Solutions of Kapustin-Witten equations for ADE-type groups

Abstract

Kapustin-Witten (KW) equations are encountered in the localization of the topological N=4 SYM theory. Mikhaylov has constructed model solutions of KW equations for the boundary 't~Hooft operators on a half space. Direct proof of the solutions boils down to check a boundary condition. There are two computational difficulties in explicitly constructing the solutions to Lie algebra of higher rank. The first one is related to the commutation of generators of Lie algebra. We derived an identity which effectively reduces this computational difficulty. The second one involves the number of ways from the highest weights to other weights in the fundamental representation. For ADE-type gauge groups, we found an amazing formula which can be used to rewrite the solutions of KW equations. This new formula of solutions bypass above two computational difficulties.

26 pages, 3 figure. typos corrected, english improved, references added, corrected mistakes and rewrote Sec. IV

References in corpus (6)

• Supersymmetric Boundary Conditions in N=4 Super Yang-Mills Theory
• Electric-Magnetic Duality And The Geometric Langlands Program
• On the singular sets of solutions to the Kapustin-Witten equations and the Vafa-Witten ones on compact Kähler surfaces
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