Exact results for perturbative Chern-Simons theory with complex gauge group.

By Tudor Dimofte, Sergei Gukov, Jonatan Lenells, Don Zagier. 2009

207 citations recorded by the source corpus.

We develop several methods that allow us to compute all-loop partition functions in perturbative Chern-Simons theory with complex gauge group G_C, sometimes in multiple ways. In the background of a non-abelian irreducible flat connection, perturbative G_C invariants turn out to be interesting topological invariants, which are very different from finite type (Vassiliev) invariants obtained in a theory with compact gauge group G. We explore various aspects of these invariants and present an example where we compute them explicitly to high loop order. We also introduce a notion of “arithmetic TQFT” and conjecture (with supporting numerical evidence) that SL(2,C) Chern-Simons theory is an example of such a theory.

Subfields: Mathematical Physics, Nuclear and High Energy Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/f7aa00a6ce0ea457dc8ecf18beff36e8c1df92ac

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Citations in the corpus

Gauge Theories Labelled by Three-Manifolds (2011, background): “Up to a factor of due to a small change of polarization, the result is then identical to the figure-eight wavefunctions described in [53, 9, 12] (see also [54, 55]).”
Analytic Continuation Of Chern-Simons Theory (2010, background): “(A useful review is [17]; partial physics-based explanations with some relevance to the present paper are [6,18,19].”
Quantum Riemann Surfaces in Chern-Simons Theory (2011, background): “corresponds to a perturbative expansion of SL(2,C) Chern-Simons theory in the background of a fixed flat connection A [1, 2].”
Framed BPS states (2010, unknown): “Moreover, the 3-manifold invariants for noncompact Chern-Simons involve the quantum dilogarithm [75].”
Chern-Simons theory and S-duality (2011, background): “J H E P 0 5 ( 2 0 1 3 ) 1 0 9 from the triangulation data [10, 11].”
Vortex Counting and Lagrangian 3-Manifolds (2010, background): “1) depends on the so-called shape parameters of M and, in the simplest case of SL(2,C) Chern-Simons theory, can be expressed as a multiple contour integral [2]: Z(M ; ~) = ∮ ∏”
Quantum modular forms (2010, background): “[7]), is of a very different nature than the Jones polynomials themselves.”
A-polynomial, B-model, and quantization (2011, background): “for the figure-8 knot complement [26] up to order n = 4, provided certain ad hoc renormalizations are made.”
On the asymptotic expansion of the Kashaev invariant of the $5_2$ knot (2016, background): “Furthermore, it is conjectured [11] (see also [3, 12, 46]) from the viewpoint of the SL(2,C) Chern-Simons theory that the asymptotic expansion of JN(K; e √ −1/k)”
3-Manifolds and 3d Indices (2011, background): “7) is well known in knot theory and Chern-Simons theory as the “quantum A-polynomial” of the figure-eight knot [29, 31, 52].”