ON THE DESCRIPTION OF SURFACE OPERATORS IN ${\mathcal N} = 2^*$ SYM

Abstract

Alday and Tachikawa [Lett. Math. Phys.94, 87 (2010)] observed that the Nekrasov partition function of [Formula: see text] superconformal gauge theories in the presence of fundamental surface operators can be associated to conformal blocks of a 2D CFT with affine sl(2) symmetry. This can be interpreted as the insertion of a fundamental surface operator changing the conformal symmetry from the Virasoro symmetry discovered in Ref. 2 to the affine Kac–Moody symmetry. A natural question arises as to how such a 2D CFT description can be extended to the case of non-fundamental surface operators. Motivated by this question, we review the results [Y. Hikida and V. Schomerus, JHEP0710, 064 (2007); S. Ribault, JHEP0805, 073 (2008)] and put them together to suggest a way to address the problem: It follows from this analysis that the expectation value of a non-fundamental surface operator in the SU(2) [Formula: see text] super Yang–Mills (YM) theory would be in correspondence with the expectation value of a single vertex operator in a two-dimensional CFT with reduced affine symmetry and whose central charge is parametrized by the integer number that labels the type of singularity of the surface operator.