Conformal blocks and generalized theta functions

We prove a conjecture about the moduli space S U C ( 3 ) \mathcal {SU}_C(3) of semi-stable rank 3 vector bundles with trivial determinant on a genus 2 curve C C , due to I. Dolgachev. Given C C a smooth projective curve of genus 2, and the embedding of the Jacobian J C JC into | 3 Θ | |3\Theta | , A. Coble proved, at the beginning of the 20th century, that there exists a unique cubic hypersurface C \mathcal {C} in | 3 Θ | ∗ ≃ P 8 |3\Theta |^* \simeq \mathbb {P}^8 , J C [ 3 ] JC[3] -invariant and singular along J C JC . On the other hand, we have a map of degree 2 from S U C ( 3 ) \mathcal {SU}_C(3) over | 3 Θ | ≃ P 8 ∗ |3\Theta | \simeq \mathbb {P}^{8*} , ramified along a sextic hypersurface B \mathcal {B} . Dolgachev’s conjecture affirms that the sextic B \mathcal {B} is the dual variety of Coble’s cubic C \mathcal {C} .

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The purpose of this paper is to carry out the abelianization program pro posed by Atiyah [1] and Hitchin [9] for the geometric quantization of SU(2) Wess-Zumino-Witten model. Let C be a Riemann surface of genus g. Let Mg be the moduli space of semi-stable rank 2 holomorphic vector bundles on C with trivial determinant. For a positive integer k, let T(Mg, Ck) be the space of holomorphic sections of the fc-th tensor product of the determinant line bundle C on Mg. An element of T(Mg, Ck) is called a rank 2 theta function of level k. The main result of our abelianization is to give an explicit representation of a base of T{Mg, Ck) as well as its transformation formula in terms of classical Riemann theta functions with automorphic form coefficients defined on the Prym variety P associated with a two-fold branched covering surface C of C. r(Mg,Ck) can be identified with the conformai block of level k of the SU(2) WZW model ([5], [15]). The abelianization procedure enables us to de duce the various known results about the conformai block in a uniform way. Firstly, we construct a projectively flat connection on the vector bundle over the Teichm?ller space with fibre T{Mg,Ck). Secondly, making use of our ex plicit representation of rank 2 theta functions we construct a Hermitian product on the vector bundle preserved by the connection. Also our explicit represen tation enables us to prove that Y(Mg,Ck) has the predicted dimension from the Quantum Clebsh-Gordan conditions. A natural connection on the said vector bundle for the SU(iV) WZW model was first constructed by Hitchin [11]. It will turn out that the connection constructed in this paper coincides with the Hitchin connection. Laszlo [16] showed that the Hitchin connection coincides with the con nection constructed by Tsuchiya, Ueno and Yamada [21] through the above identification. On the other hand Kirillov [13], [14] constructed a Hermitian product on the conformai block compatible with the Tsuchiya-Ueno-Yamada connection using the representation theory of affine Lie algebras together with the theory of hermitian modular tensor categories; cf. [22]. Laszlo's result

We are interested in studying moduli spaces of rank 2 logarithmic connections on elliptic curves having two poles. To do so, we investigate certain logarithmic rank 2 connections defined on the Riemann sphere and a transformation rule to lift such connections to an elliptic curve. The transformation is as follows: given an elliptic curve C with elliptic quotient, and the logarithmic connection, we may pullback the connection to the elliptic curve to obtain a new connection on C. After suitable birational modifications we bring the connection to a particular normal form. The whole transformation is equivariant with respect to bundle automorphisms and therefore defines a map between the corresponding moduli spaces of connections. The aim of this paper is to describe the moduli spaces involved and compute explicit expressions for the above map in the case where the target space is the moduli space of rank 2 logarithmic connections on an elliptic curve C with two simple poles and trivial determinant.

We investigate functorial properties of two hermitian line bundles over the moduli space of flat SU(n)-connections on a closed oriented surface; that is, of the Chern–Simons line bundle and the determinant line bundle. We investigate actions of cyclic subgroups of the mapping class group on them. As a consequence, we show that if we modify the determinant line bundle by the Hodge bundle over the moduli space of Riemann surfaces, then these line bundles are functorially isomorphic. This implies two quantum Hilbert spaces defined by the Chern–Simons line bundle and the modified determinant line bundle are functorially isomorphic.

This is the first in a series of papers which describe the action of an affine Lie algebra with central charge $n$ on the moduli space of $U(n)$-instantons on a four manifold $X$. This generalises work of Nakajima, who considered the case when $X$ is an ALE space. In particular, this describes the combinatorial complexity of the moduli space as being precisely that of representation theory, and thus will lead to a description of the Betti numbers of moduli space as dimensions of weight spaces. This Lie algebra acts on the space of conformal blocks (i\.e\., the cohomology of a determinant line bundle on the moduli space) generalising the ``insertion'' and ``deletion'' operations of conformal field theory, and indeed on any cohomology theory. In the particular case of $U(1)$-instantons, which is essentially the subject of this present paper, the construction produces the basic representation after Frenkel-Kac. Then the well known quadratic nature of $ch_2$, $$ch_2 = \frac{1}{2} c_1\cdot c_1 - c_2 $$ becomes precisely the formula for the eigenvalue of the degree operator, i\.e\. the well known quadratic behaviour of affine Lie algebras.

In this paper we study the constraints imposed by conformal invariance on extended objects a.k.a. defects in a conformal field theory. We identify a particularly nice class of defects that is closed under conformal transformations. Correlation function of the defect with a bulk local operator is fixed by conformal invariance up to an overall constant. This gives rise to the notion of defect expansion, where the defect itself is expanded in terms of local operators. This expansion generalizes the idea of the boundary state. We will show how one can fix the correlation function of two defects from the knowledge of the defect expansion. The defect correlator admits a number of conformal cross-ratios depending on their dimensionality. We find the differential equation obeyed by the conformal block and solve them in certain special cases.

This is the second part of the paper (the first part is published in Jour. of AMS, vol.9, 1135--1170, q-alg/9508017). In the first part, we defined for every modular tensor category (MTC) inner products on the spaces of morphisms and proved that the inner product on the space $\Hom (\bigoplus X_i\otimes X^*_i, U)$ is modular invariant. Also, we have shown that in the case of the MTC arising from the representations of the quantum group $U_q \sln$ at roots of unity and $U$ being a symmetric power of the fundamental representation, this inner product coincides with so-called Macdonald's inner product on symmetric polynomials. In this paper, we apply the same construction to the MTC coming from the integrable representations of affine Lie algebras. In this case our construction immediately gives a hermitian form on the spaces of conformal blocks, and this form is modular invariant (Warning: we cannot prove that it is positive definite). We show that this form can be rewritten in terms of asymptotics of KZ equations, and calculate it for $sl_2$, in which case the formula is a natural affine analogue of Macdonald's inner product identities. We also formulate as a conjecture similar formula for $sl_n$.

We study the generating functional, the adiabatic curvature and the adiabatic phase for the integer quantum Hall effect (QHE) on a compact Riemann surface. For the generating functional we derive its asymptotic expansion for the large flux of the magnetic field, i.e., for the large degree $k$ of the positive Hermitian line bundle $L^k$. The expansion consists of the anomalous and exact terms. The anomalous terms are the leading terms of the expansion. This part is responsible for the quantization of the adiabatic transport coefficients in QHE. We then identify the non-local (anomalous) part of the expansion with the Quillen metric on the determinant line bundle, and the subleading exact part with the asymptotics of the regularized spectral determinant of the Laplacian for the line bundle $L^k$, at large $k$. Finally, we show how the generating functional of the integer QHE is related to the gauge and gravitational (2+1)d Chern-Simons functionals. We observe the relation between the Bismut-Gillet-Soul\'e curvature formula for the Quillen metric and the adiabatic curvature for the electromagnetic and geometric adiabatic transport of the integer Quantum Hall state. Then we relate the adiabatic phase in QHE to the eta invariant and show that the geometric part of the adiabatic phase is given by the Chern-Simons functional.

We prove a local analog of the Deligne-Riemann-Roch isomorphism in the case of line bundles and relative dimension $1$. This local analog consists in computation of the class of $12$th power of the determinant central extension of a group ind-scheme $\mathcal G$ by the multiplicative group scheme over $\mathbb Q$ via the product of $2$-cocyles in the second cohomology group. These $2$-cocycles are the compositions of the Contou-Carrère symbol with the $\cup$-product of $1$-cocycles. The group ind-scheme $\mathcal{G}$ represents the functor which assigns to every commutative ring $A$ the group that is the semidirect product of the group $A((t))^*$ of invertible elements of $A((t))$ and the group of continuous $A$-automorphisms of $A$-algebra $A((t))$. The determinant central extension naturally acts on the determinant line bundle on the moduli stack of geometric data (proper quintets). A proper quintet is a collection of a proper family of curves over $\operatorname{Spec} A$, a line bundle on this family, a section of this family, a relative formal parameter at the section, a formal trivialization of the bundle at the section that satisfy further conditions.

Since the definition of $$T\bar T$$ deformation in the curved Riemann surface is obstructive in the literature, we propose a way to do the deformation in the genus two Riemann surfaces by sewing prescription. We construct the correlation functions of conformal field theories (CFTs) on genus two Riemann surfaces with the $$T\bar T$$ deformation in terms of the perturbative CFT approach. Thanks to sewing construction to form higher genus Riemann surfaces from lower genus ones and conformal symmetry, we systematically obtain the first-order $$T\bar T$$ correction to the genus two correlation functions in the $$T\bar T$$ deformed CFTs, e.g., partition function and one/higher-point correlation functions.

Let [Formula: see text] be a smooth complex projective curve of genus [Formula: see text] and let [Formula: see text] be a line bundle on [Formula: see text] with [Formula: see text]. Let [Formula: see text] be the moduli space of semistable rank 2 [Formula: see text]-twisted Higgs bundles with trivial determinant on [Formula: see text]. Let [Formula: see text] be the moduli space of stable rank 2 [Formula: see text]-twisted Higgs bundles with determinant [Formula: see text] for some [Formula: see text] on [Formula: see text]. We construct a cycle in the product of a stack of rational maps from nonsingular curves to [Formula: see text] and [Formula: see text] by using Hecke modifications of a stable [Formula: see text]-twisted Higgs bundle in [Formula: see text].

Let {Xt ↪→ Y | t ∈ M} be a family of compact complex submanifolds Xt of a complex manifold Y parametrised by a complex manifold M . In this paper we study vector bundles and linear connections on the moduli space M induced from holomorphic vector bundles on the space Y . The origin of this approach lies in the Ward transform [W] of holomorphic vector bundles E on Y which are trivial when restricted to each submanifoldXt of the family {Xt ↪→ Y | t ∈M}. We consider a more general class of holomorphic vector bundles E on twistor spaces Y which, when restricted to a submanifold Xt of the family, have the same integer h(Xt, E|Xt) for all t ∈M . With such a vector bundle E on Y there is an associated vector bundle E on M whose fibre at t ∈ M is isomorphic to H(Xt, E|Xt). It is shown that provided the cohomology groups H ( Xt, E|Xt ⊗N ∗ t ) and H ( Xt, E|Xt ⊗N ∗ t ) vanish (where Nt is the normal bundle of Xt ↪→ Y ), the vector bundle E induced on the parameter space M comes equipped with a distinguished linear connection ∇ satisfying some natural integrability conditions (cf. [L, E, B-E]). The curvature tensor of ∇ is represented, at each t ∈ M , by a cohomology class in H ( Xt, E|Xt ⊗ 2N∗ ) ⊗ ( H(Xt, E|Xt) )∗ . This theorem-construction gives a very simple way to estimate the curvature tensor of an induced connection directly from the original twistor data. As an application, we consider a pair X ↪→ Y consisting of a complex contact 3-fold (Y, L) and a Legendre submanifold X = CP such that the contact line bundle L restricts to X as O(3). Then the moduli space M of all holomorphic Legendre deformations of X inside Y is a 4-dimensional manifold which comes equipped with an induced 1-flat G3structure [Br2], and any such structure arises locally in this way. Suppose that there exists a line bundle E → Y such that L = E⊗3 (such an E always exists on a sufficiently small tubular neighbourhood of X in Y ). Since the normal bundle of X is isomorphic to JO(3) = C ⊗ O(2), the above cohomology conditions on E|Xt ⊗ N ∗ t are satisfied. Then the induced rank 2 vector bundle E comes equipped with a distinguished linear connection which, as easily follows from its integrability properties, is a “spinor” version of the torsion-free connection on M with holonomy in G3. In fact, any torsion-free connection with holonomy in G3 can be constructed, at least locally, in this way (cf. [Me1]). The point is that the way we prove the above mentioned theorem-construction gives a rather

This article deals with two topics: the first, which has a general character, is a variation formula for the determinant line bundle in non-Kahlerian geometry. This formula, which is a consequence of the non-Kahlerian version of the Grothendieck–Riemann–Roch theorem proved recently by Bismut [Bi], gives the variation of the determinant line bundle corresponding to a perturbation of a Fourier–Mukai kernel E on a product B×X by a unitary flat line bundle on the fiber X. When this fiber is a complex surface and E is a holomorphic 2-bundle, the result can be interpreted as a Donaldson invariant.

We provide a thorough construction of a system of compatible determinant line bundles over spaces of Fredholm operators, fully verify that this system satisfies a number of important properties, and include explicit formulas for all relevant isomorphisms between these line bundles. We also completely describe all possible systems of compatible determinant line bundles and compare the conventions and approaches used elsewhere in the literature.