Line Operators in Supersymmetric Gauge Theories and the 2d-4d Relation

Abstract

Four-dimensional gauge theories with N = 2 supersymmetry admit half-BPS line operators. We review the exact localization methods for analyzing these operators. We also review the roles they play in the relation between four-and two-dimensional field theories, and explain how the two-dimensional CFT can be used to obtain the quantitative results for 4d line operators. This is a contribution to the special LMP volume on the 2d-4d relation, edited by J. Teschner.

Orbifolds, defects and sphere partition function

“…The parameters of the vortex defects η N , η S and the saddle point parameters s, a now take values in Cartan subalgebra. By generalizing the result for the abelian case, we find the character for a chiral multiplet in the representation Λ and R V = q is given by 28) and the determinant is…”

“…However, this symmetry is actually anomalous in the path integral, and the formula 27) gives the correct one-loop determinant which leads to the nice factorization property of sphere partition function [34,35]. To elaborate on this anomaly, let us look into the path integral with respect to the fermion in the chiral multiplet, which is 28) with the vector multiplet fields fixed at the saddle point value (3.9). We notice that the squashed sphere geometry (2.5) is invariant under the antipodal map…”

“…For example, the inclusion of surface defects in 4D N = 2 supersymmetric gauge theories has been studied in the computation of instanton partition functions [3][4][5][6][7][8][9][10][11][12], superconformal indices [13][14][15][16][17] or sphere partition functions [18][19][20][21], and the results led to a more detailed understanding of the relation between 4D N = 2 SUSY theories and 2D conformal field theories [22][23][24], topological field theories or topological strings. The loop operators in 4D N = 2 theories were studied from a similar viewpoint in [25][26][27]; see [28] for a review. Another interesting point is that some of the defect operators can also be described alternatively by a lower-dimensional field theory on their worldvolume interacting with the fields in the bulk [18,29].…”

Resumming perturbative series in the presence of monopole bubbling effects

“…Instead of a path integral expression, we use a conjectural finite dimensional integral representation for the 't Hooft loop [18][19][20] which has passed highly nontrivial tests. There are exact results for the 't Hooft loop in the 4d N = 2 theories on round S 4 [19], S 1 × R 3 [18] and S 1 × S 3 [21] by SUSY localization [16].…”

“…Let us consider the 4d N = 2 SUSY gauge theories with a semi-simple gauge group G = G 1 × · · · × G n , which are coupled to hyper multiplets of representations (R 1 , · · · , R N f ). According to [18][19][20], the 't Hooft loop is given by [39]…”