scholarly journals On Chern-Simons theory with an inhomogeneous gauge group and BF theory knot invariants

2005 ◽  
Vol 46 (12) ◽  
pp. 122302 ◽  
Author(s):  
Gad Naot
Keyword(s):  
Gauge Group ◽  
Simons Theory ◽  
Knot Invariants ◽  
Bf Theory ◽  
Chern Simons ◽  
Chern Simons Theory

scholarly journals What Chern–Simons theory assigns to a point

2017 ◽  
Vol 114 (51) ◽  
pp. 13418-13423 ◽  
Author(s):  
André G. Henriques
Keyword(s):  
Gauge Group ◽  
Von Neumann Algebras ◽  
Mathematical Object ◽  
Loop Group ◽  
Simons Theory ◽  
Fusion Product ◽  
Positive Energy ◽  
Von Neumann ◽  
Chern Simons ◽  
Chern Simons Theory

We answer the questions, “What does Chern–Simons theory assign to a point?” and “What kind of mathematical object does Chern–Simons theory assign to a point?” Our answer to the first question is representations of the based loop group. More precisely, we identify a certain class of projective unitary representations of the based loop group 𝛀G. We define the fusion product of such representations, and we prove that, modulo certain conjectures, the Drinfel’d center of that representation category of 𝛀G is equivalent to the category of positive energy representations of the free loop group LG.† The abovementioned conjectures are known to hold when the gauge group is abelian or of type A1. Our answer to the second question is bicommutant categories. The latter are higher categorical analogs of von Neumann algebras: They are tensor categories that are equivalent to their bicommutant inside Bim(R), the category of bimodules over a hyperfinite 𝐼𝐼𝐼1 factor. We prove that, modulo certain conjectures, the category of representations of the based loop group is a bicommutant category. The relevant conjectures are known to hold when the gauge group is abelian or of type An.

scholarly journals On partition functions of refined Chern-Simons theories on S3

2021 ◽  
Vol 2021 (10) ◽  
Author(s):  
M.Y. Avetisyan ◽  
R.L. Mkrtchyan
Keyword(s):  
Partition Function ◽  
Gauge Group ◽  
Topological Strings ◽  
Simons Theory ◽  
Partition Functions ◽  
Coupling Constant ◽  
Sine Functions ◽  
Chern Simons ◽  
Arbitrary Gauge ◽  
Chern Simons Theory

Abstract We present a new expression for the partition function of the refined Chern-Simons theory on S3 with an arbitrary gauge group, which is explicitly equal to 1 when the coupling constant is zero. Using this form of the partition function we show that the previously known Krefl-Schwarz representation of the partition function of the refined Chern-Simons theory on S3 can be generalized to all simply laced algebras.For all non-simply laced gauge algebras, we derive similar representations of that partition function, which makes it possible to transform it into a product of multiple sine functions aiming at the further establishment of duality with the refined topological strings.

scholarly journals PERTURBATIVE CHERN-SIMONS THEORY

1995 ◽  
Vol 04 (04) ◽  
pp. 503-547 ◽  
Author(s):  
DROR BAR-NATAN
Keyword(s):  
Field Theory ◽  
Perturbation Theory ◽  
Integral Formula ◽  
Simons Theory ◽  
Gauge Field Theory ◽  
Complete Agreement ◽  
Knot Invariants ◽  
Arf Invariant ◽  
Chern Simons ◽  
Chern Simons Theory

We present the perturbation theory of the Chern-Simons gauge field theory and prove that to second order it indeed gives knot invariants. We identify these invariants and show that in fact we get a previously unknown integral formula for the Arf invariant of a knot, in complete agreement with earlier non-perturbative results of Witten. We outline our expectations for the behavior of the theory beyond two loops.

scholarly journals The Resurgent Structure of Quantum Knot Invariants

2021 ◽  
Author(s):  
Stavros Garoufalidis ◽  
Jie Gu ◽  
Marcos Mariño
Keyword(s):  
Asymptotic Expansion ◽  
Fundamental Solutions ◽  
Simons Theory ◽  
Knot Invariants ◽  
Integer Coefficients ◽  
Bps States ◽  
Formal Power ◽  
Chern Simons ◽  
Resurgent Functions ◽  
Chern Simons Theory

AbstractThe asymptotic expansion of quantum knot invariants in complex Chern–Simons theory gives rise to factorially divergent formal power series. We conjecture that these series are resurgent functions whose Stokes automorphism is given by a pair of matrices of q-series with integer coefficients, which are determined explicitly by the fundamental solutions of a pair of linear q-difference equations. We further conjecture that for a hyperbolic knot, a distinguished entry of those matrices equals to the Dimofte–Gaiotto–Gukov 3D-index, and thus is given by a counting of BPS states. We illustrate our conjectures explicitly by matching theoretically and numerically computed integers for the cases of the $$4_1$$ 4 1 and the $$5_2$$ 5 2 knots.

scholarly journals Exact Computations in Topological Abelian Chern-Simons and BF Theories

2017 ◽  
Vol 2017 ◽  
pp. 1-6
Author(s):  
Philippe Mathieu
Keyword(s):  
Simons Theory ◽  
Partition Functions ◽  
Topological Invariants ◽  
Deligne Cohomology ◽  
Functional Integrals ◽  
Fibre Bundles ◽  
Bf Theory ◽  
Chern Simons ◽  
Chern Simons Theory

We introduce Deligne cohomology that classifies U1 fibre bundles over 3 manifolds endowed with connections. We show how the structure of Deligne cohomology classes provides a way to perform exact (nonperturbative) computations in U1 Chern-Simons theory (BF theory, resp.) at the level of functional integrals. The partition functions (and observables) of these theories are strongly related to topological invariants well known to the mathematicians.

scholarly journals Chern-Simons theory with finite gauge group

1993 ◽  
Vol 156 (3) ◽  
pp. 435-472 ◽  
Author(s):  
Daniel S. Freed ◽  
Frank Quinn
Keyword(s):  
Gauge Group ◽  
Simons Theory ◽  
Chern Simons ◽  
Chern Simons Theory
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