S -duality and loop operators in canonical formalism

Shan Hu

doi:10.1103/physrevd.103.045010

Abstract

We study the gauge invariant 't Hooft operator in canonical formalism for Yang-Mills theory as well as the $\mathcal{N}=4$ super-Yang-Mills theory with the gauge group $U(N)$. It is shown that the spectrum of the 't Hooft operator labeled by the arbitrary irreducible representation of the gauge group is the same as the spectrum of the Wilson operator labeled by the same representation. So it is possible to construct a unitary operator $S$ making the two kinds of loop operators transformed into each other. S-duality transformation could be realized by the operator $S$. We compute the supersymmetry variations of the loop operators with the fermionic couplings turned off. The result is consistent with the expectation that the action of $S$ should make supercharges transform with a $U(1{)}_{Y}$ phase.

Highlights

It is well known that the source-free Maxwell theory exhibits the electric-magnetic duality (S-duality)
The duality can be extended to nonlinear electrodynamics such as the Born-Infeld theory describing the dynamics of a D3 brane [1,2,3,4]
VI, we study the S-transformation of loop operators

Summary

Introduction

It is well known that the source-free Maxwell theory exhibits the electric-magnetic duality (S-duality). The duality can be extended to nonlinear electrodynamics such as the Born-Infeld theory describing the dynamics of a D3 brane [1,2,3,4]. For Uð1Þ gauge theory with the coupling constant g2 and the Lagrangian Lðg; FμνÞ, the dual field strength can be obtained by adding a Lagrange multiplier (dual potential) [5]: L0ðg; Fμν; AμÞ 1⁄4 Lðg; FμνÞ − AμGμ; Gμ 1⁄4 1 2 εμνρσ ∂ νFρσ : ð1:1Þ. The saddle-point equations are δL0 δAμ −Gμ −. 1 2 εμνρσ ∂ ν Fρσ ð1:2Þ and δL0 δFμν δL δFμν þ 1 2 ερσμν∂σAρ

Objectives

Discussion

Conclusion