Donaldson Invariants of Product Ruled Surfaces¶and Two-Dimensional Gauge Theories

Using the u-plane integral of Moore and Witten, we derive a simple expression for the Donaldson invariants of product ruled surfaces Σ _g ×S ², where Σ _g is a Riemann surface of genus g. This expression generalizes a theorem of Morgan and Szabó for g=1 to any genus g. We give two applications of our results: (1) We derive Thaddeus' formulae for the intersection pairings on the moduli space of rank two stable bundles over a Riemann surface. (2) We derive the eigenvalue spectrum of the Fukaya–Floer cohomology of Σ _g ×S ¹. Received: 22 July 1999 / Accepted: 12 June 2000     

The Moduli of Flat PGL(2,ℝ) Connections on¶Riemann Surfaces

Suppose X is a compact Riemann surface with genus g>1. Each class [σ] ∈ Hom(π₁(X),PGL(2,ℝ))/PGL(2,ℝ) is associated with the first and second Stiefel–Whitney classes w ₁([σ]) and w ₂([σ]). The set of representation classes with a fixed w ₁≠ 0 has two connected components. These two connected components are characterized by w ₂ being 0 or 1. For each fixed w ₁≠ 0, we prove that the component, characterized by w ₂= 0, contains an open dense set diffeomorphic to the total space of a vector bundle of rank 2g−2 over a once punctured algebraic torus of dimension g−1. The other component, characterized by w ₂= 1, contains an open dense set diffeomorphic to the total space of a vector bundle of rank 2g−2 over an algebraic torus of dimension g−1. Received: 2 January 1997 / Accepted: 28 November 1998     

Flat connections and geometric quantization

Using the space of holomorphic symmetric tensors on the moduli space of stable bundles over a Riemann surface we construct a projectively flat connection on a vector bundle over Teichmüller space. The fibre of the vector bundle consists of the global sections of a power of the determinant bundle on the moduli space. Both Dolbeault and ech techniques are used.     

Topological Strings and (Almost) Modular Forms

The B-model topological string theory on a Calabi-Yau threefold X has a symmetry group Γ, generated by monodromies of the periods of X. This acts on the topological string wave function in a natural way, governed by the quantum mechanics of the phase space H ³(X). We show that, depending on the choice of polarization, the genus g topological string amplitude is either a holomorphic quasi-modular form or an almost holomorphic modular form of weight 0 under Γ. Moreover, at each genus, certain combinations of genus g amplitudes are both modular and holomorphic. We illustrate this for the local Calabi-Yau manifolds giving rise to Seiberg-Witten gauge theories in four dimensions and local IP ₂ and IP ₁ × IP ₁. As a byproduct, we also obtain a simple way of relating the topological string amplitudes near different points in the moduli space, which we use to give predictions for Gromov-Witten invariants of the orbifold .     

Projective Structure,Symplectic Connection and Quantization

Let X be a connected Riemann surface equipped with a projective structure . Let E be a holomorphic symplectic vector bundle over X equipped with a flat connection. There is a holomorphic symplectic structure on the total space of the pullback of E to the space of all nonzero holomorphic cotangent vectors on X. Using , this symplectic form is quantized. A moduli space of Higgs bundles on a compact Riemann surface has a natural holomorphic symplectic structure. Using , a quantization of this symplectic form over a Zariski open subset of the moduli space of Higgs bundles is constructed.     

Scaled Correlations of Critical Points of Random Sections on Riemann Surfaces

In this paper we prove that as N goes to infinity, the scaling limit of the correlation between critical points z ₁ and z ₂ of random holomorphic sections of the N-th power of a positive line bundle over a compact Riemann surface tends to 2/(3π ²) for small . The scaling limit is directly calculated using a general form of the Kac-Rice formula and formulas and theorems of Pavel Bleher, Bernard Shiffman, and Steve Zelditch.     

Geometry of Higgs and Toda fields on Riemann surfaces

We discuss geometrical aspects of Higgs systems and Toda field theory in the framework of the theory of vector bundles on Riemann surfaces of genus greater than one. We point out how Toda fields can be considered as equivalent to Higgs systems — a connection on a vector bundle E together with an End(E)-valued one form both in the standard and in the Conformal Affine case. We discuss how variations of Hodge structures can arise in such a framework and determine holomorphic embeddings of Riemann surfaces into locally homogeneous spaces, thus giving hints to possible realizations of W_n-geometries.     

Geometry of the Aharonov–Bohm Effect

We show that the connection responsible for any Abelian or non-Abelian Aharonov–Bohm effect with n parallel “magnetic” flux lines in ℝ³, lies in a trivial G-principal bundle P→M, i.e. P is isomorphic to the product M×G, where G is any path connected topological group; in particular a connected Lie group. We also show that two other bundles are involved: the universal covering space , where path integrals are computed, and the associated bundle P× _G ℂ ^m →M, where the wave function and its covariant derivative are sections.     

Symplectic Structures of Moduli Space¶of Higgs Bundles over a Curve and Hilbert Scheme¶of Points on the Canonical Bundle

The moduli space of triples of the form (E,θ,s) are considered, where (E,θ) is a Higgs bundle on a fixed Riemann surface X, and s is a nonzero holomorphic section of E. Such a moduli space admits a natural map to the moduli space of Higgs bundles simply by forgetting s. If (Y,L) is the spectral data for the Higgs bundle (E,θ), then s defines a section of the line bundle L over Y. The divisor of this section gives a point of a Hilbert scheme, parametrizing 0-dimensional subschemes of the total space of the canonical bundle K _X , since Y is a curve on K _X . The main result says that the pullback of the symplectic form on the moduli space of Higgs bundles to the moduli space of triples coincides with the pullback of the natural symplectic form on the Hilbert scheme using the map that sends any triple (E,θ,s) to the divisor of the corresponding section of the line bundle on the spectral curve. Received: 15 January 2000 / Accepted: 25 March 2001     

An Energy Gap for Yang-Mills Connections

Consider a Yang-Mills connection over a Riemann manifold M = M ⁿ , n ≥ 3, where M may be compact or complete. Then its energy must be bounded from below by some positive constant, if M satisfies certain conditions, unless the connection is flat.     

Quantum Yang-Mills on a Riemann surface

We obtain the quantum expectations of gauge-invariant functions of the connection on aG=SU(N) product bundle over a Riemann surface of genusg. We show that the spaceA/G _m of connections modulo those gauge transformations which are the identity at one point is itself a principal bundle with affine linear fiber. The base space Path^2g G consists of 2g-tuples of paths inG subject to a relation on their endpoint values. Quantum expectations are iterated path integrals over first the fiber then over Path^2g G, each with respect to the push-forward toA/G _m of the measuree ^–S(A) D A. Here,S(A) denotes the Yang-Mills action onA. We exhibit a global section ofA/G _m to define a choice of origin in each fiber, relative to which the measure on the fiber is Gaussian. The induced measure on Path^2g G is the product of Wiener measures on the component paths, conditioned to preserve the endopoint relation. Conformal transformations of the metric onM act by reparametrizing these paths. We explicitly compute the partition function in the general case and the expectations of functions of certain products of Wilson loops in the caseg=1.Research supported in part by DOE grant DE-FGO2-88ER25066     

Correlation Functions for MN/SN Orbifolds

We develop a method for computing correlation functions of twist operators in the bosonic 2-d CFT arising from orbifolds M ^N /S _N , where M is an arbitrary manifold. The path integral with twist operators is replaced by a path integral on a covering space with no operator insertions. Thus, even though the CFT is defined on the sphere, the correlators are expressed in terms of partition functions on Riemann surfaces with a finite range of genus g. For large N, this genus expansion coincides with a 1/N expansion. The contribution from the covering space of genus zero is “universal” in the sense that it depends only on the central charge of the CFT. For 3-point functions we give an explicit form for the contribution from the sphere, and for the 4-point function we do an example which has genus zero and genus one contributions. The condition for the genus zero contribution to the 3-point functions to be non-vanishing is similar to the fusion rules for an SU(2) WZW model. We observe that the 3-point coupling becomes small compared to its large N limit when the orders of the twist operators become comparable to the square root of N – this is a manifestation of the stringy exclusion principle. Received: 20 July 2000 / Accepted: 17 December 2000     

Universal Teichmüller Space and DiffS 1/S 1

We point out that the coset space DiffS ¹/S ¹ is a dense complex submanifold of the Universal Teichmüller SpaceS of compact Riemann spaces of genus g1. A holomorphic map ofS into the inifinite dimensional Segal diskD ₁ is constructed. This is the Universal analogue of the map of Teichmüller spaces into the Siegel disk provided by the period matrix. The Kähler potential for the general homogenous metric on DiffS ¹/S ¹ is computed explicitly using the map intoD ₁. Some applications to string theory are discussed.This work was supported in part by the U.S. Department of Energy Contract No. DE-AC02-76ER13065     

Witten–Reshetikhin–Turaev Invariants of¶Seifert Manifolds

For Seifert homology spheres, we derive a holomorphic function of K whose value at integer K is the sl ₂ Witten–Reshetikhin–Turaev invariant, Z _K , at q= exp 2πi/K. This function is expressed as a sum of terms, which can be naturally corresponded to the contributions of flat connections in the stationary phase expansion of the Witten–Chern–Simons path integral. The trivial connection contribution is found to have an asymptotic expansion in powers of K ⁻¹ which, for K an odd prime power, converges K-adically to the exact total value of the invariant Z _K at that root of unity. Evaluations at rational $K$ are also discussed. Using similar techniques, an expression for the coloured Jones polynomial of a torus knot is obtained, providing a trivial connection contribution which is an analytic function of the colour. This demonstrates that the stationary phase expansion of the Chern–Simons–Witten theory is exact for Seifert manifolds and for torus knots in S ³. The possibility of generalising such results is also discussed. Received: 26 October 1998 / Accepted: 1 March 1999     

Generic Metrics and Connections on Spin- and Spin c -Manifolds

We study the dependence of the dimension h ⁰(g,A) of the kernel of the Atyiah-Singer Dirac operator on a spin ^c -manifold M on the metric g and the connection A. The main result is that in the case of spin-structures the value of h ⁰(g) for the generic metric is given by the absolute value of the index provided . In dimension 2 the mod-2 index theorems have to be taken into a account and we obtain an extension of a classical result in the theory of Riemann surfaces. In the spin ^c -case we also discuss upper bounds on h ⁰(g,A) for generic metrics, and we obtain a complete result in dimension 2. The much simpler dependence on the connection A and applications to Seiberg–Witten theory are also discussed. Received: 3 July 1996 / Accepted: 27 February 1997     

On a covariant version of Caianiello’s model

Caianiello’s derivation of Quantum Geometry through an isometric embedding of the spacetime (M, g̃) in the pseudo-Riemannian structure (T*M, g* _AB ) is reconsidered. In the new derivation, using a non-linear connection and the bundle formalism, we obtain a Lorentzian-type structure in the 4-dimensional manifold M that is covariant under arbitrary local coordinate transformations in M. We obtain that if models with maximal acceleration are non-trivial, gravity should be supplied with other interactions in a unification framework.     

The Parameter Planes of λz<Superscript><Emphasis Type="Italic">m</Emphasis></Superscript> exp(z) for <Emphasis Type="Italic">m</Emphasis> ≥ 2*

We consider the families of entire transcendental maps given by F _λ,m (z) = λz ^m exp(z), where m ≥ 2. All functions F _λ,m have a superattracting fixed point at z = 0, and a critical point at z = −m. In the parameter planes we focus on the capture zones, i.e., λ values for which the critical point belongs to the basin of attraction of z = 0, denoted by A(0). In particular, we study the main capture zone (parameter values for which the critical point lies in the immediate basin, A ^*(0)) and prove that is bounded, connected and simply connected. All other capture zones are unbounded and simply connected. For each parameter λ in the main capture zone, A(0) consists of a single connected component with non-locally connected boundary. For all remaining values of λ, A ^*(0) is a quasidisk. On a different approach, we introduce some families of holomorphic maps of which serve as a model for F _λ,m , in the sense that they are related by means of quasiconformal surgery to F _λ,m . Both authors were supported by MTM2005-02139/Consolider (including a FEDER contribution) and CIRIT 2005 SGR01028. The first author was also supported by MTM2006-05849/Consolider (including a FEDER contribution).     

Gauge Theoretical Equivariant Gromov–Witten Invariants and the Full Seiberg–Witten Invariants¶of Ruled Surfaces

Let F be a differentiable manifold endowed with an almost K?hler structure (J,ω), α a J-holomorphic action of a compact Lie group on F, and K a closed normal subgroup of which leaves ω invariant. The purpose of this article is to introduce gauge theoretical invariants for such triples (F,α,K). The invariants are associated with moduli spaces of solutions of a certain vortex type equation on a Riemann surface Σ. Our main results concern the special case of the triple