SL(2,R) Chern-Simons, Liouville, and Gauge Theory on Duality Walls

TL;DR

The paper proposes a 3d/3d duality: the partition function of a noncompact $SL(2,\mathbb{R})$ Chern-Simons theory on a punctured surface $\Sigma$ times an interval matches the partition function of a 3d $\mathcal{N}=2$ theory on a deformed sphere $S^3_b$, realized as a duality domain wall inside a 4d $\mathcal{N}=2$ theory on $S^4$. This bridge is constructed through a triad of correspondences: gauge theory to Liouville theory (via AGT and Nekrasov partition functions), Liouville to quantum Teichmüller theory (where conformal blocks correspond to Teichmüller Hilbert-space overlaps and momentum labels to geodesic lengths), and Teichmüller to Chern-Simons (identifying the CS boundary Hilbert space with Teichmüller data and interpreting domain-wall amplitudes as CS matrix elements). The authors provide explicit checks for the once-punctured torus, formulate a generalized AGT interpretation in CS language, and discuss higher-rank generalizations and M5-brane realizations. This framework yields a 3+3 dimensional analog of AGT, linking 4d/3d dualities to 3d CS theory and suggesting new perspectives on quantum geometry, dualities, and topological invariants.

Abstract

We propose an equivalence of the partition functions of two different 3d gauge theories. On one side of the correspondence we consider the partition function of 3d SL(2,R) Chern-Simons theory on a 3-manifold, obtained as a punctured Riemann surface times an interval. On the other side we have a partition function of a 3d N=2 superconformal field theory on S^3, which is realized as a duality domain wall in a 4d gauge theory on S^4. We sketch the proof of this conjecture using connections with quantum Liouville theory and quantum Teichmuller theory, and study in detail the example of the once-punctured torus. Motivated by these results we advocate a direct Chern-Simons interpretation of the ingredients of (a generalization of) the Alday-Gaiotto-Tachikawa relation. We also comment on M5-brane realizations as well as on possible generalizations of our proposals.

Figures (14)

• Figure 1: The flow of logic of this section (and the paper).
• Figure 2: Our 3d $\mathcal{N}=2$ theory is realized as a duality domain wall $S^3$ inside $S^4$, where the mother 4d $\mathcal{N}=2$ theory lives. On the left figure, the complexified gauge coupling of the mother 4d theory in the northern (southern) hemisphere is given by $\tau$ ($\tau'$), where $\tau'$ is related to $\tau$ by an action of the S-duality group element $\varphi$: $\tau'=\varphi(\tau)$. Instead, we can take S-duality in the southern hemisphere to make the complexified gauge coupling $\tau$ in the whole $S^4$, but then we have a non-trivial 3d theory on the equator $S^3$.
• Figure 3: We consider Chern-Simon theory on $\Sigma\times I$, where the complex structure of the two Riemann surfaces on the boundary are specified by $l$ and $\varphi(l')$.
• Figure 4: A genus $g$ handlebody is a 3-manifold obtained by attaching attaching $g$ handles to a 3-ball. Its boundary is a genus $g$ Riemann surface.
• Figure 5: Graphical representation of $S$ and $T$ operations on 3d $\mathcal{N}=4$ theory. In these figures a square represents a global symmetry, a circle gauge symmetry, and a hexagon a Chern-Simons term, with level specified by an integer written nearby. In (a) we listed all the basic building block of our quiver. The left represents our $T[SU(2)]$ theory. This is often written as above, with only $SU(2)$ symmetry manifest. However, it also has quantum $SU(2)^{\vee}$ symmetry, and in the figure below we have also shown this symmetry explicitly. The figure on the right represents the action of T, i.e., Chern-Simons couplings. More generally, an element of the mapping class group is represented by a product of $S$ and $T$, and graphically represented by a linear quiver. The example of $\varphi=ST^k ST^l ST^m S$ is given in (b). When we gauge the remaining $SU(2)\times SU(2)$ symmetry, we have a circular quiver, given in (c). In figures (b) and (c) we neglected subtle differences between $SU(2)$ and $SU(2)^\vee$.