D-brane central charge and Landau-Ginzburg orbifolds
The current paper under review discusses the central charge of a B-type D-brane \(\mathcal{B}\) in the context of \(\mathcal{N}=(2,2)\) superconformal field theories (SCFT). The paper suggests a universal expression \(Z_{\text{SCFT}}(\mathcal{B})= \langle \widehat{\Gamma}^* \circ \mathbf{J} \mid \text{ch}(\mathcal{B})\rangle\) for the central charge and provides a convincing evidence for why this is the right definition in some crucial case. To explain the context of the evidence, note an important fact that non-anomalous (aka Calabi-Yau) gauged linear sigma models (GLSM) flow to those SCFTs in the low-energy limit under the renormalization group flow. On the other hand, it is known by Hori and Romo's earlier work (``Exact results in two-dimensional (2,2) supersymmetric gauge theories with boundary'') that the central charge of a B-brane \(\widetilde{\mathcal{B}}\) of GLSM can be understood as the hemisphere partition function \(Z_{D^2}(\widetilde{\mathcal{B}})\). This suggests that if a GLSM flows to a SCFT such that \(\widetilde{\mathcal B}\) corresponds to \(\mathcal B\), then their central charges \(Z_{D^2}(\widetilde{\mathcal{B}})\) and \(Z_{\text{SCFT}}(\mathcal{B})\) should coincide. This was known previously for a geometric phase. The current paper argues for an LG orbifold phase. Let us be more concrete. For the input data of GLSM, one considers a quadruple \((G,\rho_V, W,R)\), where \(G\) is a compact Lie group (gauge group), \(\rho_V\colon G\to GL(V)\cong GL(\mathbb{C}^N)\) a faithful finite-dimensional representation (chiral matter fields), \(W\in \text{Sym}(V^*)^G\) (superpotential), and \( R\colon U(1)_V\to GL(V)\), say \(\lambda \cdot (z_1,\cdots,z_N) = (\lambda^{q_1}z_1,\cdots,\lambda^{q_n}z_n )\) (vector R-charge). A GLSM being non-anomalous means that \(\rho_V \) factors through \(SL(V)\), i.e., the axial R-charge \(U(1)_A\) is non-anomalous. We also ask that \(W\) satisfies \(W( \lambda \cdot z ) = \lambda^2 W(z)\), i.e., the superpotential preserves the \(U(1)_V\) symmetry. A GLSM has FI-theta parameters \(\tau = \zeta - i\theta\). A B-brane of a GLSM is a quadruple \(\widetilde{\mathcal{B}}=(M,Q,\rho_M, r_*)\) for the equivariant matrix factorization data. Here \(M=M^0\oplus M^1\) is a \(\mathbb{Z}/2\)-graded free \(\text{Sym}(V^*)\)-module (Chan--Paton space), \(Q\in \text{End}_{ \text{Sym}(V^*)}^1(M)\) with \(Q^2= W\cdot \text{id}_M\) (matrix factorization of \(W\)), \(\rho_M\colon G\to GL(M)\) satisfying \(\rho_M(g)^{-1}Q(gz)\rho_M(g)=Q(z)\) for \(g\in G\) (gauge invariance), and \(R\)-action \(r_* \colon \mathfrak{u}(1)_V\to \mathfrak{gl}(M)\) satisfying \(\lambda^{r_*} Q(\lambda^R z ) \lambda^{-r_*} = \lambda Q(z)\) for \(\lambda \in U(1)_V\) (\(R\)-charge 1). In this generality, Hori and Romo explicitly computed the hemisphere partition function \(Z_{D^2}(\widetilde{\mathcal{B}})\) by supersymmetric localization. Here by the hemisphere partition function, we think of a disk where we consider the A-twist but put a B-brane \(\widetilde{\mathcal{B}}\) at the boundary. There are two things to note here to make sense of this. First, we consider a boundary state corresponding to a D-brane because we read off a disk, not as an evolution from an empty set to a closed string, but rather as a circular evolution of an open string (with one end being trivially fixed). Second, considering a B-brane in the A-twist may be understood by the use of spectral flow operator which identifies the \((a,c)\) ring and the \((c,c)\) ring as vector spaces. Let us digress a bit and discuss LG orbifolds in an independent manner. It consists of \((V, \overline{\rho}_m, W, U(1)_{L/R})\), where \(V\cong \mathbb{ C}^N\) is a vector space with a left R-symmetry \(U(1)_{L/R}\) action on \(V\) with weights \(q_j\in \mathbb Q \cap (0,1)\), \(\overline{\rho}_m\colon \Gamma \to GL(V)\) is a representation of a finite abelian group \(\Gamma \), and \(W \colon V\to \mathbb C\) is a \(G\)-invariant function, such that \(W\) is quasi-homogeneous of weight 1 under \(U(1)_{L/R}\), that is, \(W(\lambda^{q_i} z_i )= \lambda W(z_i)\). A B-brane of LG orbifolds is \(\mathcal{B} = ( \overline{M}, \sigma,\overline{Q},\overline{\rho},\overline{R})\), where the definition is similar to the case for GLSM. Indeed, as expected, given a GLSM with a B-brane \(\widetilde{\mathcal B}\), one can obtain a corresponding LG orbifold model with a B-brane \(\mathcal B\), as long as GLSM is abelian non-anomalous with an LG orbifold phase. On the other hand, to compute the central charge of B-brane of an LG orbifold theory, one has to understand deformations away from the given point in the moduli as well. Note that the space of marginal operators of interest may be written as \(H^{(a,c)}_{(-1,1)}\). In the geometric case, it is the so-called stringy Kähler moduli space. In the LG orbifold case, it is of the form \(H^{(a,c)}_{(-1,1)} = \oplus_{\gamma\in \Gamma } \mathcal H_{\gamma}\). The paper focuses on \(\mathcal H_{\gamma}\)'s with \(\dim \mathcal H_{\gamma} =1\), which form the so-called narrow sector. This is because for a broad sector with \(\dim \mathcal H_{\gamma}>1\), its GLSM lift is not known. With this restriction understood, the main part of the paper checks \(Z_{D^2}(\widetilde{\mathcal{B}}) = Z_{\text{LG}}(\mathcal{B})\) as desired, where \(Z_{\text{LG}}(\mathcal{B}) = \langle \widehat{\Gamma}^* \circ \mathbf{J} \mid \text{ch}(\mathcal{B})\rangle\) is explicitly defined purely in terms of the data of LG orbifolds. Compatibility with FJRW theory is also discussed.
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