Complex Chern-Simons theory at level \(k\) via the 3d-3d correspondence. (English) Zbl 1336.57044
Author’s abstract: We use the 3d-3d correspondence together with the DGG construction of theories \(T_n[M]\) labelled by 3-manifolds \(M\) to define a non-perturbative state-integral model for \(SL(n,\mathbb{C})\) Chern-Simons theory at any level \(k\), based on ideal triangulations. The resulting partition functions generalize a widely studied \(k=1\) state-integral, as well as the 3d index, which is \(k=0\). The Chern-Simons partition functions correspond to partition functions of \(T_n[M]\) on squashed lens spaces \(L(k,1)\). At any \(k\), they admit a holomorphic-antiholomorphic factorization, corresponding to the decomposition of \(L(k,1)\) into two solid tori, and the associated holomorphic block decomposition of the partition functions of \(T_n[M]\). A generalization to \(L(k,p)\) is also presented. Convergence of the state integrals, for any \(k\), requires triangulations to admit a positive angle structure; we propose that this is also necessary for the DGG gauge theory \(T_n[M]\) to flow to a desired IR SCFT.
Reviewer: Qi Chen (Winston-Salem)
MSC:
| 57R56 | Topological quantum field theories (aspects of differential topology) |
| 81T30 | String and superstring theories; other extended objects (e.g., branes) in quantum field theory |
Keywords:
Chern-Simons theory; 3d-3d correspondence; DGG construction; state integral; superconformal field theorySoftware:
SnapPyReferences:
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