T Hooft Operators in Four-Dimensional Gauge Theories and S-Duality

PHYSICAL REVIEW D 74, 025005 (2006) Wilson-’t Hooft operators in four-dimensional gauge theories and S-duality Anton Kapustin* California Institute of Technology, Pasadena California 91125 USA (Received 14 May 2006; published 7 July 2006) We study operators in four-dimensional gauge theories which are localized on a straight line, create electric and magnetic flux, and in the UV limit break the conformal invariance in the minimal possible way. We call them Wilson-’t Hooft operators, since in the purely electric case they reduce to the well- known Wilson loops, while in general they may carry ’t Hooft magnetic flux. We show that to any such 2 2 operator one can associate a maximally symmetric boundary condition for gauge fields on AdSE S .We show that Wilson-’t Hooft operators are classified by a pair of weights (electric and magnetic) for the gauge group and its magnetic dual, modulo the action of the Weyl group. If the magnetic weight does not belong to the coroot lattice of the gauge group, the corresponding operator is topologically nontrivial (carries nonvanishing ’t Hooft magnetic flux). We explain how the spectrum of Wilson-’t Hooft operators transforms under the shift of the -angle by 2. We show that, depending on the gauge group, either SL 2; Z or one of its congruence subgroups acts in a natural way on the set of Wilson-’t Hooft operators. This can be regarded as evidence for the S-duality of N 4 super-Yang-Mills theory. We also compute the one-point function of the stress-energy tensor in the presence of a Wilson-’t Hooft operator at weak coupling. DOI: 10.1103/PhysRevD.74.025005 PACS numbers: 11.15.ÿq I. INTRODUCTION ities. For example, a certain winding-state operator in the theory of a free periodic boson in 2d is a fermion which It is known to quantum field theory aficionados, but not satisfies the equations of motion of the massless Thirring appreciated widely enough, that local operators need not be model. Thus the massless Thirring model is dual to a free defined as local functions of the fields which are used to theory. Upon perturbing the Thirring model Lagrangian by write down the Lagrangian. The simplest example is an a mass term, one finds that it is dual to the sine-Gordon operator which creates a winding state in the theory of a model [1]. The fermion number of the Thirring model free periodic boson in 2d. Another example is a twist corresponds to the soliton charge of the sine-Gordon operator for a single Majorana fermion in 2d. The second model. example shows that it is a matter of convention which field So far all our examples have been two-dimensional. is regarded as fundamental: the massless Majorana fermion Already in one of the first papers on the sine-Gordon- can be reinterpreted as the continuum limit of the Ising Thirring duality it was proposed that similar dualities model at the critical temperature, and then it is more may exist in higher-dimensional fields theories [2]. More natural to regard one of the twist operators (the one with specifically, S. Mandelstam proposed that an Abelian fermion number 0) as the fundamental object, since it cor- gauge theory in 3d admitting Abrikosov-Nielsen-Olesen responds to the spin operator of the Ising model. Another (ANO) vortices may have a dual description where the famous example of this phenomenon is the fermion-boson ANO vortex state is created from vacuum by a fundamental equivalence in two dimensions. The unifying theme of all field. Much later, such 3d dual pairs have indeed been these examples is that one can define a local operator by discovered [3–7]. Duality of this kind is known as 3d requiring the ‘‘fundamental’’ fields in the path-integral to mirror symmetry. The operator in the gauge theory creating have a singularity of a prescribed kind at the insertion the ANO vortex has been constructed in Refs. [8,9]. It is a point. It may happen that the presence of the singularity topological disorder operator in the sense that the gauge can be detected from afar for topological reasons. For field looks like the field of Dirac monopole near the example, in the presence of the fermionic twist operator, insertion point. In other words, the operator is (partially) the fermion field changes sign as one goes around the characterized by the fact that the first Chern class of the insertion point. In such cases, one can say that the operator gauge bundle on any 2-sphere enclosing the insertion point insertion creates topological disorder. (In fact, all examples is nontrivial. In the dual theory, the same operator is a local mentioned above fall into this category). But it is important gauge-invariant function of the fundamental fields. to realize that the idea of defining local operators by means Analogous dualities exist for certain non-Abelian gauge of singularities in the fundamental fields is more general theories in 3d. These theories do not have any conserved than that of a topological disorder operator. topological charges and therefore do not have topological Local operators not expressible as local functions of the disorder operators. Nevertheless, they admit local opera- fundamental fields play an important role in various dual- tors characterized by the fact that near the insertion point the gauge field looks like a Goddard-Nuyts-Olive (GNO) *Electronic address: [email protected] monopole [10]. Such operators have been studied in 1550-7998=2006=74(2)=025005(14) 025005-1 © 2006 The American Physical Society ANTON KAPUSTIN PHYSICAL REVIEW D 74, 025005 (2006) Ref. [11], where it was argued that 3d mirror symmetry Another nice feature of line operators is that in some maps them to ordinary local operators whose vacuum cases they can be regarded as topological disorder opera- expectation values (VEVs) parametrize the Higgs branch tors. Indeed, the punctured neighborhood of a straight line of the dual theory. is homotopic to S2, and in a non-Abelian gauge theory with It is natural to try to extend these considerations to 4d gauge group G one may consider gauge bundles which gauge theories, the primary objective being a better under- have a nontrivial ’t Hooft magnetic flux [a class in 2 2 2 standing of various conjectural dualities. In 4d a punctured H S ;1 G]onS . By definition, these are ’t Hooft neighborhood of a point is homotopic to S3, and since loop operators. However, even if the gauge group is vector bundles on S3 do not have interesting characteristic simply-connected, one may still consider nontopological classes, we do not expect to find any local topological line operators which are defined by the requirement that disorder operators in such theories. But as we know from near the insertion line the gauge field looks like the field of 3d examples, this does not rule out the possibility of a local a GNO monopole. This is completely analogous to the operator which creates a singularity in the fields at the situation in 3d [11]. We will continue to call such operators insertion point. ’t Hooft operators, even though they may carry trivial ’t A more serious problem is that we are mostly interested Hooft magnetic flux. Note also that the line entering the in operators which can be studied at weak coupling. This definition of the Wilson and ’t Hooft operators can be means that the singularity in the fields that we allow at the infinite without boundary. This is why we prefer to use insertion point must be compatible with the classical equa- the terms ‘‘Wilson operator’’ and ‘‘’t Hooft operator’’ tions of motion. If we want the operator to have a well- instead of the more common names ‘‘Wilson loop opera- defined scaling dimension, we also have to require that the tor’’ and ‘‘’t Hooft loop operator.’’ allowed singularity be scale-invariant. Together these two In this paper we study general Wilson-’t Hooft operators requirements are too strong to allow nontrivial solutions in in 4d gauge theories using the approach of Refs. [8,9,11]. four-dimensional field theories. For example, in a theory of We are especially interested in the action of various dual- a free scalar field an operator insertion at the origin can be ities on these operators. In this paper we focus on a thought of as a local modification of the Klein-Gordon particular example: S-duality of the N 4 super-Yang- equation of the form Mills (SYM) theory. As usually formulated, this conjecture says that the N 4 SYM theory with a simple gauge Lie ᮀ 4 algebra g and complexified coupling 4 is iso- xA x;@ x; 2 g2 morphic to a similar theory with the Langlands-dual gauge g^ ÿ = where A is a polynomial in the field and derivatives. Lie algebra and coupling ^ 1 [14–16]. We will However, scale-invariance requires A to have dimension focus on a corollary of this conjecture, which says that for a g ÿ1, which is impossible. Similarly, in four dimensions no given gauge Lie algebra the self-duality group of N 4 local modifications of the Maxwell equations which would SYM is either SL 2; Z or one of its congruence subgroups preserve scale-invariance are possible. (The situation in ÿ0 q for q 2, 3, depending on g [17,18]. We will refer to two- and three-dimensional field theories is very different this group as the S-duality group. A natural question to ask in this respect.) is how the S-duality group acts on the Wilson loop operator To circumvent this problem, we recall that in some sense of the theory.

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