Derived equivalence and homological projective duality in GLSM

1. Introduction

The gauged linear sigma model (GLSM) with $\mathcal{𝒩}=(2,2)$ supersymmetry has been a thriving research area that spawned many topics in both physics and mathematics. The boundary conditions of GLSMs that preserve a particular subset of the supersymmetries, known as B-branes, form a category, which is an important tool in analyzing the GLSMs. The objects of this category are matrix factorizations of the superpotential. The Kähler moduli space of a GLSM can be divided into several phases. If the low-energy behavior of a phase can be described by a nonlinear sigma model (NLSM) with target space X, then the B-brane category projects onto $D^b(X)$, the derived category of X, upon imposing the empty brane relations corresponding to that phase.^1,2,3 On the other hand, if the low-energy behavior of a phase is described by a Landau–Ginzburg (LG) model, then the low-energy image of a GLSM brane in that phase is a matrix factorization of the LG model. Different phases have different sets of empty brane relations in general.

Brane transport provides a functor between the B-brane categories of different phases of a GLSM.^1,4,5 In certain cases, this functor gives rise to an equivalence between the categories of B-branes. In particular, if two phases are described by two NLSMs with different target space $X_{+}$ and $X_{-}$ and the functor induced by the brane transport is an equivalence, then we get a derived equivalence between $X_{+}$ and $X_{-}$, i.e. $D^b(X_{+})\cong D^b(X_{-})$. Therefore, GLSMs have been used to study a quite large class of manifolds and their derived equivalence. In Sec. 3, we review the derived equivalence realized by the two-parameter GLSMs studied in Ref. 7, most of which are nonabelian theories. To implement the brane transport, the small window categories for anomalous $U(2)$ theories were studied in Ref. 7, extending previous results on window categories.

In certain other cases, brane transport provides an embedding of the category of B-branes of the Higgs branch of one phase into that of another. In particular, given a GLSM ${\mathcal{𝒯}}_X$ realizing a projective variety X in one of its phases, it is possible to construct an extended GLSM ${\mathcal{𝒯}}_{\mathcal{𝒳}}$ realizing the homological projective dual⁶ category $\mathcal{𝒞}$ to $D^b(X)$ as the category of B-branes of the Higgs branch in one of its phases. In most of the cases, the models ${\mathcal{𝒯}}_X$ and ${\mathcal{𝒯}}_{\mathcal{𝒳}}$ are anomalous and the analysis of their Coulomb and mixed Coulomb/Higgs branches gives information on the semiorthogonal/Lefschetz decompositions of $\mathcal{𝒞}$ and $D^b(X)$. It is also possible to take a linear subspace and construct the models ${\mathcal{𝒯}}_{X_L}$ and ${\mathcal{𝒯}}_{{\mathcal{𝒳}}_L}$ that correspond to homological projective duality of linear sections $X_L$ of X. This explains why, in many cases, two phases of a GLSM are related by homological projective duality (HPD). In Sec. 4, we review the GLSM construction for HPD studied in Refs. 8 and 9, including some of the examples therein. The homological projective dual category constructed by this method is described by the category of B-branes (matrix factorizations) of a hybrid model.

In this paper, we focus on GLSMs with two Kähler parameters. Each GLSM under consideration contains a geometric phase, which only has a pure Higgs branch. Then, the category of B-branes of the Higgs branch of another phase can be embedded in the category of B-branes (derived category) of the geometric phase via brane transport. In the study of derived equivalence, the categories of B-branes of the Higgs branch of two phases can be identified with the same subcategory of the derived category of the geometric phase, thus fulfilling a derived equivalence. In the study of HPD, the category of B-branes of the Higgs branch of a LG phase can be identified with the subcategory of the derived category of the geometric phase, which is exactly the homological projective dual category.

The organization of this paper is as follows. In Sec. 2, we briefly review the basics of brane transport and window category. In Sec. 3, we present the idea of using GLSM to construct derived equivalence between different geometries. Section 4 is devoted to discussing the relationship between GLSM and homological projective duality.

2. Review of Brane Transport and Window Category

For a GLSM with gauge group G, matter fields in the representation ${\rho }_m:G\to \mathrm{\text{GL}}(V)$ and superpotential W, a B-brane is described by a pair of objects $(\mathcal{ℬ},L_t)$ with $\mathcal{ℬ}=(M,{\rho }_M,R_M,T)$, where

• M is the Chan–Paton vector space, which is a ${\mathbb{Z}}_2$-graded, finite-dimensional free $\mathrm{\text{Sym}}(V^{\vee })$ module denoted by $M=M_0\oplus M_1$.

• ${\rho }_M$ and $R_M$ are even representations of the gauge and the (vector) R-charge representations, respectively.

• $T$ is a matrix factorization of W, a ${\mathbb{Z}}_2$-odd endomorphism $T\in {\mathrm{\text{End}}}_{\mathrm{\text{Sym}}(V^{\vee })}^1(M)$ satisfying $T^2=W\cdot {\mathrm{\text{id}}}_M$.

They must satisfy the compatibility conditions: for all $\lambda \in U{(1)}_V$ (the vector R-symmetry) and $g\in G$

$L_t$ is a profile for the vector multiplet scalar, it consists of a gauge-invariant middle-dimensional subvariety of the complexified Lie algebra of G.

Brane transport means moving some B-brane of a GLSM smoothly along a path on the Kähler moduli space from one phase to another. There are certain constraints on brane transport. The situation depends on whether the Fayet–Iliopoulos (FI) parameter is renormalized under the RG flow along the direction of the brane transport. In the Calabi–Yau (CY) case (also known as the nonanomalous case, meaning that the axial $U(1)$ R-symmetry is anomaly free), in which the FI parameter is marginal, there are a number of singular points on the Kähler moduli space (parametrized by the FI parameters) between the two phases, so the path along which the brane is transported must avoid these singularities. Given such a path, only the branes satisfying the grade restriction rule (GRR) can be smoothly transported from one phase to the other. The branes satisfying the GRR constitute a subcategory of the category of matrix factorizations of the GLSM, which is called the window category.¹

In the non-CY case (also known as the anomalous case, meaning that the axial $U(1)$ R-symmetry has anomaly), there are no singularities in the middle, but in order to determine the low energy image of a B-brane, one has to find an IR equivalent brane that is in the big window category, namely, the category of branes satisfying the GRR. The low energy physics typically include a Higgs branch and some Coulomb/mixed branches, therefore, the low energy limit of a GLSM matrix factorization generally has components on both Higgs and Coulomb/mixed branches. The small window category is a subcategory of the big window category, the objects in the small window category only have components on the Higgs branch in the low energy limit.⁵ The GRRs depend on the choice of the theta-angle, and a shift of the theta-angle by certain amount results in a different but equivalent window category.

For a $U(1)$ theory, the small window ${𝕎}_{-}$ and the big window ${𝕎}_{+}$ are defined by the following constraints :

where $N_{\pm }:={\sum }_j{(Q_j)}^{\pm }$ and ${(x)}^{\pm }:=(|x|\pm x)/2$, $\theta$ is the theta-angle and q denotes any $U(1)$ charge carried by the B-brane. If the RG flow is from the phase with FI parameter ${\zeta }_1$ to the phase with FI parameter ${\zeta }_2$, we then expect that there is an equivalence between the categories of B-branes of the two phases^awhere ${𝕎}_{+}$ is the big window, $k=|N_{+}-N_{-}|$, and $D(Y,W)$ denotes the category of matrix factorizations of an LG model with target space Y and superpotential W so $D(Y_{{\zeta }_a},W_{{\zeta }_a})$ is the category of B-branes on the Higgs branch in the corresponding phase. The objects $E_i$ represent massive vacua from the Coulomb or mixed Coulomb/Higgs branches. When the model is nonanomalous ${𝕎}_{+}={𝕎}_{-}$ with ${𝕎}_{-}$ being the small window and (2) gives an equivalence of Higgs branch categories. For anomalous models there is a fully faithful map ${𝕎}_{-}\to {𝕎}_{+}$ and we have a refinement of (2)

The GRRs in both CY and non-CY cases have been studied and determined for abelian GLSMs.¹ However, the GRRs for nonabelian GLSMs are not well understood. At this stage, there is not a general result and one has to resort to a case-by-case study. There are a few nonabelian examples where the GRR can be deduced (see, for example, Refs. 4 and 10).

3. Derived Equivalence via Brane Transport in $U(1)\times U(2)$ Models

We study the derived equivalence between two manifolds, $X_{+}$ and $X_{-}$, by realizing them as the Higgs branches in different phases of a two-parameter nonabelian GLSM⁷ (the GLSM has two independent FI parameters) whose pure geometric phase realizes another manifold $\widehat{X}$.^b In the examples we are going to consider, $\widehat{X}$ will be either Fano or CY. Other than the pure geometric phase, there are also two other phases consisting of the Higgs branches $X_{\pm }$ and mixed branches ${\mathcal{𝒞}}_{\pm }$. The derived equivalence between $X_{+}$ and $X_{-}$ is implemented by the brane transport along a path across the phases of this GLSM. One necessary condition for $X_{+}$ and $X_{-}$ to be derived equivalent is that their Euler characteristics should be equal to each other, namely, $\chi (X_{+})=\chi (X_{-})$. This condition implies nontrivial conditions on $X_{+}$ and $X_{-}$, and also on $\widehat{X}$.

In order to implement the brane transport, we need to know the window categories of the local models at the phase boundaries, which can be either anomalous or nonanomalous. Some of these local models are $U(1)$ gauge theories, and the previous results from Refs. 1 and 5 can be directly used. Some of these local models are $U(2)$ gauge theories with fundamental and determinantal representations. So, before studying the B-branes in the $U(1)\times U(2)$ theories, we first discuss the window categories of the anomalous $U(2)$ theories in Subsec. 3.1 following Ref. 7.

3.1. Small window of $U(2)$ theories

For our purpose, let us consider a $U(2)$ gauge theory with n fundamental fields, $X_i$, $i=1,\dots ,n$, and one field P in the determinantal representation, ${det}^{-m}$, with $m\le n$, as summarized below

where $\xi$ is the associated FI parameter. For later purposes, we will only focus on the cases with $n=3$ and $n=4$, and the same analysis can be generalized to arbitrary n.

According to the D-term equations, the region $\xi \gg 0$ is a phase described by a total space of m-fold determinantal bundle over $\text{Gr}(2,n)$, i.e.

Another region, where $\xi \ll 0$, consists of a Higgs branch and some Coulomb vacua. The Higgs branch in this region is conjectured to be described by an orbifold of the affine cone of the image of the Grassmannian under Plücker embedding,¹¹ further evidence for the validity of this conjecture was provided in Refs. 7 and 12, where a one-to-one correspondence between the window category of the GLSM and the derived category of the affine cone was provided.

The window category of the theory with $m=n=4$ was studied in Ref. 12 by investigating the condition on the convergence of the disk partition function. It was shown that the window category is generated by the GLSM branes^c

up to a shift by ${\mathcal{𝒲}}_{{det}^l}$ induced by changing the theta-angle. The set of branes in (5) reduces to the semiorthogonal decomposition in the positive phase, where $K_{\text{Gr}(2,4)}$ is the canonical line bundle over $\text{Gr}(2,4)$.

Numerical analysis of the partition function generalizes the result to the cases with $0<m\le n=3$ and $0<m<n=4$,⁷ the results with an appropriate choice of the theta-angle are given as the following:

• $n=3$

  

• $n=4$

  

  where ${𝕎}_{-}$ is the small window category.

3.2. Examples with gauge group $U(2)\times U(1)$

Let’s look at two examples of GLSMs with $U(1)\times U(2)$ gauge group, from which derived equivalences emerge. In each example, the functor realizing the equivalence can be implemented by brane transport. The difference is that, in the first example the local models at the phase boundaries are nonanomalous, while the local models at the phase boundaries of the second example are anomalous. Therefore, as reviewed in Sec. 2, the procedures of the brane transport in the two examples are different.

3.2.1. Product of a projective space and a Grassmannian

The GLSM for the product of a projective space and a Grassmannian has the gauge group $U(1)\times U(2)$ and the following matter content :

with $\alpha =1,2$ the $U(2)$ color index, $a=1,\dots ,M$ and $j=1,\dots ,N$ are flavor indices. $m\le M,n\le N$.

Clearly, in the ${\xi }_1\gg 0,{\xi }_2\gg 0$ region, there is only a pure Higgs branch with target space

where $\mathcal{𝒮}$ is the tautological bundle over $\text{Gr}(2,N)$.

If either ${\xi }_1\ll 0$ or ${\xi }_2\ll 0$, the P field gets a nonzero vev due to the D-term equations and breaks the $U(1)\times detU(2)$ gauge group to a subgroup $U{(1)}_s\times {\mathbb{Z}}_{gcd(m,n)}$ with the FI parameter of $U{(1)}_s$ given by ${\xi }_s:=n^{\prime }{\xi }_1-m^{\prime }{\xi }_2$, where $m^{\prime }=m/gcd(m,n),n^{\prime }=n/gcd(m,n)$. The Higgs branch in the region ${\xi }_s\ll 0,{\xi }_1\ll 0$ is an NLSM with target space^d

The Higgs branch in the region ${\xi }_s\gg 0,{\xi }_2\ll 0$ is an NLSM with target space is an orbifold of the fiber bundle over ${\mathbb{P}}^{M-1}$ with each fiber being the affine cone of $\text{Gr}(2,N)$. Let us call this fiber bundle $\mathcal{𝒳}(2,N)$, thus the target space in this phase is

Comparing the Euler characteristics $\chi (X_{+})$ and $\chi (X_{-})$, the derived equivalence between $X_{+}$ and $X_{-}$ suggests Note that the condition with N odd is also the CY condition for $X_{\pm }$, while $\widehat{X}$ is not necessarily a CY. When N is even, if we further apply the CY condition for $X_{\pm }$, i.e. $mN=nM$, then the second equation becomes $(N-n)M=0,$ namely, we should have $M=m$ and $N=n$, which is also the CY condition for $\widehat{X}$.

The local model at the phase boundary of $\{{\xi }_1=0,{\xi }_2\gg 0\}$ is a $U(1)$ gauge theory with the following matter content. Therefore, the window category at the phase boundary $\{{\xi }_1=0,{\xi }_2\gg 0\}$ can be chosen to be generated by branes of the form $\mathcal{𝒲}(q_1,\lambda )$, where $q_1\in \{0,1,\dots ,m-1\}$ and $\lambda$ is any $U(2)$ representation.

The local model at the phase boundary of $\{{\xi }_2=0,{\xi }_1\gg 0\}$ is the $U(2)$ gauge theory with the following matter content :

For the cases with $n\le N$, $N=3$ and 4, this model has been discussed in Subsec. 3.1 and the small window is given by Eqs. (6) and (7). Therefore, the small window category at the phase boundary $\{{\xi }_2=0,{\xi }_1\gg 0\}$ can be chosen to be generated by branes of the form $\mathcal{𝒲}(q_1,\lambda )$, where $\lambda$ is given by Eqs. (6) and (7) in the cases of $N=3$ and $N=4$, respectively, and $q_1$ is any $U(1)$ charge.

In the case of $N=4$, the CY condition for $X_{\pm }$ requires that $n=N=4$ and $m=M$. In this case, $\widehat{X}$ is also CY, and we have the derived equivalence

In this case, the axial $U(1)$ R-symmetry is nonanomalous, the small window category and the big window category coincide. Moreover, because the FI parameters are not renormalized, the brane transport amounts to the following steps^e:

• For any brane $B_{+}$ in $D^b(X_{+})$, lift it to a GLSM brane ${\mathcal{𝒲}}_{+}$ such that it is in the window category of the local model at the phase boundary $\{{\xi }_1=0,{\xi }_2\gg 0\}$, so it can be smoothly transported to the phase of $\widehat{X}$.

• Once in the phase of $\widehat{X}$, we can read off its image in $D^b(\widehat{X})$.

• Use empty branes to grade restrict ${\mathcal{𝒲}}_{+}$ and obtain an IR equivalent brane ${\mathcal{𝒲}}_{-}$, which is in the window category of the local model (13).

• ${\mathcal{𝒲}}_{-}$ can be smoothly transported to the phase of $X_{-}$, where its image in $D^b(X_{-})$ can be determined.

Some concrete implementations of the above procedure can be found in Ref. 7.

3.2.2. Derived equivalence between CY fivefolds

Consider $\widehat{X}$ as the total space of a fiber bundle over the two-step symplectic flag manifold SF$(1,2,N)$. When $N=4$, this becomes the example studied in Ref. 13, which has

where $\mathcal{ℒ}$ and $\mathcal{𝒮}$ are the first and second tautological bundles over SF$(1,2,4)$. The CY fivefolds, $X_{+}$ and $X_{-}$, are defined as and where $\mathcal{𝒮}$ and $\mathcal{ℒ}$ are the tautological bundles of the symplectic Grassmannian SG$(2,4)$ and the projective space ${\mathbb{P}}^3$, respectively, ${\mathcal{ℒ}}^{\perp }$ is the symplectic orthogonal to $\mathcal{ℒ}$ with respect to the symplectic structure on ${\mathcal{𝒪}}^{\oplus 4}$.

The GLSM for $\widehat{X}$ is a $U(1)\times U(2)$ gauge theory with chiral matters ${\mathrm{\Phi }}_{\alpha }$, $X_i^{\alpha }$, $Q_{\alpha \beta }$ and $P_{\alpha \beta }$ with $\alpha =1,2$ and $i=1,2,3,4$. These fields transform under the gauge group action as follows :

and there is a superpotential where ${\mathrm{\Omega }}^{ij}$ is a symplectic form in ${ℂ}^4$. The D-term equations read In the above, we have denoted $P_{\alpha \beta }=P{ϵ}_{\alpha \beta }$ and $Q_{\alpha \beta }=Q{ϵ}_{\alpha \beta }$, $\alpha ,\beta =1,2$, where ${ϵ}_{\alpha \beta }$ is the Levi-Civita symbol.

Phase I: ${\mathit{\xi }}_1\gg \mathbf{0}$ and ${\mathit{\xi }}_2\gg \mathbf{0}$. In the phase ${\xi }_1,{\xi }_2\gg 0$, the solutions to the D-term and F-term equations tell us that the vevs of ${\mathrm{\Phi }}_{\alpha }$ and $X_i^{\alpha }$ constitute the symplectic flag manifold $SF(1,2,4)$ (the F-term equations derived from (16) imposes the symplectic condition and force the Q field to vanish), which is the base for $\widehat{X}$, while $P_{\alpha \beta }$ contributes the fibers of $\widehat{X}$.

Phase II: ${\mathit{\xi }}_1\ll \mathbf{0}$ and ${\mathit{\xi }}_2\gg \mathbf{0}$. The $U(2)$ D-term equation tells that $X_i^{\alpha }$ is nondegenerate, so they become homogeneous coordinates of $\text{Gr}(2,4)$ at low energy scale. The F-term equations impose the symplectic condition and force Q to vanish. The $U(1)$ invariant field $P{\mathrm{\Phi }}_{\alpha }$ is in the $U(2)$ representation $\overline{\square }\otimes {\wedge }^2\overline{\square }$, which become the coordinates along the fiber. Therefore, the vacuum manifold in this phase is given by (14).

Phase III: ${\mathit{\xi }}_1\gg \mathbf{0}$, ${\mathit{\xi }}_2\ll \mathbf{0}$ and ${\mathit{\xi }}_1+2{\mathit{\xi }}_2\gg 0$. If we take the trace of the second equation of (17), we get ${\sum }_{\alpha ,i}|X_i^{\alpha }{|}^2-{\sum }_{\alpha }|{\mathrm{\Phi }}_{\alpha }{|}^2-2(|P{|}^2+|Q{|}^2)=2{\xi }_2$. Adding the first equation of (17) to the equation above, we get ${\sum }_{\alpha ,i}|X_i^{\alpha }{|}^2-3|P{|}^2-2|Q{|}^2={\xi }_1+2{\xi }_2,$ therefore, $X_i^{\alpha }$ cannot vanish simultaneously in this phase. Consequently, the F-term equations force the Q field to vanish. Define the $SU(2)$ invariant fields $Z_i={\mathrm{\Phi }}_{\alpha }{X}_i^{\alpha }$ and $B_{ij}={ϵ}_{\alpha \beta }{X}_i^{\alpha }{X}_j^{\beta }$, then the remaining degrees of freedom can be summarized as

Moreover, from the D-term equations, the $Z_i$’s cannot vanish simultaneously, thus become the homogeneous coordinates of ${\mathbb{P}}^3$, and $B_{ij}$’s encode the subspace spanned by $X^1$ and $X^2$ in ${ℂ}^4$ containing $Z_i$. In addition, $X^1$ and $X^2$ must be orthogonal to each other with respect to the symplectic form on ${ℂ}^4$ due to the F-term equations, so the field $P{B}_{ij}$ give us the vector bundle^f$({\mathcal{ℒ}}^{\perp }/\mathcal{ℒ})\otimes {\mathcal{ℒ}}^2$. As such, the vacuum manifold in this phase is given by (15).

The local model at the phase boundary of $\{{\xi }_1=0,{\xi }_2\gg 0\}$ is the $U(1)$ gauge theory with the following matter content :

The small window of this model consists of only one $U(1)$ charge. By a suitable choice of theta-angle, this charge can be taken to be zero.

The local model at the phase boundary of $\{{\xi }_2=0,{\xi }_1\gg 0\}$ is the $U(2)$ gauge theory with the following matter content :

The small window is given by Eq. (7) with $m=2$. With a suitable choice of the theta-angle, we can take the small window to be

Now, we can compute the functor for the equivalence of $D^b(X_{+})$ and $D^b(X_{-})$. Let ${\pi }_{\pm }$ be the projection of the vector bundle $X_{\pm }$ onto the corresponding base space. We choose the generators of $D^b(X_{+})$ to be ${\pi }_{+}^{\ast }{\mathcal{𝒪}}_{\text{SGr}(2,4)}$, ${\pi }_{+}^{\ast }{\mathcal{𝒮}}_{\text{SGr}(2,4)}^{\vee }$, ${\pi }_{+}^{\ast }{\wedge }^2{\mathcal{𝒮}}_{\text{SGr}(2,4)}^{\vee }$ and ${\pi }_{+}^{\ast }{({\wedge }^2{\mathcal{𝒮}}_{\text{SGr}(2,4)}^{\vee })}^{\otimes 2}$. Since now we have a superpotential $W=Q\mathrm{\Omega }(X^1,X^2)$, the lift of an IR brane is a matrix factorization of W.

${\pi }_{+}^{\ast }{\mathcal{𝒪}}_{\text{SGr}(2,4)}$ can be lifted to the matrix factorization

which fits in the small windows of both (18) and (19), so it can be directly transported and projected to give the image ${\pi }_{-}^{\ast }{\mathcal{𝒪}}_{{\mathbb{P}}^3}$.

Similarly, ${\pi }_{+}^{\ast }{\wedge }^2{\mathcal{𝒮}}_{\text{SGr}(2,4)}^{\vee }$ can be transported to give the image ${\pi }_{-}^{\ast }{\mathcal{𝒪}}_{{\mathbb{P}}^3}(-1)$, and the image of ${\pi }_{+}^{\ast }{({\wedge }^2{\mathcal{𝒮}}_{\text{SGr}(2,4)}^{\vee })}^{\otimes 2}$ is ${\pi }_{-}^{\ast }{\mathcal{𝒪}}_{{\mathbb{P}}^3}(-2)$.

For ${\pi }_{+}^{\ast }{\mathcal{𝒮}}_{\text{SGr}(2,4)}^{\vee }$, we can use the IR empty brane given by the exact sequence

The lift of ${\pi }_{+}^{\ast }{\mathcal{𝒮}}_{\text{SGr}(2,4)}^{\vee }$ is

From the empty brane (20) and the equivalence

induced by the action of the P field, we see the IR image of (21) in the phase of $X_{-}$ is the complex $\mathrm{\text{Cone}}({\pi }_{-}^{\ast }{\mathcal{𝒪}}_{{\mathbb{P}}^3}(1)[-1]\to {\pi }_{-}^{\ast }{\mathcal{𝒪}}_{{\mathbb{P}}^3}(-2))$.

4. GLSM construction for Homological Projective Duality

In this section, we describe the GLSM construction^8,9 for HPD. We will see that the HPD category will appear as the Higgs branch in one of the phases of the two-parameter GLSM for the universal hyperplane section associated with a projective embedding. This can be shown by brane transport. First, let’s briefly review some basic facts about HPD.

4.1. Review of homological projective duality

Definition 1 (Ref. 6). A (right) Lefschetz decomposition of $D^b(X)$ with respect to the line bundle $\mathcal{ℒ}$ on X corresponds to a semiorthogonal decomposition¹⁴

such that ${\mathcal{𝒜}}_0\supseteq {\mathcal{𝒜}}_1\supseteq \cdots \supseteq {\mathcal{𝒜}}_k$ are a collection of admissible subcategories of $D^b(X)$ and ${\mathcal{𝒜}}_i(i):={\mathcal{𝒜}}_i\otimes {\mathcal{ℒ}}^{\otimes i}$. The Lefschetz decomposition is called rectangular if ${\mathcal{𝒜}}_0={\mathcal{𝒜}}_1=\cdots ={\mathcal{𝒜}}_k$.

The Lefschetz decomposition is completely determined by its center ${\mathcal{𝒜}}_0$ and $\mathcal{ℒ}$ via the relation

We can always construct a dual Lefschetz decomposition by setting¹⁵where ${\mathcal{𝒜}}_0{(i)}^{\perp }$ denotes the right orthogonal of ${\mathcal{𝒜}}_0(i)$. Then, we have a left Lefschetz decomposition Consider a smooth projective variety X with a morphism $f:X\to \mathbb{P}(V)$ and assume we have a Lefschetz decomposition of the form (22) with respect to the line bundle $\mathcal{ℒ}={\mathcal{𝒪}}_X(1):=f^{\ast }{\mathcal{𝒪}}_{\mathbb{P}(V)}(1)$. We define the incidence divisor $\mathcal{\mathscr{H}}\subset \mathbb{P}(V)\times \mathbb{P}(V^{\vee })$ by Then, the universal hyperplane section of X is defined by and has the following semiorthogonal decomposition⁶ :

Definition 2 (Ref. 6). $\mathcal{𝒞}$ is called the HPD category of X. In other words, $\mathcal{𝒞}$ is the right orthogonal of $〈{\mathcal{𝒜}}_1(1)⊠D(\mathbb{P}(V^{\vee })),\dots ,{\mathcal{𝒜}}_k(k)⊠D(\mathbb{P}(V^{\vee }))〉$.

If we have an algebraic variety Z with a morphism $g:Z\to \mathbb{P}(V^{\vee })$ such that there is an equivalence $D^b(Z)\cong \mathcal{𝒞}$, we call Z the HPD to $f:X\to \mathbb{P}(V)$ with respect to the Lefschetz decomposition (22). A very important property of $\mathcal{𝒞}$, that will be used for consistency checks is the following: given the embedding map $\delta :\mathcal{𝒳}\to X\times \mathbb{P}(V^{\vee })$, we have

Indeed, one can define the $\mathcal{𝒞}$ by the objects in $D(\mathcal{𝒳})$ whose image under ${\delta }_{\ast }$ belongs to ${\mathcal{𝒜}}_0⊠D(\mathbb{P}(V^{\vee }))$.

4.2. GLSM for HPD

In this section, we describe how to construct a GLSM realizing the HPD for a projective embedding. Assume the projective embedding $f:X\to \mathbb{P}(S)$ can be realized by a GLSM ${\mathcal{𝒯}}_X=(G,{\rho }_m:G\to GL(V),W)$, where G is the gauge group, ${\rho }_m$ is a representation of G on V (space of the matter fields), and W is the superpotential. It has a large volume point in the Kähler moduli space ${\mathcal{ℳ}}_K$ where we can identify a geometric phase whose category of B-branes at low energies is equivalent to $D^b(X)$. In order for the map f to be well defined in all of X it must be base point free, i.e. $\{x\in X:f(x)=0\}=\varnothing$. Since there is a G-action on S, one can see that there exists a map $\alpha :\mathrm{\text{Hom}}(G,{ℂ}^{\ast })\to \mathrm{\text{Pic}}(X)$ induced by the G-equivariant line bundle $V\times {ℂ}_{\mu }$ over V for any $\mu \in \mathrm{\text{Hom}}(G,{ℂ}^{\ast })$, and hence there exists a character $\chi$ of G such that

The character $\chi$ defines a one-dimensional representation ${ℂ}_{\chi }$ of G. In the context of the GLSM ${\mathcal{𝒯}}_X$, we can write the image of f as $[f_0(X),\dots ,f_n(X)]$, where $n+1=\mathrm{\text{dim}}(S)$. The functions $f_j(X)$ depend on some fields $X_a$, representing the coordinates of X, however they are not necessarily fundamental fields, in general they will be polynomials in the coordinates ${\varphi }_{\alpha }$, $\alpha =1,\dots ,N=\mathrm{\text{dim}}(V)$ of V. Assume WLOG that the large volume phase is located at a region $\zeta \gg 1$. Let $Y_{\zeta \gg 1}$ be the GIT quotient specified by the D-term equations. We will require that the GLSM ${\mathcal{𝒯}}_X$ satisfies the condition Therefore, we write^gwhere we identify X with $X_{\zeta \gg 1}$, at the point in the Kähler moduli determined by t with $\zeta =\mathrm{\text{Re}}(t)\gg 1$. Then, we have where $E_i$ denotes the Coulomb/mixed vacua deep in the phase where $\zeta \ll -1$.

We construct the following extended GLSM :

where the representation $V^{\prime }$ of $\widehat{G}$ is given by $V^{\prime }={ℂ}_{({\chi }^{-1},-1)}\oplus S^{\vee }$, where $\chi$ is defined by (30), $U{(1)}_s$ acts on $S^{\vee }$ with weight 1 and G acts on $S^{\vee }$ trivially. We denote the coordinates of $V^{\prime }$ as $(P,S_0,\dots ,S_n)$. The superpotential $\widehat{W}$ is given by Here, we denoted $f_j(X)$ for the components of the image of the map $f:X\to \mathbb{P}(S)$. The GLSM ${\mathcal{𝒯}}_{\mathcal{𝒳}}$ is quite analogous to the model studied in Refs. 17 and 18. We interpret ${\mathcal{𝒯}}_{\mathcal{𝒳}}$ as the GLSM of the universal hyperplane section of X. To be more precise, ${\mathcal{𝒯}}_{\mathcal{𝒳}}$ has a large volume phase at $\{\zeta \gg 1,{\zeta }_s\gg 1\}$, where ${\zeta }_s$ is the FI parameter corresponding to $U{(1)}_s$. The F-term equations $\frac{\partial \widehat{W}}{\partial S_j}=0$ implies that $f_j(X)=0$ for all j if $P\ne 0$,. Then, the moment map equation for $\zeta \gg 1$ and the condition (31) implies this is not possible and P must vanish. Therefore, we must have $P=0$ on the Higgs branch, then it is easy to see that $Ŷ\cap d{\widehat{W}}^{-1}(0)$ reduces to the universal hyperplane section $\mathcal{𝒳}$. Therefore, we identify this phase with the NLSM with target $\mathcal{𝒳}$.

Consider now the phase $\{\zeta \ll -1,{\zeta }_s\gg 1\}$. Then, we have the following correspondence of B-brane categories :

where $C_i$ denotes the Coulomb/mixed vacua deep in the phase $\{\zeta \ll -1,{\zeta }_s\gg 1)$ and ${Ŷ}_{\zeta \ll -1}$ is the appropriate symplectic quotient associated with the D-terms of the GLSM ${\mathcal{𝒯}}_{\mathcal{𝒳}}$. The vacua $C_i$, $i=1,\dots ,k^{\prime }$ can be computed using the local model at the phase boundary $\{\zeta =0,{\zeta }_s\in {ℝ}_{\ge 0}\}$. Because the subcategory $D({Ŷ}_{\zeta \ll -1},{\widehat{W}}_{\zeta \ll -1})$ is equivalent to the small window category of the local model and it can be shown that its image under brane transport to the geometric phase is exactly ${\mathcal{𝒜}}_0⊠D^b(\mathbb{P}(V^{\vee })$ with ${\mathcal{𝒜}}_0$ to be defined in (38) (see Eq. (29)), we have where $\mathcal{𝒞}$ is the HPD category of X with respect to $\mathcal{ℒ}=f^{\ast }{\mathcal{𝒪}}_{\mathbb{P}(S)}(1)$ and the Lefschetz decomposition is determined by the center In general, in our examples, we will be able to find an explicit description of $D({Ŷ}_{\zeta \ll -1},{\widehat{W}}_{\zeta \ll -1})$ in terms of a hybrid model. Denoting by ${𝕎}_{\pm }$ the big/small window categories associated with the phase boundary at $\{\zeta =0,{\zeta }_s\in {ℝ}_{\ge 0}\}$, we can rephrase this result as Generically, the phase space will look like Fig. 1.

4.3. Examples of GLSM realization of HPD

In this section, we present three examples of GLSM realization of HPD, namely, Veronese embedding, quadrics in projective spaces and Plücker embedding. Some of the cases were studied in the mathematical literature, our result agrees with what was found and proved in these cases (e.g. Veronese embedding and quadrics with odd dimension). Our GLSM construction gives rise to predictions in the other cases. The three examples differ in the degree of the embedding and the rank of the gauge group.

4.3.1. Veronese embedding

The degree-d Veronese embedding $\mathbb{P}(V)\to \mathbb{P}({\mathrm{\text{Sym}}}^{d}V)$ can be realized by abelian GLSM as discussed in Ref. 19. Upon integrating out the massive fields, this theory is equivalent to the GLSM describing $\mathbb{P}(V)$, namely, $n+1$ matter fields with charge 1 under the $U(1)$ gauge symmetry, where $\mathrm{\text{dim}}(V)=n+1$. Since $\mathrm{\text{dim}}({\mathrm{\text{Sym}}}^{d}V)=\left(\genfrac{}{}{0ex}{}{n+d}{d}\right)$, our discussion in Subsec. 4.2 shows that the GLSM ${\mathcal{𝒯}}_{\mathcal{𝒳}}$ describing the universal hyperplane section and HPD has gauge group $U(1)\times U(1)$ with matter content

and superpotential where $f_a(X)$ form a basis of the monomials in $X_i$ with degree d. Higgs branch of one of the phases gives the HPD of degree-d Veronese embedding. From the F-term and D-term constraints, one can show that the Higgs branches in different phases are as follows:

(i)	${\zeta }_1\gg 0,{\zeta }_2\gg 0$: The Higgs branch is the universal hyperplane section of the Veronese embedding, i.e. the universal degree-d hypersurface $$\mathcal{𝒳}=\left\{\sum _a{S}_a{f}_a(X)=0\right\}\subseteq {\mathbb{P}}^n\times {\mathbb{P}}^{\left(\begin{array}{c}\hfill n+d\hfill \\ \hfill d\hfill \end{array}\right)-1},$$ where $X_i$ are homogeneous coordinates of ${\mathbb{P}}^n$ and $S_a$ are homogeneous coordinates of ${\mathbb{P}}^{\left(\begin{array}{c}\hfill n+d\hfill \\ \hfill d\hfill \end{array}\right)-1}$.
(ii)	${\zeta }_1\ll 0,{\zeta }_1\ll d{\zeta }_2$: The Higgs branch is the LG model on $$\mathrm{\text{Tot}}\left(\mathcal{𝒪}{\left(-\frac{1}{d}\right)}^{\oplus (n+1)}\to {\mathbb{P}}^{\left(\begin{array}{c}\hfill n+d\hfill \\ \hfill d\hfill \end{array}\right)-1}\right)/{\mathbb{Z}}_d,$$(41) with superpotential $$W_0=\sqrt{\frac{-{\zeta }_1}{d}}\sum _a{S}_a{f}_a(X).$$(42)
(iii)	${\zeta }_1\gg d{\zeta }_2,{\zeta }_2\ll 0$: There is no Higgs branch.

To determine which phase of the GLSM (40) describes the HPD, let’s consider the local model at the boundary between phases (i) and (ii). It is a $U(1)$ GLSM with matter content

Thus, we have $N_{+}=n+1,N_{-}=d$. The small window consists of the branes satisfying We can choose the theta-angle such that the charge q of the branes in the small window satisfies $0\le q\le \mathrm{\text{min}}\{d-1,n\}$. For $d\le n+1$, the corresponding Lefschetz decomposition is where ${\mathcal{𝒜}}_0={\mathcal{𝒜}}_1=\cdots ={\mathcal{𝒜}}_{p-1}=〈\mathcal{𝒪},\dots ,\mathcal{𝒪}(d-1)〉$, ${\mathcal{𝒜}}_p=〈\mathcal{𝒪},\dots ,\mathcal{𝒪}(k)〉$ and $p\ge 0,0\le k<d$ with $n=pd+k$. Branes in the small window have charges $q=0,1,\dots ,d-1$. These branes project to the HPD category $\mathcal{𝒞}$ in phase (i) because they restrict to ${\mathcal{𝒜}}_0$ on ${\mathbb{P}}^n$. From Ref. 5, we know that this subcategory $\mathcal{𝒞}$ of $D(\mathcal{𝒳})$, which is the HPD category by definition, is equivalent to the category of matrix factorizations of the LG model corresponding to phase (ii). Therefore, we claim that the HPD of degree-d Veronese embedding with respect to the Lefschetz decomposition (43) is described by the LG orbifold defined by (41) and (42).

By taking linear sections of the dual projective space, i.e. a linear subspace L of ${ℂ}^{\left(\genfrac{}{}{0ex}{}{n+d}{d}\right)}$ and going to the strong coupling limit of $U{(1)}_s$, one can recover many of the examples previously studied in the literature.²¹

4.3.2. Quadrics

A quadric X in ${\mathbb{P}}^n$ defined by $G(X)=0$ can be realized by the positive phase of a $U(1)$ GLSM with matter content and charges

and superpotential $W=P\cdot G(X)$. The negative phase of this model is a LG model For a suitable choice of theta-angle. The small window of (44) consists of branes with gauge charges $q=-1,0$. From the resolutions of the spinor bundles $S_{\pm }$ on Xwhere and ${\psi }_{+}$, ${\psi }_{-}$ are morphisms such that ${\psi }_{+}\circ {\psi }_{-}={\psi }_{-}\circ {\psi }_{+}=G\cdot \mathrm{\text{id}}$, we see that the lift of the spinor bundles as GLSM matrix factorizations are in the small window. But the lift of ${\mathcal{𝒪}}_X(m)$ is not in the small window because its resolution reads Therefore, the category of matrix factorizations of (45) is equivalent to the subcategory of $D^b(X)$ generated by the spinor bundles. Note that $S_{+}\cong S_{-}\equiv S$ when $\mathrm{\text{dim}}X$ is odd. Then, we have the Lefschetz decomposition where ${\mathcal{𝒜}}_0=〈S,\mathcal{𝒪}(-1)〉$ if $\mathrm{\text{dim}}X$ is odd, ${\mathcal{𝒜}}_0=〈S_{-},S_{+},\mathcal{𝒪}(-1)〉$ if $\mathrm{\text{dim}}X$ is even, and ${\mathcal{𝒜}}_i=〈\mathcal{𝒪}(-1)〉$ for $i>0$. Note that this Lefschetz decomposition is different from the one adopted in Ref. 20 when $\mathrm{\text{dim}}X$ is even.

From our proposal in Subsec. 4.2, for a quadric in ${\mathbb{P}}^n$ defined by the zero locus of a quadratic polynomial G, the model describing the HPD associated with Lefschetz decomposition (47) is a $U(1)\times U(1)$ GLSM with matter content

and superpotential $\widehat{W}=P_{1}G(X)+P_2{\sum }_{i=0}^n{Y}_i{X}_i.$ This GLSM description is consistent with the mathematical description in Ref. 17. Again the phase with ${\zeta }_1>0,{\zeta }_2>0$ is the geometric phase describing the universal hyperplane section of the embedding, i.e. the target space is defined by $G(X)=0$ and ${\sum }_{i=0}^n{X}_i{Y}_i=0$ in ${\mathbb{P}}^n\times {\check{\mathbb{P}}}^n$. For ${\zeta }_2<0,{\zeta }_1>{\zeta }_2$, there is no Higgs branch. When ${\zeta }_1<0,{\zeta }_2>0$, the Higgs branch is described by the LG model on with superpotential ${\sum }_{i,j}{X}_i{Q}_{ij}{X}_j+P_2{\sum }_{i=0}^n{X}_i{Y}_i$, where we assume that $G(X)={\sum }_{i,j}{X}_i{Q}_{ij}{X}_j$ for a $(n+1)\times (n+1)$ invertible matrix Q. This is the LG model description of the HPD with respect to the Lefschetz decomposition (47).

Further analysis in Ref. 8 shows that the target space of the low energy theory for HPD is a ${\mathbb{Z}}_2$-orbifold of the double cover of ${\check{\mathbb{P}}}^n$ branched over the dual quadric ${\sum }_{i,j}{Y}_i{(Q^{-1})}_{ij}{Y}_j=0$. When n is even, ${\mathbb{Z}}_2$ acts trivially. When n is odd, the ${\mathbb{Z}}_2$ action exchanges the two sheets of the covering space so the dual quadric is the fixed locus.

4.3.3. Plücker embedding

The Grassmannian $G(k,V)$ with $\mathrm{\text{dim}}V=N$ can be implemented by the geometric phase of a nonabelian GLSM with $U(k)$ gauge group and N chiral fields in the fundamental representation. From our general construction, the HPD of the Plücker embedding $G(k,V)\to \mathbb{P}({\wedge }^{k}V)$ can be described by a nonabelian GLSM with gauge group $U(k)\times U(1)$ and the following matter content :

where $\square$ stands for fundamental representation of $U(k)$, and the indices $i_p$ satisfy $1\le i_1<i_2<\cdots <i_k\le N$. The superpotential is where the Plücker coordinates read $B_{i_1\cdots i_k}={ϵ}_{a_1\cdots a_k}{\mathrm{\Phi }}_{i_1}^{a_1}\cdots {\mathrm{\Phi }}_{i_k}^{a_k}.$ When the FI parameters satisfy ${\zeta }_1\gg 1,{\zeta }_2\gg 1$, the IR theory is an NLSM with target space the universal hyperplane section where $Y_{i_1\cdots i_k}$ serve as homogeneous coordinates of $\mathbb{P}({\wedge }^k{V}^{\vee })$.

When ${\zeta }_1\ll -1,{\zeta }_2>{\zeta }_1$, the P field receives a vev $〈P〉=\sqrt{-{\zeta }_1}$, breaking the $U(k)$ gauge symmetry to $SU(k)$. Thus, we get a family of $SU(k)$ gauge theories fibered over $\mathbb{P}({\wedge }^k{V}^{\vee })$, each has N fundamentals and a superpotential

Note that every $B_{i_1\cdots i_k}$ is a section of $\mathcal{𝒪}(-1)$ over $\mathbb{P}({\wedge }^k{V}^{\vee })$. The HPD category is expected to be equivalent to the category of matrix factorizations of this $SU(k)$ gauge theory.

The twisting bundle $\mathcal{ℒ}$, which is the pull-back of $\mathcal{𝒪}(1)$ under the Plücker embedding is $\mathcal{ℒ}={\wedge }^k{\mathcal{𝒮}}^{\vee },$ where $\mathcal{𝒮}$ is the tautological bundle over $G(k,V)$. For $k=2$, the number of Coulomb vacua suggests that the associated Lefschetz decomposition is Kapranov’s collection, namely

Again, we can take a linear section, for example, a seven-dimensional subspace of ${\wedge }^{k}V$. Then, the $X_L$ is the CY threefold $\text{Gr}(2,7)[1,1,1,1,1,1,1]$ and the HPD $Y_L$ is the intersection $\mathrm{\text{Pf}}(4,{\wedge }^2{V}^{\vee })\cap \mathbb{P}(L)$ in $\mathbb{P}({\wedge }^2{V}^{\vee })$, which recovers the result of Ref. 16.

4.4. Noncommutative geometric interpretation

The GLSM construction of Subsec. 4.2 describes the HPDs as hybrid models. Due to the relationship between LG models and $A_{\infty }$ categories, it is possible to give these hybrid models a noncommutative geometric interpretation.

These hybrid models, with target space $Y=\mathrm{\text{Tot}}(\mathcal{ℰ}\to B)$ can be viewed as (orbifold of) LG models over affine charts of B in the cases that B is smooth, or if it has the structure of a global orbifold. This gives rise to the global structure of a noncommutative resolution or more generally, the B-brane category becomes the derived category of sheaves of $\mathcal{𝒜}$-modules for some sheaf of $A_{\infty }$-algebras $\mathcal{𝒜}$.

For example, for a LG model with quadratic superpotential $W_{LG}$, it was shown in Ref. 22 that the category of B-branes (matrix factorizations) $MF(W_{LG})$ is equivalent to the derived category of finite-dimensional Clifford modules, where the Clifford algebra is defined by the Hessian of the superpotential $W_{LG}$. In addition, if a ${\mathbb{Z}}_2$ orbifold is present that leaves $W_{LG}$ invariant, then the category of B-branes of this LG orbifold $MF(W_{LG},{\mathbb{Z}}_2)$ is equivalent to the derived category of finite-dimensional modules of the even subalgebra of the corresponding Clifford algebra. Consequently, the category of matrix factorizations of a ${\mathbb{Z}}_2$-orbifold hybrid model with superpotential quadratic along the fiber coordinates is equivalent to the derived category of the sheaf of modules of the sheaf of even parts of a Clifford algebra over the base B. This is the case of the HPD category of the degree-2 Veronese embedding $\mathbb{P}(V)\hookrightarrow \mathbb{P}({\mathrm{\text{Sym}}}^{2}V)$.²⁰ Thus, the hybrid model orbifold can be viewed as a noncommutative resolution of B.

The idea was generalized and an explicit correspondence between hybrid models and noncommutative spaces was constructed in Ref. 9. Consider first the case where B can be described locally by affine charts and Y is a global orbifold such that the orbifold group G acts trivially on B. Then, at a generic point in the base $p\in B$ of the hybrid model, we can model its dynamics by a LG orbifold. Denote the category of B-branes of this LG orbifold as $MF(W,G)$, where G is the orbifold group. We first studied the $A_{\infty }$-algebra ${\mathcal{𝒜}}_{D0}=\mathrm{\text{End}}({\mathcal{ℬ}}_{D0})$ associated with the endomorphism algebra of a $D0$-brane ${\mathcal{ℬ}}_{D0}\in MF(W)$ of the LG model. The algebra ${\mathcal{𝒜}}_{D0}$ takes the form of an $A_{\infty }$ algebra with a finite number of generators (as an algebra) ${\psi }_i$ that satisfy the $A_{\infty }$ products relations¹⁸

This result then sets up the equivalence between matrix factorizations and $A_{\infty }$-modules of ${\mathcal{𝒜}}_{D0}♯G$, where $♯$ denotes the smash product (the mathematical approach toward this equivalence can be found in Ref. 23), more precisely where $D^b(\mathrm{\text{Mod-}}{\mathcal{𝒜}}_{D0}♯G)$ denotes the derived category of the $A_{\infty }$ modules of ${\mathcal{𝒜}}_{D0}♯G$. We then use this equivalence to relate a hybrid model to a noncommutative resolution i.e. the derived category of sheaf of ${\mathcal{𝒜}}_{D0}♯G$-modules over B.

The more general case, for example, when G acts on B, or more precisely, when B has orbifold singularities and/or it cannot be written as a global orbifold still has a similar structure. In such a case we have that $D(Y,W)$ is equivalent to the derived category of $\mathcal{𝒜}$-modules for some sheaf of $A_{\infty }$-algebras $\mathcal{𝒜}$, defined over the algebraic stack Y.

These results can be used to study HPD of several spaces. As shown in Subsec. 4.2, the GLSM construction realizes the HPDs as hybrid orbifold models, which can be identified with noncommutative resolutions, or derived categories of sheaves of $A_{\infty }$-modules, as the equivalence suggests. Therefore, given a projective embedding engineered by a GLSM, the hybrid model describing the HPD can be read off following Subsec. 4.2. One can then use the correspondence discussed in this section to give a noncommutative geometric description of the HPD. This idea was applied to Veronese embedding, Fano hypersurfaces and complete intersections in Ref. 9.

Acknowledgments

The author would like to thank Z. Chen, B. Lin, M. Romo, L. Smith, H. Zou for collaborations and the organizers of the workshop “GLSM@30” held at the Simons Center for Geometry and Physics.

ORCID

Jirui Guo  https://orcid.org/0000-0003-0423-3330

Notes

^a Here and in the following, $D(Y,W)$ stands for the category of matrix factorizations of the LG model with target space Y and superpotential W.

^b If the GLSM is anomalous, we should first take the gauge-decoupling limit and then consider the IR limit to see this phase. See more discussions on these two limits in Sec. 2 of Ref. 5.

^c Here and in the following, the Young diagram with a boxes in the first row and b boxes in the second row are used to denote the $U(2)$ representation with highest weight $(a,b)$. The empty diagram $\cdot$ denotes the trivial representation.

^d In fact, $\mathcal{𝒪}(-n/m)$ should be interpreted as the pullback of the orbibundle $\mathcal{𝒪}(-n/m)$ on ${\mathbb{P}}^{\left(\genfrac{}{}{0.0pt}{}{N}{2}\right)-1}$ under the Plücker embedding and here we abuse the notation to avoid clumsy symbols.

^e Here, we take the transport to be going from the phase of $X_{+}$ to the phase of $X_{-}$, for example, the transport in the other direction follows the same procedure in the reverse order.

^f The extra $\mathcal{ℒ}$ factor is due to the $U(1)$ charge carried by P.

^g Even though we assume the large volume phase is weakly coupled, our arguments should carry on for cases where the NLSM on X is realized nonperturbatively such as in Ref. 16.