Consistent Orientation of Moduli Spaces
aa r X i v : . [ m a t h . A T ] D ec CONSISTENT ORIENTATION OF MODULI SPACES
DANIEL S. FREED, MICHAEL J. HOPKINS, AND CONSTANTIN TELEMAN
For Nigel
Abstract.
We give an a priori construction of the two-dimensional reduction of three-dimensionalquantum Chern-Simons theory. This reduction is a two-dimensional topological quantum fieldtheory and so determines to a Frobenius ring, which here is the twisted equivariant K -theory of acompact Lie group. We construct the theory via correspondence diagrams of moduli spaces, whichwe “linearize” using complex K -theory. A key point in the construction is to consistently orientthese moduli spaces to define pushforwards; the consistent orientation induces twistings of complex K -theory. The Madsen-Tillmann spectra play a crucial role. In a series of papers [FHT1, FHT2, FHT3] we develop the relationship between positive en-ergy representations of the loop group of a compact Lie group G and the twisted equivariant K -theory K τ +dim GG ( G ). Here G acts on itself by conjugation. The loop group representationsdepend on a choice of “level”, and the twisting τ is derived from the level. For all levels the maintheorem is an isomorphism of abelian groups, and for special transgressed levels it is an isomor-phism of rings : the fusion ring of the loop group and K τ +dim GG ( G ) as a ring. For G connected with π G torsionfree we prove in [FHT1, §
4] and [FHT4, §
7] that the ring K τ +dim GG ( G ) is a quotientof the representation ring of G and we calculate it explicitly. In these cases it agrees with thefusion ring of the corresponding centrally extended loop group. We also treat G = SO in [FHT4,(A.10)]. In this paper we explicate the multiplication on the twisted equivariant K -theory for anarbitrary compact Lie group G . We work purely in topology; loop groups do not appear. In fact,we construct a Frobenius ring structure on K τ +dim GG ( G ). This is best expressed in the language oftopological quantum field theory: we construct a two-dimensional TQFT over the integers in whichthe abelian group attached to the circle is K τ +dim GG ( G ).At first glance the ring structure seems apparent. The multiplication map µ : G × G → G inducesa pushforward on K -theory: the Pontrjagin product. But in K -cohomology the pushforward is thewrong-way, or umkehr , map. Thus to define it we must K -orient the map µ . Furthermore, the twist-ings must be accounted for in the orientations. Finally, to ensure associativity we must consistently K -orient maps constructed from µ by iterated composition. For connected and simply connectedgroups there is essentially a unique choice, but in general one must work more. This orientationproblem is neatly formulated in the language of topological quantum field theory. Cartesian prod-ucts of G then appear as moduli spaces of flat connections on surfaces, and the maps along whichwe push forward are restriction maps of the connections to the boundary. What is required, then,is a consistent orientation of these moduli spaces and restriction maps. The existence of consistentorientations, which we prove in Theorem 3.24, is in some sense due to the Narasimhan-Seshadri Date : October 19, 2007.The work of D.S.F. is supported by NSF grant DMS-0603964.The work of M.J.H. is supported by NSF grant DMS-0306519.The work of C.T. is supported by EPSRC GR/S06165/01. theorem which identifies moduli spaces of flat connections with complex manifolds of stable bun-dles: complex manifolds carry a canonical orientation in K -theory. Our proof, though, uses onlythe much more simple linear statement that the symbol of the de Rham complex on a surface is thecomplexification of the symbol of the Dolbeault complex. As we explain in §
1, which serves as aheuristic introduction and motivation, ‘consistent orientations on moduli spaces’ is the topologicalanalog of ‘consistent measures on spaces of fields’ in quantum field theory. The latter is what onewould like to construct in the path integral approach to quantum field theory.Our topological construction, outlined in §
3, proceeds via a universal orientation (Definition 3.7).The main observation is that the problem of consistent orientations is a bordism problem, and therelevant bordism groups are those constructed by Madsen and Tillmann [MT] in their formulation ofthe Mumford conjectures; see [MW, GMTW] for proofs and generalizations. A universal orientationinduces a level (Definition 3.23). The map from universal orientations to levels is an isomorphismfor simply connected and connected compact Lie groups G , but in general it may fail to be injective,surjective, or both. The theories we construct are parametrized by universal orientations, not bylevels. It is interesting to ask whether universal orientations also appear in related topological andconformal field theories as a refinement of the level.The two-dimensional TQFT we construct here is the dimensional reduction of three-dimensionalChern-Simons theory, refined to have base ring Z in place of C . Our construction is a priori in the sense that the axioms of TQFT—the topological invariance and gluing laws—are deduceddirectly from the definition. By contrast, rigorous constructions of many other TQFTs, such asthe Chern-Simons theory, proceed via generators and relations . Such constructions are based ongeneral theorems which tell that these generators and relations generate a TQFT: gluing laws andtopological invariance are satisfied. One can ask if there is an a priori topological construction ofChern-Simons theory using twisted K -theory. We do not know of one. In another direction we canextend TQFTs to lower dimension, so look for a theory in 0-1-2 dimensions which extends the 1-2dimensional theory constructed here. Again, we do not know if there is an a priori construction ofthat extended theory.Section 2 of this paper is an exposition of twistings and orientation, beginning on familiar groundwith densities in differential geometry. Section 4 briefly considers this TQFT for families of 1- and2-manifolds. Our purpose is to highlight an extra twist which occurs: that theory is “anomalous”.As far as we know, the problem of consistently orienting moduli spaces first arises in work ofDonaldson [D, DKr]. He works with anti-self-dual connections on a 4-manifold and uses excision inindex theory to relate all of the different moduli spaces. In both his situation and ours the modulispaces in question sit inside infinite dimensional function spaces, and the virtual tangent bundle tothe moduli space extends to a virtual bundle on the function space. Thus it suffices to orient overthe function space, and this becomes a universal problem. Presumably our methods apply to hissituation as well, but we have not worked out the details.We thank Veronique Godin, Jacob Lurie, Ib Madsen, and Ulrike Tillmann for enlightening con-versations.It is a pleasure and an honor to dedicate this paper to Nigel Hitchin. We greatly admire hismathematical taste, style, and influence. ¡Feliz cumplea˜nos y que cumplas muchos m´as! ONSISTENT ORIENTATION OF MODULI SPACES 3 Push-Pull Construction of TQFT
Quantum Field Theory
The basic structure of an n -dimensional Euclidean quantum field theory may be axiomatized sim-ply. Let BR iem n be the bordism category whose objects are closed oriented ( n − X : Y → Y is a compact oriented n -dimensional Riemann-ian manifold X together with an orientation-preserving isometry of its boundary to the disjointunion − Y ⊔ Y , where − Y is the oppositely oriented manifold. We term Y the incoming boundaryand Y the outgoing boundary. A quantum field theory is a functor from BR iem n to the categoryof Hilbert spaces and trace class maps. The functoriality encodes the gluing law; there is also asymmetric monoidal structure which encodes the behavior under disjoint unions. There are manydetails and subtleties (see [S1] in this volume, for example), but our concern is a simpler topologicalversion. Thus we replace BR iem n by the bordism category BSO n of smooth oriented manifolds andconsider orientation-preserving diffeomorphisms in place of isometries. We define an n -dimensional topological quantum field theory (TQFT) to be a functor from BSO n to the category of complexvector spaces. The functor is required to be monoidal: disjoint unions map to tensor products.The functoriality expresses the usual gluing law and the structure of the domain category BSO n encodes the topological invariance. The example of interest here has an integral structure: thecodomain is the category of abelian groups rather than complex vector spaces. The integralityreflects that the theory is a dimensional reduction; see [F] for a discussion.Physicists often employ a path integral to construct a quantum field theory. Here is a cartoonversion. To each manifold M is attached a space F M of fields and so to a bordism X : Y → Y acorrespondence diagram(1.1) F Xs t F Y F Y in which s, t are restriction maps. The important property of fields is locality: in the diagram(1.2) F X ′ ◦ Xr r ′ F Xs t F X ′ s ′ t ′ F Y F Y F Y the space of fields F X ′ ◦ X on the composition of bordisms X : Y → Y and X ′ : Y → Y is thefiber product of the maps t, s ′ . Fields are really infinite dimensional stacks —for example, in gaugetheories the gauge transformations act as morphisms of fields—and the maps and fiber productsmust be understood in that sense. D. S. FREED, M. J. HOPKINS, AND C. TELEMAN
The backdrop for the path integral is measure theory. If there exist measures µ X , µ Y on thespaces F X , F Y with appropriate gluing properties, then one can construct a quantum field theory.Namely, define the Hilbert space H Y = L ( F Y , µ Y ) and the linear map attached to a bordism X : Y → Y as the push-pull Z X = t ∗ ◦ s ∗ : H Y −→ H Y . The pushforward t ∗ is integration. Thus if f ∈ L ( F Y , µ Y ) and g ∈ L ( F Y , µ Y ), then h ¯ g, Z X ( f ) i H Y = Z F X g (cid:0) t (Φ) (cid:1) f (cid:0) s (Φ) (cid:1) dµ X (Φ) . One usually postulates an action functional S X : F X → C and a measure ˜ µ X such that µ X = e − S X ˜ µ X and the action satisfies the gluing law S X ′ ◦ X (Φ) = S X (cid:0) r (Φ) (cid:1) + S X ′ (cid:0) r ′ (Φ) (cid:1) in (1.2). These measures have not been constructed in most examples of geometric interest. Topological Construction
Our idea is to replace the infinite dimensional stack F X by a finite dimensional stack M X ⊂ F X of solutions to a first order partial differential equation and to shift from measure theory to algebraictopology. Examples of finite dimensional moduli spaces M X in supersymmetric field theory includeanti-self-dual connections in four dimensions and holomorphic maps in two dimensions. From thephysical point of view the differential equations are the BPS equations of supersymmetry; from amathematical point of view they define the minima of a calculus of variations functional. In thispaper we consider pure gauge theories. Fix a compact Lie group G and for any manifold M let F M denote the stack of G -connections on M . Define M M as the stack of flat G -connections on M . Ifwe choose a set { m i } ⊂ M of “basepoints”, one for each component of M , then M M is representedby the product of groupoids Q i (cid:2) Hom (cid:0) π ( M, m i ) , G (cid:1) //G (cid:3) . A basic property of flat connections isthe gluing law (see (1.2)). Lemma 1.3.
Suppose X : Y → Y and X ′ : Y → Y are bordisms of smooth manifolds. Then M X ′ ◦ X is the fiber product of M X t M X ′ s ′ M Y Roughly speaking, this says that given flat connections on
X, X ′ and an isomorphism of theirrestrictions to Y , one can construct a flat connection on X ′ ◦ X and every flat connection on X ′ ◦ X comes this way. ONSISTENT ORIENTATION OF MODULI SPACES 5
Replace the infinite dimensional correspondence diagram (1.1) with the finite dimensional cor-respondence diagram of flat connections:(1.4) M Xs t M Y M Y Whereas the path integral linearizes (1.1) using measure theory, we propose instead to linearize (1.4)using algebraic topology. Let E be a generalized cohomology theory. To every closed ( n − A Y = E • ( M Y ) . To a morphism X : Y → Y we would like to attach a homomorphism Z X : A Y → A Y defined asthe push-pull(1.5) Z X := t ∗ ◦ s ∗ : E • ( M Y ) −→ E • ( M Y )in E -cohomology. Whereas the path integral requires measures consistent under gluing to defineintegration t ∗ , in our topological setting we require orientations of t consistent with gluing to definepushforward t ∗ . The consistency of orientations under gluing ensures that (1.5) defines a TQFTwhich satisfies the gluing law (functoriality).This, then, is the goal of the paper: we formulate the algebro-topological home for consistentorientations and study a particular example. Namely, specialize to n = 2 and require that the1-manifolds Y and 2-manifolds X be oriented. In other words, the domain category of our TQFTis BSO . For Y = S the moduli stack of flat connections is the global quotient M Y ∼ = G//G of G by its adjoint action; the isomorphism is the holonomy of a flat connection around the circle.Take the cohomology theory E to be complex K -theory. The resulting two-dimensional TQFTon oriented manifolds is the dimensional reduction of three-dimensional Chern-Simons theory forthe group G . In this case there is a map from consistent orientations to “levels” on G ; the levelis what is usually used to describe Chern-Simons theory. A two-dimensional TQFT on orientedmanifolds determines a Frobenius ring and conversely. The Frobenius ring constructed here isthe Verlinde ring attached to the loop group of G . The abelian group A S is a twisted formof K ( G//G ) = K G ( G ) and its relation to positive energy representations of the loop group isdeveloped in [FHT1, FHT2, FHT3]. In this paper we describe a topological construction of thering structure. D. S. FREED, M. J. HOPKINS, AND C. TELEMAN
Remarks • Let X be the “pair of pants” with the two legs incoming and the single waist outgoing.Then restriction to the outgoing boundary is the map t : ( G × G ) //G → G//G induced bymultiplication µ : G × G → G . So Z X = t ∗ ◦ s ∗ , which defines the ring structure in a two-dimensional TQFT, is pushforward by multiplication on G . Therefore, we do construct thePontrjagin product on K τ +dim GG ( G )—here τ is the twisting and there is a degree shift aswell—and have implicitly used an isomorphism of twistings µ ∗ τ → τ ⊗ ⊗ τ which, sincethe TQFT guarantees an associative product, satisfies a compatibility condition for tripleproducts. This isomorphism and compatibility are embedded in our consistent orientationconstruction. • We do not use the theorem [A] which constructs a two-dimensional TQFT from a Frobeniusring. Rather, our a priori construction manifestly produces a TQFT which satisfies thegluing law, and we deduce the Frobenius ring as a derived quantity. • Three-dimensional Chern-Simons theory is defined on a bordism category of manifolds whichcarry an extra topological structure. For oriented manifolds this extra structure is describedas a trivialization of p or signature, or a certain sort of framing. (For spin manifolds it isdescribed as a string structure or, since we are in sufficiently low dimensions, an ordinaryframing.) The two-dimensional reduction constructed here factors through the bordismcategories of oriented manifolds. • The topological push-pull construction extends to families of bordisms parametrized by abase manifold S . A choice of consistent orientation determines this extension to a theoryfor families of manifolds, albeit an “anomalous” theory; see § • The pushforward t ∗ is only defined if t : M X → M Y is a representable map of stacks, i.e.,only if the fibers of t are spaces—no automorphisms allowed. This happens only if eachcomponent of X has a nonempty outgoing boundary. Therefore, the push-pull constructiononly gives a partial TQFT. We complete to a full TQFT using the nondegeneracy of acertain bi-additive form; see [FHT3, § • As mentioned earlier, a standard TQFT is defined over the ring C whereas this theory,being a dimensional reduction of a 3-dimensional theory, is defined over Z . It is possible togo further and refine the push-pull construction to obtain a theory over K , where K is the K -theory ring spectrum. See [F] for further discussion. • The theory constructed here has two tiers—it concerns 1- and 2-manifolds—so could betermed a ‘1-2 theory’. Extensions to 0-1-2 theories, which have three tiers, are of greatinterest. The general structure of such theories has been much studied recently in variousguises [MS], [C], [HL]. A theory defined down to points is completely local, and so ultimatelyhas a simpler structure than less local theories. We do not know if the push-pull constructionhere can be extended to construct a 0-1-2 theory.
ONSISTENT ORIENTATION OF MODULI SPACES 7 Orientation and Twisting
Ordinary Cohomology
The first example for a differential geometer is de Rham theory. Let M be a smooth manifoldand suppose it has a dimension equal to n . An orientation on M , which is an orientation of thetangent bundle T M , enables integration Z M : Ω nc ( M ) −→ R on forms of compact support. Absent an orientation we can integrate twisted forms, or densities.The twisting is defined as follows. For any real vector space V of dimension r let B ( V ) denote the GL r R -torsor of bases of V . There is an associated Z -graded real line o ( V ) of functions f : B ( V ) → R which satisfy f ( b · A ) = sign det A · f ( b ) for b ∈ B ( V ) , A ∈ GL r R ; the degree of o ( V ) is r . Appliedfiberwise this construction yields a flat Z -graded line bundle o ( V ) → M for a real vector bundle V → M . There is a twisted de Rham complex(2.1) 0 −→ Ω o ( V ) − r ( M ) d −−→ Ω o ( V ) − r +1 ( M ) d −−→ · · · d −−→ Ω o ( V ) ( M ) −→ o ( V )+ q ( M ) is the space of smooth sections of the ungraded vector bundle V r + q T ∗ M ⊗ o ( V ).The cohomology of (2.1) is the twisted de Rham cohomology H o ( V )+ • dR ( M ). Let o ( M ) = o ( T M ).Then integration is a map(2.2) Z M : Ω o ( M ) c ( M ) −→ R . Notice this formulation-notation works if M has several components of varying dimension: thedegree of o ( M ) is then the locally constant function dim M : M → Z .A similar construction works in integer cohomology. If π : V → M is a real vector bundle overa space M (which needn’t be a manifold) we define o ( V ) as the orientation double cover of M determined by V and introduce a Z -grading according to the rank of V . (Note rank V : M → Z is alocally constant function.) There is an o ( V )-twisted singular complex analogous to (2.1): cochainsin this complex are cochains on the double cover which change sign under the deck transformation.The equivalence class of the twisting o ( V ) is (cid:2) o ( V ) (cid:3) = (cid:0) rank V, w ( V ) (cid:1) ∈ H ( M ; Z ) × H ( M ; Z / Z ) , where w is the Stiefel-Whitney class. The relationship of the twisting to integration occurs in theThom isomorphism. The Thom class U ∈ H π ∗ o ( V )cv ( V ) lies in twisted cohomology with compactvertical support. Let B ( V ) , S ( V ) be the ball and sphere bundles relative to a metric on V . The Thom space M V is the pair (cid:0) B ( V ) , S ( V ) (cid:1) or equivalently, assuming M is a CW complex, thequotient B ( V ) /S ( V ). The composite H − o ( V )+ • ( M ) π ∗ −−−→ H − π ∗ o ( V )+ • ( V ) U −−→ H • ( M V ) D. S. FREED, M. J. HOPKINS, AND C. TELEMAN is an isomorphism—the Thom isomorphism—a generalization of the suspension isomorphism. If M is a compact manifold and i : M ֒ → S n a (Whitney) embedding with normal bundle ν → M ,then the Pontrjagin-Thom collapse is the map c : S n → M ν defined by identifying ν with a tubularneighborhood of M and sending the complement of B ( ν ) in S n to the basepoint of M ν . Integrationis then the composite(2.3) H − o ( ν )+ n ( M ) Thom −−−−→ ∼ = H n ( M ν ) c ∗ −−−→ H n ( S n ) suspension −−−−−−→ ∼ = Z . Twistings obey a Whitney sum formula: there is a natural isomorphism o ( V ⊕ V ) ∼ = −−→ o ( V ) + o ( V ) . Applied to
T M ⊕ ν = n , where n is the trivial bundle of rank n , we conclude that integration (2.3)is a map (compare (2.2)) H o ( M ) ( M ) −→ Z . More generally, if p : M → N is a proper map there is a pushforward(2.4) p ∗ : H o ( p )+ • ( M ) −→ H • ( N ) , where o ( p ) = o ( M ) − p ∗ o ( N ). K -theory This discussion applies to any multiplicative cohomology theory. The only issue is to determinethe twisting of a real vector bundle in that theory. For complex K -theory there are many possiblemodels for the twisting τ V of a vector bundle V → M . In the Donovan-Karoubi [DK] picture τ V is represented by the bundle of complex Z / Z -graded Clifford algebras defined by V . A bundleof algebras A → M of this type is considered trivial if A = End( W ) for a Z / Z -graded complexvector bundle W → M , i.e., if A is Morita equivalent to the trivial bundle of algebras M × C . Theequivalence class of τ V is(2.5) [ τ V ] = (cid:0) rank V, w ( V ) , W ( V ) (cid:1) ∈ H ( M ; Z / Z ) × H ( M ; Z / Z ) × H ( M ; Z ) . Only torsion classes in H ( M ; Z ) are realized by bundles of finite dimensional algebras, but wehave in mind a larger model which includes nontorsion classes. (Such models are developed in [AS],[FHT1], [M] among other works.) There is a Whitney sum isomorphism(2.6) τ V ⊕ V ∼ = −−→ τ V + τ V ; also to a cohomology theory defined by a module over a ring spectrum. ONSISTENT ORIENTATION OF MODULI SPACES 9 the sum of twistings is realized by the tensor product of algebras. A spin c structure on V inducesan orientation , i.e., a Morita equivalence(2.7) τ rank V ∼ = −−→ τ V . An A -twisted vector bundle is a vector bundle with an A -module structure; it represents an elementof twisted K -theory.The Whitney formula (2.6) allows us to attach a twisting to any virtual real vector bundle: set(2.8) τ − V = − τ V . Since the Thom space satisfies the stability condition X V ⊕ n ∼ = Σ n X V , where ‘Σ’ denotes suspension,there is also a Thom spectrum attached to any virtual vector bundle and a corresponding Thomisomorphism theorem. An orientation, which is an isomorphism as in (2.7), is equivalently atrivialization of the twisting attached to the reduced bundle ( V − rank V ). Remark . There are also twistings of K -theory—indeed of any cohomology theory—which donot come from vector bundles.Suppose τ is any twisting on a manifold N . We can put that extra twisting into the pushforwardin K -theory associated to a proper map p : M → N (compare (2.4)):(2.10) p ∗ : K ( τ p + p ∗ τ )+ • ( M ) −→ K τ + • ( N ) . Here τ p is the twisting τ p = τ M − p ∗ τ N of the relative tangent bundle. In the next section weencounter a situation in which τ p + p ∗ τ is trivialized, and so construct a pushforward from untwisted K -theory to twisted K -theory; see (3.31).Twistings of K • (pt) form a symmetric monoidal 2-groupoid; its classifying space Pic g K is thusan infinite loop space. The notation: Pic K is the classifying space of invertible K -modules andPic g K the subspace classifying certain “geometric” invertible K -modules including twisted formsof K -theory defined by real vector bundles. As a space there is a homotopy equivalence(2.11) Pic g K ∼ K ( Z / Z , × K ( Z / Z , × K ( Z , g K is not a prod-uct. The group of equivalence classes of twistings on M is the group of homotopy classes ofmaps [ M, Pic g K ], which as a set is the product of cohomology groups in (2.5).Let pic g K denote the spectrum whose 0-space is Pic g K and which is a Postnikov section ofthe real KO -theory spectrum: the “ Z / , Z / , , Z ” bit of the “ . . . , Z / , Z / , , Z , , , , Z , . . . ”song. Thus the 1-space B Pic g K of the spectrum pic g K is a Postnikov section of BO . Also, let ko denote the connective KO -theory spectrum. Its 0-space is the group completion of the classifyingspace of the symmetric monoidal category of real vector spaces of finite dimension [S2]. The mapwhich attaches a twisting of K • (pt) to a real vector space, say via the Clifford algebra, induces aspectrum map(2.12) τ : ko −→ pic g K. Remark . If M is a smooth stack , then its tangent space is presented as a graded vector bundle.An orientation of M is then an orientation of this graded bundle, viewed as a virtual bundle bytaking the alternating sum of the homogeneous terms. The virtual tangent bundle is classified by amap M → ko whose composition with (2.12) gives the induced twisting of complex K -theory. Weapply this in § M → ko is computedfrom the Atiyah-Singer index theorem [ASi].The Z part of Pic g K , the Eilenberg-MacLane space K ( Z , g K too—one need only add Z / Z -gradings.3. Universal Orientations and Consistent Orientations
Overview
In this section, the heart of the paper, we define universal orientations (Definition 3.7) and provethat they exist (Theorem 3.24). A universal orientation simultaneously orients the maps t in (1.4)along which we pushforward classes in twisted K -theory; see (3.33) for the precise push-pull maps inthe theory. Universal orientations form a torsor for an abelian group (3.9). A universal orientationdetermines a level (Definition 3.23), which is the quantity typically used to label theories. Themap (3.27) from universal orientations to levels is not an isomorphism in general.We begin with a closed oriented surface X . The virtual tangent space to the stack M X offlat G -connections on X is the index of a twisted de Rham complex (3.1), and we construct auniversal symbol (3.4)—whence universal index—for this operator. A trivialization of the universaltwisting (3.8) is a universal orientation, and it simultaneously orients the moduli stacks M X forall closed oriented X .For a surface X with (outgoing) boundary we must orient the restriction map t : M X → M ∂X onflat connections. It turns out that a universal orientation does this, simultaneously and coherentlyfor all X , as expressed in the isomorphism (3.30). An important step in the argument is Lemma 3.19,which uses work of Atiyah-Bott [AB] to interpret the universal symbol in terms of standard localboundary conditions for the de Rham complex. Closed surfaces
Fix a compact Lie group G with Lie algebra g . Let X be a closed oriented 2-manifold and M X themoduli stack of flat G -connections on X . A point of M X is represented by a flat connection A ona principal bundle P → X , and the tangent space to M X at A by the deformation complex(3.1) 0 −→ Ω X ( g P ) d A −−−→ Ω X ( g P ) d A −−−→ Ω X ( g P ) −→ , the de Rham complex with coefficients in the adjoint bundle associated to P . This is an ellipticcomplex. Its symbol σ satisfies the reality condition σ ( − ξ ) = σ ( ξ ) for ξ ∈ T X , since (3.1) is
ONSISTENT ORIENTATION OF MODULI SPACES 11 a complex of real differential operators [ASi]. Recall that the symbol of any complex differentialoperator lies in K ( T X ) ∼ = K ( X T X ). The reality condition gives a lift σ ∈ KR ( X iT X ), where theimaginary tangent bundle iT X carries the involution of complex conjugation [At]. Bott periodicityasserts that V ⊕ iV is canonically KR -oriented for any real vector bundle V with no degree shift.In the language of twistings of KR this means(3.2) τ ( KR ) V + τ ( KR ) iV = 0 . Therefore(3.3) KR ( X iT X ) Thom −−−−→ ∼ = KR − τ ( KR ) iTX ( X ) (3.2) −−−−→ ∼ = KR τ ( KR ) TX ( X ) (2.8) −−−−→ ∼ = KR − τ ( KR ) − TX ( X ) Thom −−−−→ ∼ = KO ( X − T X )from which we locate the symbol σ ∈ KO ( X − T X ). Note that by Atiyah duality KO ( X − T X ) ∼ = KO ( X ) and the KO -homology group is well-known to be the home of real elliptic operators.Now (3.1) is a universal operator: its symbol is constructed from the exterior algebra of T X andthe adjoint representation of G ; it does not depend on details of the manifold X . Thus it is pulledback from a universal symbol. Let V n → BSO n denote the universal oriented n -plane bundle. Theuniversal symbol lives on the bundle V → BSO × BG , and by (3.3) we identify it as an element(3.4) σ univ ∈ KO ( BSO − V ∧ BG + ) . Here BG + is the space BG with a disjoint basepoint adjoined and ‘ ∧ ’ is the smash product.Introduce the notation M T SO n = BSO − V n n for this Thom spectrum and so write σ univ ∈ KO ( M T SO ∧ BG + ) . If f : X → BSO × BG is a classifying map for T X and P , and ˜ f : X − T X → M T SO × BG → M T SO ∧ BG the induced map on Thom spectra, then σ = ˜ f ∗ σ univ . It is in this sense that (3.4) isa universal symbol. Remark . We digress to explain
M T SO n in more detail. Let Gr + n ( R N ) be the Grassmannian oforiented n -planes in R N and 0 → V n → N → Q N − n → Gr + n ( R N ) in which V n is the universal subbundle and Q N − n the universal quotient bundle. Denote the Thom space of the latter as Z N := Gr + n ( R N ) Q N − n . Then the suspension Σ Z N is the Thom space of Q N − n ⊕ → Gr + n ( R N ). But Q N − n ⊕ Q N +1 − n → Gr + n ( R N +1 ) under the natural inclusion Gr + n ( R N ) ֒ → Gr + n ( R N +1 ), and in this manner we produce a map Σ Z N → Z N +1 . Whence the spectrum M T SO n = { Z N } N ≥ . The notationidentifies M T SO n as an unstable version of the Thom spectrum M SO and also alludes to itsappearance in the work of Madsen-Tillmann [MT]. There are analogous spectra
M T O n , M T
Spin n , M T
String n , etc. If F : S N → Z N is transverse to the 0-section, then X := F − (0-section) is an n -manifold and the pullback of Q N − n − N is stably isomorphic to − T X . Thus a map S → M T SO n classifies a map M → S of relative dimension n together with a rank n bundle W → M and astable isomorphism T ( M/S ) ∼ = W . An important theorem of Madsen-Weiss [MW] and Galatius-Madsen-Tillmann-Weiss [GMTW] identifies M T SO n as a bordism theory of fiber bundles ratherthan a bordism theory of arbitrary maps.If a smooth manifold M parametrizes a family of flat G -connections on X —that is, P → M × X is a G -bundle with a partial flat connection along X —then there is a classifying map M → M X and the pullback of the stable tangent bundle of M X to M is the index of the family of ellipticcomplexes (3.1). Note that if we replace the adjoint bundle g P in (3.1) by the trivial bundle ofrank dim G then the resulting elliptic complex does not vary over M ; its index is a trivializablebundle. Hence the reduced stable tangent bundle to M X is computed by the de Rham complexcoupled to the reduced adjoint bundle ¯ g P = g P − dim G .There is a corresponding reduced universal symbol (compare (3.4))(3.6) ¯ σ univ ∈ KO ( M T SO ∧ BG ) . It induces a universal twisting in K -theory and a consistent orientation is constructed by trivializingthis twisting. Definition 3.7. A universal orientation is a null homotopy of the composition (3.8) M T SO ∧ BG ¯ σ univ −−−−→ ko τ −−→ pic g K. Two universal orientations are said to be equivalent if the null homotopies are homotopy equivalent.
The set of equivalence classes of universal orientations is a torsor for the abelian group(3.9) O ( G ) := [Σ M T SO ∧ BG, pic g K ] . We prove in Theorem 3.24 below that universal orientations exist. In fact, there is a canonical uni-versal orientation, so the torsor of universal orientations may be naturally identified with the abeliangroup (3.9). Definition 3.7 is designed to orient the moduli spaces attached to closed surfaces. Inan equivalent form it leads to the pushforward maps we need for surfaces with boundary and totwisted K -theory of moduli spaces attached to the boundary; see the discussion preceding (3.20).Return now to the family of partial G -connections on P → M × X . The bundle P → M × X isclassified by a map f : M → M T SO × BG and the Atiyah-Singer index theorem [ASi] implies thatthe index of the family of operators (3.1) is f ∗ σ univ . Therefore, a universal orientation pulls backto an orientation of the index of (3.1); c.f. Remark 2.13. It follows that a universal orientationsimultaneously orients M X for all closed oriented 2-manifolds X . i.e., a stable map from the suspension spectrum of S to MT SO n Thom bordism theories, such as
MSO , retain the information of the stable normal bundle. Madsen-Tillmanntheories, such as
MT SO n , track the stable t angent bundle, which is one more justification for the ‘T’ in the notation. ONSISTENT ORIENTATION OF MODULI SPACES 13
Surfaces with boundary
As a preliminary we observe two topological facts about
M T SO n . Lemma 3.10. (i)
M T SO ≃ S − , the desuspension of the sphere spectrum.(ii) There is a cofibration (3.11) Σ − M T SO n − b M T SO n ( BSO n ) + . Proof.
For (i) simply observe(3.12)
M T SO ≃ BSO − V ≃ pt − R ≃ S − . For (ii) begin with the cofibration built from the sphere and ball bundles of the universal bundle:(3.13) S ( V n ) + B ( V n ) + (cid:0) B ( V n ) , S ( V n ) (cid:1) , Then identify
BSO n − as the unit sphere bundle S ( V n ) and write (3.13) in terms of Thom spaces:(3.14) BSO n − BSO n BSO V n n . Here 0 is the vector bundle of rank zero. Now add − V n to each of the vector bundles in (3.14) andnote that the restriction of V n to BSO n − is V n − ⊕ − M T SO ∧ BG b M T SO ∧ BG q ¯ σ univ ( M T SO , Σ − M T SO ) ∧ BG ¯ σ ′ univ ko The top row is a cofibration. From (3.11) we can replace (
M T SO , Σ − M T SO ) with ( BSO ) + . Lemma 3.16.
Define ¯ σ ′ univ in (3.15) to be the map ( BSO ) + ∧ BG → ko induced by the trivial rep-resentation of SO smash with the reduced adjoint representation of G . Then the triangle in (3.15) commutes and the diagram gives a canonical null homotopy of the composite ¯ σ univ ◦ b .Proof. Recalling the isomorphisms in (3.3), and using the fact that the universal symbol ¯ σ univ is canonically associated to a representation of SO × G , we locate ¯ σ univ ∈ KR SO × G ( − R ) c ∼ = KR SO × G ( i R ) c . (Recall that the involution on C ∼ = R ⊕ i R is complex conjugation and thesubscript ‘c’ denotes compact support.) Similarly, ¯ σ ′ univ ∈ KR SO × G (pt) ∼ = KR SO ( C ) c . Usingthe Thom isomorphism we identify ¯ σ ′ univ as the difference of classes represented by V • C ⊗ g C and V • C ⊗ C dim G , where in both summands θ ∈ C acts as ǫ ( θ ) ⊗ id. Exterior multiplication ǫ ( θ ) isexact for θ = 0, so this does represent a class with compact support. Also, as ǫ commutes withcomplex conjugation it is Real in the sense of [At]. It remains to observe that its restriction under KR SO × G ( C ) c → KR SO × G ( i R ) c is the universal symbol ¯ σ univ of the de Rham complex coupledto the reduced adjoint bundle. For any n a point of the 0-space of the pair of spectra ( M T SO n , Σ − M T SO n − ) is representedby a map from ( B N , S N − ) into the pair of Thom spaces attached to Q N − n Q N − n Gr + n − ( R N − ) Gr + n ( R N )for N sufficiently large. A map of ( B N , S N − ) into this pair which is transverse to the 0-sectiongives a compact oriented n -manifold M with boundary embedded in ( B N , S N − ), a rank n bundle W → M equipped with a stable isomorphism T M ∼ = W , and a splitting of W (cid:12)(cid:12) ∂M as the directsum of a rank ( n −
1) bundle and a trivial line bundle. The composition with the boundary map(3.17) r : ( M T SO n , Σ − M T SO n − ) → M T SO n − is the restriction of this data to ∂M .Now let M be a smooth manifold, X a compact oriented 2-manifold with boundary, and P → M × X a principal G -bundle with partial flat connection along X . This data induces a classifyingmap M → M X and, forgetting the connection, a classifying map(3.18) f : M −→ ( M T SO , Σ − M T SO ) ∧ BG.
There are induced classifying maps M → M ∂X and˙ f : M −→ M T SO ∧ BG for the boundary data; here ˙ f = r ◦ f . View X as a bordism X : ∅ → ∂X ; later we incorporateincoming boundary components. The following key result relates the universal topology above tosurfaces with boundary. Lemma 3.19.
The map ¯ σ ′ univ ◦ f : M → ko classifies the reduced tangent bundle of the restrictionmap t : M X → M ∂X —the bundle T M X − t ∗ T M ∂X reduced to rank zero—pulled back to M .Proof. At a point A ∈ M there is a flat connection on P (cid:12)(cid:12) { A }× X . The tangent space to M X atthat point is computed by the twisted de Rham complex (3.1), so is represented by the twisted deRham cohomology H • A ( X ). Similarly, the tangent space to M ∂X at the restriction t ( A ) of A tothe boundary is H • t ( A ) ( ∂X ). From the long exact sequence of the pair ( X, ∂X ) we deduce that thedifference T M X − t ∗ T M ∂X at A is the twisted relative de Rham cohomology H • A ( X, ∂X ).Now the twisted relative de Rham cohomology is the index of the deformation complex (3.1)with relative boundary conditions [G, § ∂X . This is an example of a local elliptic boundary value problem. ONSISTENT ORIENTATION OF MODULI SPACES 15
Atiyah and Bott [AB] interpret local boundary conditions in K -theory and prove an index formula.More precisely, the triple (cid:0) B ( T X ) , ∂B ( T X ) , S ( T X ) (cid:1) leads to the exact sequence KR − (cid:0) B ( T X ) (cid:12)(cid:12) ∂X , S ( T X ) (cid:12)(cid:12) ∂X (cid:1) −→ KR (cid:0) B ( T X ) , ∂B ( T X ) (cid:1) −→ KR (cid:0) B ( T X ) , S ( T X ) (cid:1) −→ KR (cid:0) B ( T X ) (cid:12)(cid:12) ∂X , S ( T X ) (cid:12)(cid:12) ∂X (cid:1) The symbol σ of an elliptic operator lives in the third group, and Atiyah-Bott construct a liftto the second group from a local boundary condition. (The image of σ in the last group is anobstruction to the existence of local boundary conditions; the image of the first group in the secondmeasures differences of boundary conditions.) The relative boundary conditions on the twisted deRham complex are universal , so the corresponding lift of the symbol occurs in the universal setting.Recall from the proof of Lemma 3.16 the exact sequence (3.15), now extended one step to the left: KR G ( i R ) c −→ KR SO × G ( C ) c −→ KR SO × G ( i R ) c −→ KR G ( i R ) c The group G acts trivially in all cases. The Atiyah-Bott procedure applied to the relative boundaryconditions gives a lift of ¯ σ univ ∈ KR SO × G ( i R )) c to KR SO × G ( C ) c . Recall that ¯ σ ′ univ , constructedin the proof of Lemma 3.16, is also a lift of ¯ σ univ . But by periodicity we find KR G ( i R ) c ∼ = KR G ( − R ) c ∼ = KO G ( − R ) c ∼ = KO G (pt) which vanishes by [An]. Thus the lift of ¯ σ univ is unique and¯ σ ′ univ computes the relative twisted de Rham cohomology. This completes the proof. The Level
A universal orientation induces a level, which is commonly used to identify the theory. Oneobservation arising from this study is that the level is a derived quantity, and it is the universalorientation which determines the theory. We explain the relationship, and deduce the existence ofuniversal orientations, in this subsection.To begin we recast Definition 3.7 of a universal orientation in a form suited for surfaces withboundary. Consider the diagram(3.20)
M T SO ∧ BG q ¯ σ univ ( M T SO , Σ − M T SO ) ∧ BG ¯ σ ′ univ r M T SO ∧ BG − λ ko τ pic g K The top row is a cofibration, the continuation of the top row of (3.15) in the Puppe sequence.Recall that a universal orientation is a null homotopy of τ ◦ ¯ σ univ = τ ◦ ¯ σ ′ univ ◦ q . Lemma 3.21.
A universal orientation is equivalent to a map − λ in (3.20) and a homotopy from τ ◦ ¯ σ ′ univ to − λ ◦ r . The proof is immediate. In view of (3.12) and adjunction we can write λ : Σ ∞ BG → Σ pic g K as amap from the suspension spectrum of BG to the spectrum pic g K , or equivalently as a map(3.22) λ : BG −→ B Pic g K on spaces. Definition 3.23.
The map (3.22) is the level induced by a universal orientation.
There is a map K ( Z , → B Pic g K (see (2.11)) and in some important cases, for example if G isconnected and simply connected, the level factors through K ( Z , λ ∈ H ( BG ).A universal orientation is more than a level: it is a map − λ : M T SO ∧ BG → pic g K togetherwith a homotopy of − λ ◦ r and τ ◦ ¯ σ ′ univ in (3.20). Our next result proves that universal orientationsexist. Theorem 3.24.
There is a canonical universal orientation µ . The corresponding level h is thenegative of BG → BO → B Pic g K , where the first map is induced from the reduced adjoint repre-sentation ¯ g and the second is projection to a Postnikov section.Proof. Since complex vector spaces have a canonical K -theory orientation, the composite map k ko τ pic g K is null, where k is the connective K -theory spectrum. (See the text pre-ceding (2.12).) Therefore, a universal orientation is given by filling in the left dotted arrow in thediagram(3.25) M T SO ∧ BG q ( M T SO , Σ − M T SO ) ∧ BG ¯ σ ′ univ r M T SO ∧ BG − λ k ko τ pic g K and specifying a homotopy which makes the left square commute. There is a natural choice:smash the K -theory Thom class U : M T U ≃ M T SO → k with the complexified reduced adjointrepresentation ¯ g C . This is the universal rewriting of de Rham on a Riemann surface in terms ofDolbeault, at least on the symbolic level. In terms of the proof of Lemma 3.16, the map ¯ σ univ ,restricted to M T SO , is the exterior algebra complex ( V • C , ǫ ) in KR SO ( i R ) c . Write R = L R for the complex line L = C and C ∼ = R ⊗ C ∼ = L ⊕ L . Then the symbol complex at θ ∈ i R , C ǫ ( θ ) L ⊕ L ǫ ( θ ) L ⊗ L , is the realification of the complex C ǫ ( θ ) L which defines the K -theory Thom class. Tensor with the complexified reduced adjoint representa-tion ¯ g C to complete the argument. ONSISTENT ORIENTATION OF MODULI SPACES 17
To compute the level of µ we factorize τ as ko η Σ − ko pic g K , where the first map ismultiplication by η : S → S − and the second is projection to a Postnikov section. The map η fitsinto the Bott sequence k → ko → Σ − ko , and so we extend (3.25) to the left:(3.26) Σ − M T SO ∧ BG α M T SO ∧ BG qU ∧ ¯ g C ( M T SO , Σ − M T SO ) ∧ BG ¯ σ ′ univ Σ − ko Σ − ko k ko The homotopy which expresses the commutativity of the right hand square induces the map α inthis diagram, and the map − λ induced in (3.25) is the suspension of α . We claim that there isa unique α , up to homotopy, which makes the left square in (3.26) commute. For the differenceof any two choices for α is a map Σ − M T SO ∧ BG → Σ − ko , and the homotopy classes of suchmaps form the group KO ( BG ) which vanishes [An]. It is easy to find a map α as follows. Since(Lemma 3.10(i)) Σ − M T SO ≃ S − , the upper left map is the inclusion of the bottom cell of M T SO ∧ BG . The composite Σ − M T SO ∧ BG ≃ Σ − BG → k factors as Σ − BG → Σ − k → k ,where the first map is the double desuspension of ¯ g C and the second Bott periodicity. Choose α to be the double desuspension of ¯ g , the real reduced adjoint representation. This completes theproof.Since equivalence classes of universal orientations form a torsor for the group O ( G ) in (3.9), thecanonical universal orientation identifies the torsor of universal orientations with O ( G ). Notice thenatural map(3.27) ℓ : O ( G ) = [Σ M T SO ∧ BG, pic g K ] −→ [ M T SO ∧ BG, pic g K ] ∼ = [ BG, B
Pic g K ]from universal orientations to levels. If g ∈ O ( G ), then the level of µ + g is ℓ ( g ) − h . If G isconnected, simply connected, and simple, then [ BG, B
Pic g K ] ∼ = H ( BG ; Z ) ∼ = Z and h is the dualCoxeter number of G times a generator. Then g ℓ ( g ) − h is a version of the ubiquitous “adjointshift”. Remark . For any G the top homotopy group of Map(Σ M T SO , pic g K ) and of B Pic g K is π ,which is infinite cyclic. So there is a homomorphism of H ( BG ; Z ) into the domain and codomainof (3.27), and on these subspaces ℓ is an isomorphism. This means that we can change a consistentorientation by an element of H ( BG ; Z ), and the level changes by the same amount. The pushforward maps
Suppose we have chosen a universal orientation with level λ . Let X be a compact oriented 2-manifold with boundary. We can work as before with a family of flat connections on X parametrizedby a smooth manifold M , but instead for simplicity we work universally on M X . As in (3.18) fixa classifying map(3.29) f : M X −→ ( M T SO , Σ − M T SO ) ∧ BG ; then there is an induced classifying map˙ f = r ◦ f : M ∂X −→ M T SO ∧ BG for r the map (3.17). Set τ = ˙ f ∗ ( λ ). Let t : M X → M ∂X be the restriction map on flatconnections. According to Lemma 3.19 the composition τ ◦ ¯ σ ′ univ ◦ f is the reduced twisting τ t − (dim M X − t ∗ dim M ∂X ). The homotopy which expresses commutativity of the square in (3.20)gives an isomorphism(3.30) τ t − (dim M X − t ∗ dim M ∂X ) ∼ = −→ − t ∗ τ. In principle, dim M X and dim M ∂X are locally constant functions which vary over the modulispace. However, the Euler characteristic of the deformation complex (3.1) is independent of theconnection, and we easily deducedim M X − t ∗ dim M ∂X ≡ (dim G ) b ( ∂X ) (mod 2) , where b is the number of connected components. (We only track degrees in K -theory modulotwo; see (2.5).) According to the discussion in § t ∗ : K ( M X ) −→ K τ + (dim G ) b ( ∂X ) ( M ∂X ) . This is the pushforward (1.5) associated to the bordism X : Y → Y with Y = ∅ and Y = ∂X .The invariant (1.5) is then t ∗ (1).Note the special case ∂X = S . Then M S = G//G is the global quotient stack of the actionof G on G by conjugation. The codomain of (3.31) is thus K τ +dim G ( M S ) ∼ = K τ +dim G ( G//G ) = K τ +dim GG ( G ) . This is the basic space of the two-dimensional TQFT we construct; see § τ in the codomain of (3.31). This is the mechanism which was envisioned in (2.10) when we discussedtwistings in general: we have constructed a pushforward from untwisted K -theory to twisted K -theory. The universal orientation neatly accounts for the construction of a Frobenius ring structureon twisted K -theory.To treat an arbitrary bordism X : Y → Y we note that the deformation complex at a flatconnection a on a principal G -bundle Q → Y is(3.32) 0 Ω Y ( g Q ) d a Ω Y ( g Q ) 0 ONSISTENT ORIENTATION OF MODULI SPACES 19
The operator d a is skew-adjoint. Therefore, there is a canonical trivialization of the K -theory classof (3.32); e.g., a canonical isomorphism ker d a ∼ = coker d a . Suppose a classifying map f is given asin (3.29) and let ˙ f , ˙ f denote its restriction to the boundary connections on Y , Y . Set τ i = ˙ f ∗ i ( λ ).Then (3.30) and the canonical trivialization of (3.32) on the incoming boundary lead to the desiredpush-pull map(3.33) K τ + (dim G ) b ( Y ) ( M Y ) s ∗ K s ∗ τ + (dim G ) b ( Y ) ( M X ) t ∗ K τ + (dim G ) b ( Y ) ( M Y )This is the map (1.5) with the twistings induced from the universal orientation.A universal orientation induces consistent orientations on the outgoing restriction maps of bor-disms. In other words, if X : Y → Y and X ′ : Y → Y are bordisms, then the push-pull mapsderived from the diagram M X ′ ◦ Xr r ′ M Xs t M X ′ s ′ t ′ M Y M Y M Y satisfy ( t ′ r ′ ) ∗ ◦ ( sr ) ∗ = [ t ′∗ ◦ s ′∗ ] ◦ [ t ∗ ◦ s ∗ ] . This follows from Lemma 1.3 and the “Fubini property”(3.34) ( t ′ r ′ ) ∗ = t ′∗ r ′∗ of pushforward. The orientation of t induces orientations of r ′ and t ′ r ′ , since the diamond is a fiberproduct. At stake in (3.34) is the consistency of the orientations, which is ensured by the use of auniversal orientation. The details of this argument will be given on another occasion.One caveat: since M X , M ∂X are stacks we can only pushforward along representable maps, andthis forces every component of X to have a nonempty outgoing boundary. As mentioned at theend of §
1, the partial topological quantum field theory obtained from the push-pull constructionextends to a full theory using the invertibility of the (co)pairing attached to the cylinder [FHT3, § Families of Surfaces, Twistings, and Anomalies
We begin with a general discussion of topological quantum field theories for families. Let F bean n -dimensional TQFT in the most naive sense. Thus F assigns a finite dimensional complex The main point is that consistent orientations themselves form an invertible topological field theory, and thesefield theories factor through the group completion of bordism, i.e., the Madsen-Tillmann space. vector space F ( Y ) to a closed oriented ( n − Y and a linear map F ( X ) : F ( Y ) → F ( Y )to a bordism X : Y → Y . In particular, F ( X ) ∈ C if X is closed. Suppose that Y → S is a fiberbundle with fiber a closed oriented ( n − F ( Y /S ) → S . If γ : [0 , → S is a path, then γ ∗ Y → [0 , Y γ (0) to the fiber Y γ (1) . The topological invariance of F shows that F ( γ ∗ Y ) : F ( Y γ (0) ) → F ( Y γ (1) ) is unchanged by a homotopy of γ , and so F ( Y /S ) → S carries anatural flat connection. Then a family of bordisms X → S from Y → S to Y → S produces asection F ( X /S ) of Hom (cid:0) F ( Y /S ) , F ( Y /S ) (cid:1) → S ; the topological invariance and gluing law of theTQFT imply that this section is flat. In other words, F ( X /S ) ∈ H (cid:0) S ; Hom (cid:0) F ( Y /S ) , F ( Y /S ) (cid:1)(cid:1) .It is natural, then, to postulate that a TQFT in families gives more, namely classes of all degrees:(4.1) F ( X /S ) ∈ H • (cid:0) S ; Hom (cid:0) F ( Y /S ) , F ( Y /S ) (cid:1)(cid:1) . These classes are required to satisfy gluing laws and topological invariance as well as naturalityunder base change.The idea of a TQFT in families—at least in two dimensions—was introduced in the mid 90s. Intwo dimensions it is often formulated in a holomorphic language (e.g. [KM]), and classes are requiredto extend to the Deligne-Mumford compactification of the moduli space of Riemann surfaces.Our push-pull construction works for families of surfaces—with a twist. The purpose of thissection is to alert the reader to the twist. Suppose X → S is a family of bordisms from Y → S to Y → S , where Y i → S are fiber bundlesof oriented 1-manifolds. Then the moduli stacks of flat connections form a correspondence diagramover S : M X /Ss t M Y /S π M Y /Sπ S The push-pull constructs a map from twisted K ( M Y /S ) to twisted K ( M Y /S ). We can also worklocally over S : the K -theory of the fibers of π i form bundles of spectra over S and the push-pullgives a map of these spectra. But for our purposes the global push-pull suffices. This constructionis a variation of (4.1): we use K -theory rather than cohomology.The discussion of § Y = ∅ so that the boundary ∂ X = Y is entirely outgoing.Fix A ∈ M X s a flat connection on a principal G -bundle P → X s . Then the KO -theory class ofthe de Rham complex coupled to the reduced adjoint bundle ¯ g P = g P − dim G computes the dif-ference T A M X s − (dim G ) H • ( X s ), where H • ( X s ) is the real cohomology of X s viewed as a class in KO -theory. In § H • ( X s ) as a trivial vector space (there S = pt), but now H • ( X s ) varieswith s ∈ S and so can give rise to a nontrivial twisting. More precisely, H • ( X s ) is the fiber at s ∈ S We thank Veronique Godin for the perspicacious sign question which prompted this exposition.
ONSISTENT ORIENTATION OF MODULI SPACES 21 of a flat vector bundle H • ( X /S ) → S . Let τ X /S denote the twisting of complex K -theory attachedto this bundle. This twisting replaces the degree shift in (3.30) and the pushforward (3.31) ismodified to include that extra twist:(4.2) t ∗ : K ( M X /S ) −→ K τ + (dim G ) π ∗ τ X /S ( M Y /S ) . The degree shift is now incorporated into the twist τ X /S , and there may be nontrivial contributionsto the twist from w and W of H • ( X /S ) → S as well. (The degree shift and twistings vanishcanonically if dim G is even.) Example 4.3.
Consider the disjoint union X of two 2-disks. The boundary circles are outgoing,as above. Suppose that dim G is odd. For a single disk the pushforward t ∗ (1) in (3.31) lands in K τ +1 G ( G ) and is the unit in the Verlinde ring. Thus for the disjoint union of two disks, t ∗ (1) is theimage of ⊗ under the external product K τ +1 G ( G ) ⊗ K τ +1 G ( G ) → K ( τ,τ ) G × G ( G × G ). Now considerthe family X → S with fiber X and base S = S in which the monodromy exchanges the two disks.The flat bundle H • ( X /S ) → S has rank 2 and nontrivial w . According to (4.2), then, t ∗ (1) forthe family lives in the twisted group K ( τ,τ )+ π ∗ τ X /S ( M ∂ X /S ). On each fiber of π : M ∂ X /S → S werecover the class ⊗ above. But upon circling the loop S = S this class changes sign in the π ∗ τ X /S -twisted K -group. Said differently, the diffeomorphism which exchanges the disks acts by asign on ⊗ . Of course, one might predict this from the sign rule in graded algebra: the Verlindering K τ +1 G ( G ) is in odd degree, so upon exchanging the factors of ⊗ one picks up a sign. Itshows up here as an extra twisting.This extra twisting is a topological analog of what is usually called an anomaly in quantum fieldtheory. In an anomalous theory in n dimensions the partition function on a closed n -manifold,rather than being a complex-valued function on a space of fields, is a section of a complex linebundle over that space of fields. Furthermore, there is a gerbe over the space of fields on a closed( n − S plays the role of the space of fields and for a family of closed n -manifolds the partitionfunction in a non-anomalous theory is an element of H • ( S ; R ). An anomalous theory would assigna flat complex line bundle L → S to the family, and the partition function would live in the twistedcohomology H • ( S ; L ) = H L + • ( S ). In the 2-dimensional TQFT we construct using push-pull on K -theory, the extra K -theory twist τ X /S is the anomaly; see (4.2). Notice that there is no gerbeattached to a family of 1-manifolds (better: it is canonically trivial). We remark that the anomalyis itself a particular example of an invertible topological quantum field theory. References [A] Lowell Abrams,
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