Superconformal Index and 3d-3d Correspondence for Mapping Cylinder/Torus

TL;DR

The paper establishes a precise 3d-3d correspondence for mapping cylinder/torus M with fiber Σ_{1,1}, showing that T_M’s superconformal index equals the SL(2,C) Chern-Simons partition function on M. It develops two coherent descriptions of the mapping torus: a duality-wall trace built from 4d N=2^* dualities and a tetrahedron-based DGG construction; both yield the same index, realized as a trace or a matrix element in a carefully constructed boundary Hilbert space. Central to the equivalence is the development of two CS-induced bases, SR and FN, and the demonstration that the matrix elements of SL(2,Z) actions in these bases reproduce the corresponding 3d indices, verified through difference equations and q-expansions. The work further connects to squashed S^3 and SL(2,R) CS theories, via quantum dilogarithm identities and operator-algebra considerations, highlighting a broad, structurally parallel framework across real and complex CS theories. Overall, the results solidify the 3d-3d dictionary for tori(φ) and illuminate the role of boundary Hilbert spaces and basis changes in unifying distinct 3d constructions of the same SCFT.

Abstract

We probe the 3d-3d correspondence for mapping cylinder/torus using the superconformal index. We focus on the case when the fiber is a once-punctured torus (Σ_{1,1}). The corresponding 3d field theories can be realized using duality domain wall theories in 4d N=2* theory. We show that the superconformal indices of the 3d theories are the SL(2,C) Chern-Simons partition function on the mapping cylinder/torus. For the mapping torus, we also consider another realization of the corresponding 3d theory associated with ideal triangulation. The equality between the indices from the two descriptions for the mapping torus theory is reduced to a basis change of the Hilbert space for the SL(2,C) Chern-Simons theory on RxΣ_{1,1}.

Figures (6)

• Figure 1: Quiver diagrams for $T[SU(2)]$
• Figure 2: Mapping cylinder $\Sigma_{1,1}\times_\varphi I$. A global symmetry in the duality wall theory is associated to each component of boundary.
• Figure 3: Triangulation of boundary torus of tori($LR$). Four triangles for each letter, $L$ or $R$, come from 'small' boundary triangles in ideal tetrahedron associated to the letter. See figure 3 in Gueritaud.
• Figure 4: Triangulation of once-punctured torus, $\Sigma_{1,1}$, represented as $(\mathbb{R}^2 - \mathbb{Z}^2 )/\mathbb{Z}^2$. Each circle in the figure denotes an image of the puncture in the covering space $\mathbb{R}^2$.
• Figure 5: Generators of $\pi_1 (\Sigma_{2,0})$