A 3d-3d appetizer

Du Pei,Ke Ye

doi:10.1007/jhep11(2016)008

Abstract

We test the 3d-3d correspondence for theories that are labeled by Lens spaces. We find a full agreement between the index of the 3d $$ \mathcal{N}=2 $$ “Lens space theory” T [L(p, 1)] and the partition function of complex Chern-Simons theory on L(p, 1). In particular, for p = 1, we show how the familiar S 3 partition function of Chern-Simons theory arises from the index of a free theory. For large p, we find that the index of T[L(p, 1)] becomes a constant independent of p. In addition, we study T[L(p, 1)] on the squashed three-sphere S 3 . This enables us to see clearly, at the level of partition function, to what extent G ℂ complex Chern-Simons theory can be thought of as two copies of Chern-Simons theory with compact gauge group G.

Highlights

The level of complex Chern-Simons theory has a real part k and an “imaginary part”2 σ, and σ is related to the squashing parameter b of Lens space L(k, 1)b = Sb3/Zk by σ
We test the 3d-3d correspondence for theories that are labeled by Lens spaces
We find a full agreement between the index of the 3d N = 2 “Lens space theory” T [L(p, 1)] and the partition function of complex Chern-Simons theory on L(p, 1)

Summary

Chern-Simons theory on S3 and free chiral multiplets

Combining (2.5) with the 3d-3d correspondence, one concludes that the index of the free theory equals the S3 partition function of Chern-Simons theory. This is what we will explicitly verify . One can obtain the partition function for GL(N, C) Chern-Simons theory by noticing that it factorizes into two copies of (2.6) at level k1 = τ /2 and k2 = τ /2. It is interesting to ask whether the Chern-Simons theory on S3 naturally admits such an N -parameter deformation and whether one can have a more general relation, IndexT [S3](q; β1, β2, . The fact that Mflat is a collection of points is important for us to compute the partition function of complex Chern-Simons theory.

Superconformal index

A Complex Chern-Simons theory on Lens spaces