Abstract
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 (Σ1,1). The corresponding 3d field theories can be realized using duality domain wall theories in 4d $ \mathcal{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 simple basis change of the Hilbert space for the SL(2, $ \mathbb{C} $ ) Chern-Simons theory on $ \mathbb{R} $ × Σ1,1.
Highlights
We show that the superconformal indices of the 3d theories are the SL(2, C) Chern-Simons partition function on the mapping cylinder/torus
We show that the superconformal index for the duality wall theory associated with φ ∈ SL(2, Z) is a matrix element of φ in a suitable basis of the Hilbertspace
According to an axiom of topological quantum field theory, the matrix element is nothing but the SL(2, C) CS partition function on the mapping cylinder, and it provides an evidence for the 3d-3d correspondence (1.1) for the mapping cylinder.For mapping torus, tori(φ), the CS partition function is given as a trace of an operator φ ∈ SL(2, Z)
Summary
A mapping torus is specified by a Riemann surface Σg,h of genus g with h punctures and an element φ of the mapping class group of Σg,h. It is a bundle with Σ fibered over an interval I = [0, 1] with Σ at one end of the interval identified with φ(Σ) at the other end. The mapping class group SL(2, Z) is the group of duality transformations in the sense of the 4d theory From this viewpoint, TM can be obtained by taking a proper “trace” action on a 3d duality wall theory associated with φ.
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