
TL;DR
This paper investigates the 3d-3d correspondence for theories labeled by Lens spaces, demonstrating a match between 3d ${ m N}=2$ theory indices and complex Chern-Simons partition functions, revealing new insights into their relationship.
Contribution
It provides a detailed analysis of the 3d-3d correspondence for Lens space theories, including explicit calculations and the behavior of indices and partition functions for various p values.
Findings
Index of $T[L(p,1)]$ matches complex Chern-Simons partition function on $L(p,1)$
For p=1, the index reproduces the $S^3$ Chern-Simons partition function
At large p, the index becomes p-independent
Abstract
We test the 3d-3d correspondence for theories that are labelled by Lens spaces. We find a full agreement between the index of the 3d "Lens space theory" and the partition function of complex Chern-Simons theory on . In particular, for , we show how the familiar partition function of Chern-Simons theory arises from the index of a free theory. For large , we find that the index of becomes a constant independent of . In addition, we study on the squashed three-sphere . This enables us to see clearly, at the level of partition function, to what extent complex Chern-Simons theory can be thought of as two copies of Chern-Simons theory with compact gauge group .
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