Twisted Supergravity and Koszul Duality: A Case Study in AdS$_3$

Abstract

In this note, we study a simplified variant of the familiar holographic duality between supergravity on $\hbox {AdS}_3\times S^3\times T^4$ and the SCFT (on the moduli space of) the symmetric orbifold theory $Sym^N(T^4)$ as $N \rightarrow \infty$. This variant arises conjecturally from a twist proposed by the first author and Si Li. We recover a number of results concerning protected subsectors of the original duality working directly in the twisted bulk theory. Moreover, we identify the symmetry algebra arising in the $N\rightarrow \infty$ limit of the twisted gravitational theory. We emphasize the role of Koszul duality—a ubiquitous mathematical notion to which we provide a friendly introduction—in field theory and string theory. After illustrating the appearance of Koszul duality in the “toy” example of holomorphic Chern-Simons theory, we describe how (a deformation of) Koszul duality relates bulk and boundary operators in our twisted setup, and explain how one can compute algebra OPEs diagrammatically using this notion. Further details, results, and computations will appear in a companion paper.

Appendix A. Koszul duality from TQFT

Here we briefly recapitulate some basic facts about Koszul duality and its natural appearance in topological field theory.

Suppose that ${\mathcal {A}}$ is a differential graded associative algebra and that $h: {\mathcal {A}} \rightarrow {\mathbb {C}}$ is homomorphism of dg algebras:

$$\begin{aligned} h(ab) = h(a)h(b), \ \ \ \ h(\mathrm {d}a) = 0 . \end{aligned}$$

(A.0.1)

The homomorphism h is called the augmentation. The augmentation makes the one-dimensional vector space ${\mathbb {C}}$ into a module for ${\mathcal {A}}$, the module structure given by multiplication with h(a). We refer to this module as ${\mathbb {C}}_{h}$. In the context of topological line defects described in section 5, if the theory on the line one obtains upon reducing the theory along the transverse directions is the trivial theory, then we have the augmentation map ${\mathcal {A}} \rightarrow {\mathbb {C}}$, viewing ${\mathbb {C}}$ as a rank-1 ${\mathcal {A}}$-module. This reduction requires a choice of vacuum, viewed as a suitable boundary condition at infinity, in the noncompact transverse directions.³¹

The mathematical definition of the Koszul dual algebra ${\mathcal {A}}^!$ is

$$\begin{aligned} {\mathcal {A}}^! = \text {Ext}^*_{{\mathcal {A}}}({\mathbb {C}}_{h},{\mathbb {C}}_{h}). \end{aligned}$$

(A.0.2)

That is, it is the self-Ext’s (roughly speaking, symmetries) of the module ${\mathbb {C}}_{h}$.

One way to interpret this more physically is the following. Using TFT axiomatics [L09, L11] (see also [BBBDN18] for recent related discussions), we can build a two-dimensional topological field theory whose category of left (or right) boundary conditions is the category ${\mathcal {A}}$-$\text {mod}$ of left (or right) ${\mathcal {A}}$-modules. Given any module M, the algebra of operators on the corresponding boundary condition is $\text {Ext}^*_{\mathcal {A}}(M,M)$, the self-Ext’s of M.

The algebra ${\mathcal {A}}$ itself is a right ${\mathcal {A}}$-module under right multiplication: therefore it defines a right boundary condition, $B_R$. The algebra of self-Ext’s of ${\mathcal {A}}$, as a right ${\mathcal {A}}$-module, is simply ${\mathcal {A}}$ itself, acting by left multiplication. Therefore ${\mathcal {A}}$ is the algebra of local operators on the boundary $B_R$.

The module ${\mathbb {C}}_{h}$ defines a left boundary condition, $B_L$. The operators on the boundary with this boundary condition are, by definition (A.0.2), the Koszul dual algebra ${\mathcal {A}}^!$.

The states of the two-dimensional TFT on a strip, with $B_L$ on one side and $B_R$ on the other, are the one-dimensional vector space ${\mathbb {C}}_{h}$. This strip configuration is one convenient physical interpretation of the augmentation.

In general, following this picture, we can propose a physical origin of Koszul duality: if we have a two-dimensional TFT with left and right boundary conditions $B_L$, $B_R$, such that the states on a strip with these boundary conditions is one dimensional, then the algebras of boundary operators for the two boundary conditions are Koszul dual.