5d/6d Wilson loops from blowups

Abstract

We generalize Nakajima-Yoshioka’s blowup formula to calculate the partition functions counting the spectrum of bound states to half-BPS Wilson loop operators in 5d (and 6d) supersymmetric field theories. The partition function in the presence of a Wilson loop operator on the Ω-background is factorized when put on the blowup $$ \hat{\mathbb{C}} $$ ℂ ̂ 2 into two Wilson loop partition functions under the localization. This structure provides a set of blowup equations for Wilson loop operators. We explain how to formulate the blowup equations and solve them to compute the partition functions of Wilson loop operators. We test this idea by explicitly calculating the Wilson loop partition functions in various 5d/6d field theories and comparing them against known results and expected dualities.

Topological strings and Wilson loops

“…In particular, the 1d line defects play an important role in the study of quantum Seiberg-Witten curve and the non-perturbative Dyson-Schwinger equation [22,26]. More recently, the blowup equation for the instanton contributions with 5d/6d Wilson loops/surfaces were suggested in [32], and the Wilson loop expectation values for many theories which may not admit ADHM descriptions were predicted there.…”

“…In M-theory, the heavy stationary quark can be realized as particles from M2-branes winding on additional stationary non-compact holomorphic 2-cycles [32]. In general, these JHEP08(2022)207 additional curve classes can always be decomposed to non-decomposable curve classes C i , which are called primitive curves.…”

“…For a single primitive curve C insertion, the generated Wilson loop is in the non-decomposable representation r C . It was conjectured in [32] that the expectation value of the Wilson loop operator in the non-decomposable representation r C has a BPS expansion…”

“…Recently, based on geometric descriptions and inspirations from the blowup equation for gauge theories [49][50][51][52], the blowup equation for refined topological strings was proposed in [53], which was further generalized to generic 5d/6d SQFTs and the blowup equation play an important role to compute the partition functions and elliptic genera for 5d/6d SQFTs [53][54][55][56][57][58][59][60]. The generalization of the blowup equation to Wilson loop expectation values can be found in [32,50,52]. In particular, in [32], the blowup equation for Wilson loops in generic gauge theories and non-Lagrangian theories was suggested, and the Wilson loop BPS invariants for many 5d/6d theories were calculated there.…”

Refined topological vertex with ON-planes

“…In fact, there has been various progress on computing partition functions of 5d/6d N = 1 supersymmetric gauge theories. For instance, the ADHM method [15][16][17][18][19][20][21], the Ding-Iohara-Miki algebra [22][23][24][25], topological vertex [26][27][28][29][30] and blowup equation [31][32][33][34][35][36][37][38][39][40]. In particular, the blowup method can be applied to theories without 5-brane configurations or any Lagrangian descriptions [39,40].…”