Interface Minimal Model Holography and Topological String Theory

Abstract

We study the dynamics of 2d fermions coupled to 3d Chern-Simons gauge fields. For $SU(N)$ gauge group and fermions in the fundamental representation, the resulting interfaces are closely related to $W_N$ minimal models. We give an holographic description of the interfaces within the A-model Topological String Theory. The model has exotic integrability properties, which allow us to propose an exact holographic match of all sphere correlation functions of meson operators. This construction embeds Minimal Model Holography in String Theory.

Figures (11)

• Figure 1: Two variants of the minimal model RCFT. Top: any RCFT can be "resolved" into a pair of chiral and anti-chiral boundary conditions for a 3d TFT. Up to a topological manipulation, we can use a $SU(N)_\kappa \times SU(N)_{1} \times SU(N)_{-\kappa-1}$ 3d Chern-Simons theory and represent each boundary condition as a trivalent junction. Bottom: we replace the $SU(N)_1$ chiral algebra in the coset with $N$ chiral complex fermions $\mathrm{Ff}_\mathbb{C}$, simplifying the associated 3d TFT to $SU(N)_\kappa \times SU(N)_{-\kappa-1}$ and the junctions to interfaces. We can denote the resulting 2d theory as $M_{N;k}$. Individual interfaces can be engineered by canonically coupling the 3d Chern-Simons gauge fields to the (anti)chiral 2d fermions.
• Figure 2: Two interfaces related to $M_{N;k}$ by topological manipulations. Top: the interface $M^+_{N;\kappa}$ connects the chiral and anti-chiral interfaces through the $SU(N)_\kappa$ TFT to produce an interface in $SU(N)_{\kappa+1}$ CS theory. Bottom: the interface $M^-_{N;\kappa}$ connects the chiral and anti-chiral interfaces through the $SU(N)_{\kappa+1}$ TFT to produce an interface in $SU(N)_{\kappa}$ CS theory. If we shrink the intermediate slabs to zero size, both $M^+_{N;\kappa-1}$ and $M^-_{N;\kappa}$ become non-chiral RCFT interfaces in $SU(N)_\kappa$ Chern-Simons theory. They are scale-invariant fixed points for a system of non-chiral 2d fermions coupled to $SU(N)_\kappa$ Chern-Simons gauge fields. This and later figures omit the 2d space-time directions.
• Figure 3: A collection of "slab" configurations producing a 2d system from a 3d $G_k$ Chern-Simons theory. The 2d directions are not drawn. Top left: a 2d theory $T_k$ with chiral Kac-Moody $G$ symmetry at level $k>0$ can be decomposed into a slab with a chiral Dirichlet boundary supporting a $G_k$ WZW chiral algebra and enriched Neumann supporting a coset $T_k/G_k$. Top right: the analogous resolution of a 2d theory $\overline T_k$ with anti-chiral Kac-Moody $G$ symmetry. Bottom left: two Dirichlet b.c. engineer a non-chiral WZW model. Bottom right: two enriched Neumann engineer a non-chiral coset. We expect the coset to be the IR limit of both a 2d $G$ gauge theory with matter $T_k \times \overline T_k$ and of an asymptotically free $J \cdot \bar{J}$ deformation of $T_k \times \overline T_k$.
• Figure 4: A proposed resolution of the interface between $G_\kappa$ and $G_{\kappa+k}$ Chern-Simons theories defined by coupling the 3d gauge fields to $T_k$. We assume $\kappa>0$ here. Shrinking the segment gives a $\frac{G_{\kappa}\times T_k}{G_{\kappa + k}}$ coset at the interface.
• Figure 5: A resolution of an enriched Neumann b.c. for $G_{k_1+k_2}$ CS theory coupled to $T^{(1)}_{k_1}\times T^{(2)}_{k_2}$. Shrinking the segments reproduces the coset $\frac{T^{(1)}_{k_1}\times T^{(2)}_{k_2}}{G_{k_1 + k_2}}$.