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Title Bundle and moduli spaces

Author(s) Bouwknegt, P; Mathai, V; Wu, S

Citation Journal of and Physics, 2012, v. 62 n. 1, p. 1-10

Issued Date 2012

URL http://hdl.handle.net/10722/156273

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Σ M M to fafml fself-adjoint of family a of M cae ocmatRie- compact to ociated h ouispace moduli The . oevr h rtChern first the Moreover, . fgnsgetrta one. than greater genus of G hc ewl eoeby denote will we which , eoetemdl space moduli the denote tdidxbundle index cted ooopi sections holomorphic f f emta connection hermitian M ofra blocks conformal orsodn to corresponding soeyProjects iscovery determinant M plays ators in a standard conformal field theory, the WZW-model [7], on which there exists an extensive literature. For a family of self-adjoint Dirac-type operators, an index gerbe was introduced by Carey- Mickelsson-Murray [12, 13] and by Lott [23]. The gerbe was described by local data, which can be viewed as the analogues of the transition functions of the determinant line bundle. The index gerbe is related to Hamiltonian anomalies. In [34], Segal constructs a projective Hilbert bundle; its Dixmier-Douady invariant is the obstruction to lifting it to a Hilbert bundle. In this paper, we construct the index bundle gerbe for such a family, which is a global version of prior constructions. Our construction of the index bundle gerbe is inspired by these papers and by the work on determinant line bundles [9, 17, 32]. Melrose-Rochon [25] have a completely different construction of an index bundle gerbe using pseudodifferential operators. In the special case of a circle fibration, we also construct the caloron bundle gerbe using geometric data and show that it is isomorphic to the analytically defined index bundle gerbe. We apply these constructions to the moduli space M and obtain on it natural bundle gerbes with connections and curvings. The 3-curvatures represent Dixmier-Douady classes that are generators of the third de Rham cohomology group of M. Applications of moduli spaces of Riemann surfaces are ubiquitous. The construction of explicit geometric realizations of the degree 3 classes on moduli spaces through bundle gerbes was motivated by the desire to generalize the geometric Langlands correspondence [18, 19, 21] by including background fluxes. Here we outline the contents of the paper. In Section 2, we construct the index bundle gerbe associated to a set of geometric data using the spectrum of a family of Dirac-type operators. In Section 3, given a unitary representation of the structure group, we construct the caloron bundle gerbe using the projective Hilbert space bundle obtained from the caloron correspondence of Murray-Stevenson [29]. In Section 4, we establish an isomorphism (not just a stable isomorphism) between the geometrically constructed caloron bundle gerbe and the analytically defined index bundle gerbe for a circle fibration. Section 5 contains the application of these constructions to the moduli space M, obtaining natural bundle gerbes with connections and curvings, whose Dixmier-Douady classes generate the third de Rham cohomology group of M. We end with a section of conclusions, where we outline future work on constructing bundle gerbes on other moduli spaces such as the moduli space of anti-self-dual connections, on a compact four dimensional Riemannian manifold.

2 2. Index bundle gerbe

In this section we will construct an index bundle gerbe associated to the following set of geometric data. For an introduction to bundle gerbes, see [27].

Basic setup: Let Z X be a smooth fibre bundle whose typical fibre is a compact odd-dimensional man- → ifold. Let T (Z/X) Z denote the vertical tangent bundle, which is a sub-bundle of the → tangent bundle TZ. We assume that there is a Riemannian metric on T (Z/X). Suppose that T (Z/X) has a spin structure and let S Z denote the corresponding bundle of spinors. → Let E Z be a hermitian vector bundle with connection E. → ∇ In this case, there is a smooth family DE : x X of self-adjoint Dirac-type operators { x ∈ } DE : Γ (Z ,S E) Γ (Z ,S E) acting on sections over the fibres Z (x X) of Z X. x x ⊗ → x ⊗ x ∈ → The L2-completions H of Γ (Z ,S E) form a Hilbert bundle H over X. We denote by the x x ⊗ same symbol DE the operator on H . Recall that for any x X, spec(DE) R is a closed x x ∈ x ⊂ countable set with accumulation point only at infinity. For λ Q,1 consider the open subset ∈ U = x X λ / spec(DE) (2.1) λ { ∈ ∈ x } + of X. For x U , let the Hilbert space H (H− , respectively) be the span of eigenspaces ∈ λ λ,x λ,x E of Dx with eigenvalues greater than (less than, respectively) λ. They form Hilbert bundles

Hλ± over Uλ. Suppose µ>λ and λ,µ Q. Observe that for x U U , we have ∈ ∈ λ ∩ µ + + + + H = H (H H− ), H− = H− (H H− ), λ,x µ,x ⊕ λ,x ∩ µ,x µ,x λ,x ⊕ λ,x ∩ µ,x + + where H H− is a finite dimensional space. Let L = det(H H−), which is a line λ,x ∩ µ,x λµ λ ∩ µ bundle over U U . Since λ ∩ µ + + + (H H−) (H H−)= H H− λ ∩ µ ⊕ µ ∩ τ λ ∩ τ on U U U for any λ<µ<τ in Q, we have a cocycle condition λ ∩ µ ∩ µ L L = L . (2.2) λµ ⊗ µτ ∼ λτ The collection L defines the index gerbe [12, 13, 23] over X. { λµ}

1We restrict to λ Q to have a countable open cover of X. ∈ 3 + Over U , we construct the Fock bundle F = (H H−). Hodge duality asserts that λ λ λ ⊕ λ for any finite dimensional complex vector space VVthere is a canonical isomorphism

V det V = V. ⊗ ∼ ^ ^ Thus on the overlap U U , we get λ ∩ µ

+ F = H H− λ λ ⊗ λ ^ + ^ + = H (H H−) H− ∼ µ ⊗ λ ∩ µ ⊗ λ ^ + ^ + ^ = H H− (H H ) Lλµ ∼ µ ⊗ λ ⊗ λ ∩ µ− ⊗ = ^ H+ ^ H L^ ∼ µ ⊗ µ− ⊗ λµ = F^ L .^ µ ⊗ λµ F F Therefore there is a canonical isomorphism of the projectivizations P( λ) ∼= P( µ) on the overlap U U and hence a well defined projectivized Fock bundle π : P(F) X. λ ∩ µ → To construct the index bundle gerbe, we now define a line bundle L P(F)[2] on the fibre → 1 product of P(F) with itself. Let Lλ be the universal line bundle over P(Fλ)= π− (Uλ). That is, over any ℓ P(F ), the fibre is ∈ λ (L ) = (ℓ, v): v ℓ ℓ (F ) . λ ℓ { ∈ }⊂{ } × λ π(ℓ)

Lλ is a hermitian line bundle, but does not glue properly on the overlaps in P(F) for the 1 1 1 same reason that F doesn’t on X. In fact, on π− (U ) π− (U )= π− (U U ), we have λ λ ∩ µ λ ∩ µ

L = L π∗L . (2.3) λ ∼ µ ⊗ λµ

Now, δ(L ) = L∗ ⊠ L is a line bundle on P(F ) P(F ) which restricts to the total λ λ λ λ × λ space of the bundle π[2] : P(F )[2] U . However, this time, δ(L ) = δ(L ) on the overlap λ λ → λ λ µ [2] 1 (π )− (U U ), since the fudge factor for each cancels, thereby defining a global hermitian λ λ ∩ µ line bundle L P(F)[2]. There is a line bundle δ(L) P(F)[3] defined by → →

1 δ(L)= π∗ L (π∗ L)− π∗ L, 12 ⊗ 13 ⊗ 23 where π : P(F)[3] P(F)[2] for 1 i < j 3 is the projection onto the ith and jth factors. ij → ≤ ≤ It is trivial precisely because of (2.2). Therefore we have established the following. 4 Theorem 2.1. Given the Basic setup, there is a well-defined global fibre bundle of projective Hilbert spaces π : P(F) X and a line bundle L P(F)[2] such that δ(L) P(F)[3] is → → → trivial.

Thus we have a bundle gerbe over X, which we call the index bundle gerbe. Note that the line bundle L P(F ) provides a local trivialization of the index bundle gerbe over λ → λ U X. λ ⊂ We remark that the considerations in this section can straightforwardly be generalized to more general open coverings U of X so that for each U there exists a spectral cut { α} α λ / spec(DE), for all x U . We restrict our discussion to the cover (2.1) for notational α ∈ x ∈ α simplicity. Finally, we remark that the construction in this section works for a family of general elliptic self-adjoint operators on a compact manifold.

3. The caloron bundle gerbe

We begin by reviewing the caloron correspondence of Murray-Stevenson [29]. We then study the behavior of this construction under a unitary representation of the structure group. Finally, we obtain a fibre bundle whose typical fibre is a projective Hilbert space; we will call the associated bundle gerbe the caloron bundle gerbe.

3.1. The caloron correspondence. Let X be a manifold and let S1 denote the unit circle. Let P S1 X be a principal G-bundle, where G is a connected and simply-connected → × compact Lie group. Then there exists a canonical identification between equivalence classes of principal G-bundles over S1 X and principal LG-bundles over X, × G P LG Q −−−→ −−−→ , (3.1)   ⇔   1    S  X   X   y×   y  as follows. Given P , the bundle Q is the push-forward of P under the projection map S1 X X. That is, the fibre of Q over x X is Q = Γ (S1 x ,P ). Clearly, × → ∈ x × { } the free loop group LG acts freely and transitively on Qx and hence Q is a principal LG- bundle. The total space Q can be constructed globally as follows. Consider the LG-bundle LP L(S1 X). Then Q X is the pullback of LP by the map η : X L(S1 X) → × → → × given by η(x)=(θ (θ, x)). Conversely, given an LG-bundle Q X, we can define a 7→ → principal G-bundle P over S1 X by P =(Q G S1)/LG, where the right LG-action on × × × 1 1 Q G S is given by (p,g,θ)γ =(pγ,γ(θ)− g, θ) and the G-action is by right multiplication × × 5 in the second factor. Clearly these actions commute, establishing the canonical equivalence in (3.1). For a generalization of the caloron correspondence to principal G-bundles over non-trivial circle bundles or over more general fibrations, see [8, 11, 30, 20].

Given a connection A˜ on P , we define a connection A = η∗LA˜ on Q by pulling back the connection LA˜ on LP . In addition, A˜ determines a Higgs field Φ = η∗ιΞ (LA˜), where Ξ is a canonical vector field on LP generating the translation on S1. The map Φ : Q Lg is → smooth and satisfies 1 1 Φ(pγ) = ad(γ− )Φ(p)+ γ− ∂θγ (3.2) for γ LG. Conversely, given a connection A on Q and a Higgs field Φ for Q satisfying ∈ Eqn. (3.2), we have a 1-form

1 1 A˜ = ad(g− )A(θ)+ Θ + ad(g− )Φ dθ on Q G S1 which descends to a connection 1-form on P . Here Θ denotes the Cartan- × × Maurer 1-form on G. We summerize the above in the following

Proposition 3.1 (caloron correspondence [29]). The canonical equivalence in (3.1) deter- mines a bijection between isomorphism classes of G-bundles with a connection over S1 X × and isomorphism classes of LG-bundles with a connection and a Higgs field over X.

Recall that for a simple group G, there is a basic central extension (cf. [31])

p 1 U(1) LG LG 1 −→ −→ −→ −→ corresponding to a cocycle that generates the second cohomology of LG. Denote by ξ : Q[2] c → LG the map given by (q , q ) γ, where γ LG is the unique element such that q = q γ. 1 2 7→ ∈ 2 1 Let PQ Q[2] be the pullback of the U(1)-bundle LG LG via ξ and let LQ Q[2] be → → → the line bundle associated to PQ. Equivalently, if LLG LG is the line bundle associated c → Q LG Q to LG, then L = ξ∗(L ). Since ξ is a homomorphism of , the line bundle L determines a bundle gerbe called the lifting bundle gerbe of Q. c Theorem 5.1 of [29] asserts that a connection A on Q and a Higgs field for Q determine a curving (or B-field) √ 1 1 B = − A, ∂θA FA, Φ dθ, 2π 1 2h i−h ∇ i ZS   where F is the curvature of the connection A and Φ = dΦ + [A, Φ] ∂ A. Here and A ∇ − θ below, the inner product , on the Lie algebra g is chosen such that the length of a long h· ·i root is √2. It induces an inner product (with the same notation) on Lg. The 3-curvature 6 H = dB Ω 3(X), associated to B, represents the Dixmier-Douady class of the bundle gerbe ∈ Q √ 1 (Q, L ) in de Rham cohomology. It pulls back to − F , Φ dθ on Q. Let F ˜ denote − 4π2 S1 h A ∇ i A 1 the curvature of the connection A˜ on P . Then the first Pontryagin form of P is F ˜, F ˜ , R 8π2 h A Ai and Theorem 6.1 in [29] asserts that 1 2 FA˜, FA˜ = H. (3.3) − 8π 1 h i ZS If ρ: G SU(V ) is a unitary representation of G on a finite dimensional vector space → V , we can study the behavior of the caloron correspondence under ρ. First, we have a principal SU(V )-bundle P ρ = P SU(V ) S1 X; let r : P P ρ be the map induced ×G → × P → by ρ that changes the structure group. Correspondingly, there is a principal LSU(V )- ρ ρ ρ bundle Q = η∗(LP ) = Q LSU(V ) over X. Let r : Q Q be the map induced ×LG Q → by Lρ: LG LSU(V ). Using the map ξ : (Qρ)[2] LSU(V ), we have a line bundle → ρ → ρ LSU(V ) ρ ρ L = ξρ∗(L ) and hence a bundle gerbe (Q , L ). We recall that the line bundles

LG LSU(V ) LSU(V ) LG ιρ L LG and L LSU(V ) satisfy the relation (Lρ)∗(L ) = (L )⊗ , where → → ∼ ιρ is the Dynkin index of the representation ρ [5]. Using the commutative diagram

ξ Q[2] / LG

[2] rQ Lρ   ξρ (Qρ)[2] / LSU(V ),

[2] Lρ LQ ιρ ρ Lρ we get (rQ )∗( ) ∼= ( )⊗ . Therefore, the bundle gerbe (Q , ) on X is the ιρ-th power of (Q, LQ). 1 ρ Given a connection A˜ of P S X, the induced connection A˜ on P satisfies r∗ (A˜ )= → × ρ P ρ ρ˙(A˜), whereρ ˙ : g su(V ) is the representation of the Lie algebra g. The connection on Qρ is → ˜ Aρ = η∗(LAρ); it satisfies rQ∗ (Aρ)= Lρ˙(A), and therefore we have rQ∗ (FAρ )= Lρ˙(FA) for the ρ corresponding curvatures. Similarly, the Higgs field Φ = η∗ιΞ (LA˜ ): Q Lsu(V ) satisfies ρ ρ → rQ∗ (Φρ)= Lρ˙(Φ). Since the pull-back byρ ˙ of the primitive bilinear form on su(V ) is ιρ times ρ the form , on g, the curving (B-field) B on Q is related to B on Q by r∗ (B ) = ι B. h· ·i ρ Q ρ ρ Therefore, the 3-curvature Hρ on X associated to Bρ is Hρ = ιρ H. This is compatible with ρ ρ Q an earlier result that the bundle gerbe (Q , L ) is the ιρ-th power (Q, L ). We summarize the results in the following

Proposition 3.2. Given a representation ρ: G SU(V ) in the above setting, the induced → ρ ρ Q bundle gerbe (Q , L ) on X is the ιρ-th power of (Q, L ). Furthermore, its B-field and the 7 3-curvature are, respectively,

rQ∗ Bρ = ιρ B, Hρ = ιρ H.

3.2. Projective Hilbert bundle and the caloron bundle gerbe. As before, G is a simple, connected and simply-connected compact Lie group and ρ: G SU(V ) is a unitary → representation of G. Let H = L2(S1,V ). Then there is an induced unitary representation ρ˜: LG U(H) of LG defined byρ ˜(γ)(θ) = ρ(γ(θ)), where γ LG and θ S1. For any → ∈ ∈ + + λ R, we have a decomposition H = H H−, where H (H−, respectively) denotes the ∈ λ ⊕ λ λ λ Hilbert space span of the Fourier modes not less than (less than, respectively) λ. Actually, we haveρ ˜: LG U (H), where the restricted unitary group is → res U (H)= S U(H):[S,I ] is a Hilbert-Schmidt operator . res { ∈ λ } + Here I is an involution on H such that I = id on H and I = id on H− (cf. [31]). It is λ λ λ λ − λ clear that Ures(H) does not depend on the choice of λ.

Let Fλ denote the associated Fock (Hilbert) space

+ F = (H H−) . λ λ ⊕ λ Then we have the Shale-Stinespring embedding^ σ : U (H) ֒ P U(F ) (cf. [37]). Let λ res → λ ρˆ = σ ρ˜: LG P U(F ) be the composition. It is a positive energy representation of λ λ ◦ → λ LG. In the next section, we will compare the representations for different values of λ, but for the remainder of this section we will only need λ = 0. Given a principal G-bundle P S1 X, we have the principal LG-bundle Q X by the → × → caloron correspondence. Together with a representation ρ of G on V , we form the associated fibre bundle P(Fρ)= Q P(F ) X, ×LG 0 → where LG acts on the typical fibre P(F ) viaρ ˆ . We will next define a line bundle Lρ 0 0 → P(Fρ)[2] induced by PQ or LQ Q[2]. Since → PQ = (q , q ,g) Q[2] LG q = q p(g) , { 1 2 ∈ × | 2 1 } there is a right action of LG LG on PQ given by × c 1 (g ,g ): (q , q ,g) (q p(g ), q p(g ),g− gg ). 1c2 c1 2 7→ 1 1 2 2 1 2

On the other hand, if we denote the universal line bundle over P(F0) by L, the left LG-action Lρ PQ ⊠ on P(F0) (viaρ ˆ0) lifts to an action of LG on L. We set = LGd LGd (L L); this is 8 × × c [2] ρ [2] Q [3] a line bundle over Q (LG LG) (P(F0) P(F0)) = P(F ) . The triviality of δ(L ) Q × × × → implies that of δ(Lρ) P(Fρ)[3]. Thus we have a bundle gerbe which we call the caloron → bundle gerbe. We explain the above construction using local data. Let U be an open cover of X which { α} locally trivializes the principal LG-bundle Q. Then the restriction of Q to U , Q U , α α → α has a lift to a principal LG-bundle Qˆ U . On the overlap U U , the two bundles α → α α ∩ β Qˆα and Qˆβ differ by a principal U(1)-bundle whose associated line bundle we denote by c LQ U U . They satisfy LQ LQ = LQ on U U U and the collection αβ → α ∩ β αβ ⊗ βγ ∼ αγ α ∩ β ∩ γ LQ is a local description of the bundle gerbe (Q, LQ). Given a unitary representation { αβ} ρ: G SU(V ), the principal LSU(V )-bundle Qρ = Q LSU(V ) lifts to a principal → α α ×LG \ ρ \ ρ LSU(V )-bundle Qα = Qˆ d LSU(V ) over U . On the overlap U U , the two lifts Qα α ×LG α α ∩ β ρ ρ Q ιρ and Q differ by a U(1)-bundle whose associated line bundle is L =(L )⊗ . β c αβ αβ c ρ ρ Let F = Qˆα d F0 Uα be the associated Fock bundles; the projectivizations P(F ) c α ×LG → α glue properly to form the bundle π : P(Fρ) X. On the overlap U U , the Fock bundles ρ → α ∩ β ρ Q ιρ ρ are related by F = (L )⊗ F . As in the construction of the index bundle gerbe, let α ∼ αβ ⊗ β ρ ρ ρ ρ ρ L be the universal bundle over P(F ). Then L = L π∗(L ). So the line bundles α α α β ⊗ ρ αβ ρ 1 ρ ρ [2] ρ ρ (L )− ⊠ L over P(F ) P(F ) P(F ) glue properly to define globally the line bundle α α α ⊂ α × α Lρ P(Fρ)[2]. This defines the same caloron bundle gerbe. The line bundles Lρ are its → α local trivializations under which the bundle gerbe is described locally by a collection of line

ρ Q ιρ bundles L =(L )⊗ over U U . αβ αβ α ∩ β

4. Comparison of the index bundle gerbe and the caloron bundle gerbe

Recall that P is a principal G-bundle over S1 X and Q is the corresponding principal × LG-bundle over X. The fibre of Q at x X is Q = Γ (S1 x ,P ) with the obvious ∈ x ×{ } right LG-action. Given a finite dimensional unitary representation ρ of G on V , there is an associated hermitian vector bundle E = P V S1 X. Consider the Hilbert space bundle ×G → × H X whose fibre H over x X is the L2-completion of the space Γ (S1 x , E). There → x ∈ ×{ } is a family of Dirac operators D on S1 coupled to E acting on the fibres of H. By Sect. 2, { x} there is a projectivized Fock bundle P(F) X and a hermitian line bundle L P(F)[2] that → → defines the index bundle gerbe. By Sect. 3.2, there is a another projectivized Fock bundle P(Fρ) X constructed by using the data from caloron correspondence and the spectral cut → at 0 and a hermitian line bundle Lρ P(Fρ)[2] that defines the caloron bundle gerbe. The → purpose of this section is to show that the two bundle gerbes are isomorphic. 9 4.1. Spectral cuts, equivalent representations and Bogoliubov transformations. In this section, we show that the projective loop group representation the projectivized Fock space P(F ), corresponding to a spectral cut λ, is equivalent to P(F ), for µ = λ. This is λ µ 6 achieved by using a Bogoliubov transformation (see also [26], Sect. 12.3). Let ψi : n Z, 1 i dim V be a basis for H. Then ψ¯i : n Z, 1 i dim V { n ∈ ≤ ≤ } { n ∈ ≤ ≤ } ¯i i + is a basis for H, where ψn is the complex conjugation of ψ n. They are chosen so that Hλ − i i is spanned by ψ : n Z, n λ, 1 i dim V and H− is spanned by ψ : n Z,n < { n ∈ ≥ ≤ ≤ } λ { n ∈ λ, 1 i dim V . Similarly, H+ is spanned by ψ¯i : n Z, n λ, 1 i dim V and ≤ ≤ } λ { n ∈ ≤ − ≤ ≤ } i H− is spanned by ψ¯ : n Z,n> λ, 1 i dim V . The Fock space F is spanned by λ { n ∈ − ≤ ≤ } λ ψi1 ...ψiN ψ¯j1 ... ψ¯jM λ , n ,...,n λ, m ,...,m > λ , where the ‘vacuum’ λ { n1 nN m1 mM | i 1 N ≥ 1 M − } | i has the properties

ψi λ =0 , n<λ and ψ¯i λ =0 , n λ . n | i n | i ≤ − i ¯j and ψm and ψn are operators act on Fλ satisfying

ψi , ψj =0= ψ¯i , ψ¯j , { m n} { m n} ψi , ψ¯j = δ δi,j . { m n} m+n,0 If µ>λ, then

µ = ψi λ , λ = ψ¯i µ . | i n | i | i n | i λ

Just as Ures(H), we have the restricted general linear group

GL (H)= U GL(H):[U, I ] is a Hilbert-Schmidt operator , res { ∈ λ } ij ij k j,k i which is also independent of λ. Define the bounded operator en on H by en (ψm)= δ ψm+n. Then id + eij GL (H). On the Fock space F we define the operators σ (eij) = : n ∈ res λ λ n m i ¯j ψmψn m :λ , where the λ-normal ordering is defined as − P ψi ψ¯j , if m λ , i ¯j m n :ψmψn :λ= ≤ ( ψ¯j ψi , if m < λ . − n m A standard computation then gives

ij kl j,k il i,l kj i,l [σλ(em), σλ(en )] = δ σλ(em+n) δ σλ(em+n)+ δj,kδ mδm+n,0 , 10− \ which shows that σλ is a representation on Fλ of the central extension glres(H) of the Lie algebra glres(H) of GLres(H). Now let µ>λ. Then one computes

ij ij i ¯j σµ(en )= σλ(en ) ψm, ψn m − { − } λ

nλµ σµ(U)= σλ(U) det(U)− for any U GL\(H). That is, the projective representations on P(F ) and P(F ) of ∈ res λ µ U (H) GL (H), and hence those of LG are equal. res ⊂ res We note that in this particular case, not only do we have an action of a central extension of GLres(H) on Fλ, but also an action of the Virasoro algebra (the central extension of the 2-dimensional conformal algebra). The calculations in this section are of course well-known to the experts in conformal field theory (see, e.g., [14] and references therein). It is an interesting question whether our construction of the caloron bundle gerbe has applications in the context of conformal field theory as well.

4.2. Isomorphism between the index and caloron bundle gerbes. In this section, we will show that the geometrically obtained caloron bundle gerbe is isomorphic (not just stably isomorphic) to the analytically defined index bundle gerbe in this context. For the notion of stable isomorphism of bundle gerbes, see [28]. Consider the vector bundle E = P V over S1 X, which itself fibers over X with circle ×G × fibres. We first show that Hilbert bundle H defined in Sect. 2 is the associated bundle of Q byρ ˜, i.e., H = Q H. Suppose X has an open covering U such that the bundle ∼ ×LG { α} Q is trivial over each U . Then P is also trivial on S1 U ; let g : S1 (U U ) G α × α αβ × α ∩ β → be the transition functions. Then the transition functionsg ˜ : U U LG of Q are αβ α ∩ β → given byg ˜ (x) = g ( , x), x U U . Under the trivialisation over U , the fibre H αβ αβ · ∈ α ∩ β α x (x U ) can be identified with H. Using the local trivialisations of P , if x U U , ∈ α ∈ α ∩ β the two identifications of Hx with H from Uα and Uβ are related byρ ˜(˜gαβ(x)) U(H). 11 ∈ This shows the result that H is associated to Q and thus we have the bundle isomorphism H = Q H. ∼ ×LG We now show that P(F) defined in Sect. 2 is an associated bundle of Q byρ ˆ0. We assume that on each U , we can choose λ spec(D ) for all x U . We then have a bundle of α α 6∈ x ∈ α

Fock spaces Fλα over Uα. Since H is an associated bundle of Q by the representationρ ˜ of LG on the typical fibre H, P(F ) is an associated bundle of Q U by the representation λα α → α ρˆ : LG P U(F ), where F is the Fock space of H with a polarization corresponding λα → λα λα to λα. Under the natural identification of P(Fλα ) and P(F0), the representationρ ˆλα is the

same asρ ˆ0. This crucial step follows from the independence of representation of LG on the cuts, as explained in Sect 4.1. Therefore, P(F) is an associated bundle of Q; in fact it is isomorphic to P(Fρ). Since the line bundles L P(F)[2] in Sect. 2 and LQ P(Fρ)[2] in Sect. 3.2 are both → → constructed from the universal line bundle, we have the following results.

Theorem 4.1. Let P S1 X be a principal G-bundle and let Q X be the corresponding → × → principal LG-bundle. Let ρ be a finite dimensional unitary representation of G on V and let E = P V . Let ρˆ be the projective representation of LG on F , the Fock space of ×G 0 0 H = L2(S1,V ) with polarization at 0. Let H be the Hilbert bundle whose fiber H at x X x ∈ is the L2-completion of Γ (S1 x , E). Then the projectivized Fock bundle P(F), constructed ×{ } from H, is isomorphic to the associated bundle Q P(F )= P(Fρ). Furthermore, the index ×ρˆ0 0 bundle gerbe of the family of Dirac operators on S1 coupled to E is isomorphic to the caloron bundle gerbe.

An alternative, less explicit argument for the same result goes as follows. For simplicity, we will assume for the rest of the section that H3(X, Z) is torsion free, i.e., the torsion subgroup of H3(X, Z) is trivial. It is well known that fibre bundles over X whose typical fibre is a projective (infinite dimensional) Hilbert space are classified up to isomorphism by their Dixmier-Douady classes in H3(X, Z). Conversely, every class in H3(X, Z) is the Dixmier-Douady class of a fibre bundles over X whose typical fibre is a projective (infinite dimensional) Hilbert space. This is essentially contained in Proposition 2.1 of [2]. Given a unitary representation ρ: G U(V ), E = P V is a hermitian vector bundle → ×G over S1 X. A connection A˜ on P induces a hermitian connection E on E, whose cur- × ∇ vature is F E. For the family of self-adjoint Dirac operators on S1 coupled to E, the index bundle gerbe (P(F), L) is constructed in Sect. 2, and its Dixmier-Douady class in H3(X, Z) 12 is represented by the 3-curvature form on X [12, 13, 23]

1 E E 2 trV (F F ), 8π 1 ∧ ZS where F E is the curvature of the induced connection E on E. This is equal to H = ι H ∇ ρ ρ according to Proposition 3.2. Therefore the Dixmier-Douady class of P(F) is equal to that of the bundle gerbe (Qρ, Lρ). On the other hand, the homomorphism LSU(V ) U (H) pulls back the basic central → res extension of Ures(H) to that of LSU(V ) (see [31], Sect. 6.7), whereas the Shale-Stinespring embedding σ : U (H) P U(F ) pulls back the basic central extension U(F ) P U(F ) 0 res → 0 0 → 0 of P U(F0) to that of Ures(H). The latter can be argued from the literature as follows. One has the commutative diagram

/ Ures(H) P U(F0) (4.1)

  / Grres(H) P(F0), where Grres(H) denotes the restricted grassmannian, which is the quotient of Ures(H) by a contractible subgroup (cf. page 115 in [31]). The vertical arrows in equation (4.1) are the projection maps, which have degree one since the fibres are contractible. Similarly, since the fibre of P U(F ) P(F ) is also contractible, the Dixmier-Douady class of the P(F ) bundle 0 → 0 0 is equal to that of the P U(F0) bundle. The bottom horizontal arrow in (4.1) is the Pl¨ucker embedding. The sentence just after equation (7.7.4) on page 116 in [31] says that the pullback of the tautological line bundle on projective Fock space P(F0) (under the Pl¨ucker embedding) is the determinant line bundle on the restricted grassmannian Grres(H). Page 115 in [31] identifies the determinant line bundle on the restricted grassmannian Grres(H) with the central extension of restricted unitary group Ures(H). Therefore the Dixmier-Douady class ρ ρ of the bundle gerbe (Q , L ) is equal to that of the induced P U(F0) bundle and that of the projective Hilbert space bundle P(Fρ). Thus, both the projective bundle P(F) on X associated to the index bundle gerbe and the projective bundle P(Fρ) on X associated to the caloron bundle gerbe have the same Dixmier-Douady class in de Rham cohomology. By our assumption on H3(X, Z) and by the classification theorem for projective Hilbert bundles over X, there is an isomorphism of projective Hilbert bundles over X, P(Fρ) = P(F). In particular, there is an induced 13∼ Fρ [2] F [2] isomorphism of fibred products P( ) ∼= P( ) . Finally, we conclude that the caloron bundle gerbe (P(Fρ), Lρ) and the index bundle gerbe (P(F), L) are isomorphic.

5. Bundle gerbes on moduli spaces associated to Riemann surfaces

In this section we apply the constructions of bundle gerbes in Sect. 2 and Sect. 3.2 to construct bundle gerbes on moduli spaces associated to Riemann surfaces whose Dixmier- Douady classes are generators of the third integral cohomology groups of these moduli spaces. We first review the construction of moduli spaces and universal bundles.

Let G be a compact, connected, semisimple real Lie group whose centre is Z(G). Let Gad be the quotient G/Z(G), which has a trivial centre. Let Σ be a compact, connected Riemann surface of genus g > 1, and let Σ˜ be its universal

covering space, with a right action of the fundamental group π1Σ. There is a left G-action on the set of homomorphisms Hom(π Σ,G) given by G h: φ Ad φ. An element 1 ∋ 7→ h ◦ φ is irreducible if the isotropy subgroup is Z(G), that is, Ad φ = φ for h G implies h ◦ ∈ h Z(G). Let Homirr(π Σ,G) be the subset of such. The quotient M = Homirr(π Σ,G)/G ∈ 1 1 is the moduli space of flat G-connections on Σ. More generally, fix an element z Z(G). Let ∈ Σ ′ be the surface Σ with a puncture. Its fundamental group π1Σ ′ is the central extension of π1Σ by Z. Let Homz(π1Σ ′,G) be the set of homomorphisms such that the generator of Z irr is mapped to z. Again there is a G-action on it and we have the subset Homz (π1Σ ′,G) of irr irreducible homomorphisms. The quotient Mz = Homz (π1Σ ′,G)/G is the moduli space of

flat G-connections on Σ ′ whose holonomy around the puncture is z. irr For any z Z(G), there is a left G-action on Σ˜ Hom (π Σ ′,G) G given by G ∈ × z 1 × ∋ h: (p,φ,g) (p, Ad φ,hg). There is also a left π Σ-action on the same space given 7→ h ◦ 1 1 by π Σ γ : (p,φ,g) (pγ− ,φ,φ(γ)g). It is easy to check that the two actions com- 1 ∋ 7→ irr mute and hence there is a left action of G π Σ on Σ˜ Hom (π Σ ′,G) G. Let × 1 × z 1 × irr U = (Σ˜ Hom (π Σ ′,G) G)/G π Σ be the quotient space and let π : U (Σ˜ × z 1 × × 1 → × irr Hom (π Σ ′,G))/G π Σ = Σ M be the projection given by π : [(p,φ,g)] [(p,φ)] = z 1 × 1 × 7→ irr ([p], [φ]), where φ Hom (π Σ ′,G), p Σ˜ and g G. Then U is a principal G - ∈ z 1 ∈ ∈ ad bundle over Σ M . To see this, we note that there is a right G-action on each fibre × z 1 π− ([p], [φ]) = [(p,φ,g)] g G given by G h: [(p,φ,g)] [(p,φ,gh)]. The isotropic { | ∈ } ∋ 7→ 1 subgroup is Z(G) because if [(p,φ,g)] = [(p,φ,gh)] = [(p, Ad −1 φ,g)], then ghg− Z(G) ghg ◦ ∈ or h Z(G) by the irreducibility of φ. The right G-action thus descends to a free action of ∈ Gad on U. 14 The bundle π : U Σ M is called the universal bundle [22] because for each [φ] M , → × z ∈ z the restriction of U to Σ [φ] is the flat G -bundle over Σ defined by the homomorphism × ad α φ Homirr(π Σ,G ), where α: G G is the quotient map. In fact, there is a ◦ ∈ 1 ad → ad 1 Gad-equivariant 1-1 map from U Σ [φ] = π− (Σ [φ]) = [(p,φ,g)] : p Σ˜ ,g G to | × × { ∈ ∈ } Σ˜ α φ Gad = [(p,g)] : p Σ˜ ,g Gad given by [(p,φ,g)] [(p,α(g))]. Moreover, for any × ◦ { ∈ ∈ } 7→ irr σ Σ, the restriction of U to σ M is isomorphic to the G -bundle Hom (π Σ ′,G) M . ∈ × z ad z 1 → z 1 irr To see this, we note that U σ Mz = π− (σ Mz)= [(p0,φ,g)] φ Homz (π1Σ ′,G),g G | × × { | ∈ ∈ } 1 for a fixed p Σ˜ such that [p ]= σ. Under the bijection [(p ,φ,g)] = [(p , Ad− φ, 1)] 0 ∈ 0 0 0 g ◦ 7→ 1 1 Adg− φ, the right G-action G h: [(p0, φ, 1)] [(p0,φ,h)] = [(p0, Adh− φ, 1)] on U σ Mz ◦ ∋ 7→ ◦ | × 1 irr corresponds to G h: φ Ad− φ on Hom (π Σ ′,G). ∋ 7→ h ◦ z 1 We assume that the moduli space Mz is closed. When G = SU(n), Mz is closed if and only if z generates the center Z(SU(n)) = Z . Then the first Pontryagin class x of AdU Σ M ∼ n → × z is a generator of H4(Σ M , Q). For G = SU(n), H2(M, Q) is generated by the slant × z product [Σ] x while H3(M , Q) is generated by [γ] x, where γ : S1 Σ are smooth loops \ z \ → 2g whose homology classes generate H1(Σ, Z) ∼= Z when g > 1 [15]. There is no torsion 3 for H (Mz, Z) if z generates Z(SU(n)) [6]. Given a Riemannian metric on Σ, the universal bundle has a canonical connection [3] whose curvature is denoted by F U Ω 2(Σ M , AdU). ∈ × z In de Rham cohomology, x is represented by 1 F U, F U . Therefore, the generators [Σ] x − 8π2 h i \ and [γ] x are represented by \

1 U U 1 U U ∗ 2 F , F and Hρ,γ = 2 (γ id) F , F , −8π Σ h i −8π 1 × h i Z ZS respectively. When G = SU(n) and z generates Z(G), we apply the constructions of the index and

caloron bundle gerbes to obtain bundle gerbes on Mz whose Dixmier-Douady invariants are the generators of the third de Rham cohomology group. Given a smooth loop γ : S1 Σ, → 1 let P = (γ id)∗U be the principal G -bundle over S M . With a finite dimensional γ × ad × z unitary representation ρ of G on V , we have a principal SU(V )-bundle P ρ S1 M ad γ → × z and hence an LSU(V )-bundle Qρ M by caloron correspondence, with a line bundle Lρ γ → z γ ρ [2] ρ over (Qγ) . The universal connection on U pulls back to Pγ and induces one on Pγ . The ρ latter determines a connection and a Higgs field for Qγ. In particular, the Dixmier-Douady ρ ρ class of bundle gerbe (Qγ , Lγ) is represented by ιρ Hρ,γ, which is one of the 2g generators of H3(M, R) if g > 1. 15 Moreover, there are two ways to construct a projective Fock space bundles over Mz of the same Dixmier-Douady class as above. First, by considering the the family of self-adjoint Dirac operators on S1 coupled to the vector bundle E = P V , we get the index bundle γ γ ×G ρ gerbe (P(Fγ), Lγ ) from Sect. 2. Second, using the LSU(V ) bundle Qγ , we have the caloron ρ ρ bundle gerbe (P(Fγ), Lγ) from Sect. 3.2. They are isomorphic by Sect. 4.2 and their Dixmier- ρ ρ Douady class is equal to that of (Qγ , Lγ). The construction of natural bundle gerbes also applies to other moduli spaces associated to Riemann surfaces, such as the Hitchin moduli space and the monopole moduli space.

6. Conclusions and outlook

In this paper, we have constructed bundle gerbes on moduli spaces M of flat G-bundles on a compact Riemann surface Σ of genus g > 1, where G is a compact connected simply- connected Lie group. These are the index bundle gerbe and the caloron bundle gerbe, which we show are isomorphic (not just stably isomorphic). We have constructed a bundle gerbe connection and curving (or B-field), and have computed the 3-curvature which represents the Dixmier-Douady class of the bundle gerbe in de Rham cohomology. The construction is such that it extends without change to other moduli spaces, such as the moduli space of principal U(n)-bundles with fixed determinant bundle and the moduli space of Higgs bundles. In Sect. 2, given the basic setup, we have constructed the index bundle gerbe, refining a construction of Segal. It remains to define a natural bundle gerbe connection and curving on it, and to compute the 3-curvature form. This has been established in the paper in a special case using Sect. 4.2, which is the case that applies to the moduli spaces considered here. In the setting of Sect. 3, let Q be a principal LG-bundle over X andρ ˆ : LG P U(F ) 0 → 0 a positive energy representation of LG. Then we can form the algebra bundle

K = Q K(F ) X Q ×LG 0 −→ with fibre the algebra of compact operators on Fock space, K(F0). The space of continuous sections of KQ,ρˆ0 vanishing at infinity, C0(X, KQ,ρˆ0 ), is a C∗-algebra, which is a continuous trace algebra with spectrum equal to X. This non-commutative algebra is only locally Morita equivalent to continuous functions on X. The operator K-theory K (C0(X, KQ,ρˆ0 )) is the • ρ twisted K-theory K•(X,H) (cf. [33]), where H is the Dixmier-Douady class of P(F ) (see [10] for a alternate description of twisted K-theory). In particular, for each of the 2g bundle 16 gerbes that we have constructed on the moduli space M, we can form the twisted K-theory, and it would be interesting to compute these. Another interesting problem is to give natural constructions of holomorphic bundle gerbes on the moduli space stable holomorphic bundles over a Riemann surface Σ. If the degree and the rank are coprime, then the 2g generators of the third cohomology are of types (2, 1) and (1, 2) for g > 1 [16]. According to section 7 in [24], there are hermitian holomorphic bundle gerbes that have these as Dixmier-Douady classes, but the challenge is to find natural constructions for them. Next we outline a construction of bundle gerbes on other moduli spaces such as the moduli space of anti-self-dual (ASD) connections on a compact four dimensional Riemannian man- ifold M such that dim(H1(M, R)) > 0. More precisely, let P denote a principal G-bundle over M, where G is a compact semisimple Lie group, and let AP− denote the space of all ASD connections on P . We will assume that A− = , which occurs under various hypotheses on P 6 ∅ P and M, cf. the introduction in [35].

If GP denotes the gauge group of P , then one has the universal principal Gad bundle

G (P A−)/G M A−/G . ad −→ × P P −→ × P P Using this, we may also construct our bundle gerbes, for instance as follows. For γ : S1 M → a generator of H1(M, Z), consider the restricted bundle

1 G (γ∗P A−)/G S A−/G , (6.1) ad −→ × P P −→ × P P which as before determines the caloron bundle gerbe over the ASD moduli space.

LG Q A−/G . ad −→ γ −→ P P One can similarly define the index bundle gerbe and show that these bundle gerbes are isomorphic to each other and that their Dixmier-Douady is class given by

3 p1 γ∗(P ) AP−)/GP H (AP−/GP , Z). 1 × ∈ ZS  Here p γ∗P A−)/G denotes the first Pontryagin class of the principal bundle in equation 1 × P P (6.1). 

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19 (P Bouwknegt) Department of Theoretical Physics, Research School of Physics and Engi- neering and Department of Mathematics, Mathematical Sciences Institute, The Australian National University, Canberra, ACT 0200, Australia E-mail address: [email protected]

(V Mathai) Department of Pure Mathematics, University of Adelaide, Adelaide, SA 5005, Australia E-mail address: [email protected]

(S Wu) Department of Mathematics, University of Hong Kong, Pokfulam Road, Hong Kong, China E-mail address: [email protected]

20