Abstract
We compute expectation values of Wilson loops in q-deformed 2d Yang–Mills on a Riemann surface and show that they give invariants of knots in 3-manifolds which are circle bundles over the Riemann surface. The areas of the loops play an essential role in encoding topological information about the extra dimension, and they are quantized to integer or half-integer values.
Highlights
There has been a lot of interest in q-deformed 2d Yang–Mills theory [1,2,3,4,5,6,7]
We have shown that q-deformed 2d Yang–Mills theory on a Riemann surface Σ gives topological invariants in one dimension higher, namely in a Seifert fibered manifold
Expectation values of Wilson loops on the Riemann surface give knot invariants of the Seifert manifold
Summary
There has been a lot of interest in q-deformed 2d Yang–Mills theory [1,2,3,4,5,6,7]. Where as usual ρ denotes the trivial representation This formula allows us to perform most of the sums in the partition function (9). This is precisely the Chern–Simons result, where the product of the n unlinked loops factorizes in this case This is easy to generalize to a surface of genus g. Setting n = 0 we just get the partition function of Chern–Simons on S1 × Σ : ZCS S1 × Σ = dimq (λ) 2−2g This is the well-known Verlinde formula for the dimension of the space of conformal blocks on a surface of genus g. It is obtained from the partition function of the q-deformed 2dYM on Σ
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
