On monopole bubbling contributions to ’t Hooft loops

We study half-BPS 't Hooft line operators in 4d N = 2 U (N ) gauge theories on S 1 ×R 3 with an Ω-deformation. The recently proposed brane construction of 't Hooft operators shows that non-perturbative contributions to their correlator are identified with the Witten indices of quiver supersymmetric quantum mechanics. For the products of minimal 't Hooft operators, a chamber in the space of Fayet-Iliopoulos parameters in the quantum mechanics corresponds to an ordering of the operators inserted along a line. These considerations lead us to conjecture that the Witten indices can be read off from the Moyal products of the expectation values of the minimal 't Hooft operators, and also that wall-crossing occurs in the quantum mechanics only when the ordering of the operators changes. We confirm the conjectures by explicitly computing the Witten indices for the products of two and three minimal 't Hooft operators in all possible chambers.2 In [18] the authors also observed the correspondence between the ordering and wall-crossing for the product of T and T . We complete their analysis by including the product of two minimal operators of the same type. We further investigate the products of three minimal operators of all types, which exhibit a richer structure.3 A primitive version of the conjectures and some evidence were presented by T.O. in the workshop "Representation Theory, Gauge Theory, and Integrable Systems" held at the Kavli IPMU in February, 2019. 5 The same symbol e i (or e a ) will denote an orthonormal basis of a Euclidean space other than R N . 6 To compare wtih [12,14] note the following. Λ cort in (2.3) is also the coroot lattice of SU (N ). The cocharacter lattice of SU (N ) is the sublattice { i z i = 0} of Λ cochar in (2.2). Defining z U (1) := (1/N ) i z i j e j and z SU (N ) := z − z U (1) for z = i z i e i , we have |v| ≤ |B| ⇔ |v SU (N ) | ≤ |B SU (N ) | for v ∈ B + Λ cort .7 Since the Langlands dual group of SU (N ) is SU (N )/Z N , 't Hooft operators corresponding to B = ±e i do not exist in an SU (N ) gauge theory.8 Section 5 discusses subtleties for N F odd.12 The D3-branes are assumed to be at generic positions in the (x 4 , x 5 )-space. 13 Due to the s-rule a single NS5-brane, with K D1-branes (1 ≤ K ≤ N ) attached and placed on the right of the stack of all the D3-and D7-branes, gives an 't Hooft operator with magnetic charge K i=1 e i , which seems to correspond to the K-th exterior power of the fundamental representation. Similarly an NS5-brane

Section: Introductionmentioning

confidence: 76%

Section: Conclusion and Discussionmentioning

confidence: 92%

“…For the product of two operators, the case of one T and one T was studied earlier in [18]. In addition we analyzed the product of T and T .…”

Section: Conclusion and Discussionmentioning

confidence: 99%

Section: U (N ) Sqcd With N F < 2n Flavorsmentioning

confidence: 99%

See 2 more Smart Citations

Wall-crossing and operator ordering for ’t Hooft operators in $$ \mathcal{N} $$ = 2 gauge theories

Hayashi¹,

Okuda²,

Yoshida³

2019

“…On the right hand side the ζdependence is captured by the use of complexified Fenchel-Nielsen coordinates on M ζ . 2 Here the expectation value above is expressed as a sum over monopole bubbling configurations where cosh(v, b)F (a) v encodes the contribution of bulk fields and Z mono (a, m, ǫ; P, v) describes the contribution from the SQM that arises on the 't Hooft defect from bubbling [4]. See [4][5][6] for more background and explanation of notation.…”

Section: Technical Summarymentioning

confidence: 99%

Index-like theorems from line defect vevs

Brennan

Moore

2019

In this paper we investigate the relation between complexified Fenchel-Nielsen coordinates and spectral network coordinates on Seiberg-Witten moduli space. The main technique is the comparison of exact expressions for the expectation value of 't Hooft defects in certain 4D SU (2) N = 2 gauge theories. We derive an index-like theorem for a class of Dirac operators on singular monopole moduli spaces. Our expression determines the indices of Dirac operators on singular monopole moduli spaces in terms of characteristic numbers for vector bundles over certain Kronheimer-Nakajima quiver varieties.

Wilson-’t Hooft lines as transfer matrices

Maruyoshi

Ota

Yagi

2021

We establish a correspondence between a class of Wilson-’t Hooft lines in four-dimensional $$ \mathcal{N} $$ N = 2 supersymmetric gauge theories described by circular quivers and transfer matrices constructed from dynamical L-operators for trigonometric quantum integrable systems. We compute the vacuum expectation values of the Wilson-’t Hooft lines in a twisted product space S1 × ϵ ℝ2 × ℝ by supersymmetric localization and show that they are equal to the Wigner transforms of the transfer matrices. A variant of the AGT correspondence implies an identification of the transfer matrices with Verlinde operators in Toda theory, which we also verify. We explain how these field theory setups are related to four-dimensional Chern-Simons theory via embedding into string theory and dualities.