The Categorical 't Hooft Expansion

TL;DR

The paper develops a categorical framework for the large $N$ 't Hooft expansion by modeling the proposed worldsheet dual as a 2d dg-TFT governed by an $A_\infty$-category of boundary conditions. It shows how planar deformations of fundamental modifications of a large $N$ QFT correspond to deformations of D-brane boundary data, encoding the genus expansion through a 3d Calabi–Yau $A_\infty$-category and its wedge algebra of boundary operators. The authors analyze a concrete 2d chiral gauge theory with target $SL(2,\mathbb{C})$, derive the D-brane boundary structure and its deformations, and connect these to Schur-like algebras and holographic duals in Chern–Simons, BF, and 1d systems, illustrating a broad categorical strategy for holography. This approach aims to formalize a wide class of large $N$ dualities by reconstructing worldsheet data from boundary-category calculations, potentially enabling rigorous treatment of loop corrections and non-geometric backgrounds. Overall, the work proposes a unifying, mathematically precise route to connect large $N$ QFTs to string theories via dg-TFTs and $A_\infty$-categories of D-branes.

Abstract

We review categorical aspects of 't Hooft's large $N$ expansion, which is expected to map any Quantum Field Theory of large matrices to a string theory. Our goal is to describe a general strategy to derive the string theory dual to given QFT, at least at the leading order in the 't Hooft expansion. The basic idea is to characterize the underlying worldsheet theory of the dual string theory as an extended two-dimensional differential graded Topological Field Theory (dg-TFT), i.e. present an $A_\infty$-category of boundary conditions ("D-branes"). A basic aspect of the 't Hooft expansion is that D-branes arise from the addition of vector-valued degrees of freedom to the QFT. We propose that formal deformations of such "fundamental modifications" must match the formal deformations of the dual D-branes, which in turn capture the $A_\infty$-category structure and thus the worldsheet dg-TFT. We discuss several systems for which a rigorous analysis along these lines is or should be possible.

Figures (3)

• Figure 1.1: Some order $\lambda^2$ contributions to the perturbative expansion of the quartic matrix integral. Left: the leading (aka planar) contribution has $g=0$. Middle and right: the sub-leading contributions have $g=1$.
• Figure 1.2: Some order $\mu^2$ contributions to the perturbative expansion of the quartic matrix integral with a fundamental modification. Left: the leading (aka planar) contribution has $g=0$, $b=1$. Right: the sub-leading contribution has $g=1$, $b=1$.
• Figure 1.3: Schematic depiction of D-branes which can occur in a large $N$ expansion. The depiction refers to "holographic" examples of large $N$ dualities, where a $d$-dimensional QFT on a manifold $M_d$ is dual to a string theory with a 2d worldsheet theory of maps into a $(d+1)$-dimensional manifold with boundary $M_d$ (not to be confused with boundaries for the worldsheet itself). Some D-branes can emerge from a choice of large $N$ saddle in the absence of a fundamental modification. We refer to these as "compact" D-branes: in a holographic setup they do not reach the $M_d$ boundary. Other D-branes emerge from explicit fundamental modifications of the QFT. The same modification may correspond to multiple "non-compact" D-branes depending on a choice of large $N$ saddle. In a holographic setup, the asymptotic shape near $M_d$ of the non-compact D-branes is determined by the choice of modification.

Theorems & Definitions (2)

• Claim 1.1
• Definition 1.2