The moduli space M(n,d) is an algebraic variety parametrizing those representations of the fundamental group of a punctured Riemann surface into the Lie group SU(n) for which a loop around the boundary is sent to the n-th root of unity exp (2 πi d/n) multiplied by the identity matrix. If n and d are coprime it is in fact a Kähler manifold. One may relax the constraint and study moduli spaces M(L) parametrizing those representations for which the loop around the boundary is sent to an element conjugate to L, if L is some element in SU(n), and these are also Kähler manifolds for a suitable class of L.
The Verlinde formula calculates the dimension of the space of holomorphic sections of certain line bundles over the spaces M(n,d) and M(L). We recall how a new proof of the Verlinde formula for M(n,d) given in joint work with Frances Kirwan may be obtained, and show how to modify this proof to obtain a proof of the variant of the Verlinde formula which applies to M(L) (the moduli space of parabolic bundles with one marked point) and to the analogous object with b marked points.
This paper computes the quantization of the moduli space of flat SO(3)-bundles over an oriented surface with boundary, with prescribed holonomies around the boundary circles. The result agrees with the generalized Verlinde formula conjectured by Fuchs and Schweigert.
We study the splitting properties of the Verlinde bundles over elliptic curves. Our methods rely on the explicit description of the moduli space of semistable vector bundles on elliptic curves, and on the analysis of the symmetric powers of the Schrodinger representation of the Theta group.
We determine the splitting type of the Verlinde vector bundles in higher genus in terms of simple semihomogeneous factors. In agreement with strange duality, the simple factors are interchanged by the Fourier-Mukai transform, and their spaces of sections are naturally dual.
A Verlinde space of level $k$ is the space of global sections of the $k$-th power of the determinant line bundle on the moduli space $\cSU_C(r)$ of semi-stable bundles of rank $r$ on a curve $C$. The aim of this note is to make accessible some remarks on the action of the Theta group on the Verlinde spaces of higher level. This gives a decomposition of the bundle of Verlinde spaces over the moduli space of curves and we indicate how to compute the rank of the isotypical components in the decomposition.
Verlinde bundles are vector bundles $V_k$ arising as the direct image $π_*(\mathcal L^{\otimes k})$ of polarizations of a proper family of schemes $π\colon \mathfrak X \to S$. We study the splitting behavior of Verlinde bundles in the case where $π$ is the universal family $\mathfrak X \to |\mathcal O(d)|$ of hypersurfaces of degree $d$ in $\mathbb P^n$ and calculate the cohomology class of the locus of jumping lines of the Verlinde bundles $V_{d+1}$ in the cases $n=2,3$.
We prove an explicit formula for the total Chern character of the Verlinde bundle over the moduli space of pointed stable curves in terms of tautological classes. The Chern characters of the Verlinde bundles define a semisimple CohFT (the ranks, given by the Verlinde formula, determine a semisimple fusion algebra). According to Teleman's classification of semisimple CohFTs, there exists an element of Givental's group transforming the fusion algebra into the CohFT. We determine the element using the first Chern class of the Verlinde bundle on the moduli space of nonsingular curves and the proje…
In this paper we approach the study of generalized theta linear series on moduli of vector bundles on curves via vector bundle techniques on abelian varieties. We study what are called the Verlinde bundles in order to obtain information about duality between theta functions and effective global and normal generation on these moduli spaces.
A formula for the first Chern class of the Verlinde bundle over the moduli space of smooth genus g curves is given. A finite-dimensional argument is presented in rank 2 using geometric symmetries obtained from strange duality, relative Serre duality, and Wirtinger duality together with the projective flatness of the Hitchin connection. A derivation using conformal-block methods is presented in higher rank. An expression for the first Chern class over the compact moduli space of curves is obtained.
We prove that moduli spaces of semistable parabolic bundles and generalized parabolic sheaves (GPS) with a fixed determinant on a smooth projective curve are globally F-regular type. As an application, we prove vanishing theorems on the moduli spaces of semistable parabolic sheaves on a singular curve, which combining with Factorization theorems in [24] and [25] give two recurrence relations among dimensions of spaces of generalized theta functions. By using of these recurrence relations, we prove an explicit formula (Verlinde formula) for the dimension of spaces of generalized theta functions…
We propose and prove the Verlinde formula for the quantization of the Higgs bundle moduli spaces and stacks for any simple and simply-connected group. This generalizes the equivariant Verlinde formula for the case of $SU(n)$ proposed previously by the second and third author. We further establish a Verlinde formula for the quantization of parabolic Higgs bundle moduli spaces and stacks. Finally, we prove that these dimensions form a one-parameter family of $1+1$-dimensional TQFT, uniquely classified by the complex Verlinde algebra, which is a one-parameter family of Frobenius algebras. We cons…
We prove an analog of the Verlinde formula on the moduli space of semistable meromorphic G-Higgs bundles over a smooth curve for a reductive group G whose fundamental group is free. The formula expresses the graded dimension of the space of sections of a positive line bundle as a finite sum whose terms are indexed by formal solutions of a generalized Bethe ansatz equation on the maximal torus of G. In an appendix, Constantin Teleman proves a vanishing theorem for the higher cohomology of positive line bundles on the stack of Higgs bundles.
We prove canonical isomorphisms between Spin Verlinde spaces, i.e, spaces of global sections of a determinant line bundle over the moduli space of semistable Spin-bundles over a smooth projective curve C, and the dual spaces of theta functions over Prym varieties of unramified double covers of C.
I conjecture that index formulas for $K$-theory classes on the moduli of holomorphic $G$-bundles over a compact Riemann surface $Σ$ are controlled, in a precise way, by Frobenius algebra deformations of the Verlinde algebra of $G$. The Frobenius algebras in question are twisted $K$-theories of $G$, equivariant under the conjugation action, and the controlling device is the equivariant Gysin map along the "product of commutators" from $G^{2g}$ to $G$. The conjecture is compatible with naive virtual localization of holomorphic bundles, from $G$ to its maximal torus; this follows by localization …
Theta functions of level n on the principally polarised Prym varieties of an algebraic curve are dual to sections of the orthogonal theta line bundle on the moduli space of Spin(n)-bundles over the curve. As a by-product of our computations we also note that when n is odd the pfaffian line bundle on moduli space has a basis of sections labelled by the even theta characteristics of the curve.
Let G be a complex semi-simple group, X a Riemann surface, M_G the moduli space of principal G-bundles on X. When G is simply-connected, there exists a closed formula expressing the dimension of the space H^0(M_G,L) for any line bundle L on M_G (this is usually called among mathematicians, somewhat incorrectly, the "Verlinde formula").In this paper we find an analogous formula for G = PGL(r) where r is prime.
The Verlinde formula computes the dimension of certain vector spaces ("conformal blocks") associated to a Rational Conformal Field Theory. In this paper we show how this can be made rigorous for one particular such theory, the WZW model. Thanks to the results of [B-L], [F] and [T-U-Y], this gives the dimension of the space of global sections of the determinant line bundles (and its multiples) on the moduli space of vector bundles with fixed rank and determinant.
We study the moduli problem of pairs consisting of a rank 2 vector bundle and a nonzero section over a fixed smooth curve. The stability condition involves a parameter; as it varies, we show that the moduli space undergoes a sequence of flips in the sense of Mori. As applications, we prove several results about moduli spaces of rank 2 bundles, including the Harder-Narasimhan formula and the SU(2) Verlinde formula. Indeed, we prove a general result on the space of sections of powers of the ideal sheaf of a curve in projective space, which includes the Verlinde formula.
Given a compact Riemann surface $X$ of genus at least $2$ with automorphism group $G$ we provide formulae that enable us to compute traces of automorphisms of X on the space of global sections of $G$-linearized line bundles defined on certain blow-ups of proyective spaces along the curve $X$. The method is an adaptation of one used by Thaddeus to compute the dimensions of those spaces. In particular we can compute the traces of automorphisms of $X$ on the Verlinde spaces corresponding to the moduli space $SU_{X}(2,ξ)$ when $ξ$ is a line bundle $G$-linearized of suitable degree.
A residue formula is given for the Verlinde formula, which allows one to calculate its coefficients as a polynomial in the level and connects it to the Riemann-Roch formula on the moduli space of vector bundles on a curve.
We find an agreement of equivariant indices of semi-classical homomorphisms between pairwise mirror branes in the GL(2) Higgs moduli space on a Riemann surface. On one side we have the components of the Lagrangian brane of U(1,1) Higgs bundles whose mirror was proposed by Nigel Hitchin to be certain even exterior powers of the hyperholomorphic Dirac bundle on the SL(2) Higgs moduli space. The agreement arises from a mysterious functional equation. This gives strong computational evidence for Hitchin's proposal.
We prove that the pull back of the canonical theta divisor for E_8-bundles at level one induces a strange duality between Verlinde spaces for G_2 and F_4 at level one on smooth curves of genus g. We also prove a parabolic generalization in terms of conformal blocks and write down identities between conformal blocks divisors in the Picard group of \bar-M_{g,n}.
Let $\mathrm{Hilb}_nS$ be the Hilbert scheme of $n$ points on a smooth projective surface $S$. To a class $α\in K^0(S)$ correspond a tautological vector bundle $α^{[n]}$ on $\mathrm{Hilb}_nS$ and line bundle $L_{(n)}\otimes E^{\otimes r}$ with $L=\det(α)$, $r=\mathrm{rk}(α)$. In this paper we give closed formulas for the generating functions for the Segre classes $\int_{\mathrm{Hilb}_nS} s(α^{[n]})$, and the Verlinde numbers $χ(\mathrm{Hilb}_nS,L_{(n)}\otimes E^{\otimes r})$, for any surface $S$ and any class $α\in K^0(S)$. In fact we determine a more general generating function for $K$-theore…
K-theoretic Donaldson invariants are holomorphic Euler characteristics of determinant line bundles on moduli spaces of rank 2 sheaves on surfaces. We develop an algorithm which determines the generating functions of K-theoretic Donaldson invariants on the projective plane and more generally rational surfaces, and apply it in many cases to get explicit formulas. We relate the results to Le Potier's strange duality conjecture.
In this paper, we give a Verlinde formula for computing the ranks of the bundles of twisted conformal blocks associated with a simple Lie algebra equipped with an action of a finite group $Γ$ and a positive integral level $\ell$ under the assumption that "$Γ$ preserves a Borel". As a motivation for this Verlinde formula, we prove a categorical Verlinde formula which computes the fusion coefficients for any $Γ$-crossed modular fusion category as defined by Turaev. To relate these two versions of the Verlinde formula, we formulate the notion of a $Γ$-crossed modular functor and show that it is v…
The aim of this paper is to construct the parabolic version of the Donaldson--Uhlenbeck compactification for the moduli space of parabolic stable bundles on an algenraic surface with parabolic structures along a divisor with normal crossing singularities. We prove the non--emptiness of the moduli space of parabolic stable bundles of rank 2 and also prove the existence of components with smooth points.
We generalize the classical Beauville-Narasimhan-Ramanan correspondence to the case of parabolic Higgs bundles with regular singularities and Higgs $V$-bundles. Using this correspondence along with Bott-Morse theoretic techniques we provide an exact component count for moduli spaces of maximal parabolic $\text{Sp}\left( 2n,\mathbb{R} \right)$-Higgs bundles with fixed parabolic structure.
We introduce a notion of admissible Hermitian metrics on parabolic bundles and define positivity properties for the same. We develop Chern-Weil theory for parabolic bundles and prove that our metric notions coincide with the already existing algebro-geometric versions of parabolic Chern classes. We also formulate a Griffiths conjecture in the parabolic setting and prove some results that provide evidence in its favour for certain kinds of parabolic bundles. For these kinds of parabolic structures, we prove that the conjecture holds on Riemann surfaces. We also prove that a Berndtsson-type resu…
We build compact moduli spaces of Grassmannian framed bundles over a Riemann surface, essentially replacing a group by its bi-invariant compactification. We do this both in the algebraic and symplectic settings, and prove a Hitchin-Kobayashi correspondence between the two. The spaces are universal spaces for parabolic bundles, and the reduction to parabolic bundles commutes with the correspondence. An analogous correspondence is outlined for the generalized parabolic bundles of Bhosle.
There is a well-known correspondence between the symplectic variety of representations of the fundamental group of a punctured Riemann surface into a compact Lie group G, with fixed conjugacy classes at the punctures, and a complex variety of holomorphic bundles on the unpunctured surface with a parabolic structure at the puncture points. For G=SU(2), we build a symplectic variety of pairs (representations of the fundamental group into G, `weighted frame' at the puncture points), and a corresponding complex variety of moduli of `framed parabolic bundles', which encompass respectively all of th…
Parabolic triples of the form $(E_*,θ,σ)$ are considered, where $(E_*,θ)$ is a parabolic Higgs bundle on a given compact Riemann surface $X$ with parabolic structure on a fixed divisor $S$, and $σ$ is a nonzero section of the underlying vector bundle. Sending such a triple to the Higgs bundle $(E_*,θ)$ a map from the moduli space of stable parabolic triples to the moduli space of stable parabolic Higgs bundles is obtained. The pull back, by this map, of the symplectic form on the moduli space of stable parabolic Higgs bundles will be denoted by $\text{d}Ω'$. On the other hand, there is a map f…
Given a smooth complex projective variety X and a smooth divisor D on X, we prove the existence of Hermitian-Einstein connections, with respect to a Poincaré-type metric on X - D, on polystable parabolic principal Higgs bundles with parabolic structure over D, satisfying certain conditions on its restriction to D.
We study moduli spaces of parabolic Higgs bundles on a curve and their dependence on the choice of weights. We describe the chamber structure on the space of weights and show that, when a wall is crossed, the moduli space undergoes an elementary transformation in the sense of Mukai.
Let X be a smooth complex projective curve and S a finite subset of X. We show that an orthogonal or symplectic parabolic Higgs bundle on X with parabolic structure over S admits a Hermitian-Einstein connection if and only if it is polystable.
The moduli stack M_X(E_8) of principal E_8-bundles over a smooth projective curve X carries a natural divisor Delta. We study the pull-back of the divisor Delta to the moduli stack M_X(P), where P is a semi-simple and simply connected group such that its Lie algebra Lie(P) is a maximal conformal subalgebra of Lie(E_8). We show that the divisor Delta induces "Strange Duality"-type isomorphisms between the Verlinde spaces at level one of the following pairs of groups (SL(5), SL(5)), (Spin(8), Spin(8)), (SL(3), E_6) and (SL(2), E_7).
We introduce the moduli space of quasi-parabolic $SL(2,\mathbb{C})$-Higgs bundles over a compact Riemann surface $Σ$ and consider a natural involution, studying its fixed point locus when $Σ$ is $\mathbb{C} \mathbb{P}^1$ and establishing an identification with a moduli space of null polygons in Minkowski $3$-space.
We study natural families of d-bar operators on the moduli space of stable parabolic vector bundles. Applying a families index theorem for hyperbolic cusp operators from our previous work, we find formulae for the Chern characters of the associated index bundles. The contributions from the cusps are explicitly expressed in terms of the Chern characters of natural vector bundles related to the parabolic structure. We show that our result implies formulae for the Chern classes of the associated determinant bundles consistent with a result of Takhtajan and Zograf.
The moduli space $M$ of semi-stable rank 2 bundles with trivial determinant over a complex curve carries involutions naturally associated to 2-torsion points on the Jacobian of the curve. For every lift of a 2-torsion point to a 4-torsion point, we define a lift of the involution to the determinant line bundle $Ł$. We obtain an explicit presentation of the group generated by these lifts in terms of the order 4 Weil pairing. This is related to the triple intersections of the components of the fixed point sets in $M$, which we also determine completely using the order 4 Weil pairing. The lifted …
We construct projectivization of a parabolic vector bundle and a tautological line bundle over it. It is shown that a parabolic vector bundle is ample if and only if the tautological line bundle is ample. This allows us to generalize the notion of a k-ample bundle, introduced by Sommese, to the context of parabolic bundles. A parabolic vector bundle $E_*$ is defined to be k-ample if the tautological line bundle ${\mathcal O}_{{\mathbb P}(E_*)}(1)$ is $k$--ample. We establish some properties of parabolic k-ample bundles.
In this paper, we study rank 2 (quasi) parabolic bundles over the Riemann sphere with an effective divisor and these moduli spaces. First we consider a criterium when a parabolic bundle admits a unramified irregular singular parabolic connection. Second, to give a good compactification of the moduli space of semistable parabolic bundles, we introduce a generalization of parabolic bundles, which is called refined parabolic bundles. Third, we discuss a stability condition of refined parabolic bundles and define elementary transformations of the refined parabolic bundles. Finally, we describe the…
We compute the Betti numbers of the moduli space of rank 3 parabolic Higgs bundles, using Morse theory. A key point is that certain critical submanifolds of the Morse function can be identified with moduli spaces of parabolic triples. These moduli spaces come in families depending on a real parameter and we study their variation with this parameter.
We prove a Torelli theorem for the moduli space of semistable parabolic Higgs bundles over a smooth complex projective algebraic curve under the assumption that the parabolic weight system is generic. When the genus is at least two, using this result we also prove a Torelli theorem for the moduli space of semistable parabolic bundles of rank at least two with generic parabolic weights. The key input in the proofs is a method of J.C. Hurtubise, Integrable systems and algebraic surfaces, Duke Math. Jour. 83 (1996), 19--49.
Our aim in this article is to produce new examples of semistable Lazarsfeld- Mukai bundles on smooth projective surfaces $X$ using the notion of parabolic vector bundles. In particular, we associate natural parabolic structures to any rank two (dual) Lazarsfeld-Mukai bundle and study the parabolic stability of these parabolic bundles. We also show that the orbifold bundles on Kawamata coverings of $X$ corresponding to the above parabolic bundles are themselves certain (dual) Lazarsfeld-Mukai bundles. This gives semistable Lazarsfeld-Mukai bundles on Kawamata covers of the projective plane and …
The moduli space of $G$-bundles on an elliptic curve with additional flag structure admits a Poisson structure. The bivector can be defined using double loop group, loop group and sheaf cohomology constructions. We investigate the links between these methods and for the case $\SL_2$ perform explicit computations, describing the bracket and its leaves in detail.
In this version referee's comments have been incorporated. Besides minor corrections, new material has been added on irrational markings. To appear in Ann. Inst. Fourier. We prove a condition for the existence of flat bundles on the punctured two-sphere with prescribed holonomies around the punctures, involving Gromov-Witten invariants of generalized flag varieties. This generalizes the case of special unitary bundles described by S. Agnihotri and the second author and independently P. Belkale.
Let $φ: Y \rightarrow X$ be a finite surjective morphism between smooth complex projective curves, where $X$ is irreducible but $Y$ need not be so. Let $V_*$ be a parabolic vector bundle on $Y$. We construct a parabolic structure on the direct image $φ_* V$ on $X$, where $V$ is the vector bundle underlying $V_*$. The parabolic vector bundle $φ_* V_*$ on $X$ obtained this way has a ramified torus sub-bundle; it is a torus bundle of $\text{Ad}(φ_* V)$ outside the parabolic divisor for $φ_* V_*$ that satisfies certain conditions at the parabolic points. Conversely, given a parabolic vector bundle…
Algebraic parabolic bundles on smooth projective curves over algebraically closed field of positive characteristic is defined. It is shown that the category of algebraic parabolic bundles is equivalent to the category of orbifold bundles defined in \cite{KP}. Tensor, dual, pullback and pushforward operations are also defined for parabolic bundles.
We give a cohomological criterion for a parabolic vector bundle on a curve to be semistable. It says that a parabolic vector bundle $E$ with rational parabolic weights is semistable if and only if there is another parabolic vector bundle $F$ with rational parabolic weights such that the cohomologies of the vector bundle underlying the para- bolic tensor product $E \otimes F$ vanish. This criterion generalizes the known semistability criterion of Faltings for vector bundles on curves and significantly improves the result in [Bis07].
Take an irreducible smooth projective curve $X$ defined over an algebraically closed field of characteristic zero, and fix finitely many distinct point $D\, =\, \{x_1,\, \cdots,\, x_n\}$ of it; for each point $x\, \in\, D$ fix a positive integer $N_x$. Take a nonconstant map $f\, :\, Y\, \longrightarrow \, X$ from an irreducible smooth projective curve. We construct a natural subbundle $\mathcal{F}\, \subset\, f_*{\mathcal O}_Y$ using $(D,\, \{N_x\}_{x\in D})$. Let $E_*$ be a stable parabolic vector bundle whose parabolic weights at each $x\, \in\, D$ are integral multiples of $\frac{1}{N_x}$.…
