A GLSM View on Homological Projective Duality

Abstract

Given a gauged linear sigma model (GLSM) ${\mathcal {T}}_{X}$ realizing a projective variety X in one of its phases, i.e. its quantum Kähler moduli has a geometric point, we propose an extended GLSM ${\mathcal {T}}_{{\mathcal {X}}}$ realizing the homological projective dual category ${\mathcal {C}}$ to $D^{b}Coh(X)$ as the category of B-branes of the Higgs branch of 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 $D^{b}Coh(X)$. We also study 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. We study mostly abelian examples: linear and Veronese embeddings of ${\mathbb {P}}^{n}$ and Fano complete intersections in ${\mathbb {P}}^{n}$. In such cases, we are able to reproduce known results as well as produce some new conjectures. In addition, we comment on the construction of the HPD to a nonabelian GLSM for the Plücker embedding of the Grassmannian G(k, N).

Appendices

Matrix Factorization on Gerbes

As we have seen that the HPD category of a geometric embedding is usually described by matrix factorizations of orbifold of LG model defined on gerbes (quotient stack). In this appendix, we discuss the definition of matrix factorization on gerbes. The definition requires the notion of orbibundles. The reader may refer to [47] for more details.

1.1 Orbibundle

Let X be a smooth manifold admitting a G-action, where G is a group. An orbibundle on the quotient stack [X/G] is a fiber bundle $E {\mathop {\rightarrow }\limits ^{\pi }} X/G$ with each fiber an orbifold. Explicitly, let V be a vector space admitting a representation of G:

$$\begin{aligned} \rho :\quad G \rightarrow GL(V). \end{aligned}$$

The fibre of E is $V/\rho (G)$. If $\{U_\alpha :~\alpha \in I\}$ is an open cover of X/G and

$$\begin{aligned} \phi _\alpha : U_\alpha \times V/\rho (G) \rightarrow \pi ^{-1}(U_\alpha ) \end{aligned}$$

are the corresponding local trivializations. Then the transition functions $g_{\alpha \beta } = \phi ^{-1}_\alpha \circ \phi _\beta$ take values in $GL(V)/\rho (G)$. A local section of E is given by a $\rho (G)$-invariant function $s_\alpha : U_\alpha \rightarrow V$ so the relation

$$\begin{aligned} s_\alpha = g_{\alpha \beta }\cdot s_\beta \end{aligned}$$

is well defined on $U_\alpha \cap U_\beta$. Given a representation of G as above, the orbibundles on [X/G] are classified by $H^1(X,GL(V)/\rho (G))$. When the representation is trivial, the orbibundle is just an ordinary vector bundle. When $\dim _{{\mathbb {C}}}V=1$, we call it a line bundle.

A morphism between two orbibundles $E_1{\mathop {\rightarrow }\limits ^{\pi _1}} X/G$ and $E_2{\mathop {\rightarrow }\limits ^{\pi _2}} X/G$ is a bundle map $f: E_1 \rightarrow E_2$, i.e. $\pi _2 \circ f = \pi _1$. Given local trivializations of $E_1$ and $E_2$ in an open set U:

$$\begin{aligned}&\phi _1: U \times V_1/\rho _1(G) \rightarrow \pi _1^{-1}(U), \\&\phi _2: U \times V_2/\rho _2(G) \rightarrow \pi _2^{-1}(U), \end{aligned}$$

and for each $x \in U$, $f_U(x) := \phi _2^{-1} \circ f \circ \phi _1|_x$ is a linear map from $V_1$ to $V_2$ satisfying

$$\begin{aligned} f_U(x) \circ \rho _1(g) = \rho _2(g) \circ f_U(x) \end{aligned}$$

for all $g \in G$.

Example 1

$[{\mathbb {C}}/{\mathbb {Z}}_k]$, where ${\mathbb {Z}}_k$ acts on ${\mathbb {C}}$ by

$$\begin{aligned} z \mapsto \exp {\left( \frac{2 \pi i}{k} \right) } z. \end{aligned}$$

The representation $\rho _m$ of ${\mathbb {Z}}_k$ is defined by

$$\begin{aligned} \rho _m \left( \exp {\frac{2 \pi i}{k}} \right) = \exp {\left( \frac{2 \pi m i}{k}\right) } \end{aligned}$$

for $m =0,\ldots ,k-1$. Because $H^1({\mathbb {C}},{\mathbb {C}}^*/\rho _m({\mathbb {Z}}_k))=0$, the line bundles on $[{\mathbb {C}}/{\mathbb {Z}}_k]$ are in one to one correspondence with the representation of ${\mathbb {Z}}_k$. Let’s denote by ${\mathcal {L}}_m$ the line bundle determined by $\rho _m$. It is easy to see that the complex

$$\begin{aligned} {\mathcal {L}}_{m-1} {\mathop {\rightarrow }\limits ^{z}} {\mathcal {L}}_m \end{aligned}$$

is quasi-isomorphic to the skyscraper sheaf ${\mathcal {O}}_0(m)$ at the origin $z=0$, carrying the representation $\rho _m$ along its fiber.

Example 2

$[{\mathbb {P}}^n/{\mathbb {Z}}_2]$, where ${\mathbb {Z}}_2$ acts as the identity map. There are two irreducible representations for ${\mathbb {Z}}_2$,

$$\begin{aligned} \rho _0(1)=\rho _0(-1)=1,\quad \rho _1(1)=1,~\rho _1(-1)=-1. \end{aligned}$$

The line bundles defined by $\rho _0$ are just ordinary line bundles on ${\mathbb {P}}^n$, they are of the form ${\mathcal {O}}(m)$ for some integer m.

The line bundles defined by $\rho _1$ are twisted line bundles. Let’s denote by ${\mathcal {O}}(m/2)$ for some odd integer m the orbibundle whose transition functions are given by $g_{\alpha \beta }^{m/2}$, where $g_{\alpha \beta }$ are the transition functions of ${\mathcal {O}}(1)$. The square root makes sense because the transition functions take values in ${\mathbb {C}}^*/{\mathbb {Z}}_2$. For example, the line bundle ${\mathcal {O}}(1/2)$ on ${\mathbb {P}}^1$ has the transition function

$$\begin{aligned} g_{12} = \underline{z_1^{\frac{1}{2}}} = \underline{z_2^{-\frac{1}{2}}} \end{aligned}$$

where ${\underline{z}}$ is the class of $\pm z$ in ${\mathbb {C}}^*/{\mathbb {Z}}_2$.

1.2 Derived category

We consider coherent sheaves on [X/G] as sheaves with finite resolutions by orbibundles (see for example [48] for a review of sheaves on stacks). A sheaf ${\mathcal {F}}$ is called coherent if there is an exact sequence

$$\begin{aligned} 0 \rightarrow {\mathcal {E}}_1 \rightarrow {\mathcal {E}}_2 \rightarrow \cdots \rightarrow {\mathcal {E}}_n \rightarrow {\mathcal {F}}, \end{aligned}$$

where each ${\mathcal {E}}_i$ is the sheaf of sections of an orbibundle. Then the derived category of [X/G] is defined to be the derived category of the category of coherent sheaves with morphisms being given by G-equivariant chain maps.

1.3 Matrix factorization

Let W be a holomorphic G-invariant function on X, we can define a matrix factorization as being given by bundle maps $F_1 \in \mathrm {Mor}({\mathcal {E}}_1,{\mathcal {E}}_2)$ and $F_2 \in \mathrm {Mor}({\mathcal {E}}_2,{\mathcal {E}}_1)$, where ${\mathcal {E}}_1$ and ${\mathcal {E}}_2$ are orbibundles, such that

$$\begin{aligned} F_2 \circ F_1 = W \cdot id_{{\mathcal {E}}_1},\quad F_1 \circ F_2 = W \cdot id_{{\mathcal {E}}_2}. \end{aligned}$$

Morphisms between two matrix factorizations are required to be G-equivariant.

Example

Let X be the space ${\mathbb {C}}^2_z \oplus {\mathbb {C}}^2_p - \{p_1=p_2=0\}$. ${\mathbb {C}}^*$ acts on X with the following charges

$$\begin{aligned} \begin{array}{cccc} z_1 &{} z_2 &{} p_1 &{} p_2 \\ -1 &{} -1 &{} 2 &{} 2 \end{array} \end{aligned}$$

Let’s consider matrix factorization for the function

$$\begin{aligned} W = z_1^2 p_1 + z_2^2 p_2. \end{aligned}$$

Clearly, there is a projection

$$\begin{aligned}{}[X/{\mathbb {C}}^*] {\mathop {\rightarrow }\limits ^{\phi }} [{\mathbb {P}}^1/{\mathbb {Z}}_2], \end{aligned}$$

where ${\mathbb {Z}}_2$ acts on ${\mathbb {P}}^1$ trivially. For any integer Q, define the representation $\rho _Q$ to be the one-dimensional representation with charge Q, i.e.

$$\begin{aligned} \rho _{Q}(\lambda ) = \lambda ^Q \end{aligned}$$

for all $\lambda \in {\mathbb {C}}^*$, then the line bundle on $[X/{\mathbb {C}}^*]$ determined by the representation $\rho _Q$ is

$$\begin{aligned} {\mathcal {L}}(Q) = \phi ^* {\mathcal {O}}\left( \frac{Q}{2} \right) . \end{aligned}$$

It is easy to check that

is a matrix factorization for W.

Analysis of Coulomb Vacua

The phase diagrams in Figs. 3 and 7 can be determined by examining the asymptotic behavior of the equations of motion on the Coulomb branch. In this appendix, we perform this analysis in detail. For ease of notation, we present the computation for double Veronese embedding of ${\mathbb {P}}^2$ and quadric in ${\mathbb {P}}^3$. The general cases can be analyzed in the same way. In the following, $q_a = \exp (-t_a)$, where $t_a = \zeta _a - i \theta _a$ is the complexified FI parameter.

1.1 ${\mathbb {P}}^2 \rightarrow {\mathbb {P}}^5$

The equations of motion for the Coulomb vacua read

$$\begin{aligned} \left\{ \begin{array}{l} \sigma _1^3 = q_1 (2 \sigma _1+ \sigma _2)^2, \\ \sigma _2^6 = -q_2 (2 \sigma _1+ \sigma _2). \end{array} \right. \end{aligned}$$

(B.1)

Nonzero solutions satisfy

$$\begin{aligned} \sigma _2^{15}+3 q_2 \sigma _2^{10} + 8 q_1 q_2 \sigma _2^9 + 3 q_2^2 \sigma _2^5 + q_2^3 = 0 \end{aligned}$$

(B.2)

and

$$\begin{aligned} \sigma _1^3 = q_1 q_2^{-2} \sigma _2^{12}. \end{aligned}$$

(B.3)

For generic $q_1$ and $q_2$, (B.2) cannot be solved exactly, but in order to investigate the asymptotic behavior of the solutions, we can use asymptotic approximations of the equation in different phases.

(i) $\zeta _1 \gg 1,\zeta _2 \gg 1$: $q_1 \rightarrow 0$ and $q_2 \rightarrow 0$, so (B.2) can be approximated by

$$\begin{aligned} \sigma _2^{15} + 3 q_2 \sigma _2^{10} = 0. \end{aligned}$$

Nonzero solutions tend to zero as $\sigma _2 \sim q_2^{1/5}$, then $\sigma _1 \propto \sigma _2 (q_2^{-1} \sigma _2^5 + 1) \rightarrow 0$ in the asymptotic region. Therefore, all solutions of (B.1) approach $(\sigma _1,\sigma _2)=(0,0)$ as $\zeta _1 \rightarrow \infty , \zeta _2 \rightarrow \infty$, which means that we only have Higgs branch in this phase, described by the universal quadric in ${\mathbb {P}}^2$.

(ii) $\zeta _1 \ll -1, \zeta _2 \gg 1$: $q_1 \rightarrow \infty$ and $q_2 \rightarrow 0$, so (B.2) can be approximated by

$$\begin{aligned} \sigma _2^{15} + 8 q_1 q_2 \sigma _2^{9} = 0. \end{aligned}$$

Aside from the nine zero solutions, which correspond to the Higgs branch (the HPD), there are six solutions satisfying

$$\begin{aligned} \sigma _2^6 = -8 q_1 q_2. \end{aligned}$$

Let’s look at the line $\zeta _2 = -\lambda \zeta _1$ for $\lambda >0$. Then

$$\begin{aligned} \sigma _2^6 \rightarrow \left\{ \begin{array}{ll} \infty , &{} \lambda <1 \\ 0, &{} \lambda >1 \end{array} \right. \end{aligned}$$

and

$$\begin{aligned} \sigma _1^3 = q_1 q_2^{-2} \sigma _2^{12} \sim q_1^3 \rightarrow \infty . \end{aligned}$$

Therefore, other than the Higgs branch, there is a mixed branch described by ${\mathbb {P}}^5$ for $\lambda >1$, and there are six Coulomb vacua for $\lambda <1$.

(iii) $\zeta _1 \ll -1,\zeta _2 \ll -1$: $q_1 \rightarrow \infty$ and $q_2 \rightarrow \infty$. Again, let’s look at the line $\zeta _2 = \lambda \zeta _1$, i.e. $q_2 = q_1^\lambda$. When $\lambda > 1/2$, (B.2) can be approximated by

$$\begin{aligned} \sigma _2^{15} + q_1^{3 \lambda } = 0. \end{aligned}$$

Consequently, $\sigma _2 \sim - q_1^{\lambda /5} \rightarrow \infty$ and $\sigma _1^3 \sim q_1^{1+2 \lambda /5} \rightarrow \infty$. So all the fifteen solutions approach infinity and we have fifteen Coulomb vacua. On the other hand, when $\lambda < 1/2$, (B.2) can be approximated by

$$\begin{aligned} \sigma _2^{15} + 8 q_1^{1+\lambda } \sigma _2^9 = 0. \end{aligned}$$

Nonzero solutions satisfy

$$\begin{aligned} \sigma _2^6 \sim q_1^{1+\lambda } \rightarrow \infty ,\quad \sigma _1^3 \sim q_1^3 \rightarrow \infty . \end{aligned}$$

Therefore we have a Higgs branch described by the HPD and six Coulomb vacua.

(iv) $\zeta _1 \gg 1,\zeta _2 \ll -1$: $q_1 \rightarrow 0, q_2 \rightarrow \infty$, then (B.2) can be approximated by

$$\begin{aligned} \sigma _2^{15} + q_2^3 = 0, \end{aligned}$$

and all solutions satisfy $\sigma _2 \sim q_2^{1/5} \rightarrow \infty$. Assume that $\zeta _2 = -\lambda \zeta _1$, then (B.3) implies

$$\begin{aligned} \sigma _1^3 \sim q_1^{1-2 \lambda /5} \rightarrow \left\{ \begin{array}{ll} 0, &{} \lambda <5/2 \\ \infty , &{} \lambda >5/2 \end{array} \right. \end{aligned}$$

Thus there are five mixed branches, each of which is described by ${\mathbb {P}}^2$, when $\lambda <5/2$. When $\lambda >5/2$, each ${\mathbb {P}}^2$ split into three Coulomb vacua, and we are left with fifteen Coulomb vacua in total.

1.2 ${\mathbb {P}}^3[2]$

The equations of motion for the Coulomb vacua read

$$\begin{aligned} \left\{ \begin{array}{l} \sigma _1^4 = -4 q_1 \sigma ^2_1 (\sigma _1+ \sigma _2), \\ \sigma _2^4 = -q_2 (\sigma _1 + \sigma _2). \end{array} \right. \end{aligned}$$

(B.4)

Nonzero solutions satisfy

$$\begin{aligned} (\sigma _2^3 + q_2)^2=0, \end{aligned}$$

(B.5)

or

$$\begin{aligned} \sigma _2^6 + 2 q_2 \sigma _2^3 + q_2^2 - 4 q_1 q_2 \sigma _2^2 = 0, \end{aligned}$$

(B.6)

and

$$\begin{aligned} \sigma _1 = -q_2^{-1} \sigma _2^4 - \sigma _2. \end{aligned}$$

(B.7)

As $q_2 \rightarrow 0$ for $\zeta _2 \gg 1$ and $q_2 \rightarrow \infty$ for $\zeta _2 \ll -1$, solutions to (B.5) and (B.7) contribute to Higgs branch on the upper half plane of the real FI-space, and contribute to mixed branches on the lower half plane. Now let us analyze the asymptotic behavior of solutions to equations (B.6) and (B.7).

(i) $\zeta _1 \gg 1,\zeta _2 \gg 1$: $q_1 \rightarrow 0, q_2 \rightarrow 0$, all solutions approach (0, 0) as $\zeta _1 \rightarrow \infty , \zeta _2 \rightarrow \infty$. Therefore we only have Higgs branch in this phase, which is described by the universal hyperplane section in ${\mathbb {P}}^3[2] \times {\mathbb {P}}^3$.

(ii) $\zeta _1 \ll -1,\zeta _2 \gg 1$: $q_1 \rightarrow \infty$ and $q_2 \rightarrow 0$, so (B.6) can be approximated by

$$\begin{aligned} \sigma _2^6 - 4 q_1 q_2 \sigma _2^2 = 0. \end{aligned}$$

Other than the two zero solutions, which contribute to the Higgs branch (the HPD), there are four solutions satisfying

$$\begin{aligned} \sigma _2^4 = 4 q_1 q_2. \end{aligned}$$

Consequently,

$$\begin{aligned} \sigma _2^4 \rightarrow \left\{ \begin{array}{ll} \infty , &{} \zeta _1+\zeta _2<0 \\ 0, &{} \zeta _1+\zeta _2>0 \end{array} \right. \end{aligned}$$

and

$$\begin{aligned} \sigma _1^2 = 4 q_1 q_2^{-1} \sigma _2^4 = 16 q_1^2 \rightarrow \infty . \end{aligned}$$

Therefore, other than the Higgs branch, there is a mixed branch described by ${\mathbb {P}}^3$ for $\zeta _2 > -\zeta _1$, and there are four Coulomb vacua for $\zeta _2 < -\zeta _1$.

(iii) $\zeta _1 \ll -1,\zeta _2 \ll -1$: $q_1 \rightarrow \infty$ and $q_2 \rightarrow \infty$. Let’s take the semi-infinite line $\zeta _1 = \lambda \zeta _2$, i.e. $q_1 = q_2^\lambda$. When $\lambda < 1$, (B.6) can be approximated by

$$\begin{aligned} \sigma _2^6 + q_2^2 = 0. \end{aligned}$$

Thus $\sigma _2^6 = -q_2^2 \rightarrow \infty$ and $\sigma _1^2 = 4 q_1 q_2^{-1} \sigma _2^4 \sim q_1 q_2^{1/3} \rightarrow \infty$. Then we have six Coulomb vacua. When $\lambda > 1$, (B.6) can be approximated by

$$\begin{aligned} \sigma _2^6 - 4 q_2^{1+\lambda } \sigma _2^2 = 0. \end{aligned}$$

There are four nonzero solutions satisfying

$$\begin{aligned} \sigma _2^4 = 4 q_2^{1+\lambda } \rightarrow \infty , \quad \sigma _1^2 \sim 16 q_1 q_2^\lambda \rightarrow \infty , \end{aligned}$$

which contribute to four Coulomb vacua. The two zero solutions contribute to the Higgs branch.

(iv) $\zeta _1 \gg 1,\zeta _2 \ll -1$: $q_1 \rightarrow 0, q_2 \rightarrow \infty$, then (B.6) can be approximated by

$$\begin{aligned} \sigma _2^6 + q_2^2 = 0. \end{aligned}$$

Thus $\sigma _2 \rightarrow \infty$, and

$$\begin{aligned} \sigma _1^2 \sim 4 q_1 q_2^{1/3} \rightarrow \left\{ \begin{array}{ll} 0, &{} \zeta _1+\zeta _2/3>0 \\ \infty , &{} \zeta _1+\zeta _2/3<0 \end{array} \right. \end{aligned}$$

Therefore, we have mixed branch when $\zeta _2 > -3 \zeta _1$, and Coulomb vacua when $\zeta _2 < -3 \zeta _1$. The full phase diagram is given by Fig. 7 with $n=3, k=1, d_1=2$.

D0-Brane Probes

We have seen that the HPD category can be described by the category of matrix factorizations in general. However, the geometric meaning of the HPD is vague in this description. In some cases, we can completely or partially recover a geometric entity from the matrix factorizations by studying the so-called point-like branes or D0-brane probes. In this appendix, we provide a careful analysis of the point-like branes. A similar analysis was performed in [19] for LG models, here we take the GLSM point of view. We take the ${\mathbb {P}}^3[2,2]$ model as a demonstrative example explaining our construction of a GLSM D0-brane, where the moduli space of this D0-brane describes a branched double cover.

The corresponding GLSM has the matter content with the U(1) charges

$$\begin{aligned} \begin{array}{ccccccc} X_0 &{} X_1 &{} X_2 &{} X_3 &{} P_0 &{} P_1 \\ 1 &{} 1 &{}1 &{} 1 &{} -2 &{} -2 \end{array} \end{aligned}$$

and superpotential

$$\begin{aligned} W = P_0 G_0(X) + P_1 G_1(X), \end{aligned}$$

where $G_1(X)$ and $G_2(X)$ are quadratic polynomials of $X_i$. It is easy to see that the geometric phase $( \zeta \gg 1 )$ of this model is a NLSM on ${\mathbb {P}}^3[2,2]$. The LG phase $( \zeta \ll -1 )$ is better understood if we rewrite the superpotential as

$$\begin{aligned} W = \sum _{ij} X_i A_{ij}(P) X_j, \end{aligned}$$

where $A_{ij}$ is a $4 \times 4$ matrix with entries linear in $P_i$. In this phase, $p_1$ and $p_2$ expand a ${\mathbb {Z}}_2$-gerby ${\mathbb {P}}^1$ with branch points at

$$\begin{aligned} \det A_{ij} = 0, \end{aligned}$$

where some of $X_i$ become massless. In other words, it is a branched double cover ramified over four points, which is a torus.

Since $A_{ij}$ is a symmetric matrix, we can always diagonalize it with suitable field redefinitions of $X_i$. Therefore, without loss of generality, we can work with the superpotential

$$\begin{aligned} W = f_0 X_0^2 + f_1 X_1^2 + f_2 X_2^2 + f_3 X_3^2, \end{aligned}$$

(C.1)

where $f_i(P_0,P_1)$ are linear polynomials and they determine the branch points. The superpotential can be written as

$$\begin{aligned} W =&(P_0 + \lambda P_1) G_0 + P_1(G_1 - \lambda G_0), \end{aligned}$$

(C.2)

$$\begin{aligned} =&(P_0 + \lambda P_1) G_0 + P_1( L_1 F_1 + L_2 F_2), \end{aligned}$$

(C.3)

with

$$\begin{aligned} L_1&= \sqrt{f_0'} X_0 + \sqrt{-f_1'} X_1, \quad F_1 = \sqrt{f_0'} X_0 - \sqrt{-f_1'} X_1,\\ L_2&= \sqrt{f_2'} X_2 + \sqrt{-f_3'} X_3, \quad F_2 = \sqrt{f_2'} X_2 - \sqrt{-f_3'} X_3,\\ \end{aligned}$$

where

$$\begin{aligned} f_i' = f_i(-\lambda , 1). \end{aligned}$$

Our proposal of the matrix factorization is

$$\begin{aligned} Q = (P_0 + \lambda P_1) {\bar{\eta }}_0 + G_0 \eta _0 + P_1 F_1 {\bar{\eta }}_1 + L_1 \eta _1 + P_1 F_2 {\bar{\eta }}_2 + L_2 \eta _2. \end{aligned}$$

(C.4)

This brane is supported on

$$\begin{aligned} P_0 + \lambda P_1 = 0, \end{aligned}$$

(C.5)

$$\begin{aligned} L_1 =L_2 = 0, \end{aligned}$$

(C.6)

$$\begin{aligned} G_0 =0, \end{aligned}$$

(C.7)

$$\begin{aligned} P_1 F_1 = P_1 F_2 = 0. \end{aligned}$$

(C.8)

The first constraint restricts the brane to a specific point on the base ${\mathbb {P}}^1$; the second equation describes the isotropic submanifold of the fiber as in [19]; the last equation restricts the brane support to the origin of the fiber, so indeed we get a D0-brane. The corresponding B-brane is described by a ${\mathbb {Z}}_2$ complex,

(C.9)

with the morphisms

$$\begin{aligned} f&= \begin{pmatrix} P_1 F_1 &{} -L_2 &{} G_0 &{} 0 \\ P_1 F_2 &{} L_1 &{} 0 &{} G_0 \\ P_0 + \lambda P_1 &{} 0 &{} -L_1 &{} - L_2 \\ 0 &{} P_0 + \lambda P_1 &{} P_1 F_2 &{} -P_1 F_1 \end{pmatrix}, \\ \\ g&= \begin{pmatrix} L_1 &{} L_2 &{} G_0 &{} 0 \\ -P_1 F_2 &{} P_1 F_1 &{} 0 &{} G_0 \\ P_0 + \lambda P_1 &{} 0 &{} - P_1 F_1 &{} L_2\\ 0 &{} P_0 + \lambda P_1 &{} -P_1 F_2 &{} -L_1 \end{pmatrix}. \end{aligned}$$

First, notice that ${{{\mathcal {B}}}}(n)$ is quasi-isomorphic to ${{{\mathcal {B}}}}(n+2)$ which can be shown by the cone

where we denote the branes ${{{\mathcal {B}}}}(n)$ and ${{{\mathcal {B}}}}(n+2)$ as

Notice that the chain maps $\phi _0$ and $\phi _1$ satisfying the following properties:

• 1. All entries are holomorphic;

• 2. All entries should have the correct gauge charges;

• 3. The commutative relations: $\phi _0 g = g \phi _1$, $f \phi _0 = \phi _1 f$.

In order to show the quasi-isomorphism, one need to find morphisms $\phi _0$ and $\phi _1$ such that at every point the cone potential $\{ Q_c, Q_c^{\dagger } \} > 0$ in the Landau-Ginzburg phase where $P_0$ and $P_1$ can not vanish simultaneously. The matrix factorization here is given by

$$\begin{aligned} Q_c = \begin{pmatrix} 0 &{} {\tilde{g}} \\ {\tilde{f}} &{} 0 \end{pmatrix}, \end{aligned}$$

(C.10)

with

$$\begin{aligned} {\tilde{f}} = \begin{pmatrix} f &{} 0 \\ \phi _0 &{} -g \end{pmatrix}, \quad {\tilde{g}} = \begin{pmatrix} g &{} 0 \\ \phi _1 &{} -f \end{pmatrix}. \end{aligned}$$

(C.11)

The potential for the cone is

$$\begin{aligned} \{&Q_c, Q_c^{\dagger } \} = \\&\begin{pmatrix} gg^{\dagger } + f^{\dagger }f+ \phi _0^{\dagger } \phi _0 &{} g \phi _1^{\dagger } - \phi _0^{\dagger } g &{} 0 &{} 0 \\ \phi _1 g^{\dagger } - g^{\dagger } \phi _0 &{} gg^{\dagger } + f^{\dagger } f + \phi _1 \phi _1^{\dagger } &{} 0 &{} 0 \\ 0 &{} 0 &{} f f^{\dagger }+ g^{\dagger } g + \phi _1^{\dagger } \phi _1 &{} f \phi _0^{\dagger } - \phi _1^{\dagger } f \\ 0 &{} 0 &{} \phi _0 f^{\dagger } - f^{\dagger } \phi _1 &{} f f^{\dagger } + g^{\dagger } g + \phi _0 \phi _0^{\dagger } \end{pmatrix}. \end{aligned}$$

Taking

$$\begin{aligned} \phi _0 = \phi _1 = P_1 \, \mathrm {Id}_{4 \times 4}, \end{aligned}$$

(C.12)

one can easily see the cone potential only vanishes at $P_0 = P_1 = 0$ which means that the cone potential always greater than zero at Landau-Ginzburg phase. Therefore, there are two equivalent families of branes ${{{\mathcal {B}}}}(n)$ with n even or n odd. WLOG, one can focus on ${{{\mathcal {B}}}}(0)$ and ${{{\mathcal {B}}}}(1)$. Let’s denote ${{{\mathcal {B}}}}(0)$ and ${{{\mathcal {B}}}}(1)$ as

To study the relations between ${{{\mathcal {B}}}}(0)$ and ${{{\mathcal {B}}}}(1)$, we take the cone

with the chain maps ${\tilde{\phi }}_0$ and ${\tilde{\phi }}_1$. Notice that the brane ${{{\mathcal {B}}}}(1)$ has been shifted by 1, and the commutative relations for the diagram becomes $g \phi _0 = \phi _1 f$, $\phi _0 g = f \phi _1$. The matrix factorization for the cone is given by

$$\begin{aligned} {\tilde{Q}}_c = \begin{pmatrix} 0 &{} {\tilde{g}} \\ {\tilde{f}} &{} 0 \end{pmatrix}, \end{aligned}$$

(C.13)

with

$$\begin{aligned} {\tilde{f}} = \begin{pmatrix} f &{} 0 \\ {\tilde{\phi }}_0 &{} -f \end{pmatrix}, \quad {\tilde{g}} = \begin{pmatrix} g &{} 0 \\ {\tilde{\phi }}_1 &{} -g \end{pmatrix}. \end{aligned}$$

(C.14)

The potential for the cone is

$$\begin{aligned} \{&{\tilde{Q}}_c, {\tilde{Q}}_c^{\dagger } \} = \\&\begin{pmatrix} gg^{\dagger } + f^{\dagger }f+ {\tilde{\phi }}_0^{\dagger } {\tilde{\phi }}_0 &{} g {\tilde{\phi }}_1^{\dagger } - {\tilde{\phi }}_0^{\dagger } f &{} 0 &{} 0 \\ {\tilde{\phi }}_1 g^{\dagger } - f^{\dagger } {\tilde{\phi }}_0 &{} gg^{\dagger } + f^{\dagger } f + {\tilde{\phi }}_1 {\tilde{\phi }}_1^{\dagger } &{} 0 &{} 0 \\ 0 &{} 0 &{} f f^{\dagger }+ g^{\dagger } g + {\tilde{\phi }}_1^{\dagger } {\tilde{\phi }}_1 &{} f {\tilde{\phi }}_0^{\dagger } - {\tilde{\phi }}_1^{\dagger } g \\ 0 &{} 0 &{} {\tilde{\phi }}_0 f^{\dagger } - g^{\dagger } {\tilde{\phi }}_1 &{} f f^{\dagger } + g^{\dagger } g + {\tilde{\phi }}_0 {\tilde{\phi }}_0^{\dagger } \end{pmatrix}. \end{aligned}$$

Away from the branch points, it is not hard to find morphisms $\phi _0$ and $\phi _1$ such that the cone potential vanishes at some points. Thus, the brane ${{{\mathcal {B}}}}(0)$ and brane ${{{\mathcal {B}}}}(1)$ are not quasi-isomorphic to each other. We have two different sets of branes away from branch points corresponding to two copies of ${\mathbb {P}}^1$.

To show that ${{{\mathcal {B}}}}(0)$ and ${{{\mathcal {B}}}}(1)$ are quasi-isomorphic at branch points, one needs to find a special pair of morphisms $\phi _0$ and $\phi _1$ such that at every point the potential $\{Q_c, Q_c^{\dagger } \} > 0$. What is special about the branch points is that some of the polynomials defining the matrix factorization become the same,

$$\begin{aligned} L_1 = \pm F_1 , \quad L_2 = \pm F_2. \end{aligned}$$

If $L_1 = F_1$, we find the chain maps can be

$$\begin{aligned} \phi _0 = c \cdot \mathrm {diag}\{ -P_1, 1, -1, P_1 \}, \quad \phi _1 = c \cdot \mathrm {diag} \{ -1, P_1, -P_1, 1\}, \end{aligned}$$

(C.15)

where c is a constant. The determinant of the potential is

$$\begin{aligned} \det \{Q_c, Q_c^{\dagger }\} = \Big ( (A+|c P_1|^2)(A+|c|^2) - |c|^2(|P_1|^2-1)^2 |L_1|^2\Big )^8, \end{aligned}$$

(C.16)

with

$$\begin{aligned} A =|P_0 + \lambda P_1|^2 + |L_1|^2+ |L_2|^2 + |G_0|^2 + |P_1 F_1|^2 + |P_1 F_2|^2. \end{aligned}$$

Vanishing of the determinant implies that $P_0 = P_1 = 0$, which means the potential is always greater than zero at Landau-Ginzburg phase. Therefore, we showed that the branes ${{{\mathcal {B}}}}(0)$ and ${{{\mathcal {B}}}}(1)$ are quasi-isomorphic at the branch point corresponding to $L_1 =F_1$.

Similarly, when $L_1 = - F_1$, we find

$$\begin{aligned} \phi _0 = c \cdot \mathrm {diag}\{ P_1, 1, -1, -P_1 \}, \quad \phi _1 = c \cdot \mathrm {diag} \{ -1, -P_1, P_1, 1\}, \end{aligned}$$

(C.17)

such that the cone potential is always greater than zero on the Landau-Ginzburg phase. For the branch points corresponding to $L_2 = \pm F_2$, the branes are also quasi-isomorphic since $L_2, F_2$ and $L_1, F_1$ are on the same footing. One can see this by constructing ${{{\mathcal {B}}}}'(0)$ and ${{{\mathcal {B}}}}'(1)$ with $L_2, F_2$ and $L_1, F_1$ exchanged which are quasi-isomorphic to ${{{\mathcal {B}}}}(0)$ and ${{{\mathcal {B}}}}(1)$.

In extreme case that $L_1 = F_1 = L_2 = F_2 = 0$ at branch locus, the morphisms of matrix factorization become

$$\begin{aligned} f = g = \begin{pmatrix} 0 &{} 0 &{} G_0 &{} 0 \\ 0 &{} 0 &{} 0 &{} G_0 \\ P_0 + \lambda P_1 &{} 0 &{} 0 &{} 0 \\ 0 &{} P_0 + \lambda P_1 &{} 0 &{} 0 \end{pmatrix}. \end{aligned}$$

(C.18)

The brane ${{{\mathcal {B}}}}(0)$ and ${{{\mathcal {B}}}}(1)$ are obviously quasi-isomorphic.

It is also possible that $L_1 = F_1 = 0$ or $L_2 = F_2 = 0$ at the branch point which means that two of the chiral fields $X_i$ are always massive and can be integrated out. In this case, the matrix factorization can be reduced to

$$\begin{aligned} Q = (P_0 + \lambda P_1) {\bar{\eta }}_0 + G_0 \eta _0 + P_1 F_1 {\bar{\eta }}_1 + L_1 \eta _1, \end{aligned}$$

(C.19)

with the assumption that $X_2$ and $X_3$ are massive. The brane complex is

(C.20)

with

$$\begin{aligned} f' = \begin{pmatrix} P_1F_1 &{} G_0 \\ P_0 + \lambda P_1 &{} -L_1 \end{pmatrix}, \quad g' = \begin{pmatrix} L_1 &{} G_0 \\ P_1 + \lambda P_1 &{} -P_1 F_1 \end{pmatrix}. \end{aligned}$$

(C.21)

Following the same procedure above, one can easily show that the brane ${{{\mathcal {B}}}}'(0)$ and ${{{\mathcal {B}}}}'(1)$ are quasi-isomorphic. On the LG phase, the Wilson line brane descends to orbibundle supporting on the $[{\mathbb {P}}^1]_{{\mathbb {Z}}_2}$,

$$\begin{aligned} W(q) \rightarrow {\mathcal {O}}(q/2). \end{aligned}$$

(C.22)

Then, ${{{\mathcal {B}}}}'(0)$ reduces to

(C.23)

and ${{{\mathcal {B}}}}'(1)$ reduces to

(C.24)

Notice that on the local patch of ${\mathbb {P}}^1$ matrix factorization (C.23) and (C.24) reduce to the local matrix factorization studied in [19]. Those two local matrix factorizations are obviously quasi-isomorphic.

In summary, we have found morphisms $\phi _0$, $\phi _1$ such that the cone potential $\{ Q_c, Q_c^{\dagger } \} > 0$ for all possible scenarios. Also, our proposal recovers the local model studied in the literature. In this way, we have shown that the LG phase of ${\mathbb {P}}^3[2,2]$ model is indeed a branched double cover and our proposed matrix factorization (C.4) gives a global description of the D0-brane on the branched double cover geometry.