How Wilson Operators Act On Branes

8. Electric Eigenbranes

8.1. How Wilson Operators Act On Branes

In particular, when we apply S-duality to branes of specified e₀ and m₀, we must exchange the two methods of quantization in addition to exchanging the two adjoint groups G_ad and ^LG_ad.

Let us recall how Chan-Paton bundles enter sigma-models. We consider a sigma- model of maps Φ : Σ → M_H, and a brane B that is endowed with a Chan-Paton bundle U → M_H with connection α. The quantum theory is defined by an integral over the possible maps Φ, along with certain fermionic variables. One factor in the path integral comes from the bulk action I and takes the form exp −R

ΣI

. There also is a boundary factor that involves parallel transport in the Chan-Paton bundle. Let Qbe the part of the boundary of Σ that is labeled by the brane B, and write ΦQ for the restriction of Φ toQ.

The boundary factor in the path integral involving Q is given by the parallel transport or holonomy along Qof the bundle Φ^∗_Q(U):

Pexp

− Z

Q

Φ^∗_Q(α) +. . .

. (8.2)

IfQis a closed circle, we take a trace of this holonomy, and otherwise this factor combines at the endpoints of Q with other factors, depending on the precise calculation that one chooses to perform, to make a gauge-invariant expression. The ellipses in (8.2) are fermionic corrections to the connection Φ^∗_Q(α) on Φ^∗_Q(U). They are required by supersymmetry, rather as the shift A → A = A+iφ was needed in section 6.1 to define supersymmetric Wilson operators.

There is an obvious analogy between the factor (8.2) by which Chan-Paton bundles enter in sigma-models and the factor (8.1) by which a Wilson line operator influences the underlying four-dimensional gauge theory. The analogy is even closer because to define the action of the Wilson operator on the brane B, we must take the limit as S (or rather its projection from Σ×C to Σ) approaches Q.

To get something precise from this analogy, we begin with the following observation.

When gauge theory on a G-bundle E → M = Σ ×C is described in terms of a map Φ : Σ → M_H, the G-bundle E can be identified as (Φ×1)^∗(E), where E is the “bundle”

part of the universal Higgs bundle (E,ϕ) overb M_H, and (Φ×1)^∗(E) is its pullback via the map Φ× 1 : Σ× C → M_H ×C. This statement just means that, to the extent that the sigma-model is a good description, for each point q ∈ Σ, the bosonic fields A, φ of the gauge theory, when restricted to q × C, are given by the solution of Hitchin’s equations corresponding to the point Φ(q)∈ M_H. This solution is simply, up to a gauge transformation, the restriction of the universal Higgs bundle (E,ϕ) to Φ(q)b ×C.

To interpret the connectionA =A+iφ in (8.1) in terms of the sigma-model, we note that in general, this connection involves both AΣ, the part of the connection tangent to

Σ, and AC, the part tangent to C. Here in the low energy theory, we can assume that AC

obeys Hitchin’s equations, and as long as we avoid singularities in M_H, the fields AΣ are massive in the sigma-model. They can therefore be integrated out in favor of the sigma- model fields AC. For very large Imτ, it is sufficient to integrate out AΣ at the classical level. The part of the gauge theory action which depends on only A and φ was written in (3.46). Assuming that A_C and φ_C satisfy Hitchin equations and dropping the terms which vanish as the volume of C goes to zero, we find a quadratic action for AΣ. The corresponding equations of motion read

DC†DCAΣ =DC†d_ΣAC +. . . , (8.3) whereDC is the covariant differential with respect to the connectionAC. The ellipses refer to terms involving zero modes of the fermions ψ, ψ, etc., of the four-dimensional gaugee theory; we will not write these terms explicitly. A map Φ : Σ →M_H determines AC and hence alsod_ΣAC, and then, assuming we keep away from singularities ofM_H, the equation (8.3) has a unique solution for AΣ.

So once Φ : Σ → M_H is given (and assuming that we keep away from singularities of M_H), the connection A = (AΣ,AC) is determined. A is, of course, a connection on the bundle E = (Φ×1)^∗(E). The connection A is actually the pullback by Φ×1 of a connection Ab on E → M_H ×C. In fact, to define Ab, we must specify its components A^MH,AC tangent to M_H and C. AC is the appropriate solution of Hitchin’s equations, and A^MH is defined by generalizing (8.3) in an obvious way:³⁷

DC†DCA^MH =DC†d^M_HAC +. . . . (8.4) Now let us specialize to the case that S = γ ×p, with γ a curve in Σ and p a point in C. We write E_p(R) for the restriction of E(R) to M_H ×p. We also write Φ_p for the restriction to Σ× {p} ⊂ Σ×C of the map Φ×1 : Σ×C → M_H ×C. We can replace the connection A =A+iφ in (8.1) by Φ^∗_p(Ab). Hence the factor in the path integral that comes from the inclusion of a Wilson operator on the contour S in the representation R can be written as

W_R(S) =Pexp

− Z

γ

Φ^∗_p(A)

. (8.5)

37 The ellipses in (8.3) involve fermionic zero modes, which represent tangent vectors to M_H and so have analogs in the case of M_{H ×C.}

In the limit that γ approaches the boundary Q of Σ, this has the same form as the term that comes anyway from the Chan-Paton bundle U of the original brane B:³⁸

Pexp

− Z

Q

(Φ^∗_Q(α) +. . .)

. (8.6)

So we learn how Wilson lines act on branes. A Wilson line in the representation R and supported at a point p∈C transforms the Chan-Paton bundle U of a brane B by

U →U ⊗E_p(R). (8.7)

Transformation Of The B-Field

As we have discussed in section 7.1, E(R), and hence alsoE_p(R), in general does not exist as a vector bundle. But it always exists as a twisted vector bundle, twisted by the flat B-field θR(ζ), where θR is the character of the center of the gauge group determined by R.

The category of branes depends on a choice of B-field, a fact that we exploited in section 7.2. For a given discrete electric fielde₀, the backgroundB-field in the sigma-model on M_H is b^e₀ = e₀(ζ), where ζ = ξ(E_ad) is the obstruction to existence of a universal G Higgs bundle. Tensoring with a twisted bundle that is twisted byθ_R(ζ) maps a brane that is twisted by a flatB-field bto a brane that is twisted by b+θ_R(ζ).Therefore, the action of a Wilson line operator on branes changes theB-field, byb→b+θ_R(ζ). In other words, it changes the discrete electric field studied in section 7 by e₀ →e₀+θ_R.

We want to understand what this result means for S-duality. So we write it as a statement about the dual gauge theory, with gauge group ^LG, and a Wilson line W(^LR) determined by a representation ^LR. This Wilson line transforms the discrete electric field by

e₀ →e₀+θ(^LR), (8.8)

where, for convenience, we write θ(^LR) rather than θ^L_R.

38 The ellipses in (8.6) represent fermionic terms whose analog in (8.5) arises from the ellipses in (8.3), which reflect fermionic contributions toAΣ. All these terms are uniquely determined by the topological symmetry, so we do not need to worry about comparing them.

m0

C^′ C

m +₀ S

ξ

ξ

Fig. 11: Insertion of an ’t Hooft operator changes the topology of a G-bundle, as shown here. Sketched is a three manifold with boundary components consisting of two Riemann surfaces C and C^′ and a small two-sphere S enclosing a point at which an ’t Hooft operator is inserted. Cobordism invariance of the characteristic class implies that ifm0is the characteristic class of theG-bundleE →C, then the characteristic class ofE →C^′must bem0+ξ, whereξ=ξ(^LR) is the characteristic class associated with the ’t Hooft operator.

Under S-duality, the fact that a Wilson line operator can change the discrete electric field e₀ maps to the fact that an ’t Hooft line operator can change the characteristic class m₀ ∈ H²(C, π₁(G)) which classifies the topology of the G-bundle E → C. Indeed, as we explained in section 6.2, an ’t Hooft operator T(^LR) is constructed from a G-bundle, which we may call E(^LR), over S² ∼= CP¹. This G-bundle has a characteristic class ξ(^LR) =ξ(E(^LR)). The action of the ’t Hooft operator on m₀ is

m₀ →m₀+ξ(^LR). (8.9)

This statement, which is the S-dual of (8.8), just comes from the behavior of the charac- teristic class under cobordism, as in fig. 11.

Wilson Operators And Supersymmetry

Let us see what kind of supersymmetry the operation (8.7) preserves. For a generic choice of curve S ⊂ Σ×C, the supersymmetric Wilson line W(R,S) preserves B-type supersymmetry in the complex structure J of M_H and nothing else. However, if we take S =γ×p, for γ a curve in Σ andpa point inC, then the Wilson operator preserves more supersymetry. In fact, as we showed at the end of section 6.4, it preserves supersymmetry of type (B, B, B), that is, it preserves B-type supersymmetry in each complex structure.

We can now see explicitly how this is reflected in the action of the Wilson operator on branes. For a point p ∈ C, the bundle E_p(R) is holomorphic in each of the com- plex structures on M_H. (This can be naturally proved using the hyper-Kahler quotient construction of M_H.) So the operation (8.7) preserves B-type supersymmetry for each complex structure.