4d Index to 3d Index and 2d TQFT

TL;DR

This work analyzes the 4d superconformal index for $\mathcal{N}=1,2$ theories on $S^1\times L(p,1)$, deriving explicit orbifold-index expressions and showing two key dimensional reductions: to the 3d index on $S^1\times S^2$ in the large-$p$ limit and to the 3d partition function on $L(p,1)$ as the temporal circle shrinks. It connects these 4d observables to 3d quantities and applies the framework to $\mathcal{N}=2$ theories from the 6d $\mathcal{N}=(2,0)$ A1 theory on punctured Riemann surfaces, proposing a 2d TQFT on the surface whose correlators reproduce the 4d index, with a consistent gluing/duality structure that hints at a unifying higher-dimensional/topological picture. The results emphasize a robust 4d–3d–2d web, including refinements by flavor holonomies and potential links to $q$-deformed Yang–Mills and AGT-type correspondences. Overall, the paper provides new exact connections across dimensions and a concrete TQFT flavor for class $\mathcal{S}$ theories on lens spaces.

Abstract

We compute the 4d superconformal index for N=1,2 gauge theories on S^1 x L(p,1), where L(p,1) is a lens space. We find that the 4d N=1,2 index on S^1 x L(p,1) reduces to a 3d N=2,4 index on S^1 x S^2 in the large p limit, and to a 3d partition function on a squashed L(p,1) when the size of temporal S^1 shrinks to zero. As an application of our index, we study 4d N=2 superconformal field theories arising from the 6d N=(2,0) A_1 theory on a punctured Riemann surface, and conjecture the existence of a 2d Topological Quantum Field Theory on the Riemann surface whose correlation function coincides with the 4d N=2 index on S^1 x L(p,1).

Figures (2)

• Figure 1: A schematic summary of the relations obtained in section 2, 3 and 4.
• Figure 2: The dimensional reduction from the 4d $\mathcal{N}=2$ index to the 3d $\mathcal{N}=4$ index, as discussed in this paper, should correspond to a dimensional oxidation from the 2d TQFT to a one higher dimensional theory.