Review

7. Fluxes and S-Duality

7.1. Review

iǫ and Γ2367ǫ = ǫ. This system is physically sensible (if we interpret it with Lorentz signature); it is described by a real, positive-definite Hilbert space with a positive-definite, hermitian Hamiltonian H. The four unbroken supersymmetries are the two topological supercharges Q_i, i = 1,2 described earlier, and their hermitian adjoints Q^†_i. They obey a physical supersymmetry algebra {Q_i, Q^†_j} = 2δ_ijH, {Q_i, Q_j} = 0 = {Q^†_i, Q^†_j}, for i, j = 1,2.

Group Center

SU(N) Z_N

Spin(2n+ 1) Z₂ Spin(4n+ 2) Z₄

Spin(4n) Z₂ ×Z₂

Sp(n) Z₂

E₆ Z₃

E₇ Z₂

Table 3. Simple and simply-connected Lie groups with non-trivial centers. F2, G4, and E₈, which are omitted, have trivial center.

An important property is that the group Z is naturally selfdual. The dual of an abelian group B isB^∨= Hom(B, U(1)). A finite abelian group is always isomorphic to its own dual, but not naturally. The centerZ of Galways has, however, a natural selfduality.

This duality is best expressed as a homomorphism

Υ :Z × Z →U(1), (7.1) which is symmetric, Υ(a, b) = Υ(b, a), and such that any homomorphism of Z to U(1) is a → Υ(a, b) for a unique b. Such a Υ is called a “perfect pairing.” Υ is constructed as follows. If G is simply-laced, then Z = Λ^∨/Λ, where Λ is the root lattice and Λ^∨ is the weight lattice of G. The pairing Υ is then defined as

Υ(a, b) = exp(2πiha, bi), (7.2)

where h , i is the usual quadratic form on Λ^∨ (which is even and integral when restricted to Λ and takes rational values on Λ^∨). If G is not simply-laced, its center is trivial or Z₂, and so admits precisely one selfduality, which we call Υ.

A G_ad-bundle E onM has a characteristic class ξ(E) that takes values inH²(M,Z).

For example, if G_ad = SO(3), ξ(E) is the second Stieffel-Whitney class w₂(E). One can define a partition function Zξ for every choice of ξ by restricting the path integral to bundles with givenξ. The Z_ξ transform in a unitary representation of theS-duality group [45,26], but this is something that we will not explain here.

Instead, we concentrate on the Hamiltonian description, and specialize toM =S¹×W for a three-manifoldW. In this case, we haveH²(M,Z) =H¹(W,Z)⊕H²(W,Z), soξ(E) can be decomposed asξ =a⊕m, with a∈H¹(W,Z), m∈H²(W,Z).

Consider the pairing

Υ :b H²(M,Z)×H²(M,Z)→H⁴(M, U(1)) =U(1) (7.3) obtained by composing the cup productH²(M,Z)×H²(M,Z)→H⁴(M,Z × Z) with the map Υ : Z × Z → U(1). According to Poincar´e duality, Υ is a perfect pairing, makingb H²(M,Z) a selfdual abelian group. More generally, for a closed oriented manifold Y of dimension n, Poincar´e duality together with selfduality ofZ gives perfect pairings

H^d(Y,Z)×H^n−d(Y,Z)→U(1), 0≤d≤n. (7.4) In particular, we have the perfect pairing

Υ :H¹(W,Z)×H²(W,Z)→U(1). (7.5) So these are dual abelian groups, and in particular

H²(W,Z) = Hom(H¹(W,Z), U(1)). (7.6) Let us now recall how S-duality is implemented in a Hamiltonian framework. The partition function of the gauge theory with m specified has a natural Hilbert space in- terpretation, because m is part of the data at an initial time and is independent of time.

However, becausea cannot be expressed in terms of the data at a fixed time, the partition function with fixed a cannot be given a Hamiltonian interpretation. Instead one must introduce a character e of the finite abelian group H¹(M,Z), that is, a homomorphism

e :H¹(W,Z)→U(1). (7.7)

For each choice of e and m, the sum P

ae(a)Z^a_,^m can be interpreted in terms of a trace in a Hilbert space H^e,m. This sum has a Hilbert space interpretation because summing over a with the weight factor e(a) is compatible with cutting and pasting (or what in quantum field theory is usually called cluster decomposition) in the S¹ direction. e and m are called respectively the discrete electric and magnetic flux [97]. Because of (7.6),e can be alternatively viewed as an element of H²(W,Z), just like m.

So we get a Hilbert space H^m,e for each choice of

e∈E= Hom(H¹(W,Z), U(1))

m∈M=H²(W,Z). (7.8)

Happily, the finite groups E and M are isomorphic according to (7.6). This makes it possible to have S-duality symmetry. The duality transformation S exchanges E and M and maps (e,m)→(−m,e):

S :H^e,m → H−m,e. (7.9)

It also, of course, maps τ → −1/τ. (Eqn. (7.9) is a discrete analog of (2.23), with (m, ~~ n) replaced by their discrete counterparts (m,e).)

While it is possible to define all the Hilbert spacesH^e,m, a given gauge theory construc- tion may not use all of them. If we do gauge theory onM =S¹×W with simply-connected gauge group G, then we must set to zero the characteristic classes a and m that enter in the partition function Z^a,m(M). For the Hilbert space, this means thate, which arises by a Fourier transform with respect to a, is arbitrary, whilem= 0. Hence

H(G, W) =⊕^eH^e,0. (7.10)

Taking into account the assumed transformation law for e,m underS-duality, we see that the Hilbert space of the dual theory must be

⊕^mH0,m. (7.11)

How is this Hilbert space related to a path integral on M = S¹ ×W? Setting e to zero means summing over a. So in terms of the variablesa,m, (7.11) means one must sum over all a,m with equal weight. In other words, in the dual theory we sum over ^LGad-bundles with all possible ξ. Thus the dual theory has gauge group ^LGad, as expected.

The Universal Bundle

We need another piece of background, which is the concept of the universal bundle.

Consider first the moduli space M(G, C) ofG-bundles over a Riemann surfaceC. For every point p∈M(G, C), there is a corresponding G-bundleE_p →C parametrized byp; it is determined up to isomorphism. A universal bundle, if it exists, is aG-bundleE→M×C such that for any p ∈ M, E restricted to p×C is isomorphic to Ep. Gauge theory gives

[88] a very direct method to analyze the universal bundle, though we will not explain this here.

Naively, we just defineE so that its restriction top×C is “the” bundle overC that is determined byp. The reason that a problem arises is that, to begin with,ponly determines an isomorphism class of G-bundle over C, not an actual bundle. This causes no difficulty locally³³; we just pick a bundle in the right isomorphism class at a particular p and then deform it in a small neighborhood. So for each small open set U_i ⊂ M, we can pick a universal bundle E_i→Ui×C, and moreover E_i is unique up to isomorphism.

The next step is to glue together the E_i over (U_i ∩U_j)×C. The bundles E_i and E_j are isomorphic over (Ui ∩Uj)×C, so we pick an isomorphism Θij : E_i ∼= E_j over this intersection, with Θ_ji = Θ⁻¹_ij .

Now if on triple intersections (U_i∩U_j∩U_k)×C, the composition Θ_kiΘ_jkΘ_ij is equal to 1, then we can consistently glue together the E_i to get the desired universal bundle E.

If the gauge group is the adjoint group G_ad, there is no problem in the gluing. A generic G_ad bundle has no automorphisms (exceptions occur at singularities of M). So Θ_kiΘ_jkΘ_ij, just because it is an isomorphism of E_i (restricted to (U_i∩U_j ∩U_k)×C) to itself, is the identity at a generic point of the triple intersection and hence everywhere.

Thus, the universal G_ad-bundle E_ad does exist. Now let us repeat this discussion, taking the gauge group to be the simply-connected cover G (or more generally, any cover of G_ad with a non-trivial center). We start in the same way. For each small open set U_i ∈ M, we pick a universal bundle E_i over U_i ×C, and try to glue to make a universal G-bundle E.

The reason that the result is different is that a generic G-bundle does have a non- trivial group of automorphisms; the centerZ is a group of automorphisms of anyG-bundle E.

Hence, in the above argument, the composite map Θ_kiΘ_jkΘ_ij over a triple intersection is not necessarily the identity; instead

Θ_kiΘ_jkΘ_ij =f_ijk, (7.12)

33 This statement holds near a smooth point inM. A more precise analysis than we will give shows that the class ζ(E_ad) that obstructs the universal bundle can have a local contribution at singularities.

with fijk ∈ Z. The fijk combine to a two-cycle defining an element ζ ∈ H²(M,Z). The element ζ is known to be non-trivial; it is the obstruction to the existence of the universal G-bundle E.

There is another way to explain the existence ofζ. LetM(G_ad, C)₀ be the component ofM(Gad, C) that parametrizes topologically trivial bundles – the ones that when restricted top×C, forp∈M(G_ad, C), can be lifted toG-bundles. Consider the universal bundleE_ad over M(G_ad, C)₀×C. If we could lift E to a G-bundle E, this (after being pulled back to M(G, C)×C, which is a finite cover ofM(Gad, C)0×C) would be the universalG-bundle.

In general, the obstruction to lifting aG_ad-bundleE_adto aG-bundleEis the characteristic class ξ(E_ad). So the obstruction to existence of the universalG-bundle isζ =ξ(E_ad).

There is a similar story for Higgs bundles. The universal Higgs bundle is a pair (E,ϕ),b with E being a G-bundle over M_H(G, C)×C, and ϕb∈H⁰(M_H(G, C)×C,ad(E)⊗KC), obeying the following condition. For each p∈M_H(G, C), the restriction of (E,ϕ) tob p×C should be isomorphic to the Hitchin pair (E_p, ϕ_p) parametrized by p.

The arguments that we have already given can be carried over with no essential change to show that for gauge group G_ad, the universal Higgs bundle (E_ad,ϕ) does exist. But forb the simply-connected gauge group G (or any nontrivial cover of G_ad), the universal Higgs bundle does not exist. It is obstructed by the fact that a generic Hitchin pair (E, ϕ), with E aG-bundle, has the groupZ of automorphisms. The obstruction is just the obstruction to lifting the “bundle” part of the universal G_ad Higgs bundle (E_ad,ϕ) to ab G-bundle. It is therefore

ζ =ξ(E_ad). (7.13)

The Universal Bundle As A Twisted Vector Bundle

Although for G not of adjoint type, the universalG-bundle does not exist as a vector bundle or as a principal bundle, it does exist as a twisted vector bundle. We will describe this concept in a very pedestrian way.

One way to construct an ordinary vector bundle V of rank N over a manifold X is to cover X with small open sets U_i, on each of which we pick a rank N trivial bundle V_i ∼= U_i ×C^N. Then we glue V_i to V_j on the intersection U_i ∩U_j via a gluing map vij :Vi∩Vj →U(N), with vji=v⁻¹_ij . If on triple intersections we have

vkivjkvij = 1, (7.14)

then the Vi can be glued together consistently to make a rank N vector bundle V →X.

Now suppose that we are givenb∈H²(X, U(1)). Then bcan be represented explicitly by a U(1)-valued cocycleb_ijk defined on triple intersectionsU_i∩U_j∩U_k. A twisted vector bundle is defined by the same sort of data v_ij : V_i ∩V_j → U(N). But now, instead of (7.14), we ask for

v_kiv_jkv_ij =b_ijk. (7.15)

Thus, a b-twisted vector bundle V is not a vector bundle in the usual sense. However, the associated adjoint bundle ad(V) is an ordinary vector bundle, since the phase bijk in (7.15) disappears if we pass to the adjoint representation.

Now let us return to the problem of constructing a universal bundle. In this paper, we are generally a little imprecise about whether by a G-bundle, we mean a principal G-bundle, whose fiber is a copy of G, or an associated vector bundle in some faithful representation of G. Principal bundles make possible a uniform analysis good for any G, but for a group likeU(N) that has a convenient faithful representation (theN-dimensional representation) it is useful to think in terms of vector bundles.

In discussing the universal bundle, it is helpful to be more precise. We interpret the transition functions Θij of our discussion above as G-valued functions, with no particular choice of representation. Their projection to G_ad gives us transition functions for a uni- versal principal Gad-bundle E_ad, but we cannot lift this to a principal G-bundle, because of the relation

Θ_kiΘ_jkΘ_ij =f_ijk, (7.16)

where the f_ijk define the class ζ = ξ(E_ad) ∈ H²(M_H(G_ad, C),Z). Now let us pick an irreducible representation ̺ of G. We write Θ^̺_ij for the transition functions Θ_ij evaluated in the representation ̺. Likewise we write ̺(f) for f evaluated in the representation ̺.

In the irreducible representation ̺, the center of G acts by scalar multiplication, so ̺(f) takes values in U(1). We have

Θ^̺_kiΘ^̺_jkΘ^̺_ij =̺(f_ijk). (7.17) The quantities ̺(fijk) are a cocycle defining the element ̺(ζ)∈H²(M_H(Gad, C), U(1)).

If ̺(f) = 1, the objects Θ^̺ are transition functions that define a vector bundle E_̺ → M_H(G, C) that we may call the universal bundle in the representation ̺. In general this is not the case. However, comparing eqns. (7.16) and (7.15), we see that while E_̺ may not exist as an ordinary vector bundle, it does always exist as a twisted vector bundle, twisted by ̺(ζ)∈H²(M_H(Gad, C), U(1)).