Compactification To Two Dimensions

7. Fluxes and S-Duality

7.2. Compactification To Two Dimensions

We can similarly decompose the character e that is dual to a. Clearly, we have e = e₀ +e₁, where e₀ is a character of a₀ ∈ Z and e₁ is a character of a₁ ∈ H¹(C,Z).

Hence, using the selfduality of Z and Poincar´e duality, e₀ ∈ Z^∨=Z

e₁ ∈H¹(C,Z)^∨=H¹(C,Z). (7.23) In view of (7.9), the transformation under S-duality is

(e₀,m₀)→(−m₀,e₀)

(e₁,m₁)→(−m₁,e₁). (7.24) Our next goal is to interprete₀,e₁,m₀, andm₁in the effective two-dimensional sigma- model with target M_H(G_ad). We will consider both the case of a closed Riemann surface Σ, and the case when Σ has a boundary. The latter case will enable us to understand the implications of (7.24) for the geometric Langlands program.

Interpretation Of m₀

The easiest to interpret ism₀ =ξ(E)^0,2. The target space M_H(G_ad, C) of our sigma- model is not connected. Its components are labeled by the topological type of the G_ad- bundle E →C. But this is exactly what is measured by m₀.

Interpretation Of e₀

Consider in general a sigma-model of maps Φ : Σ → X, for some X. A flat B-field is an element b ∈ H²(X, U(1)). A flat B-field is incorporated in the sigma-model path integral as follows, for the case that Σ has no boundary. Given a map Φ : Σ → X, one pulls back b to Φ^∗(b) ∈ H²(Σ, U(1)) = U(1), and then one includes in the path integral a factor of Φ^∗(b). Thus, in this situation, incorporating the flat B-field has the effect of weighting by phases the different components of maps of Σ to X.

Now e₀ determines a flat B-field in the sigma-model of maps Φ : Σ → M_H(G_ad, C).

Indeed, there is as we have explained a natural class ζ ∈ H²(M_H(Gad, C),Z), which expresses the obstruction to lifting the universal G_ad bundle E_ad → M_H(G_ad, C) to a G-bundle. By composing ζ with e₀ : Z → U(1), we get a flat B-field b^e₀ = e₀(ζ) ∈ H²(M_H(Gad, C), U(1)). We claim that the role of e₀ in the effective sigma-model of maps Σ → M_H(G_ad) is precisely to weight every map in the way that one would expect for a sigma-model with the flat B-field b^e₀.

In fact, the definition of e₀ is that the path integral includes a phase factor e₀(a₀) = e₀(ξ(E)^2,0). Butξ(E)^2,0 = Φ^∗(ζ); this follows upon composing (7.19) with the projection Σ ×C → Σ. So the phase factor induced in the path integral by e₀ is e₀(Φ^∗(ζ)) = Φ^∗(e₀(ζ)) = Φ^∗(b^e₀). This justifies our claim that the effect of e₀ in the sigma-model is to generate a flat B-field b^e0.

(7.24) therefore means that S-duality exchanges the topological class m₀ of a flat G_ad-bundle with the flat B-field determined by e₀.

Incorporation Of Branes

For applications to the geometric Langlands program, the real payoff is to understand the implications of all this for branes.

So we take Σ to be a Riemann surface with boundary. M = Σ×C is therefore a four-manifold with boundary ∂M = ∂Σ×C. On ∂M, we place some supersymmetric boundary condition.

The effective two-dimensional description is a sigma-model of maps Φ : Σ→M_H(Gad), with boundary condition corresponding to some brane. We would like to understand the role of e₀ and m₀ in this description.

There is little new to say about m₀. It labels the components of M_H(G_ad), whether Σ has a boundary or not.

The role ofe₀ is more subtle. As we have seen, the sigma-model with target M_H(Gad) is endowed with a flat B-field e₀(ζ). A flat B-field has a very interesting effect on branes [98]. In the absence of a B-field, a brane on M_H(G_ad) has a Chan-Paton bundle, which is a vector bundle over M_H(G_ad) (or more generally a sheaf, perhaps supported on a submanifold, that defines a K-theory class of M_H(Gad)). However, in the presence of a flat B-field, associated with an element b∈ H²(M_H(Gad), U(1)), the Chan-Paton bundle becomes a twisted vector bundle (or more generally a twisted sheaf related to an element of the twisted K-theory of M_H(G_ad)), twisted by b in the sense of eqn. (7.15). It is because of this that we introduced the concept of a twisted vector bundle.

So in short, for e₀ 6= 0, the Chan-Paton bundle of a brane is a twisted vector bundle, twisted by b^e0 = e₀(ζ). Luckily, from the analysis at the end of section 7.1, we have a plentiful supply of such twisted vector bundles. If ̺ is any irreducible representation of G such that the character of the center of G defined by ̺ is equal to e₀, then the universal bundle E_̺ in the representation ̺ is an example of a twisted vector bundle for the flat B-fieldb^e₀.

Other important examples of twisted branes can be constructed by picking a sub- manifold Y ⊂ M_H(G_ad) such that b^e₀ is trivial when restricted to Y. In this case, the definition of a brane supported on Y is independent of e₀ (up to a not quite canonical isomorphism). For example, Y could be a point in M_H(G_ad). Certainly b^e₀ is trivial when restricted to a point, so zerobranes exist for any e₀. From such zerobranes, we can form electric eigenbranes, as we explain in section 8. A slightly more subtle example is a mag- netic eigenbrane, a brane of rank one supported on a fiberF of the Hitchin fibration. The Chan-Paton bundle of such a brane should be a flat³⁴ twisted line bundle. In [21], it is shown that the flatb-fieldb^e0 is trivial when restricted toF, as a result of which the space of flat twisted line bundles is isomorphic, but not canonically isomorphic, to the space of ordinary flat line bundles on F.

We can now deduce from S-duality a statement about branes. The duality trans- formation S maps (e₀,m₀) → (−m₀,e₀), so it exchanges the topology of the component of M_H on which a brane is supported with the flat B-field b^e₀ by which its Chan-Paton bundle is twisted. It also, of course, exchanges G_ad with^{35 L}G_ad, and (as we explain in the concluding remark of this section), exchanges M_H(G_ad) withMb_H(^LG_ad), which we define to be the universal cover of M_H(^LG_ad).

As a special case of this duality, a point on one side, contained in a fiber F of the Hitchin fibration, is mapped on the other side to a brane of rank one supported on the corresponding fiber ^LF = Ξ(F) of the Hitchin fibration, and endowed with a flat twisted line bundle. F is a union of complex tori, labeled by the characteristic classm₀ =ξ(E) of the Higgs bundle. The choice of a component ofF on one side determines on the other side the discrete electric fielde₀ and hence the twist. The choice of a point on F determines a flat twisted line bundle on ^LF (to which it maps under the duality transformation S). Of course, this relationship between F and ^LF is reciprocal. This twisted duality between F and ^LF is in fact one of the main results of Hausel and Thaddeus [21].

Interpretation Of e₁ And m₁

34 One can here interpret flatness to mean, just as for ordinary line bundles, that the transition functions are constants.

35 By ^LGad, we mean the adjoint form of the group ^LG. The statement we are describing here is best expressed in terms of adjoint bundles on both sides to allow all possible topologies.

Momentarily we indicate explicitly whether we want a given component of M_H or its universal cover.

Finally, let us discuss the interpretation of e₁ and m₁ in two-dimensional terms. It will be helpful to begin by comparing the theories with gauge groups G and Gad.

M_H(G) is simply-connected. It has a natural group of symmetries E_C = H¹(C,Z).

Indeed, E_C parametrizes Z-bundles over C. A G Higgs bundle (E, ϕ) can be tensored with aZ-bundle to make a newGHiggs bundle. (Concretely, this operation multiplies the holonomies of E around noncontractible loops in C by elements of the center of G.) So this gives a group E_C of symmetries of the sigma-model with targetM_H(G). The Hilbert space of this sigma-model can be decomposed in characters of E_C.

On the other hand, M_H(G_ad) has no such geometrical symmetries. But it has a fundamental group M_C = H¹(C,Z), which is isomorphic to E_C. To understand where this fundamental group comes from, a shortcut is to note that one component M_H(G_ad)₀ of M_H(G_ad), namely the component that parametrizes Higgs bundles that can be lifted to G, is simply M_H(Gad)0 = M_H(G)/EC. Dividing by E_C eliminates the geometrical symmetries ofM_H(G), but of course it creates a fundamental group.³⁶ Soπ₁(M_H(G_ad)₀)∼= E_C.

Actually, the fundamental group is the same for any component of M_H(G_ad).

M_H(G_ad) is defined by dividing the space of all solutions of Hitchin’s equations, for gauge groupGad, by the groupGad(C) of allGad-valued gauge transformations onC. If one were to divide only by the connected component of Gad(C), one would get the universal cover

b

M_H(G_ad, C). The fundamental group of M_H(G_ad) is therefore the group of components of Gad(C), and this is E_C =H¹(C,Z), for any component of M_H(G_ad).

In sum, strings moving onM_H(G_ad) have a discrete group of conserved winding num- bersM_C =π₁(M_H(G_ad)) =H¹(C,Z). Likewise strings moving onM_H(G) have a discrete group of conserved momenta E_C =H¹(C,Z).

Let us compare the symmetries E_C and M_C to what we can see in the underlying gauge theory. In G_ad gauge theory on M = Σ×C = S¹ ×Se¹ ×C, the bundle E and other data determine a map Φ : S¹ ×Se¹ → M_H(Gad). In analyzing the topology of this situation, we expanded the characteristic class ξ(E) as ξ(E) = a⊕m, where m is the restriction of ξ(E) to Se¹ ×C (more precisely, to p×Se¹×C, for a point p ∈ S¹). In the

36 EC does not act freely, so M_H(Gad) has orbifold singularities. (It also has more severe singularities from reducible Higgs bundles.) As is familiar in sigma models, the fundamental group of M_H(Gad) must be understood in an orbifold sense. Alternatively, one can rely on the four-dimensional gauge theory instead of reducing to the sigma model with its singularities.

low energy sigma-model, a and m are invariants of Φ : Σ →M_H(Gad). In particular, m only depends on the restriction Φ|^Se¹ of Φ to Se¹.

In (7.21), we further expanded m = m₀ ⊕m₁ with m₁ ∈ H¹(Se¹, H¹(C,Z)). m₁ is a topological invariant of Φ|^Se¹ that vanishes for constant maps of Se¹ to M_H(G_ad). So it measures the homotopy class of the map Φ|^Se¹ in π₁(M_H(G_ad)) =H¹(C,Z).

Similarly, we could exchange the role of S¹ and Se¹. The restriction of ξ(E) to S¹×C (that is, toS¹×q×C, for a point q∈Se¹), is in the above notationm₀⊕a₁. In particular, a₁ ∈ H¹(S¹, H¹(C,Z)) measures the winding of Φ in the S¹ or “time” direction. The charactere₁ which is dual toa₁ therefore measures the conserved momentum of the strings.

An Example

For an important illustration of all this, consider the Langlands dual pair Gad and

LG. In G_ad gauge theory, we set e = 0 and consider a sigma-model with target M_H(G_ad).

In this model, there are no conserved momenta, but there is a symmetry M_C of string windings.

In the dual picture, the gauge group is ^LG, we set m= 0, and the sigma-model with target M_H(G) has no conserved string windings, but a symmetry group E_C of discrete conserved momenta.

S-duality or Montonen-Olive or Langlands duality exchanges the two pictures, ex- changing E_C(^LG) with M_C(Gad). The fact that the duality exchanges the discrete con- served momenta and windings of strings is an aspect of its relation to T-duality in two dimensions.

A Concluding Comment

More generally, we can simply specify m₀ and e₀ as we please. Then we consider branes on a component of M_H(Gad) labeled topologically bym₀; the Chan-Paton bundles of the branes are twisted by e₀.

We still have two ways to proceed with the quantization. If we divide by allG_ad-valued gauge transformations, then the target space is the component of M_H(G_ad, C) labeled by m₀. In this case, there is a finite groupM₁ that classifies the string winding numbers, but there is no group E₁ of geometrical symmetries. If we divide by only the connected gauge transformations, then the target is the universal cover Mb_H(G_ad, C). In this case, there is a group E₁ of geometrical symmetries, but no group M₁ of string winding numbers.

S-duality exchanges E₁ and M₁, so (in addition to exchanging G and ^LG) it exchanges the two methods of quantization.

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.