Chern-Simons theory and S-duality

By Tudor Dimofte, Sergei Gukov. 2011

148 citations recorded by the source corpus.

A bstractWe study S-dualities in analytically continued SL(2) Chern-Simons theory on a 3-manifold M. By realizing Chern-Simons theory via a compactification of a 6d five-brane theory on M, various objects and symmetries in Chern-Simons theory become related to objects and operations in dual 2d, 3d, and 4d theories. For example, the space of flat SL(2,$ \mathbb{C} $) connections on M is identified with the space of supersymmetric vacua in a dual 3d gauge theory. The hidden symmetry of SL(2) Chern-Simons theory can be identified as the S-duality transformation of $ \mathcal{N}=4 $ super-Yang-Mills theory (obtained by compactifying the five-brane theory on a torus); whereas the mapping class group action in Chern-Simons theory on a three-manifold M with boundary C is realized as S-duality in 4d $ \mathcal{N}=2 $ super-Yang-Mills theory associated with the Riemann surface C. We illustrate these symmetries by considering simple examples of 3-manifolds that include knot complements and punctured torus bundles, on the one hand, and mapping cylinders associated with mapping class group transformations, on the other. A generalization of mapping class group actions further allows us to study the transformations between several distinguished coordinate systems on the phase space of Chern-Simons theory, the SL(2) Hitchin moduli space.

Subfields: Geometry and Topology, Mathematical Physics, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/3e14f81b1e806dc722b071d530d1bc694c738d2a

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Citations in the corpus

2d index and surface operators (2013, methodology): “8) that we used for identifying indices of dual 2d gauge theories have a similar form to the q-difference operators which annihilate various partition functions of 3d gauge theories with the same amount of supersymmetry (namely, four real supercharges) [14, 39].”
4d Index to 3d Index and 2d TQFT (2011, methodology): “Similarly, in the limit S(1) T → 0 (γ → 0 in the notation of section 4), we expect to recover a Chern-Simons theory with a non-compact gauge group [42, 43, 44, 45, 46, 47].”
Quantum Riemann Surfaces in Chern-Simons Theory (2011, unknown): “can be considered a 3d lift of the well-known 2d S-duality, often written b$b 1 in quantum Teichmuller [58, 82, 80] and Liouville [89] theories. This connection will be more fully explored elsewhere [90]. 6.2 Wavefunctions and the Weil representation The gluing methods of Section 3 show that the con”
Comments on twisted indices in 3d supersymmetric gauge theories (2016, unknown): “The ‘holomorphic’ properties of the low-energy theory on M are determined by the effective twisted superpotential, which can be obtained by integrating out all the massive fields at generic values of ua, including all the Kaluza-Klein (KK) modes on S 1 [10, 46].”