairing (if $\mathcal{𝒳}$ is proper) : Orbifold Gamma classes²⁶ are given by as elements of $\mathrm{\text{End}}(H_{\mathrm{\text{orb}}}^{\ast }(\mathcal{𝒳}))$.

In the orbifold case, the gamma real structure on the cohomology gets modified by the involution on the twisted sectors²⁷ :

3.2. I-function and $t{t}^{\ast }$

In this section, we introduce equivariant GLSM I-functions³ that are closely related to the I-functions of the toric DM stack $\mathcal{𝒳}$.

Let $R_1,\dots ,R_N\in \mathbb{Q},\widehat{R}:={\sum }_i{R}_i/2$ and $ĉ:=n-2\widehat{R}$. In the GLSM section below, they will stand for the R-charges and central charge (or effective dimension), respectively. We define

Let $y=(y_1,\dots ,y_{\kappa })$ be a formal Novikov variable. The equivariant GLSM I-function is given by

$H^{\ast }({\mathcal{𝒳}}_v)$ is naturally a $H^{\ast }(\mathcal{𝒳})$-module via a pullback, that is for $\alpha \in H^{\ast }(\mathcal{𝒳})$ and $\beta \in H^{\ast }({\mathcal{𝒳}}_v)$ we have $\alpha \cdot \beta :={\iota }_v^{\ast }(\alpha )\cup \beta$. Using this module structure we write In the Calabi–Yau case setting $z=1$, we get Let $\mathrm{\text{Gr}}=deg/2-n/2$ as above, then We also compute (note the signs) Note that $(1-\{-x\})+(1-\{x\})=1$ if $x\notin \mathbb{Z}$ and both are equal to 1 otherwise. In particular, and We also use ${\sum }_i{d}_i(\beta )=\widehat{R}$ and ${\sum }_i\{-d_i(\beta )\}=\mathrm{\text{age}}(v(\beta ))$.

Finally,

Each of the integrals in the last expression can be evaluated by equivariant localization. Let ${\theta }_a:=-log{y}_a$ and $〈\theta ,p^{\tilde{T}}〉={\sum }_a{p}_a^{\tilde{T}}log{y}_a$. The integral in the last line of (9) can be represented as a residue integral (6) : where we introduced ${\alpha }_i=R_i/2+{\lambda }_i$, and ${\mathcal{𝒞}}_0\subset 𝔤=\mathrm{\text{Lie}}(G)$ is a product of $\kappa$ positively oriented circles around

Concluding this computation we obtain the following formula.

Proposition 3.1.

This formula is the main result of this section.

3.3. Sphere partition function

The sphere partition function of the abelian GLSM^4,13 is defined (up to a prefactor) as

for a particular choice of integration contours ${\mathcal{𝒞}}_m$ that we will discuss later. We can rewrite it as

Let $\mathrm{\Gamma }\mathrm{\Gamma }(\sigma ):={\prod }_{i=1}^N{\pi }^{-1}sin(\pi (〈D_i,\sigma +m/2〉+{\alpha }_i))\mathrm{\Gamma }(〈D_i,\sigma +m/2〉+{\alpha }_i)\mathrm{\Gamma }(〈D_i,\sigma -m/2〉+{\alpha }_i)$ be the gamma part of the integrand. This is a meromorphic function on $𝔤$ with simple poles at the hyperplanes :

This divisor is a countable union of simple normal crossings divisors for generic $\alpha$. Let $m_i:=〈D_i,m〉$. We can rewrite the polar divisor as Let ${\mathcal{𝒞}}_m$ be the same tilted contour ${\widetilde{\mathcal{𝒞}}}_0$ from the appendix for each m. Then, repeating Lemma 5.13 from Ref. 3 we get the following.

Lemma 3.1.

where ${\mathcal{𝒞}}_{k,m}$ is a product of $\kappa$ positively oriented circles around

Remark 3.1. The proof of Ref. 3 applies due to Proposition A.2.

Proposition 3.2. If $\zeta$ is far enough in the phase $C_{\zeta }$, the sphere partition function is equal to the pairing of the I-function with its $\kappa$-conjugate :

Proof. By Lemma 3.1

We can change the order of the summation ${\sum }_m{\sum }_I\to {\sum }_I{\sum }_m$ (we shall see that the right-hand side is absolutely convergent). Let $F(\sigma )=e^{〈\theta ,\sigma +m/2〉}{e}^{〈\overline{\theta },\sigma -m/2〉}\mathrm{\Gamma }\mathrm{\Gamma }(\sigma )$. Then we can regroup the terms in the last expression according to whether $m_i\ge 0$ or not : where ${𝕃}_J=\{m\in 𝕃|m_j\ge 0,j\in J,m_j<0,j\in J^{\prime }\}$. For $m\in {𝕃}_J$ we have $〈D_i,\sigma 〉+m_j/2+{\alpha }_j\ge 〈D_i,\sigma 〉+m_j/2+{\alpha }_j$, $j\in J$ and $〈D_i,\sigma 〉+m_j/2+{\alpha }_j<〈D_i,\sigma 〉+m_j/2+{\alpha }_j,j\in J^{\prime }$. Joining the contributions for all $J\subset I$ we can write (3.3) as where ${\mathcal{𝒞}}_{k,\tilde{k}}$ is a product of $\kappa$ positively oriented circles around By Proposition 3.1, we conclude that The power series is absolutely convergent, which justifies the resummation in the beginning of the proof. □

4. GLSM

An abelian GLSM data is a 5-tuple $(V,G,{ℂ}_R^{\ast },W,\zeta )$, where $V,G$ and $\zeta$ define a toric GIT quotient DM stack, ${ℂ}_R^{\ast }\to GL(V)$ is the R-charge action that descends to the action on $\mathcal{𝒳}$, and W is a weighted homogenous polynomial with respect to the R-charge action. See details and different versions in Refs. 12, 18 and 19 in the algebraic setting.

From the toric perspective, an abelian GLSM is a semi-projective toric DM stack with a choice of a one-parametric subgroup of the torus and a regular function that has weight one with respect to the subgroup.

Intersection $G\cap {ℂ}_R^{\ast }$ is necessarily a cyclic group which we denote ${\mu }_r$. We also choose a generator $J\in {\mu }_r$. Let ${\{r{R}_i/2\}}_{i=1}^N$ denote the weights^b of ${ℂ}_R^{\ast }$ on V. Recall that $\widehat{R}:={\sum }_i{R}_i/2$. The GLSM state space is¹⁹ a version of a relative cohomology group :

The grading operator $deg$ (true grading) on GLSM state space denotes the grading in (12). We use ${deg}_0$ to denote the natural grading on ${ℍ}^{\ast }(\mathcal{ℐ}\mathcal{𝒳},({\mathrm{\Omega }}_{\mathcal{ℐ}\mathcal{𝒳}}^{\ast },\pm \mathrm{\text{d}}W))$.

The state space is endowed with an involution by composing the action of $J^{1/2}$ and the involution on the twisted sectors :

The pairing on the state space is defined as where the integral is understood as the composition¹⁹$:$ The integration map is defined by integrating over the analytic Borel–Moore fundamental class of ${\mathcal{𝒳}}_v$ (see Appendix A of Ref. 19).

Under some conditions on the GLSM data, Favero and Kim¹⁹ proved that GLSM form a homogeneous cohomological field theory (CohFT) with a unit $1$. Genus zero part of such a CohFT defines a Frobenius manifold and, in particular, the quantum connection on ${ℍ}^{\ast }(\mathcal{ℐ}\mathcal{𝒳},({\mathrm{\Omega }}_{\mathcal{ℐ}\mathcal{𝒳}}^{\ast },\mathrm{\text{d}}W))\times {\mathcal{𝒜}}_z^1$ :

where the quantum product defined using 3-point GLSM invariants, the Euler vector field is and the grading operator is where $n_{\mathrm{\text{eff}}}=ĉ:=n-2R$ is the central charge or effective dimension. Note that R-charge dependence in $\mathrm{\text{Gr}}$ cancels out.

We can define the fundamental solution to the QDE (1) :

with an appropriate asymptotic, and the J-function The details of the J-function construction will be published elsewhere.

In Ref. 3 we defined a GLSM small I-function :

It is expected that the I-function is related to the J-function (14) by a version of quasimap $ϵ$-wall-crossing.¹¹

Gamma integral structure on the space of solutions to QDE on $H^{\ast }(X)$ is defined using central charges of coherent sheaves. Matrix factorizations play the role of coherent sheaves in GLSM, so we recall some basic notions about matrix factorizations.

Matrix factorizations. Let $DMF(\mathcal{𝒳},W)$ denote the derived category of matrix factorizations of W on $\mathcal{𝒳}$.^16,31 An object of $DMF(\mathcal{𝒳},W)$ is a pair of vector bundles $E_0,E_1$ on $\mathcal{𝒳}$ together with maps

such that Let $B\to \mathcal{𝒳}$ be a vector bundle and $\beta :{\mathcal{𝒪}}_{\mathcal{𝒳}}\to \mathrm{\Gamma }(B)$ be a regular section. Let $\alpha :\mathrm{\Gamma }(B)\to {\mathcal{𝒪}}_{\mathcal{𝒳}}$ be a cosection such that The Koszul matrix factorization $\{\alpha ,\beta \}$ is defined as and the maps are

The Koszul matrix factorization can be thought of as a deformation of the Koszul complex

Let $𝔅\in DMF(\mathcal{𝒳},W)$. Kim and Polishchuk³⁰ defined a version of Chern character for global matrix factorizations :

which was generalized to the stacky version in Ref. 10.

Integral and real structures on the GLSM state space. Extending constructions of Iritani^26,27 we define an integral structure on the GLSM state space.

Let $\mathcal{𝒮}(\mathcal{𝒳},W)$ denote the space of flat sections of the quantum connection. We define the algebraic Gamma integral structure as a $\mathbb{Z}$-submodule ${\mathcal{𝒮}}_{\mathbb{Z}}(\mathcal{𝒳},W)\subset \mathcal{𝒮}(\mathcal{𝒳},W)$ generated by

where ${\widehat{\mathrm{\Gamma }}}_{\mathcal{𝒳}}$ is the orbifold Gamma class of $\mathcal{𝒳}$,²⁶ ${deg}_0$ denotes the degree without the age-shift.

The factor ${(-1)}^{{deg}_0/2}$ differs from ${(2\pi i)}^{{deg}_0/2}$ the one in (3) due to the difference in normalization between topological and algebraic Chern classes, cf. Ref. 19.

Remark 4.1.

(1)	The submodule introduced above is in general not of the full rank, that is ${\mathcal{𝒮}}_{\mathbb{Z}}(\mathcal{𝒳},W)\otimes ℂ\subset \mathcal{𝒮}(\mathcal{𝒳},W)$ can be a proper subspace due to possible existence of nonalgebraic classes in the state space, see Remark 2.11 in Ref. 26.c
(2)	Let us assume that numerical equivalence implies homological equivalence, that is $\mathrm{\text{ch}}i(𝔅,-)=0\Rightarrow \mathrm{\text{ch}}(𝔅)=0$. Then every relation in $\mathcal{𝒮}(\mathcal{𝒳},W)$ comes from a relation in ${\mathcal{𝒮}}_{\mathbb{Z}}(\mathcal{𝒳},W)$. In particular $\mathrm{\text{rank}}{\mathcal{𝒮}}_{\mathbb{Z}}(\mathcal{𝒳},W)=dim{\mathcal{𝒮}}_{\mathbb{Z}}(\mathcal{𝒳},W)\otimes ℂ$.
(3)	The algebraic integral structure is compatible with the Euler pairing in the following sense : $$\begin{array}{c}\mathrm{\Psi }(𝔅):={(2\pi )}^{-n/2}{\widehat{\mathrm{\Gamma }}}_{\mathcal{𝒳}}{(-1)}^{{deg}_0/2}{\mathrm{\text{inv}}}_W^{\ast }\mathrm{\text{ch}}(ℭ),\hfill \\ \chi (ℭ\otimes {𝔅}^{\vee })={〈e^{\pi i{c}_1(\mathcal{𝒳})}\mathrm{\Psi }(ℭ),C{(-1)}^{\mathrm{\text{Gr}}}\mathrm{\Psi }(𝔅)〉}_W,\hfill \end{array}$$(15) where C is a linear involution on (the even part of) the state space ${ℍ}^{\ast }({\mathcal{𝒳}}_v,({\mathrm{\Omega }}_{{\mathcal{𝒳}}_v}^{\ast },\mathrm{\text{d}}W))$. This follows from a combination of Proposition 2.10 in Ref. 26 and Theorem 1.2 in Ref. 10.

GLSM central charges and (hemi-)sphere partition functions. The sesquilinear pairing is defined on the space of (multivalued) functions of z valued in the state space ${\mathcal{\mathscr{H}}}_W\{z\}$ :

We define GLSM quasimap central charges by

In Ref. 3, we showed that GLSM quasi-central charges have hemisphere partition function integral representation.²⁴

In the Calabi–Yau case the sphere partition function of Refs. 4 and 13 is equal to the hermitian pairing of the GLSM I-function with itself:

Theorem 4.1.

Proof. Favero and Kim¹⁹ defined the Todd–Chern class of a matrix factorization :

Let $\mathcal{ℱ}\subset \mathcal{𝒳}$ be a zero locus of a regular section $\beta$ of a vector bundle B on $\mathcal{𝒳}$. □

Lemma 4.1. Let $I,J\subset [1\cdots N]$ and $F_I\stackrel{{\iota }_I}{\to }\mathcal{𝒳}$, $F_J\stackrel{{\iota }_J}{\to }\mathcal{𝒳}$be proper toric substacks defined by equations ${\{x_i=0\}}_{i\in I}$ and ${\{x_i=0\}}_{i\in J}$, respectively. They are zero loci of regular sections ${\beta }_{I,J}$ of $\tilde{T}$-equivariant tautological vector bundles $B_{I,J}:=[V^{I,J}\times V^{\zeta -ss}/G]\to X$ on $\mathcal{𝒳}$. Let ${\alpha }_{I,J}$ be cosections ${\alpha }_{I,J}\in \mathrm{\Gamma }(B_{I,J}^{\vee })$ such that ${\alpha }_I{\beta }_I={\alpha }_J{\beta }_J=W$. Then for any $s\in H^{\ast }(\mathcal{𝒳})$ the following equality holds true :

where the last integral is the map ${\int }_{\mathcal{𝒳}}:H_c^{\ast }(\mathcal{𝒳})\to ℂ$.

Proof. We can write the following series of equalities :

The first one is true by Corollary 4.11 in Ref. 19, the second one is by Lemma B.13 in Ref. 19, and the third one is true because $F_I$ is proper. □

Now, the proposition follows from the relations between $I_W$ and I³ and Lemma 4.1.

Notes

^a In addition, one requires existence of a flat homogeneous identity vector field.

^b This notation is consistent with the physical convention of $R_i$ as R-charges.

^c The author is grateful to the referee for pointing this out.

^d In physics notation, the central charge is three times the mathematical convention.

Acknowledgments

The author is grateful to Chiu-Chu Melissa Liu for multiple discussions and to the referee for valuable comments.