The Atiyah bundle and connections on a principal bundle

Proc. Indian Acad. Sci. (Math. Sci.) Vol. 120, No. 3, June 2010, pp. 299 316. Indian Academy of Sciences The Atiyah bundle and connections on a principal bundle INDRANIL BISWAS School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400 005, India E-mail: indranil@math.tifr.res.in MS received 17 December 2008; revised 26 February 2009 Abstract. M be a C manifold and G a Lie a group. E G be a C principal G-bundle over M. There is a fiber bundle C(E G ) over M whose smooth sections correspond to the connections on E G. The pull back of E G to C(E G ) has a tautological connection. We investigate the curvature of this tautological connection. Keywords. Principal bundle; connection; Atiyah bundle. 1. Introduction Fix a Lie group G. Its Lie algebra will be denoted by g. M be a connected C manifold. E G M be a C principal G-bundle. The adjoint vector bundle, which is the one associated to E G for the adjoint action of G on g, is denoted by ad(e G ). The sheaf of G-invariant smooth vector fields on E G defines a C vector bundle over M of rank dim M + dim G. This vector bundle is known as the Atiyah bundle, and is denoted by At(E G ). The Atiyah bundle fits in a short exact sequence of vector bundles over M, 0 ad(e G ) At(E G ) TM 0, which is a known as the Atiyah exact sequence. A connection on the principal G-bundle E G is a C splitting of the Atiyah exact sequence. The sheaf of splittings of the Atiyah exact sequence defines a fiber bundle δ: C(E G ) M (1.1) (see 3). The pulled back principal G-bundle δ E G C(E G ) has a tautological connection (see 3). This connection is denoted by D 0. The curvature of D 0 is a two-form on C(E G ) with values in δ ad(e G ). We investigate the curvature of D 0. E G B G be the universal principal G-bundle. C(E G ) B G be the fiber bundle constructed as in (1.1) for the universal principal G-bundle. In a work in progress, we hope to show that the universal G-connection can be realized as a fiber bundle over C(E G ). Turning this around, we hope to get an alternative construction of the universal G-connection. Also, this approach may yield a better understanding of the universal G-connection. 299

300 Indranil Biswas 2. The Atiyah bundle As before, G is a Lie group with Lie algebra g, and M is a connected C manifold. p: E G M (2.1) be a C principal G-bundle over M. We recall that this means that E G is a C manifold equipped with a C action ψ: E G G E G (2.2) of G, satisfying the following two conditions: p ψ coincides with p p 1, where p 1 is the natural projection of E G G to E G, and the map to the fiber product over M, (ψ, p 1 ): E G G E G M E G is a diffeomorphism. (We recall that E G M E G is the submanifold of E G E G consisting of all points (x, y) with p(x) = p(y).) (See [4, 5].) FC (M) denote the sheaf of all locally defined C functions on M. In other words, for any nonempty open subset U M, FC (U) = C (U) is the space of all C functions on U. Using E G we will define a sheaf on M which is locally free over FC (M). For any nonempty open subset U M, let A(E G )(U) := Ɣ(p 1 (U), Tp 1 (U)) G be the space of all G-invariant vector fields on p 1 (U) (the action of G on p 1 (U) is given by ψ in (2.2)). Note that C (U) acts on A(E G )(U). The action of any smooth function f on U sends a vector field θ to (f p) θ, where p is the projection in (2.1). Take any point x M. U M be an open subset containing x such that the restriction E G U of E G to U is trivializable, and the tangent bundle TU is trivializable. Fix a trivialization of E G U by choosing a C isomorphism of principal G-bundles β: U G E G U. (2.3) Also, fix a trivialization of the tangent bundle of U γ : TU U R d, (2.4) where d is the dimension of M. θ: p 1 (U) Tp 1 (U)

Atiyah bundle of a principal bundle 301 be any G invariant C vector field on p 1 (U). Using the trivialization β in (2.3), the vector field θ gives a vector field on U G. θ: U G T(U G) (2.5) be the vector field defined by θ. Note that using the left invariant vector fields, we have TG= G g, where g is the Lie algebra of G. Therefore, using the trivialization γ in (2.4), we have T(U G) = (U G) (g R d ). Consequently, the restriction of θ (see (2.5)) to U {e} U G, where e G is the identity element, defines a C function θ : U g R d. Conversely, given any C function η 0 : U g R d, using γ (see (2.4)), the function η 0 defines a smooth G-invariant vector field on U G. Now, using the trivialization β in (2.3), this G-invariant vector field on U G produces a G-invariant vector field on E G U. In other words, we get a bijective linear map between A(E G )(U) (the space of smooth G-invariant vector fields on E G U ) and Ɣ(U, g R d ) (the space of smooth maps from U to g R d ). This map A(E G )(U) Ɣ(U, g R d ) clearly commutes with the actions of C (U) on A(E G )(U) and Ɣ(U, g R d ) defined by multiplication. Therefore, the sheaf A(E G ) is locally free over FC (M) of rank dim(g R d ). Hence A(E G ) defines a C vector bundle over M of rank dim(g R d ). At(E G ) M (2.6) be the vector bundle of rank dim(g R d ) defined by the sheaf A(E G ). The vector bundle At(E G ) in (2.6) is known as the Atiyah bundle for E G (see [1]). There is a natural homomorphism of vector bundles μ: p At(E G ) TE G (2.7) which we will describe. Take any point x M. Fix a point y p 1 (x) E G. The homomorphism μ(y) sends a G-invariant vector field ξ, defined around p 1 (x) E G, to the evaluation ξ(y) T y E G of ξ at y. It is easy to see that μ is a C isomorphism of vector bundles.

302 Indranil Biswas ad(e G ) M (2.8) be the adjoint vector bundle of E G. We recall that ad(e G ) is the quotient of E G g constructed using the adjoint action of G on its own Lie algebra g. More precisely, two points (z, v) and (z,v ) of E G g are identified in ad(e G ) if there is some g 0 G such that z = zg 0 and v = Ad(g 1 0 )(v) (we recall that Ad(g 1 0 )(v) is the Lie algebra automorphism of g corresponding to the automorphism of G defined by ζ g 1 0 ζg 0). Since Ad(h) is a Lie algebra automorphism of g for all h G, each fiber of ad(e G ) is a Lie algebra isomorphic to g. Lemma 2.1. The vector bundle ad(e G ) M is identified with the subbundle of the Atiyah bundle At(E G ) defined by the sheaf of G-invariant vector fields on E G that lie in the kernel of the differential dp: TE G p TM (2.9) of the projection p in (2.1). Proof. Take any point (z, v) E G g. φ(z,v): p 1 (p(z)) Tp 1 (p(z)) be the smooth vector field on the fiber p 1 (p(z)) E G that sends any z = zg p 1 (p(z)), where g G, to the element in T z p 1 (p(z)) given by the curve R E G, based at the point z, defined by t z exp(tv)g. It is straightforward to check that for any g 0 G, the two vector fields φ(z,v) and φ(zg 0, Ad(g 1 0 )(v)) on p 1 (p(z)) coincide. Consequently, for any x M, we get a homomorphism φ(x):ad(e G ) x Ɣ(p 1 (x), Tp 1 (x)) G Ɣ(p 1 (x), (T E G ) p 1 (x) )G from the fiber ad(e G ) x of ad(e G ) over x to the space of all G-invariant vector fields on the fiber p 1 (x). This homomorphism φ(x) is clearly injective. Consequently, we get a C injective homomorphism of vector bundles defined by φ:ad(e G ) At(E G ) x φ(x) Ɣ(p 1 (x), (T E G ) p 1 (x) )G = At(E G ) x. Clearly, we have dp φ = 0. Also, rank(ad(e G )) = dim g = (dim M + dim g) dim M = rank(at(e G )) rank(t M). Hence φ identifies ad(e G ) with the kernel of the differential dp in the statement of the lemma. This completes the proof of the lemma.

Atiyah bundle of a principal bundle 303 Using Lemma 2.1, we have a short exact sequence of vector bundles over M, 0 ad(e G ) ι 0 η At(E G ) TM 0, (2.10) where the projection η is given by dp in (2.9). This exact sequence of vector bundles is known as the Atiyah exact sequence for E G. A connection on E G is a C splitting of the Atiyah exact sequence for E G. In other words, a connection on E G is a C homomorphism of vector bundles D: TM At(E G ) (2.11) such that η D = Id TM, where η is the projection in (2.10). D: TM At(E G ) (2.12) be a homomorphism defining a connection on E G. Consider the composition homomorphism p TM D p μ At(E G ) TE G, where μ is the isomorphism in (2.7). Its image H(D) := (μ D)(p TM) TE G (2.13) is known as the horizontal subbundle of TE G for the connection D. Since μ is an isomorphism, and the splitting homomorphism D in (2.12) is uniquely determined by its image D(T M) At(E G ), it follows that the horizontal subbundle H(D) determines the connection D uniquely. The quotient bundle Q := TE G /H(D) E G is the trivial vector bundle E G g. The identification of Q with E G g is given by the action of G on E G. Note that the projection to Q from the kernel of dp in (2.9) is an isomorphism. The natural projection TE G TE G /H(D) = Q = E G g (2.14) defines a g-valued smooth one-form on E G. This g-valued one-form on E G determines D uniquely because it determines H(D) uniquely. 2.1 The curvature s and t be two G-invariant smooth vector fields on p 1 (U), where U M is an open subset, and p is the projection in (2.1). Then the Lie bracket [s, t] is also a G-invariant vector field on p 1 (U). Consequently, the locally defined smooth sections of the Atiyah bundle At(E G ) are equipped with the Lie bracket operation. The Lie bracket of the locally defined sections s 1 and t 1 of At(E G ) will be denoted by [s 1,t 1 ]. D: TM At(E G ) be a homomorphism defining a connection on E G. The curvature of D, which we will denote by K(D), measures the failure of D to be Lie bracket preserving.

304 Indranil Biswas More precisely, let v 1 and v 2 be two smooth vector fields on some open subset U M. Therefore, D(v 1 ) and D(v 2 ) are G-invariant smooth vector fields on p 1 (U). Now consider D(v 1,v 2 ) := [D(v 1 ), D(v 2 )] D([v 1,v 2 ]), (2.15) which is also a G-invariant vector field on p 1 (U). Therefore, D(v 1,v 2 ) is a smooth section of At(E G ) over U. D (v 1,v 2 ): U At(E G ) U (2.16) be the section of At(E G ) over U given by D(v 1,v 2 ) in (2.15). Consider the projection η in (2.10). Note that for any two smooth sections s and t of At(E G ) over U, the equality η([s, t]) = [η(s), η(t)] holds. Therefore, for the section D (v 1,v 2 ) in (2.16), we have η( D (v 1,v 2 )) = η([d(v 1 ), D(v 2 )]) η(d([v 1,v 2 ])) = [η(d(v 1 )), η(d(v 2 ))] η(d([v 1,v 2 ])) = [v 1,v 2 ] [v 1,v 2 ] = 0 (2.17) because η D = Id TM. Using the Atiyah exact sequence (see (2.10)), from (2.17) we conclude that D (v 1,v 2 ) is a section of ad(e G ) over U. D (v 1,v 2 ): U ad(e G ) U (2.18) be the smooth section given by D (v 1,v 2 ). f be any smooth function defined on U. Wehave [D(f v 1 ), D(v 2 )] D([fv 1,v 2 ]) = [fd(v 1 ), D(v 2 )] D(f [v 1,v 2 ] v 2 (f ) v 1 ) = f [D(v 1 ), D(v 2 )] η D(v 2 )(f ) D(v 1 ) f D([v 1,v 2 ]) + v 2 (f ) D(v 1 ). (2.19) Since η D = Id TM, the identity in (2.19) implies that D (f v 1,v 2 ) = f D (v 1,v 2 ), where D is constructed in (2.18). Also, note that D(v 1,v 2 ) = D(v 2,v 1 ), where D is constructed in (2.15). Hence we have D (v 1,v 2 ) = D (v 2,v 1 ). Consequently, D defines a smooth section ( 2 K(D): M T M) ad(e G ), (2.20) which is called the curvature of D. Remark 2.2. We note that K(D) = 0 if and only if D takes the Lie bracket operation on the sections of TM to the bracket operation on the sections of At(E G ) (see [1, 3].)

Atiyah bundle of a principal bundle 305 3. Sheaf of connections Tensoring the Atiyah exact sequence (see (2.10)) with the cotangent bundle T M M we get the short exact sequence of vector bundles 0 ad(e G ) T M At(E G ) T M η Id T M TM T M = End(T M) 0 (3.1) over M. Id TM denote the identity automorphism of TM. C(E G ) := (η Id T M) 1 (Id TM ) At(E G ) T M M (3.2) be the fiber bundle over M, where η Id T M is the surjective homomorphism in (3.1) (see also [2]). We recall that a connection on E G is a splitting of the Atiyah exact sequence. A homomorphism D as in (2.11) with η D = Id TM gives a smooth section of the fiber bundle C(E G ) M in (3.2). Conversely, any smooth section of C(E G ) gives a homomorphism D as in (2.11) with η D = Id TM. Therefore, we have the following lemma: Lemma 3.1. The space of connections on E G is in bijective correspondence with the space of smooth sections of the fiber bundle constructed in (3.2). δ: C(E G ) M Consider the projection δ in Lemma 3.1. We will show that the principal G-bundle δ E G C(E G ) has a tautological connection. T C(E G ) be the tangent bundle of the manifold C(E G ). T δ T C(E G ) (3.3) be the relative tangent bundle for the projection δ in Lemma 3.1. In other words, T δ is the kernel of the differential dδ: T C(E G ) δ TM (3.4) of the smooth map δ. ι: T δ T C(E G ) (3.5) be the inclusion map. At(δ E G ) C(E G ) be the Atiyah bundle for the principal G-bundle δ E G over C(E G ).

306 Indranil Biswas PROPOSITION 3.2 There is a natural short exact sequence of vector bundles over C(E G ). 0 T δ At(δ E G ) δ At(E G ) 0 Proof. We recall that the total space of the pull back δ E G coincides with the submanifold of C(E G ) E G consisting of all points (y 1,y 2 ) such that δ(y 1 ) = p(y 2 ), where p is the projection in (2.1). In other words, δ E G is the fiber product C(E G ) M E G. Therefore, we have a commutative diagram δ φ 0 E G E G q p C(E G ) δ M (3.6) which is Cartesian. dφ 0 : Tδ E G φ 0 TE G (3.7) be the differential of the projection φ 0 in (3.6). Since δ is a submersion, the map φ 0 is also a submersion. Hence the kernel W := kernel(dφ 0 ) Tδ E G (3.8) is a C subbundle of the tangent bundle Tδ E G. The rank of W is dim M + dim G. dq: Tδ E G q T C(E G ) (3.9) be the differential of the projection q in (3.6). Consider the restriction of dq to the subbundle W defined in (3.8). It is easy to see that the image of the homomorphism dq W : W q T C(E G ) coincides with the subbundle q T δ q T C(E G ), where T δ is the relative tangent bundle defined in (3.3). Furthermore, the homomorphism q := dq W : W q T δ (3.10) is an isomorphism. s: U T δ be a smooth section of T δ defined over some open subset U C(E G ). q s: q 1 (U) q T δ be the pull back of the section s, where q is the projection in (3.6). q 1 q s: q 1 (U) W (3.11)

Atiyah bundle of a principal bundle 307 be the section of W, where q is the isomorphism in (3.10). The section q 1 q s in (3.11) is clearly left invariant by the action of G on the principal G-bundle δ E G. Consequently, the isomorphism q defines an injective homomorphism of vector bundles f 0 : T δ At(δ E G ) (3.12) that sends any section s to q 1 q s. On the other hand, the differential dφ 0 in (3.7) induces a surjective homomorphism of vector bundles g:at(δ E G ) δ At(E G ) (3.13) by sending a G-invariant vector field on δ E G to its image by the homomorphism dφ 0. The kernel of the homomorphism g in (3.13) evidently coincides with the image of the homomorphism f 0 in (3.12). Indeed, this follows immediately from the construction of f 0. Therefore, we get a short exact sequence of vector bundles f 0 0 T δ At(δ g E G ) δ At(E G ) 0 over C(E G ). This completes the proof of the proposition. ad(δ E G ) C(E G ) be the adjoint vector bundle of the principal G-bundle δ E G C(E G ); see (2.8) for its definition. Clearly, we have ad(δ E G ) = δ ad(e G ), where ad(e G ) M is the adjoint vector bundle of E G. Furthermore, we have the following commutative diagram of homomorphisms of vector bundles on C(E G ): 0 0 T δ = T δ ι f 0 0 ad(δ E G ) At(δ E G ) T C(E G ) 0, (3.14) g dδ δ ι 0 0 δ ad(e G ) δ At(E G ) δ TM 0 0 0 0 where the top short exact sequence is the Atiyah exact sequence for the principal G-bundle δ E G, the bottom exact sequence is the pull back of the Atiyah exact sequence constructed in (2.10), the projection g is constructed in (3.13), the homomorphism f 0 is constructed in (3.12), the homomorphism ι is defined in (3.5), and dδ is defined in (3.4). We note that the vertical sequences in (3.14) are also exact. Now, from the construction of C(E G ) (see (3.2)) it follows that we have a tautological C homomorphism of vector bundles I δ η β 0 : δ TM δ At(E G ) (3.15)

308 Indranil Biswas such that (δ η) β 0 = Id δ TM, where δ η is the homomorphism in (3.14). To explain the homomorphism β 0, take any point y C(E G ). Recall that y corresponds to a homomorphism λ y : T δ(y) M At(E G ) δ(y) (3.16) such that η(δ(y)) λ y = Id Tδ(y) M, where δ is the projection in Lemma 3.1, the homomorphism η(δ(y)):at(e G ) δ(y) T δ(y) M is the projection in (2.10) with At(E G ) δ(y) being the fiber of At(E G ) over the point δ(y). The homomorphism β 0 in (3.15) is defined by the condition that for each point y C(E G ), the restriction of β 0 to the fiber (δ TM) y coincides with the homomorphism λ y in (3.16). W 0 := g 1 (β 0 (δ TM)) At(δ E G ) (3.17) be the subbundle of At(δ E G ), where g is the projection in (3.14), and β 0 is the homomorphism in (3.15). Consider the homomorphism I in (3.14). Using Proposition 3.2 and the diagram (3.14) it follows that the restriction I 0 := I W0 : W 0 T C(E G ) (3.18) is an isomorphism, where W 0 is constructed in (3.17). Therefore, there is an unique C homomorphism of vector bundles such that S 0 : T C(E G ) At(δ E G ) (3.19) S 0 (T C(E G )) = W 0 At(δ E G ), and I S 0 = Id T C(EG ), where I is the homomorphism in (3.14). Consequently, the homomorphism S 0 in (3.19) gives a C splitting of the Atiyah exact sequence for δ E G (which is the top exact sequence in (3.14)). Therefore, S 0 defines a connection on the principal G-bundle δ E G. The connection on the principal G-bundle δ E G defined by S 0 will be denoted by D 0. We will investigate the above connection D 0 on δ E G. First we will describe the horizontal subbundle of Tδ E G for the connection D 0. Take any point z δ E G. (3.20) y = q(z) C(E G ) be the image of z, where q is the projection in (3.6). Consider the differential dφ 0 defined in (3.7). dφ 0 (z): T z δ E G T φ0 (z)e G

Atiyah bundle of a principal bundle 309 be its restriction over the point z in (3.20). The horizontal subspace of T z δ E G for the connection D 0 on δ E G coincides with the inverse image (dφ 0 (z)) 1 (μ(φ 0 (z))(λ y (T δ(y) M))) T z δ E G, where λ y is constructed in (3.16), and μ(φ 0 (z)):at(e G ) δ(y) T φ0 (z)e G is the isomorphism in (2.7). In Lemma 3.1 we noted that the connections on E G are in bijective correspondence with the sections of C(E G ). Take any smooth section χ: M C(E G ) (3.21) of the fiber bundle C(E G ) M. D χ be the corresponding connection on the principal G-bundle E G. We note that χ δ E G = E G because δ χ = Id M. Lemma 3.3. The connection D χ on E G coincides with the pulled back connection χ D 0 on the principal G-bundle χ δ E G = E G. Proof. This follows immediately from the construction of the connection D 0 on δ E G. 4. The curvature of D 0 K(D 0 ): C(E G ) ( 2 T C(E G )) δ ad(e G ) (4.1) be the curvature of the connection D 0 on the principal G-bundle δ E G. (Note that δ ad(e G ) = ad(δ E G ).) Take any point x M. ( K(D 0 ) x : Z x := δ 1 2 (x) T Z x) δ ad(e G ) (4.2) be the δ ad(e G )-valued two form on Z x obtained by restricting the two-form K(D 0 ) in (4.1). Lemma 4.1. The two-form K(D 0 ) x in (4.2) vanishes identically. Proof. Consider the restriction (δ E G ) x := (δ E G ) δ 1 (x) δ 1 (x) of the principal G-bundle δ E G to the submanifold δ 1 (x) C(E G ). The connection D 0 on δ E G defines a connection on (δ E G ) x ; this connection on (δ E G ) x will be denoted by D x 0. The curvature of Dx 0 is K(D 0) x defined in (4.2). From the construction of D 0 it follows that the horizontal subbundle of T(δ E G ) x for the connection D x 0 on (δ E G ) x coincides with the subbundle W (δ E G ) x, where W is constructed in (3.8). The subbundle W Tδ E G is clearly integrable. Therefore, the curvature K(D 0 ) x vanishes; see Remark 2.2. This completes the proof of the lemma.

310 Indranil Biswas As in (3.21), let χ: M C(E G ) be a section of the fiber bundle C(E G ) over M. As before, let D χ be the corresponding connection on the principal G-bundle E G. Then Lemma 3.3 has the following corollary: COROLLARY 4.2 The curvature of the connection D χ coincides with the pull back χ K(D 0 ). For any g 0 G, let Ad(g 0 ): G G be the automorphism defined by h g 0 hg 1 0. This defines the adjoint action of G on itself. ϕ:ad(e G ) := E G (G) M (4.3) be the fiber bundle associated to E G for the adjoint action of G on itself. Therefore, Ad(E G ) is a quotient of E G G, and two points (z, h) and (z,h ) of E G G are identified in Ad(E G ) if there is some g 0 G such that z = zg 0 and h = Ad(g 1 0 )(h). Since the adjoint action of G on itself preserves the group structure of G, each fiber of Ad(E G ) is a Lie group isomorphic to G. It is easy to see that the Lie algebra bundle over M corresponding to Ad(E G ) coincides with the adjoint vector bundle ad(e G ). Take any point x M. The fiber Ad(E G ) x of Ad(E G ) over x acts on the fiber (E G ) x of E G over x. To explain this action, take any point (z 0,g 0 ) (E G ) x G. f (z0,g 0 ): (E G ) x (E G ) x be the map defined by z 0 h z 0 g 0 h, h G. Note that f (z0,g 0 ) = f (z0 h 0,h 1 0 g 0h 0 ) for all h 0 G. Therefore, the map f (z0,g 0 ) depends only on the image of (z 0,g 0 ) in Ad(E G ) x. In this way we get an action of the group Ad(E G ) x on (E G ) x. Note that the diffeomorphism f (z0,g 0 ) of (E G ) x commutes with the action of G on (E G ) x. Consequently, a smooth section of Ad(E G ) gives a diffeomorphism of E G which commutes with the projection p in (2.1). Furthermore, such a diffeomorphism of E G commutes with the action of G on E G. An automorphism of the principal G-bundle E G is a diffeomorphism such that f : E G E G p f = p, where p is the projection in (2.1), and f commutes with the action of G on E G. Note that any diffeomorphism of G that commutes with all the right translations of G must be a left translation. Using this it follows that all the automorphisms of the principal G-bundle E G are given by the smooth sections of Ad(E G ).

Atiyah bundle of a principal bundle 311 β: M Ad(E G ) (4.4) be a smooth section. β 1 : M Ad(E G ) (4.5) be the section defined by y β(y) 1. As we noted above, the section β gives an automorphism β : E G E G (4.6) of the principal G-bundle E G. Given any G-invariant vector field ω on E G, the pull back of ω by the diffeomorphism β in (4.6) remains G-invariant. Indeed, this follows from the fact that β commutes with the action of G on E G. Therefore, β gives an automorphism β:at(e G ) At(E G ) (4.7) of the Atiyah vector bundle. The automorphism β in (4.7) preserves the subbundle ad(e G ) At(E G ) in (2.10). The automorphism of ad(e G ) obtained by restricting β coincides with the adjoint action of β on ad(e G ) (recall that ad(e G ) is the Lie algebra bundle for the bundle Ad(E G ) of Lie groups). Also, note that the action of β on the quotient bundle TM = At(E G )/ad(e G ) (see (2.10)) is the trivial one. The automorphism β in (4.7) gives a diffeomorphism β 0 : C(E G ) C(E G ) (4.8) of the fiber bundle C(E G ) in (3.2). More precisely, the automorphism β of At(E G ) and the identity map of T M together define an automorphism of At(E G ) T M. This automorphism of At(E G ) T M clearly preserve the submanifold C(E G ) in (3.2). The automorphism β 0 in (4.8) is defined to be the restriction of this automorphism of At(E G ) T M. PROPOSITION 4.3 β0 K(D 0) be the δ ad(e G )-valued two-form on C(E G ) obtained by pulling back K(D 0 ) (defined in (4.1)) by the map β 0 constructed in (4.8). Then the following equality holds: β 0 K(D 0) = Ad(β 1 )(K(D 0 )), where β 1 is the section in (4.5), and Ad(β 1 ) is the adjoint action of β 1 on ad(e G ). Proof. Consider the automorphism β of E G constructed in (4.6). It gives a diffeomorphism of δ E G which we will describe. Take any point z δ E G. So z is a pair (y, t), where y C(E G ) and t (E G ) δ(y). Note that y = q(z), where q is the projection in (3.6). t := β (t) (E G ) δ(y) (4.9) be the image of t, where β is the automorphism in (4.6).

312 Indranil Biswas Now we have a diffeomorphism ˆβ: δ E G δ E G (4.10) defined by z (β 0 (y), t ), where β 0 and t are constructed in (4.8) and (4.9) respectively. From the construction of ˆβ in (4.10) we now conclude that the following diagram δ E G q C(E G ) ˆβ δ E G q β 0 C(E G ) is commutative, where q is the projection in (3.6). Furthermore, ˆβ commutes with the action of G on the principal G-bundle δ E G. In other words, ˆβ is an isomorphism of the principal G-bundle δ E G with its pull back (β 1 0 ) δ E G. It is easy to see that the isomorphism ˆβ of δ E G with (β 1 0 ) δ E G preserves the connection D 0 on the principal G-bundle δ E G. This means that ˆβ takes the connection D 0 on δ E G to the connection on (β 1 0 ) δ E G defined by D 0. Consequently, ˆβ pulls back the curvature of D 0 to itself. Therefore, from the action of ˆβ on the fibers of δ E G see (4.9) we conclude that β 0 K(D 0) = Ad(β 1 )(K(D 0 )). This completes the proof of the proposition. C(E G ) := C(E G ) M C(E G ) C(E G ) C(E G ) (4.11) be the submanifold consisting of all (y, z) C(E G ) C(E G ) such that δ(y) = δ(z), where δ is the projection in Lemma 3.1. Therefore, C(E G ) has a natural projection δ: C(E G ) M (4.12) defined by (y, z) δ(y) = δ(z). This projection δ makes C(E G ) a fiber bundle over M. Consider the vector bundle ν: T M ad(e G ) M. B C(E G ) (T M ad(e G )) be the submanifold defined by all (v, w) C(E G ) (T M ad(e G )), where v C(E G ) and w T M ad(e G ),

Atiyah bundle of a principal bundle 313 such that δ(v) = ν(w). Therefore, ν: B M (4.13) is a fiber bundle, where the projection ν is defined by (v, w) δ(v) = ν(w). We will show that there is a natural isomorphism between the fiber bundles C(E G ) and B constructed in (4.12) and (4.13) respectively. Take any (z, w) B C(E G ) (T M ad(e G )). (4.14) x := ν((z, w)) M be the image, where ν is the projection in (4.13). Therefore, z gives a homomorphism λ z : T x M At(E G ) x (4.15) (see (3.16)) such that η(x) λ z = Id Tx M, where η(x):at(e G ) x T x M (4.16) is the projection in (2.10). Now consider the section w in (4.14). Its evaluation at x gives a homomorphism w x : T x M ad(e G ) x (4.17) defined by w x (α) = i α w(x) ad(e G ) x, where i α is the contraction of T x M by α T xm. ζ (z,w) := λ z + ι 0 (x) w x : T x M At(E G ) x (4.18) be the homomorphism, where the homomorphism ι 0 (x):ad(e G ) x At(E G ) x is the one in (2.10); the homomorphisms λ z and w x are constructed in (4.15) and (4.17) respectively. It is easy to see that η(x) ζ (z,w) = Id Tx M, (4.19) where η(x) is the homomorphism in (4.16). In view of the identity in (4.19), we conclude that ζ (z,w) δ 1 (x) C(E G ), where δ is the projection in Lemma 3.1.

314 Indranil Biswas Consider the fiber bundles B and C(E G ) constructed in (4.13) and (4.12) respectively. F : B C(E G ) (4.20) be the map defined by (z, w) (z, ζ (z,w) ), where ζ (z,w) is constructed in (4.18) from (z, w) in (4.14). It is straight forward to check that F is an isomorphism of fiber bundles over M. θ: M T M ad(e G ) (4.21) be a smooth one-form on M with values in the adjoint vector bundle ad(e G ). Consider the fiber bundle δ: C(E G ) M in Lemma 3.1. T θ : C(E G ) C(E G ) (4.22) be the automorphism of it defined by z (p 2 F )(z, θ), where F is constructed in (4.20), and p 2 : C(E G ) C(E G ) (4.23) is the restriction of the projection of C(E G ) C(E G ) to the second factor (recall that C(E G ) C(E G ) C(E G )). A connection on a principal bundle induces a connection on each associated vector bundle. In particular, a connection on a principal bundle induces a connection on the adjoint vector bundle. The connection on the adjoint vector bundle δ ad(e G ) induced by the connection D 0 on δ E G will be denoted by D 0. So for each nonnegative integer i,we have a first order differential operator D 0 : Ɣ(C(E G), i T C(E G ) δ ad(e G )) Ɣ(C(E G ), i+1 T C(E G ) δ ad(e G )) (4.24) given by the connection D 0 on δ ad(e G ) (here Ɣ(C(E G ), V ) stands for the space of C sections of V C(E G )). Also, the Lie algebra structure of the fibers of ad(e G ) and the exterior algebra structure of the fibers of i 0 i T C(E G ) together define a homomorphism Ɣ(C(E G ), T C(E G ) δ ad(e G )) Ɣ(C(E G ), T C(E G ) δ ad(e G )) Ɣ(C(E G ), 2 T C(E G ) δ ad(e G )). (4.25) For θ 1,θ 2 Ɣ(C(E G ), T C(E G ) δ ad(e G )), the image of (θ 1,θ 2 ) in Ɣ(C(E G ), 2 T C(E G ) δ ad(e G )) by the above pairing will be denoted by [θ 1,θ 2 ].

Atiyah bundle of a principal bundle 315 PROPOSITION 4.4 The pulled back form Tθ K(D 0), where T θ is constructed in (4.22) and K(D 0 ) is the curvature form in (4.1), satisfies the identity T θ K(D 0) = K(D 0 ) + D 0 (δ θ)+ 1 2 [θ,θ] (see (4.24) and (4.25) for D 0 (δ θ) and [θ,θ] respectively). The proof of the above proposition is a straight forward computation. 5. The case of abelian groups In this section we assume the group G to be abelian. First we assume G to be a connected abelian of dimension one. So G is either R or S 1 = U(1). We first take E G to be the trivial principal G-bundle M G.ForE G = M G,wehave At(E G ) = T M (M R) (note that the Lie algebra of G is R). Hence C(E G ) constructed in (3.2) is the cotangent bundle T M M. The identification of T M with C(E G ) can also seen using the map F in (4.20). For this, first note that the trivial connection on the trivial principal G-bundle E G defines a section τ 0 : M C(E G ) of the fiber bundle C(E G ) M. Now we have a map F : T M C(E G ) (5.1) that sends any α T x M to p 2 F(τ 0 (x), α), where p 2 is the projection in (4.23), and F is the map in (4.20). It is easy to see that F in (5.1) is an isomorphism of fiber bundles over M. Since ad(e G ) is the trivial line bundle M R, the curvature K(D 0 ) in (4.1) is a usual two-form on C(E G ). Now from the properties of K(D 0 ) (see Lemma 4.1, Corollary 4.2 and Proposition 4.4) it follows that the diffeomorphism F in (5.1) takes K(D 0 ) to the canonical symplectic form on T M. Now take E G to be an arbitrary principal G-bundle over M (as before, G is either R or S 1 ). We can locally trivialize E G, hence locally we reduce to the above situation. Therefore, the above observation has the following corollary: COROLLARY 5.1 The curvature two-form K(D 0 ) on C(E G ) is symplectic. The fibers of the projection δ: C(E G ) M are Lagrangian. Now assume that n G = G i, i=1

316 Indranil Biswas where each G i is either R or S 1. Then any principal G-bundle E G over M is of the form E G = E G1 M E G2 M M E Gn, where each f i : E Gi M is a principal G i bundle, and E G1 M E G2 M M E Gn n E Gi i=1 is the submanifold consisting of all (z 1,...,z n ) such that Clearly, we have f 1 (z 1 ) = f 2 (z 2 ) = =f n (z n ). C(E G ) = C(E G1 ) M C(E G2 ) M M C(E Gn ) n C(E Gi ), where C(E G1 ) M C(E G2 ) M M C(E Gn ) is the submanifold consisting of all (z 1,z 2,...,z n ) n C(E Gi ) i=1 that project to some common point in M. It is now straight forward to check that the curvature form K(D 0 ) on C(E G ) is the direct sum of the pull backs of the curvature forms on C(E Gi ),1 i n. i=1 References [1] Atiyah M F, Complex analytic connections in fibre bundles, Trans. Am. Math. Soc. 85 (1957) 181 207 [2] Ben-Zvi D and Biswas I, Theta functions and Szegö kernels, Inter. Math. Res. Not. 24 (2003) 1305 1340 [3] Bleecker D, Gauge theory and variational principles, Global Analysis Pure and Applied Series A, 1 (Addison-Wesley Publishing Co., Reading, Mass.) (1981) [4] Kobayashi S and Nomizu K, Foundations of differential geometry, Vol. I (New York: John Wiley & Sons) (1963) [5] Kobayashi S and Nomizu K, Foundations of differential geometry, Vol. II (New York: John Wiley & Sons) (1969) Note added in the Proof. This has been achieved in a paper entitled A construction of a universal connection by I Biswas, J Hurtubise and J D Stasheff, to appear in Forum Mathematicum.