Exact Results for 't Hooft Loops in Gauge Theories ... - Solvay Institutes

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Introduction Problem Definitions Method Solution Result To find the ratio of determinants (equivariant Euler character) we consider the index of operator D (equivariant Chern character) ind D = tr KerD e R − tr CokerD e R read the weights of R and combine them into determinant using the rule ∑ c j e w j(ε 1 ,ε 2 ,â, ˆm f ) → ∏ w j (ε 1 , ε 2 , â, ˆm f ) c j j j Recall that R = J + R + [Φ, ·] J is the SD spatial rotation R is the R-symmetry rotation [Φ, ·] is the gauge transformation. Vasily Pestun Localization for ’t Hooft operators on S 4 24/35
Introduction Problem Definitions Method Solution Result Excision property of the index/Atiya-Singer theory To find index for transversally elliptic operator D on a manifold M under equivariant U(1) action R we can cut M into three pieces • neighborhood of the north pole • neighborhood of the equator • neighborhood of the south pole and apply Atiyah-Singer fixed point formula for the index ind D(R) = ∑ p∈F tr E0 (p) R − tr E1 (p) R . det T Mp (1 − R) Vasily Pestun Localization for ’t Hooft operators on S 4 25/35
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Exact Results for 't Hooft Loops in Gauge Theories ... - Solvay Institutes

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