# Three Descriptions of the Cohomology of Bun G (X) (Lecture 4)

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1 Three Descriptions of the Cohomology of Bun G (X) (Lecture 4) February 5, 2014 Let k be an algebraically closed field, let X be a algebraic curve over k (always assumed to be smooth and complete), and let G be a smooth affine group scheme over X. We let Bun G (X) denote the moduli stack of G-bundles on X (denoted by Bun G in the previous lecture), and let l denote a prime number which is invertible in k. Our main goal in this course is to give a convenient description of the l-adic cohomology ring H (Bun G (X); Q l ). In the special case where G and X are defined over some finite field F q, this will allow us to compute the trace of ϕ 1 on H (Bun G (X); Q l ) (where ϕ denotes the geometric Frobenius morphism from Bun G (X) to itself), which we will use to prove Weil s conjecture following the outline provided in the last lecture. However, the problem of describing H (Bun G (X); Q l ) makes sense over an arbitrary algebraically closed field k. In this lecture, we will specialize to the case k = C and describe several topological approaches to the problem (which we will later adapt to the setting of algebraic geometry). In this case, we do not need to work l-adically: the algebraic stack Bun G (X) has a well-defined homotopy type (namely, the homotopy type of the associated analytic stack). We will abuse notation by identifying X with the compact Riemann surface X(C). To simplify the discussion, let us assume that the group scheme G is semisimple and simply connected at each point. Fix a G-bundle P sm in the category of smooth manifolds. The tangent bundle of P sm is a G-equivariant vector bundle on P sm, and can therefore be written the pullback of a smooth vector bundle E on X. This vector bundle fits into an exact sequence 0 E 0 E T X 0, (1) where E 0 denotes the vector bundle associated by P sm to the adjoint representation of G. In particular, we can regard E 0 as a complex vector bundle on X. A -connection on P sm is a choice of complex structure on the vector bundle E for which (1) is an exact sequence of complex vector bundles on X. Let Ω denote the collection of all -connections on X. Then Ω can be regarded as a torsor for the infinite-dimensional vector space of C-antilinear bundle maps from T X into E 0 : in particular, it is an infinite-dimensional affine space, and therefore contractible. Proposition 1. Let G = Aut(P sm ) denote the group of all automorphisms of the smooth G-bundle P sm. Then the moduli stack Bun G (X) can be identified, as a differentiable stack, with the quotient of the contractible space Ω by the action of the G. In particular, Bun G (X) has the homotopy type of the classifying space BG. Sketch. Every G-bundle ρ : P X is a fiber bundle with simply connected fibers, and is therefore trivial in the category of smooth G-bundles (since X is real manifold of dimension 2). In particular, for every complex-analytic G-bundle P on X, we can choose an isomorphism of smooth G-bundles α : P sm P. We can identify isomorphism classes of pairs (P, α) with complex-analytic structures on the bundle P sm ; since X has dimension 1, these are in bijection with points of Ω. Then G acts on the space Ω, and the homotopy quotient of Ω by G classifies complex-analytic G-bundles on X. Since X is a projective algebraic variety, the category of complex-analytic G-bundles on X is equivalent to the category of algebraic vector bundles on X. (For a related discussion, see [1].) Remark 2. The argument we sketched above really proves that the groupoid Bun G (X)(C) of C-valued points of Bun G (X) can be identified with the groupoid quotient of Ω (regarded as a set) by G (regarded as a 1

2 discrete group). To formulate a stronger claim, we would need to be more precise about the procedure which associates a homotopy type to an algebraic stack over C. A reader who is concerned with this technical point should feel free to take Principle 1 as a definition of the homotopy type of Bun G (X). Warning 3. The validity of Principle 1 relies crucially on the fact that X is an algebraic curve. If X is a smooth projective variety of higher dimension, then smooth G-bundles on X need not be trivial, and -connections on a smooth G-bundle P sm need not be integrable. Consequently, the homotopy type of Bun G (X) is not so easy to describe. Note that since the G-bundle P sm is trivial, we can identify the gauge group G with the space of all smooth sections of the projection map G X. We would like to use this information to describe the homotopy type of the classifying space of BG in terms of the individual classifying spaces {BG x } x X. We next outline three approaches to this problem: the first allows us to express H (BG; Q) as the cohomology of a certain differential graded Lie algebra (Theorem 4), while the remaining two express H (BG; Q) and H (BG; Q) as the homology of certain factorization algebras on X (Theorems 9 and 13). 1 First Approach: Rational Homotopy Theory Let H be a path-connected topological group. Then the homology H (H; Q) has the structure of a cocommutative Hopf algebra: the multiplication on H (H; Q) is given by pushforward along the product map H H H, and the comultiplication on H (H; Q) is given by pushforward along the diagonal map δ : H H H. With more effort, one can construct an analogue of this Hopf algebra structure at the level of chains, rather than homology. More precisely, Quillen s work on rational homotopy theory gives a functorial procedure for associating to each topological group H a differential graded Lie algebra g(h) (defined over the field of rational numbers) with the following properties: (a) Let H (g(h)) denote the homology groups of the underlying chain complex of g(h). Then we have a canonical isomorphism Q π H H (g(h)). Under this isomorphism, the Whitehead product on π +1 BH π H corresponds to the Lie bracket on H (g(h)). (b) The singular chain complex C (H; Q) is canonically quasi-isomorphic to the universal enveloping algebra U(g(H)). This quasi-isomorphism induces a Hopf algebra isomorphism H (U(g(H))) H (H; Q). (c) The differential graded Lie algebra g(h) is a complete invariant of the rational homotopy type of the classifying space BH. More precisely, from g(h) one can functorially construct a pointed topological space Z for which there exists a pointed map BH Z which induces an isomorphism on rational cohomology. In particular, the cohomology ring H (BH; Q) can be functorially recovered as the Lie algebra cohomology of g(h). Let us now apply the above reasoning to our situation. For every open subset U X, let G U denote the (topological) group of all smooth sections of the projection map G X U U, and let g(g U ) be the associated differential graded Lie algebras. The construction U g(g U ) is contravariantly functorial in U. For each integer n, let F n denote the presheaf of rational vector spaces on X given by F n (U) = g(g U ) n, and let F n be the associated sheaf. Ignoring the Lie algebra structures on the differential graded Lie algebras g(g U ) and remember only the underlying chain complexes, we obtain a chain complex of presheaves F 2 F 1 F 0 F 1 F 2, 2

3 hence a chain complex of sheaves F 2 F 1 F 0 F 1 F 2. In this language, we can formulate a local-to-global principle as follows: Theorem 4. The canonical map g(g) = Γ(X; F ) Γ(X; F ) RΓ(X; F ) is a quasi-isomorphism of differential graded Lie algebras. In other words, the cohomology groups of the differential graded Lie algebra g(g) can be identified with the hypercohomology groups of the chain complex F of sheaves on X. Proof. This follows from the compatibility of the construction H g(h) with (suitable) homotopy inverse limits. Remark 5. Fix a point x X. If U X is an open disk containing x, then evaluation at x induces a homotopy equivalence of topological groups G U G x. Passing to the direct limit, we obtain a quasiisomorphism of chain complexes F,x F,x g(g x ). In particular, the nth homology of the complex F is a locally constant sheaf on X. Theorem 4 then supplies a convergent spectral sequence H s (X; Q π t (G )) Q π t s G, where Q π t (G ) denotes the local system of rational vector spaces on X given by x Q π t (G x ). Example 6 (Atiyah-Bott). Suppose that G is constant: that is, it is the product of X with a simply connected semisimple algebraic group G 0 over C. In this case, the chain complex F is quasi-isomorphic to the chain complex of constant sheaves with value g(g 0 ). In this case, Theorem 4 supplies a quasi-isomorphism g(g) C (X; Q) Q g(g 0 ). The rational cohomology of the classifying space BG 0 is isomorphic to a polynomial ring Q[t 1,..., t r ], where r is the rank of the semisimple algebraic group G 0 and each t i is a homogeneous element of H (BG 0 ; Q) of some even degree e i. From this, one can deduce that the differential graded Lie algebra g(g 0 ) is formal: that is, it is quasi-isomorphic to a graded vector space V on generators t i of (homological) degree e i 1, where the differential and the Lie bracket vanish. It follows that g(g) is quasi-isomorphic to the tensor product H (X; Q) Q V, where the differential and Lie bracket vanish. From this, one can deduce that H (Bun G (X); Q) is isomorphic to a (graded) symmetric algebra on the graded vector space H (X; Q) Q V [ 1]. In other words, H (M; Q) is a tensor product of a polynomial ring on 2r generators in even degrees with an exterior algebra on 2gr generators in odd degrees. We refer the reader to [1] for more details. 2 Second Approach: Factorization Homology Theorem 4 asserts that that the differential graded Lie algebra g(g) can be recovered as the hypercohomology of a local system of differential graded Lie algebras given by x g(g x ). Roughly speaking, this reflects the idea that the gauge group G can be identified with a continuous product of the groups G x, and that the construction H g(h) is compatible with continuous products (at least in good cases). Our ultimate goal is to formulate a local-to-global principle which will allow us to compute the rational cohomology ring H (Bun G (X); Q) H (BG; Q). It is possible to formulate such a principle directly, without making a detour through the theory of differential graded Lie algebras. However, the basic mechanism of the local-to-global principle takes a more complicated form. 3

4 Definition 7. For each open set U X, let B(U) denote the rational cochain complex C (B G U ; Q). Then the construction U B(U) determines a covariant functor from the partially ordered set of open subsets of X to the category of chain complexes of rational vector spaces. Let U denote the collection of all open subsets of X which can be written as a disjoint union of disks. We let B denote a homotopy colimit of the diagram {B(U)} U U (in the category of chain complexes of rational vector spaces). We refer to the homology of the chain complex B as the factorization homology of B. Example 8. Suppose that U X is an open set which can be written as a disjoint union U 1 U n, where each U i is an open disk. Choose a point x i U i for 1 i n. Then G U is homeomorphic to a product G U i, and evaluation at the points x i determine homotopy equivalences G Ui G xi. Consequently, there is a canonical quasi-isomorphism of chain complexes C (BG xi ; Q) C (B G Ui ; Q) C (BG U ; Q) = B(U). In other words, each term in the diagram {B(U)} U U can be identified with a tensor product C (BG x ; Q), x S where S is some finite subset of X. We can therefore think of the factorization homology B as a kind of continuous tensor product x X C (BG x ; Q). We refer the reader to [3] for more details. We can now formulate a second local-to-global principle for describing the cohomology of Bun G (X): Theorem 9. If the fibers of G are semisimple and simply connected, then the canonical map B = hocolim U U B(U) B(X) = C (BG; Q) = C (Bun G (X); Q) is a quasi-isomorphism. In other words, we can identify the cohomology of the moduli stack Bun G (X) with the factorization homology of B. 3 Third Approach: Nonabelian Poincare Duality The local-to-global principle expressed by Theorem 9 is based on the idea of approximating the moduli stack Bun G (X) BG from the right. For any finite set S X, evaluation at the points of S defines a map of classifying spaces BG x S BG x, hence a map of cochain complexes µ S : x S C (BG x ; Q) C (BG; Q). Roughly speaking, Theorem 9 asserts that if we allow S to vary continuously over all finite subsets of X, then we can use these maps to recover the chain complex C (Bun G (X); Q) up to quasi-isomorphism. We now explore an parallel approach, which is based on the idea of realizing BG as direct limit, rather than an inverse limit. 4

5 Notation 10. For each open set U X, let G c U denote the subgroup of G consisting of those automorphisms of P sm which are the identity outside of a compact subset of U, and let A(U) denote the chain complex C (BG c U ; Q). Note that G c U G c V whenever U V, so that we can regard the construction U A(U) is a covariant functor from the partially ordered set of open subsets of X to the category of chain complexes. Let U denote the collection of all open subsets of X which can be written as a disjoint union of disks. We let A denote a homotopy colimit of the diagram {A(U)} U U (in the category of chain complexes of rational vector spaces). We refer to the homology of the chain complex A as the factorization homology of A. Example 11. Let U X be an open disk containing a point x X. The U X G is diffeomorphic to a product U G x, so that that G c U can be identified with the space of compactly supported maps from U into G x. A choice of homeomorphism U R 2 then determines a homotopy equivalence of G c U with the two-fold loop space Ω 2 (G x ), so that BG c U can be identified with Ω 2 (BG x ) Ω(G x ). More generally, if U can be written as a disjoint union of disks U 1 U n containing points x i U i, then G c U is homeomorphic to a product Gc U i, so we obtain a quasi-isomorphism of chain complexes C (Ω 2 BG xi ; Q) C (BG c U i ; Q) C (BG c U ; Q) = A(U). In other words, each term in the diagram {A(U)} U U can be identified with a tensor product C (Ω 2 (BG x ); Q), x S where S is some finite subset of X. We can therefore think of the factorization homology A as a kind of continuous tensor product x X C (Ω 2 (BG x ); Q). Remark 12. The double loop space Ω 2 (BG x ) is homotopy equivalent to quotient G(K x )/G(O x ), where O x denotes the completed local ring of X at x, and K x denotes its fraction field. We will denote this quotient by Gr x G and refer to it as the affine Grassmannian of the group G at the point x. This paper depends crucially on the fact that Gr x G admits an algebro-geometric incarnation (as the direct limit of a sequence of algebraic varieties) and can be defined over ground fields different from C. We have the following analogue of Theorem 9: Theorem 13 (Nonabelian Poincare Duality). The canonical map A = hocolim U U A(U) A(X) = C (BG; Q) = C (Bun G (X); Q) is a quasi-isomorphism. In other words, we can identify the homology of the moduli stack Bun G (X) with the factorization homology of A. Remark 14. Theorem 13 can be regarded as version of Poincare duality for the manifold X with coefficients in the nonabelian group G. We will explain this idea in more detail in??. Let us now outline the relationship between Theorems 4, 9, and 13. Theorem 4 is the weakest of the three results. It only gives information about the rational homotopy type of the moduli stack Bun G (X), while Theorems 9 and 13 remain valid with integral coefficients. In fact, Theorem 13 is even true at the unstable level: that is, it gives a procedure for reconstructing the space BG itself, rather than just the singular chain complex of BG). However, Theorem 4 gives information in a form which is most amenable to further calculation, since it articulates a local-toglobal principle using the familiar language of sheaf cohomology, rather than the comparatively exotic language of factorization homology. 5

6 Theorem 13 can also be regarded as the strongest of the three results because it requires the weakest hypotheses: if it is formulated correctly, we only need to assume that the fibers of the map G X are connected, rather than simply connected. Theorems 13 and 9 can be regarded as duals of one another. More precisely, the construction x C (BG x ; Q) determines a factorization algebra on X which is Koszul dual to the factorization algebra x C (Ω 2 BG x ; Q). Using this duality, one can construct a duality pairing between the chain complexes A and B, which identifies each with the Q-linear dual of the other (under the assumption that the fibers of G are simply connected). Theorems 4 and 9 can also be regarded as duals of one another, but in a different sense. Namely, each of the differential graded Lie algebras g(g x ) can be regarded as the Koszul dual of C (BG x ; Q), which we regard as an E -algebra over Q. One can exploit this to deduce Theorem 4 from Theorem 9 and vice versa. We can now outline the contents of this course. Our first step will be to formulate and prove an l-adic version of Proposition 13, which is valid over an arbitrary algebraically closed ground field k. The proof is based on a calculation of the homology of the space of generic trivializations of an arbitrary G-bundle P. Our next step will be to give a precise articulation of duality between Propositions 9 and 13. The basic mechanism is a local duality calculation which allows us to recover the l-adic cohomology of a classifying space BG x from the l-adic cohomology of Ω 2 BG x (which arises in its algebro-geometric incarnation as the affine Grassmannian of the group G). We will then use this result to deduce an l-adic version of Proposition 4. Combining this with the usual Grothendieck-Lefschetz trace formula, this supplies information about the action of Frobenius on the differential graded Lie algebra g (M), which will in turn yield the desired information about the action of Frobenius on H (Bun G (X); Q l ). Many steps in the preceding outline will require us to carry out sophisticated chain-level constructions which are difficult to describe using the language of classical homological algebra. In the next lecture, we give a brief introduction to the theory of -categories, which provides a more convenient formalism for our purposes. References [1] Atiyah, M. F. and R. Bott. The Yang-Mills equations over Riemann surfaces. Philos. Trans. Roy. Soc. London Ser. A 308 (1983), no. 1505, [2] Beilinson, A. and V. Drinfeld. Chiral algebras. American Mathematical Society Colloquium Publications 51. American Mathematical Society, Providence, RI, [3] Lurie, J. Higher Algebra. Freely available from author s webpage. 6

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