Complex Algebraic Geometry MSRI Publications Volume 28, 1995

Vector Bundles on Curves and Generalized

Theta Functions: Recent Results and Open Problems

ARNAUD BEAUVILLE

Abstract. The moduli spaces of vector bundles on a compact Riemann surface carry a natural line bundle, the determinant bundle. The sections of this line bundle and its multiples constitute a non-abelian generalization of the classical theta functions. New ideas coming from mathematical physics have shed a new light on these spaces of sections—allowing notably to compute their dimension (Verlinde’s formula). This survey paper is devoted to giving an overview of these ideas and of the most important recent results on the subject.

Introduction

It has been known essentially since Riemann that one can associate to any compact Riemann surface X an abelian variety, the Jacobian JX, together with a divisor Θ (well-defined up to translation) that can be defined both in a geometric way and as the zero locus of an explicit function, the Riemann theta function. The geometry of the pair (JX, Θ) is intricately (and beautifully) related to the geometry of X.

The idea that higher-rank vector bundles should provide a non-abelian analogue of the Jacobian appears already in the influential paper [We] of A. Weil (though the notion of vector bundle does not appear as such in that paper!). The construction of the moduli spaces was achieved in the 1960’s, mainly by D. Mumford and the mathematicians of the Tata Institute. However it is only recently that the study of the determinant line bundles on these moduli spaces and of their spaces of sections has made clear the analogy with the Jacobian. This is largely due to the intrusion of Conformal Field Theory, where these spaces have appeared (quite surprisingly for us!) as fundamental objects.

Partially supported by the European HCM project “Algebraic Geometry in Europe”

(AGE), Contract CHRXCT 940557.

In these notes (based on a few lectures given in the Fall of 1992 at MSRI, UCLA and University of Utah), I will try to give an overview of these new ideas. I must warn the reader that this is by no means intended to be a complete account. I have mainly focused on the determinant line bundles and their spaces of sections, ignoring deliberately important areas like cohomology of the moduli spaces, moduli of Higgs bundles, relations with integrable systems, Langlands’ geometric correspondence …, simply because I felt it would have taken me too far afield. For the same reason I haven’t even tried to explain why the mathematical physicists are so interested in these moduli spaces.

1. The moduli space SU_{X}(r)

Let X be a compact Riemann surface of genus g. Recall that the Jacobian JX parametrizes line bundles of degree 0 on X. We will also consider the variety J^{g−1}(X) that parametrizes line bundles of degree g − 1 on X; it carries a canonical Theta divisor

Θ = {M ∈ J^{g−1}(X) | H^{0}(X, M) ≠ 0} .

For each line bundle L on X of degree g − 1, the map M ↦ M ⊗ L^{−1} induces an isomorphism of (J^{g−1}(X), Θ) onto (JX, Θ_{L}), where Θ_{L} is the divisor on JX defined by

Θ_{L} = {E ∈ JX | H^{0}(X, E ⊗ L) ≠ 0} .

We know a great deal about the spaces H^{0}(JX, O(kΘ)). One of the key points is that the sections of O(kΘ) can be identified with certain quasi-periodic functions on the universal cover of JX, the theta functions of order k. In this way one gets for instance that the dimension of H^{0}(JX, O(kΘ)) is k^{g}, that the linear system |kΘ| is base-point free for k ≥ 2 and very ample for k ≥ 3, and so on. One even obtains a rather precise description of the ring

⊕_{k≥0}H^{0}(JX, O(kΘ)),

the graded ring of theta functions.

VECTOR BUNDLES ON CURVES AND GENERALIZED THETA FUNCTIONS 19

The character who will play the role of the Jacobian in these lectures is the moduli space SU_{X}(r) of (semistable) rank-r vector bundles on X with trivial determinant. It is an irreducible projective variety, whose points are isomorphism classes of vector bundles that are direct sums of stable vector bundles of degree 0 (a degree-0 vector bundle E is said to be stable if every proper subbundle of E has degree < 0). By the theorem of Narasimhan and Seshadri, the points of SU_{X}(r) are also the isomorphism classes of representations π_{1}(X) ⟶ SU(r) (hence the notation SU_{X}(r)). The stable bundles form a smooth open subset of SU_{X}(r), whose complement—which parametrizes decomposable bundles—is singular (except in the cases g ≤ 1 and g = r = 2, where the moduli space is smooth).

The reason for fixing the determinant is that the moduli space U_{X}(r) of vector bundles of rank r and degree 0 is, up to a finite étale covering, the product of SU_{X}(r) with JX, so the study of U_{X}(r) is essentially reduced to that of SU_{X}(r). Of course the moduli spaces SU_{X}(r, L) of semistable vector bundles with a fixed determinant L ∈ Pic(X) are also of interest; for simplicity, in these lectures I will concentrate on the most central case L = O_{X}.

Observe that when g ≤ 1 the spaces SU_{X}(r) consist only of direct sums of line bundles. Since these cases are quite easy to deal with directly, I will usually assume implicitly g ≥ 2 in what follows.

2. The determinant bundle

The geometric definition of the theta divisor extends in a natural way to the higher-rank case. For any line bundle L ∈ J^{g−1}(X), define

Θ_{L} = {E ∈ SU_{X}(r) | h^{0}(X, E ⊗ L) ≥ 1} .

This turns out to be a Cartier divisor on SU_{X}(r) [D-N] (the key point here is that the degrees are chosen so that χ(E ⊗ L) = 0). The associated line bundle L := O(Θ_{L}) does not depend on the choice of L. It is called the determinant bundle, and will play a central role in our story. It is in fact canonical, because of the following result (proved in [B1] for r = 2 and in [D-N] in general):

Theorem 1. Pic SU_{X}(r) = ZL.

By analogy with the rank-one case, the global sections of the line bundles L^{k} are sometimes called generalized theta functions—we will briefly discuss this terminology in §10.

Can we describe H^{0}(SU_{X}(r), L) and the map φ_{L} : SU_{X}(r) −−→ |L|^{∗} associated to L? Let us observe that we can define a natural (rational) map of SU_{X}(r) to the linear system |rΘ|, where Θ denotes the canonical Theta divisor on J^{g−1}(X): for E ∈ SU_{X}(r), define

θ(E) := {L ∈ J^{g−1} | h^{0}(E ⊗ L) ≥ 1} .

It is easy to see that θ(E) either is a divisor in J^{g−1}(X) that belongs to the linear system |rΘ|, or is equal to J^{g−1}(X). This last case can unfortunately occur (see §3 below), but only for special E’s, so we get a rational map

θ : SU_{X}(r) −−→ |rΘ|.

Theorem 2. There is a canonical isomorphism

H^{0}(SU_{X}(r), L) ⟶^{∼} H^{0}(J^{g−1}(X), O(rΘ))^{∗} ,

making the following diagram commutative:

[figure: |L|^{∗}; ≀] [figure: SU_{X}(r); φ_{L}; θ]

|rΘ|

This is proved in [B1] for the rank-two case and in [B-N-R] in general. Let me say a few words about the proof. For L in J^{g−1}(X) denote by H_{L} the hyperplane in |rΘ| consisting of divisors passing through L. One has θ^{∗}H_{L} = Θ_{L}, so we get a linear map θ^{∗} : H^{0}(J^{g−1}(X), O(rΘ))^{∗} ⟶ H^{0}(SU_{X}(r), L) whose transpose makes the above diagram commutative. It is easy to show that θ^{∗} is injective, hence the whole problem is to prove that dim H^{0}(SU_{X}(r), L) = r^{g}. This was done by constructing an r-to-one covering π : Y → X such that the pushforward map π_{∗} : JY −−→ SU_{X}(r) is dominant, which gives an injective map of H^{0}(SU_{X}(r), L) into H^{0}(JY, O(rΘ)). Note that surjectivity of θ^{∗} means that the linear system |L| is spanned by the divisors Θ_{L} for L in J^{g−1}(X).

This theorem provides a relatively concrete description of the map φ_{L}, and gives one some hope of being able to analyze the nature of this map—whether it is a morphism, an embedding, and so on. As we will see, this is a rather intriguing question, which is far from being completely understood. We first consider whether this map is everywhere defined or not.

3. Base points

It follows from Theorem 2 (more precisely, from the fact that the divisors Θ_{L} span the linear system |L|) that the base points of |L| are the elements E of SU_{X}(r) such that θ(E) = J^{g−1}(X), that is, H^{0}(E ⊗ L) ≠ 0 for all line bundles L of degree g − 1. The existence of such vector bundles was first observed by Raynaud [R]. Let me summarize his results in our language:

Theorem 3. a) For r = 2, the linear system |L| has no base points. b) For r = 3, |L| has no base points if g = 2, or if g ≥ 3 and X is generic.

c) Let n be an integer ≥ 2 dividing g. For r = n^{g}, the system |L| has base points.

In case (c), Raynaud’s construction gives only finitely many base points. This leaves open a number of questions, which I will regroup under the same heading:

Question 1. Can one find more examples (say for other values of r)? Is the base locus of dimension > 0? On the opposite side, can one find a reasonable bound on the dimension of the base locus?

Since the linear system |L| has (or may have) base points, we have to turn to its multiples. Here we have the following result of Le Potier [LP], improving an idea of [F1]:

Proposition 1. For k > \frac{1}{4}r^{3}(g − 1), the linear system |L^{k}| is base-point free.

In fact Le Potier proves a slightly stronger statement: given E ∈ SU_{X}(r) and k > \frac{1}{4}r^{3}(g − 1), there exists a vector bundle F on X of rank k and degree k(g − 1) such that H^{0}(X, E ⊗ F) = 0 (in other words, what may fail with a line bundle always works with a rank-k vector bundle). Then

Θ_{F} := {E ∈ SU_{X}(r) | H^{0}(X, E ⊗ F) ≠ 0}

is a divisor of the linear system |L^{k}| which does not pass through E, hence the proposition.

The bound on k is certainly far from optimal; in view of Proposition 1 the most optimistic guess is

Question 2. Is |L^{2}| base-point free?

Let me also mention the following question of Raynaud [R]:

Question 3. Given E ∈ SU_{X}(r), does there exist an étale covering π : Y → X such that θ(π^{∗}E) ≠ J^{g−1}(Y)?

4. Rank 2

The rank-two case is of course the simplest one; it has two special features. On one hand, by Theorem 3(a) (which is quite easy), we know that in this case φ_{L} is a morphism; we also know that this morphism is finite because L is ample. On the other hand, the linear system |2Θ| on J^{g−1}(X) is particularly interesting because it contains the Kummer variety K_{X} of X. Recall that K_{X} is the quotient of the Jacobian JX by the involution a ↦ −a, and that the map a ↦ Θ_{a} + Θ_{−a} of JX to |2Θ| (where as usual Θ_{a} denotes the translate of Θ by a) factors through an embedding κ : K_{X} ⟶ |2Θ|. The non-stable part of SU_{X}(2) consists of vector bundles of the form L ⊕ L^{−1}, for L in JX, and can therefore be identified with K_{X}; recall that for g ≥ 3 this is the singular locus of SU_{X}(2). Theorem 2 thus gives the following commutative diagram

[figure: |L|^{∗}; ≀] [figure: K; κ; ; SU_{X}(2); φ_{L}; θ]

|2Θ|

Let me summarize what is known about the structure of φ_{L} (or, what amounts to the same, of θ). Remember that the dimension of |2Θ| is 2^{g} − 1.

Theorem 4. a) For g = 2, θ is an isomorphism of SU_{X}(2) onto |2Θ| ≅ P^{3}

[N-R1].

b) For g ≥ 3, X hyperelliptic, θ is 2-to-1 onto a subvariety of |2Θ| that can be described in an explicit way [D-R].

c) For g ≥ 3, X not hyperelliptic, θ is of degree one onto its image [B1]. Moreover if g(X) = 3 or if X is generic, θ is an embedding ([N-R2]; [L], [B-V]).

The genus 3 (non hyperelliptic) case deserves a special mention: in this case Narasimhan and Ramanan prove that θ is an isomorphism of SU_{X}(2) onto a quartic hypersurface Q_{4} in |2Θ| (≅ P^{7}). By the above remark, this quartic is singular along the Kummer variety K_{X}. Now it had been observed a long time ago by Coble [C] that there exists a unique quartic hypersurface in |2Θ| passing doubly through K_{X}! Therefore Q_{4} is nothing but Coble’s hypersurface.

Part (c) of the theorem leaves open an obvious question:

Question 4. Is θ always an embedding for X non hyperelliptic?

The case of a generic curve was proved first by Laszlo [L]; Brivio and Verra have recently developed a more geometric approach [B-V], which might hopefully lead to a complete answer to the question—though some serious technical difficulties remain at this moment.

Laszlo’s method is to look at the canonical maps

μ_{k} : S^{k}H^{0}(SU_{X}(2), L) ⟶ H^{0}(SU_{X}(2), L^{k}) .

Since we know that L^{k} is very ample for some (unknown!) integer k, surjectivity of μ_{k} for k large enough would imply that L itself is very ample. We can even dream of getting surjectivity for all k, which would mean that the image of SU_{X}(2) in |2Θ| is projectively normal. In [B2] the situation is completely analyzed for μ_{2}. Recall that a vanishing thetanull on X can be defined as a line bundle L on X with L^{⊗2} ≅ ω_{X} and h^{0}(L) even ≥ 2—this means that the corresponding theta function on JX vanishes at the origin, hence the name. Such a line bundle exists only on a special curve (more precisely on a divisor in the moduli space of curves). Then:

Proposition 2. If X has no vanishing thetanull, the map μ_{2} is an isomorphism of S^{2}H^{0}(SU_{X}(2), L) onto H^{0}(SU_{X}(2), L^{2}).

More generally, if X has v vanishing thetanulls, one has

dim Ker μ_{2} = dim Coker μ_{2} = v.

This is only half encouraging (it shows that SU_{X}(2) is not projectively normal for curves with vanishing thetanulls), but note that the case k = 2 should be somehow the most difficult. On the positive side we have the following results:

Proposition 3. a) If X has no vanishing thetanull, the map

μ_{4} : S^{4}H^{0}(SU_{X}(2), L) ⟶ H^{0}(SU_{X}(2), L^{4})

is surjective [vG-P].

b) If X is generic, the map μ_{k} : S^{k}H^{0}(SU_{X}(2), L) ⟶ H^{0}(SU_{X}(2), L^{k}) is surjective for k even and ≥ 2g − 4 [L].

As already mentioned, (b) implies that φ_{L} is an embedding for generic X.

5. The Verlinde formula

Trying to understand the maps μ_{k} raises inevitably the question of the dimension of the spaces H^{0}(SU_{X}(r), L^{k}). We have seen that even the case k = 1 is far from trivial—this is the essential part of [B-N-R]. So it came as a great surprise when the mathematical physicists claimed to have a general (and remarkable) formula for dim H^{0}(SU_{X}(r), L^{k}), called the Verlinde formula [V] (there is actually a more general formula for the moduli space of principal bundles under a semisimple group, but we will stick to the case of SU_{X}(r)):

Theorem 5.

dim H^{0}(SU_{X}(r), L^{k}) = (\frac{r}{r + k} )^{g} ∑_{\begin{matrix}S⨿T =[1,r+k] \\ |S|=r \end{matrix}} ∏_{\begin{matrix}s∈S \\ t∈T \end{matrix}} | 2 sin π \frac{s − t}{r + k} |^{g−1}.

This form of the formula (shown to me by D. Zagier) is the simplest for an arbitrary rank; for small r or k I leave as a pleasant exercise to the reader to simplify it (hint: use ∏_{p=1}^{n−1}(2 sin(pπ/n)) = n). One gets r^{g} in the case k = 1, thus confirming Theorem 2, and in the rank-two case:

Corollary. dim H^{0}(SU_{X}(2), L^{k}) = (\frac{1}{2}k + 1)^{g−1} ∑_{i=1}^{k+1} \frac{1}{(sin(iπ/(k + 2)))^{2g−2}} ·

Note that the spaces H^{i}(SU_{X}(r), L^{k}) vanish for i > 0, by the Kodaira vanishing theorem (or rather its extension by Grauert and Riemenschneider), since the canonical bundle of SU_{X}(r) is equal to L^{−2r} [D-N]. Hence Theorem 5 gives actually χ(L^{k}). The right-hand side must therefore be a polynomial in k, and take integral values, which is certainly not apparent from the formula! In fact I know no direct proof of these properties, except in the case r = 2.

The leading coefficient of this polynomial is

(c_{1}(L)^{n})/n!,

where n = (r^{2} − 1)(g − 1) is the dimension of SU_{X}(r). This number, which is the volume of SU_{X}(r) for any Kähler metric with Kähler class c_{1}(L), has been computed in a beautiful way by Witten [W1], using the properties of the Reidemeister torsion of a flat connection. The result is

\frac{c_{1}(L)^{n}}{n!} = r (2π)^{−2n}Vol (SU(r))^{2g−2} ∑_{V} \frac{1}{(dim V)^{2g−2}} ,

where V runs over all irreducible representations of SU(r), and the volume of SU(r) is computed with respect to a suitably normalized Haar measure. One should be able to deduce this formula from Theorem 5, but I don’t know how to do that except in rank 2.

6. The Verlinde formula: finite-dimensional proofs

As soon as the Verlinde formula became known to mathematicians, it became a challenge for them to give a rigorous proof, so a wealth of proofs have appeared in the last few years. I will try to describe the ones I am aware of. The basic distinction is between the proofs using standard algebraic geometry, which up to now work only in the case r = 2, and the proofs that use infinite-dimensional algebraic geometry to mimic the heuristic approach of the physicists—these work for all r. Let me start with the “finite-dimensional” proofs.

The first proof of this kind is due to Bertram and Szenes [B-S], who use the explicit description of the moduli space in the hyperelliptic case (cf. Theorem 4(b)) to compute χ(L^{k})—which is the same for all smooth curves. (Actually they work with the moduli space SU_{X}(2, 1) of vector bundles of rank 2 and fixed determinant of degree 1, which has the advantage of being smooth; they show that χ(L^{k}) is the Euler–Poincaré characteristic of a certain vector bundle E_{k} on SU_{X}(2, 1).)

A more instructive proof has been obtained by Thaddeus [T], building on ideas of Bertram and Bradlow–Daskalopoulos. The idea is to look at pairs consisting of a rank-two vector bundle E of fixed, sufficiently high degree, say 2d, together with a nonzero section s of E. There is a notion of stability for these pairs—in fact there are various such notions, depending on an integer i with 0 ≤ i ≤ d. For each of these values one gets a moduli space M_{i}, which is projective and smooth; the key point is that one passes from M_{i−1} to M_{i} (for i ≥ 2) by a very simple procedure called a flip—blowing up a smooth subvariety and blowing down the exceptional divisor in another direction. Moreover, M_{0} is just a projective space, M_{1} is obtained by blowing up a smooth subvariety in M_{0}, while M_{d−1} maps surjectively to SU_{X}(2). In short one gets the following diagram

M̃_{2}  M̃_{3}  M̃_{d−1}
↙  ↘  ↙  ↘  ↙  ↘
M_{1}  M_{2}  · · · · · ·  M_{d−1}
↓  ↓
M_{0}  SU_{X}(2) ,

from which one deduces (with some highly non-trivial computations) the Verlinde formula.

Another completely different proof has been obtained by Zagier (unpublished). Not surprisingly, it is purely computational. Building on the work of Atiyah–Bott, Mumford and others, Zagier gives a complete description of the cohomology ring of SU_{X}(2, 1); he is then able to write down explicitly the Riemann–Roch formula for χ(E_{k}) (see above).

Another approach, due, I believe, to Donaldson and Witten, starts from Witten’s formula for the volume of SU_{X}(r) (§5). More precisely, Witten gives also a formula for the volume of the moduli space of stable parabolic bundles; here again the stability depends on certain rational numbers, so we get a collection of volumes indexed by these rational numbers. It turns out that one can recover from these volumes all the coefficients of the polynomial χ(L^{k}).

The last approach of this type I’d like to mention has been developed in [N-Rs] and [D-W1], and carried out successfully in [D-W2]. These authors attempt to prove directly that the spaces H^{0}(SU_{X}(2), L^{k}) and the analogous spaces defined using parabolic vector bundles obey the so-called factorization rules (see below). Though this approach looks quite promising, the details are unfortunately quite technical, and an extension to higher rank seems out of reach.

7. The Verlinde formula: infinite-dimensional proofs

The idea here is to translate in algebro-geometric terms the methods of the physicists. Actually what the physicists are interested in is a vector space that plays a central role in Conformal Field Theory, the space of conformal blocks B_{X}^{k}(r). This is defined as follows: let C((z)) be the field of formal Laurent series in one variable. There is a canonical representation V_{k} of the Lie algebra 𝔰𝔩_{r}(C((z))) (more precisely, of its universal central extension), called the basic representation of level k. Let p ∈ X; the affine algebra A_{X} := O(X p) embeds into C((z)) (by associating to a function its Laurent expansion at p). Then

B_{X}^{k}(r) := {ℓ ∈ V_{k}^{∗} | ℓ(Mv) = 0 for allM ∈ 𝔰𝔩_{r}(A_{X}) and v ∈ V_{k}} .

Theorem 6. a) There is a canonical isomorphism H^{0}(SU_{X}(r), L^{k}) ⟶^{∼} B_{X}^{k}(r).

b) The dimension of both spaces is given by the Verlinde formula (given in Theorem 5).

There are by now several available proofs of these results. The fact that the dimension of B_{X}^{k}(r) is given by the Verlinde formula follows from the work of Tsuchiya, Ueno and Yamada [T-U-Y]. They show that the dimension of B_{X}^{k}(r) is independent of the curve X, even if X is allowed to have double points. Then it is not too difficult to express B_{X}^{k}(r) in terms of analogous spaces for the normalization of X (this is called the factorization rules by the physicists). One is thus reduced to the genus-0 case (with marked points), that is, to a problem in the theory of representations of semisimple Lie algebras, which is non-trivial in general (actually I know no proof for the case of an arbitrary semisimple Lie algebra), but rather easy for the case of 𝔰𝔩_{r}(C).

Part (a) is proved (independently) in [B-L] and [F2]; actually Faltings proves both (a) and (b). He considers a smooth curve X degenerating to a stable curve X_{s}. It is not too difficult to show that H^{0}(SU_{X}(r), L^{k}) embeds into B_{X}^{k}(r), and that the B_{X}^{k}(r)’s are semicontinuous so that dim B_{X}^{k}(r) ≤ dim B_{X_{s}}^{k}(r). Therefore the heart of [F2] is the proof of the inequality

dim B_{X_{s}}^{k}(r) ≤ dim H^{0}(SU_{X}(r), L^{k}) ,

of which I cannot say much, since I don’t really understand it (note that the proof, as well as that of [T-U-Y], works in the more general set-up of principal bundles).

I would like to explain in a few words how we construct in [B-L] the isomorphism H^{0}(SU_{X}(r), L^{k}) ⟶^{∼} B_{X}^{k}(r), because I believe its importance goes far beyond the Verlinde formula. The basic object in the proof is not SU_{X}(r), but the moduli stack SL_{X}(r) parametrizing vector bundles E on X together with a trivialization of ⋀^{r} E. Though it appears at first glance as a rather frightening object, it is both more natural and easier to work with than the moduli space: basically, working with the moduli stack eliminates all the artificial problems of non-representability due to the fact that vector bundles have non-trivial automorphisms. The proof (which is entirely algebraic) has three steps:

1) We show that the moduli stack SL_{X}(r) is isomorphic to the quotient stack SL_{r}(A_{X})\ SL_{r}(C((z)))/ SL_{r}(C[[z]]). The key point here is that a vector bundle with trivial determinant is algebraically trivial over X p. (Hint: show that such a bundle has always a nowhere vanishing section, and use induction on the rank). We choose a small disk D ⊂ X around p (actually, to avoid convergence problems we take D = Spec (O), where O is the completed local ring of X at p, but this makes essentially no difference). We then consider triples (E, ρ, σ), where E is a vector bundle on X, ρ an algebraic trivialization of E over X p and σ a trivialization of E over D. Over D p these two trivializations differ by a holomorphic map D p ⟶ GL_{r}(C) that is meromorphic at p, that is, given by a Laurent series γ ∈ GL_{r}(C((z))). Conversely, given such a matrix γ, one can use it to glue together the trivial bundles on X p and D and recover the triple (E, ρ, σ). Since we want γ in SL_{r}(C((z))) we impose moreover that ⋀^{r} ρ and ⋀^{r} σ coincide over D p. This gives a bijection of the set of triples (E, ρ, σ) (up to isomorphism) onto SL_{r}(C((z))).

To get rid of the the trivializations, we have to mod out by the automorphism group of the trivial bundle over D and X p. We get the following diagram:

{E, ρ, σ} ⟷  SL_{r}(C((z)))
↓  ↓
{E, ρ} ⟷  Q := SL_{r}(C((z)))/ SL_{r}(C[[z]])
↓  ↓ π
{E}  ⟷ SL_{r}(A_{X})\ SL_{r}(C((z)))/ SL_{r}(C[[z]]) .

Of course I have only constructed a bijection between the set of isomorphism classes of vector bundles on X with trivial determinant and the set of double classes SL_{r}(A_{X})\ SL_{r}(C((z)))/ SL_{r}(C[[z]]); with some technical work one shows that the construction actually gives an isomorphism of stacks.

2) Recall that, if Q = G/H is a homogeneous space, one associates to any character χ : H → C^{∗} a line bundle L_{χ} on Q: it is the quotient of the trivial bundle G×C on G by the action of H defined by h(g, λ) = (gh, χ(h)λ). We apply this to the homogeneous space Q = SL_{r}(C((z)))/ SL_{r}(C[[z]]) (this is actually an ind-variety, i.e., the direct limit of an increasing sequence of projective varieties). By (1) we have a quotient map π : Q ⟶ SL_{X}(r). The line bundle π^{∗}L does not admit an action of SL_{r}(C((z))), but of a group ŜL_{r}(C((z))) that is a central C^{∗}- extension of SL_{r}(C((z))). This extension splits over the subgroup SL_{r}(C[[z]]), so that Q is isomorphic to ŜL_{r}(C((z)))/(C^{∗} × SL_{r}(C[[z]])). Then π^{∗}L is the line bundle L_{χ}, where χ : C^{∗} × SL_{r}(C[[z]]) ⟶ C^{∗} is the first projection.

3) A theorem of Kumar and Mathieu provides an isomorphism H^{0}(Q, L_{χ}^{k}) ≅ V_{k}^{∗}. From this and the definition of a quotient stack one can identify

H^{0}(SL_{X}(r), L^{k})

with the subspace of V_{k}^{∗} invariant under SL_{r}(A_{X}). This turns out to coincide with the subspace of V_{k}^{∗} invariant under the Lie algebra 𝔰𝔩_{r}(A_{X}), which is by definition B_{X}^{k}(r). Finally a Hartogs type argument gives H^{0}(SU_{X}(r), L^{k}) ≅ H^{0}(SL_{X}(r), L^{k}).

To conclude let me observe that all the proofs I have mentioned are rather indirect, in the sense that they involve either degeneration arguments or sophisticated computations. The simplicity of the formula itself suggests the following question:

Question 5. Can one find a direct proof of Theorem 5?

What I have in mind is for instance a computation of χ(L^{k}) by simply applying the Riemann–Roch formula; this requires the knowledge of the Chern numbers of the moduli space. In [W2], Witten proposes some very general conjectures that should give the required Chern numbers for SU_{X}(r): the preprint [S] sketches how the Verlinde formula follows from these conjectures. Jeffrey and Kirwan have proved some of the Witten’s conjectures, and I understand that they are very close to a proof of the Verlinde formula along these lines.

8. The strange duality

Let me denote by U_{X}^{∗}(k) the moduli space of semistable vector bundles of rank k and degree k(g − 1) on X; it is isomorphic (non canonically) to U_{X}(k). A special feature of this moduli space is that it carries a canonical theta divisor Θ_{k}: set-theoretically one has

Θ_{k} = {E ∈ U_{X}^{∗}(k) | H^{0}(X, E) ≠ 0} .

Put M := O(Θ_{k}). Consider the morphism τ_{k,r} : SU_{X}(r) × U_{X}^{∗}(k) ⟶ U_{X}^{∗}(kr) defined by τ_{k,r}(E, F) = E ⊗F. An easy application of the theorem of the square shows that τ_{k,r}^{∗}(O(Θ_{kr})) is isomorphic to pr_{1}^{∗}L^{k} ⊗ pr_{2}^{∗}M^{r}. Now τ_{k,r}^{∗}Θ_{kr} is the divisor of a section of this line bundle, well-defined up to a scalar; by the Künneth theorem we get a linear map ϑ_{k,r} : H^{0}(U_{X}^{∗}(k), M^{r})^{∗} ⟶ H^{0}(SU_{X}(r), L^{k}), welldefined up to a scalar. In this section I want to discuss the following conjecture:

Question 6. Conjecture: The map ϑ_{k,r} is an isomorphism.

I heard of this statement three or four years ago, as being well-known to the physicists. The conjecture is discussed at length, and extended to vector bundles of arbitrary degree, in [D-T].

Let me discuss a few arguments in favor of the conjecture.

a) The case k = 1 is exactly Theorem 2.

b) The two spaces have the same dimension. To prove this one needs to compute the dimension of H^{0}(U_{X}^{∗}(k), M^{r}); this is easy (assuming the Verlinde formula!) because the map τ_{1,k} : SU_{X}(k) × J^{g−1}(X) ⟶ U_{X}^{∗}(k) is an étale (Galois) covering of degree k^{2g}, and τ_{1,k}^{∗}(M) ≅ pr_{1}^{∗}L ⊗ pr_{2}^{∗}O_{J}(kΘ). Therefore we get

dim H^{0}((U_{X}^{∗}(k), M^{r}) = χ(M^{r}) = \frac{1}{k^{2g}} χ(L^{r}) χ(O_{J}(krΘ))

= \frac{r^{g}}{k^{g}} dim H^{0}((SU_{X}(k), L^{r}) .

Now, Theorem 5 shows that k^{−g} dim H^{0}((SU_{X}(k), L^{r}) is symmetric in k and r, which proves our assertion.

c) Therefore it is enough to prove, for example, the surjectivity of the map ϑ_{k,r}, which has the following geometric meaning:

Question 6′. The linear system |L^{k}| in SU_{X}(r) is spanned by the divisors Θ_{F}, for F in U_{X}^{∗}(k).

(Recall from §3 that Θ_{F} is the locus of vector bundles E ∈ SU_{X}(r) such that H^{0}(X, E ⊗ F) ≠ 0). As an application, taking vector bundles F of the form L_{1} ⊕ ⋯ ⊕ L_{k} with L_{i} ∈ J^{g−1}(X), one deduces from Proposition 3(b) that the conjecture holds for r = 2 and k even and ≥ 2g − 4 (in this way we get the result for a generic curve only, but using the methods in §9 below I can extend it to every curve).

9. The projective connection

So far we have considered the moduli space SU_{X}(r) for a fixed curve X. What can we say about the vector spaces H^{0}(SU_{X}(r), L^{k}) when the curve X is allowed to vary? It is again a remarkable discovery of the mathematical physicists that these vector spaces are essentially independent of the curve. To explain this in mathematical terms, consider a family of (smooth) curves (X_{t})_{t∈T}, parametrized by a variety T; for t in T, let us denote by L_{t} the determinant line bundle on SU_{X_{t}}(r). Then:

Theorem 7. The linear systems |L_{t}^{k}| define a flat projective bundle over T.

Here again we have by now a number of proofs for this result. The first mathematical proof is due to Hitchin [H], following the method used by Welters in the rank-one case; I understand that Beilinson and Kazhdan had a similar proof (unpublished). A different approach, inspired by the work of the physicists, appears in [F1]. Finally, one of the main ingredients in [T-U-Y] is the construction of a flat vector bundle over T whose fibre at t ∈ T is the space of conformal blocks B_{X_{t}}^{k}(r) (the curves of the family are required to have a marked point p_{t} ∈ X_{t}, together with a distinguished tangent vector v_{t} ∈ T_{p_{t}}(X_{t})). Thanks to Theorem 6 this provides still another construction of our flat projective bundle. I have no doubt that all these constructions give the same object, but I must confess that I haven’t checked it.

Let us take for T the moduli space M_{g} of curves of genus g (here again, the correct object to consider is the moduli stack, but let me ignore this). We get a flat projective bundle over M_{g}, which corresponds to a projective representation

ρ_{r,k} : Γ_{g} ⟶ PGL(H^{0}(SU_{X}(r), L^{k}))

of the fundamental group Γ_{g} = π_{1}(M_{g}, X). This group, called the modular group by the physicists and the mapping class group by the topologists, is a fundamental object: it carries all the topology of M_{g}. So a natural question is

Question 7. What is the representation ρ_{r,k}?

This is a rather intriguing question. In the rank-one case, the analogue of H^{0}(SU_{X}(r), L^{k}) is the space V_{k} of k-th order theta functions; the group Γ_{g} acts on V_{k} through its quotient Sp (2g, Z) (take this with a grain of salt when k is odd), and this action is explicitly described by the classical “transformation formula” for theta functions—which shows in particular that the action factors through a finite quotient of Sp (2g, Z). Using Theorem 2 we get an analogous description for arbitrary rank in the case k = 1. I expect that the general case is far more complicated, and in particular that ρ_{r,k} doesn’t factor through Sp (2g, Z), but I know no concrete example where this happens.

Conformal Field Theory predicts that the connection should be (projectively) unitary. This is one of the remaining challenges for mathematicians:

Question 8. Find a flat hermitian metric H_{X} on H^{0}(SU_{X}(r), L^{k}) (i.e., one such that the image of ρ_{r,k} is contained in P U(H_{X})).

Here again the rank-one case is well-known, and Theorem 2 gives an answer in the case k = 1—though not a very explicit one.

10. Are there generalized theta functions?

I believe that the above results give some evidence that the spaces

H^{0}(SU_{X}(r), L^{k})

are non-abelian analogues of the spaces H^{0}(JX, O(kΘ)). There is however one aspect of the picture that is missing so far in higher rank, namely the analytic description of the sections of O(kΘ) as holomorphic functions. Clearly the theory of theta functions cannot be extended in a straightforward way, if only because SU_{X}(r) is simply connected.

One possible approach is provided by our description of the moduli stack as a double quotient SL_{r}(A_{X})\ SL_{r}(C((z)))/ SL_{r}(C[[z]]) (§7). The pullback of the determinant line bundle L to the group ŜL_{r}(C((z))), which is a C^{∗}-extension of SL_{r}(C((z))), is trivial, so we should be able to express sections of L^{k} as functions on ŜL_{r}(C((z))). This is done in [B-L] in one particular case: we prove that the pullback of the divisor Θ_{(g−1)p} is the divisor of a certain algebraic function on ŜL_{r}(C((z))) known as the τ function. (This idea appears already in [Br], with a slightly different language.) However, it is not clear how to express other sections of L (and even less of L^{k}) in a similar way.

There are other possible ways of describing sections of L^{k} by holomorphic functions. One of these, explored by D. Bennequin, is to pull back L to SL_{r}(C)^{g} by the dominant map SL_{r}(C)^{g} −−→ SU_{X}(r) that maps a g-tuple (M_{1}, …, M_{g}) to the flat vector bundle E(ρ_{M}) ∈ SU_{X}(r) associated to the representation ρ_{M} : π_{1}(X) ⟶ SL_{r}(C) with ρ(a_{i}) = I, ρ(b_{j}) = M_{j} (here (a_{1}, …, a_{g}, b_{1}, …, b_{g}) are the standard generators of π_{1}(X)).

Despite these attempts I am afraid we are still far of having a satisfactory theory of generalized theta functions. So I will end up this survey with a loosely formulated question:

Question 9. Is there a sufficiently simple and flexible way of expressing the elements of H^{0}(SU_{X}(r), L^{k}) as holomorphic functions?

References

[B1] A. Beauville, Fibrés de rang 2 sur les courbes, fibré déterminant et fonctions thêta, Bull. Soc. Math. France 116 (1988), 431–448.

[B2] , Fibrés de rang 2 sur les courbes, fibré déterminant et fonctions thêta II, Bull. Soc. Math. France 119 (1991), 259–291.

[B-L] A. Beauville and Y. Laszlo, Conformal blocks and generalized theta functions, Comm.

Math. Phys. 164 (1994), 385–419.

[B-N-R] A. Beauville, M. S. Narasimhan and S. Ramanan, Spectral curves and the generalized theta divisor, J. Reine Agnew. Math. 398 (1989), 169–179.

[B-S] A. Bertram and A. Szenes, Hilbert polynomials of moduli spaces of rank 2 vector

bundles II, Topology 32 (1993), 599–609.

[B-V] S. Brivio and A. Verra, The Theta divisor of SU_{C}(2) is very ample if C is not

hyperelliptic and Noether-Lefschetz general, preprint (1993).

[Br] J. -L. Brylinski, Loop groups and non-commutative theta-functions, preprint (1989).

[C] A. Coble, Algebraic geometry and theta functions, Amer. Math. Soc. Colloq. Publ., vol. 10, Providence, 1929, third edition, (1969).

[D-W1] G. Daskalopoulos and R. Wentworth, Local degenerations of the moduli space of vector bundles and factorization of rank 2 theta functions I, Math. Ann. 297 (1993), 417–466.

[D-W2] , Factorization of rank 2 theta functions II: proof of the Verlinde formula,

preprint (1994).

[D-R] U. V. Desale and S. Ramanan, Classification of vector bundles of rank 2 on hyper-

elliptic curves, Invent. Math. 38 (1976), 161–185.

[D-T] R. Donagi and L. Tu, Theta functions for SL(n) versus GL(n), Math. Res. Let. 1

(1994), 345–357.

[D-N] J. M. Drezet and M. S. Narasimhan, Groupe de Picard des variétés de modules de

fibrés semi-stables sur les courbes algébriques, Invent. Math. 97 (1989), 53–94.

[F1] G. Faltings, Stable G-bundles and projective connections, J. Algebraic Geom. 2 (1993), 507–568.

[F2] , A proof for the Verlinde formula, J. Algebraic Geom. 3 (1994), 347–374.

[H] N. Hitchin, Flat connections and geometric quantization, Comm. Math. Phys. 131 (1990), 347–380.

[L] Y. Laszlo, A propos de l’espace des modules des fibrés de rang 2 sur une courbe, Math. Annalen 299 (1994), 597–608.

[N-R1] M. S. Narasimhan and S. Ramanan, Moduli of vector bundles on a compact Riemann

surface, Ann. of Math. 89 (1969), 19–51.

[N-R2] , 2θ-linear systems on Abelian varieties, Vector bundles on algebraic varieties,

Oxford University Press, 1987, pp. 415–427.

[N-Rs] M. S. Narasimhan and T. R. Ramadas, Factorization of generalized Theta functions

I, Invent. Math. 114 (1993), 565–623.

[LP] J. Le Potier, Module des fibrés semi-stables et fonctions thêta, preprint (1993).

[R] M. Raynaud, Sections des fibrés vectoriels sur une courbe, Bull. Soc. Math. France 110 (1982), 103–125.

[S] A. Szenes, The combinatorics of the Verlinde formula, preprint (1994).

[T] M. Thaddeus, Stable pairs, linear systems and the Verlinde formula, Invent. Math. 117 (1994), 317–353.

[T-U-Y] A. Tsuchiya, K. Ueno, and Y. Yamada, Conformal field theory on universal family

of stable curves with gauge symmetries, Adv. Stud. Pure Math. 19 (1989), 459–566.

[V] E. Verlinde, Fusion rules and modular transformations in 2d conformal field theory, Nuclear Phys. B 300 (1988), 360-376.

[vG-P] B. van Geemen and E. Previato, Prym varieties and the Verlinde formula, Math.

Annalen 294 (1992), 741–754.

[We] A. Weil, Généralisation des fonctions abéliennes, J. de Math. P. et App. 17 (1938), no. 9^{ème} sér., 47–87.

[W1] E. Witten, On quantum gauge theories in two dimensions, Comm. Math. Phys. 141 (1991), 153–209.

[W2] , Two-dimensional gauge theories revisited, J. Geom. Phys. 9 (1992), 308– 368.

Arnaud Beauville URA 752 du CNRS Mathématiques – Bât. 425 Université Paris-Sud 91 405 Orsay Cedex, France E-mail address: arnaud@matups.matups.fr