Exact Results for Topological Strings on Resolved <Emphasis Type="Italic">Y</Emphasis><Superscript><Emphasis Type="Italic"> p</Emphasis>,<Emphasis Type="Italic">q</Emphasis></Superscript> Singularities

We obtain exact results in α′ for open and closed A-model topological string amplitudes on a large class of toric Calabi-Yau threefolds by using their correspondence with five dimensional gauge theories. The toric Calabi-Yaus that we analyze are obtained as minimal resolution of cones over Y ^p,q manifolds and give rise via M-theory compactification to SU(p) gauge theories on . As an application we present a detailed study of the local case and compute open and closed genus zero Gromov-Witten invariants of the orbifold. We also display the modular structure of the topological wave function and give predictions for higher genus amplitudes. The mirror curve in this case is the spectral curve of the relativistic A ₁ Toda chain. Our results also indicate the existence of a wider class of relativistic integrable systems associated to generic Y ^p,q geometries.     

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 .     

Quantisation of Twistor Theory by Cocycle Twist

We present the main ingredients of twistor theory leading up to and including the Penrose-Ward transform in a coordinate algebra form which we can then ‘quantise’ by means of a functorial cocycle twist. The quantum algebras for the conformal group, twistor space , compactified Minkowski space and the twistor correspondence space are obtained along with their canonical quantum differential calculi, both in a local form and in a global *-algebra formulation which even in the classical commutative case provides a useful alternative to the formulation in terms of projective varieties. We outline how the Penrose-Ward transform then quantises. As an example, we show that the pull-back of the tautological bundle on pulls back to the basic instanton on and that this observation quantises to obtain the Connes-Landi instanton on θ-deformed S ⁴ as the pull-back of the tautological bundle on our θ-deformed . We likewise quantise the fibration and use it to construct the bundle on θ-deformed that maps over under the transform to the θ-deformed instanton. The work was mainly completed while S.M. was visiting July-December 2006 at the Isaac Newton Institute, Cambridge, which both authors thank for support.     

Convergence of the Wick Star Product

We construct a Fréchet space as a subspace of where the Wick star product converges and is continuous. The resulting Fréchet algebra _ħ is studied in detail including a ^*-representation of _ħ in the Bargmann-Fock space and a discussion of star exponentials and coherent states.     

A Construction of Einstein-Weyl Spaces via LeBrun-Mason Type Twistor Correspondence

We construct infinitely many Einstein-Weyl structures on of signature (− + +) which is sufficiently close to the model case of constant curvature, and on which the space-like geodesics are all closed. Such a structure is obtained as a parameter space of a family of holomorphic disks which is associated to a small perturbation of the diagonal of . The geometry of constructed Einstein-Weyl spaces is well understood from the configuration of holomorphic disks. We also review Einstein-Weyl structures and their properties in the former half of this article. This work is partially supported by Grant-in-Aid for Scientific Research of the Japan Society for the Promotion of Science.     

Torelli Theorem for the Deligne–Hitchin Moduli Space

Fix integers g ≥ 3 and r ≥ 2, with r ≥ 3 if g = 3. Given a compact connected Riemann surface X of genus g, let denote the corresponding Deligne–Hitchin moduli space. We prove that the complex analytic space determines (up to an isomorphism) the unordered pair , where is the Riemann surface defined by the opposite almost complex structure on X.     

Parametric Representation of Noncommutative Field Theory

In this paper we investigate the Schwinger parametric representation for the Feynman amplitudes of the recently discovered renormalizable quantum field theory on the Moyal non commutative space. This representation involves new hyperbolic polynomials which are the non-commutative analogs of the usual “Kirchoff” or “Symanzik” polynomials of commutative field theory, but contain richer topological information. Work supported by ANR grant NT05-3-43374 “GenoPhy”.     

A Kolmogorov Extension Theorem for POVMs

We prove a theorem about positive-operator-valued measures (POVMs) that is an analog of the Kolmogorov extension theorem, a standard theorem of probability theory. According to our theorem, if a sequence of POVMs G _n on satisfies the consistency (or projectivity) condition then there is a POVM G on the space of infinite sequences that has G _n as its marginal for the first n entries of the sequence. We also describe an application in quantum theory. The main proof in this article was first formulated in my habilitation thesis [6].     

Space-like hypersurfaces with positive constant r-mean curvature in Lorentzian product spaces

In this paper we obtain a height estimate concerning compact space-like hypersurfaces Σ ⁿ immersed with some positive constant r-mean curvature into an (n + 1)-dimensional Lorentzian product space , and whose boundary is contained into a slice {t} × M ⁿ . By considering the hyperbolic caps of the Lorentz–Minkowski space , we show that our estimate is sharp. Furthermore, we apply this estimate to study the complete space-like hypersurfaces immersed with some positive constant r-mean curvature into a Lorentzian product space. For instance, when the ambient space–time is spatially closed, we show that such hypersurfaces must satisfy the topological property of having more than one end which constitutes a necessary condition for their existence.     

Baxter Operator and Archimedean Hecke Algebra

In this paper we introduce Baxter integral -operators for finite-dimensional Lie algebras and . Whittaker functions corresponding to these algebras are eigenfunctions of the -operators with the eigenvalues expressed in terms of Gamma-functions. The appearance of the Gamma-functions is one of the manifestations of an interesting connection between Mellin-Barnes and Givental integral representations of Whittaker functions, which are in a sense dual to each other. We define a dual Baxter operator and derive a family of mixed Mellin-Barnes-Givental integral representations. Givental and Mellin-Barnes integral representations are used to provide a short proof of the Friedberg-Bump and Bump conjectures for G = GL(ℓ + 1) proved earlier by Stade. We also identify eigenvalues of the Baxter -operator acting on Whittaker functions with local Archimedean L-factors. The Baxter -operator introduced in this paper is then described as a particular realization of the explicitly defined universal Baxter operator in the spherical Hecke algebra , K being a maximal compact subgroup of G. Finally we stress an analogy between -operators and certain elements of the non-Archimedean Hecke algebra .     

Hyperbolic Calorons, Monopoles, and Instantons

We construct families of SO(3)-symmetric charge 1 instantons and calorons on the space . We show how the calorons include instantons and hyperbolic monopoles as limiting cases. We show how Euclidean calorons are the flat space limit of this family. Supported by PPARC.     

Cardy Condition for Open-Closed Field Algebras

Let V be a vertex operator algebra satisfying certain reductivity and finiteness conditions such that , the category of V-modules, is a modular tensor category. We study open-closed field algebras over V equipped with nondegenerate invariant bilinear forms for both open and closed sectors. We show that they give algebras over a certain -extension of the so-called Swiss-cheese partial dioperad, and we can obtain Ishibashi states easily in such algebras. The Cardy condition can be formulated as an additional condition on such open-closed field algebras in terms of the action of the modular transformation on the space of intertwining operators of V. We then derive a graphical representation of S in the modular tensor category . This result enables us to give a categorical formulation of the Cardy condition and the modular invariance condition for 1-point correlation functions on the torus. Then we incorporate these two conditions and the axioms of the open-closed field algebra over V equipped with nondegenerate invariant bilinear forms into a tensor-categorical notion called the Cardy -algebra. In the end, we give a categorical construction of the Cardy -algebra in the Cardy case.     

WZW Orientifolds and Finite Group Cohomology

The simplest orientifolds of the WZW models are obtained by gauging a symmetry group generated by a combined involution of the target Lie group G and of the worldsheet. The action of the involution on the target is by a twisted inversion , where ζ is an element of the center of G. It reverses the sign of the Kalb-Ramond torsion field H given by a bi-invariant closed 3-form on G. The action on the worldsheet reverses its orientation. An unambiguous definition of Feynman amplitudes of the orientifold theory requires a choice of a gerbe with curvature H on the target group G, together with a so-called Jandl structure introduced in [31]. More generally, one may gauge orientifold symmetry groups that combine the -action described above with the target symmetry induced by a subgroup Z of the center of G. To define the orientifold theory in such a situation, one needs a gerbe on G with a Z-equivariant Jandl structure. We reduce the study of the existence of such structures and of their inequivalent choices to a problem in group-Γ cohomology that we solve for all simple simply connected compact Lie groups G and all orientifold groups . Membre du C.N.R.S.     

Moment Bounds for the Smoluchowski Equation and their Consequences

We prove bounds on moments of the Smoluchowski coagulation equations with diffusion, in any dimension d ≥ 1. If the collision propensities α(n, m) of mass n and mass m particles grow more slowly than , and the diffusion rate is non-increasing and satisfies for some b ₁ and b ₂ satisfying 0 ≤ b ₂ < b ₁ < ∞, then any weak solution satisfies for every and T ∈(0, ∞), (provided that certain moments of the initial data are finite). As a consequence, we infer that these conditions are sufficient to ensure uniqueness of a weak solution and its conservation of mass. This work was performed while A.H. held a postdoctoral fellowship in the Department of Mathematics at U.B.C. This work is supported in part by NSF grant DMS0307021.     

A Uniform Quantum Version of the Cherry Theorem

Consider in the operator family . P ₀ is the quantum harmonic oscillator with diophantine frequency vector ω, F ₀ a bounded pseudodifferential operator with symbol decreasing to zero at infinity in phase space, and . Then there exist independent of and an open set such that if and , the quantum normal form near P ₀ converges uniformly with respect to . This yields an exact quantization formula for the eigenvalues, and for the classical Cherry theorem on convergence of Birkhoff’s normal form for complex frequencies is recovered. Partially supported by PAPIIT-UNAM IN106106-2.     

On Genus Two Riemann Surfaces Formed from Sewn Tori

We describe the period matrix and other data on a higher genus Riemann surface in terms of data coming from lower genus surfaces via an explicit sewing procedure. We consider in detail the construction of a genus two Riemann surface by either sewing two punctured tori together or by sewing a twice-punctured torus to itself. In each case the genus two period matrix is explicitly described as a holomorphic map from a suitable domain (parameterized by genus one moduli and sewing parameters) to the Siegel upper half plane . Equivariance of these maps under certain subgroups of is shown. The invertibility of both maps in a particular domain of is also shown. Support provided by the National Science Foundation DMS-0245225, and the Committee on Research at the University of California, Santa Cruz. Supported by The Millenium Fund, National University of Ireland, Galway.     

Random Graph Asymptotics on High-Dimensional Tori

We investigate the scaling of the largest critical percolation cluster on a large d-dimensional torus, for nearest-neighbor percolation in sufficiently high dimensions, or when d > 6 for sufficiently spread-out percolation. We use a relatively simple coupling argument to show that this largest critical cluster is, with high probability, bounded above by a large constant times V ^2/3 and below by a small constant times , where V is the volume of the torus. We also give a simple criterion in terms of the subcritical percolation two-point function on under which the lower bound can be improved to small constant times , i.e. we prove random graph asymptotics for the largest critical cluster on the high-dimensional torus. This establishes a conjecture by [1], apart from logarithmic corrections. We discuss implications of these results on the dependence on boundary conditions for high-dimensional percolation. Our method is crucially based on the results in [11, 12], where the scaling was proved subject to the assumption that a suitably defined critical window contains the percolation threshold on . We also strongly rely on mean-field results for percolation on proved in [17–20].     

Multipoint Series of Gromov–Witten Invariants of CP1

We derive explicit formulas for the multipoint series of in degree 0 from the Toda hierarchy, using the recursions of the Toda hierarchy. The Toda equation then yields inductive formulas for the higher degree multipoint series of . We also obtain explicit formulas for the Hodge integrals , in the cases i=0 and 1.     

On the Uniqueness of Weak Weyl Representations of the Canonical Commutation Relation

Let (T, H) be a weak Weyl representation of the canonical commutation relation (CCR) with one degree of freedom. Namely T is a symmetric operator and H is a self-adjoint operator on a complex Hilbert space satisfying the weak Weyl relation: for all (the set of real numbers), e^−itH D(T) ⊂ D(T) (i is the imaginary unit and D(T) denotes the domain of T) and . In the context of quantum theory where H is a Hamiltonian, T is called a strong time operator of H. In this paper we prove the following theorem on uniqueness of weak Weyl representations: Let be separable. Assume that H is bounded below with and , where is the set of complex numbers and, for a linear operator A on a Hilbert space, σ(A) denotes the spectrum of A. Then ( is the closure of T) is unitarily equivalent to a direct sum of the weak Weyl representation on the Hilbert space , where is the multiplication operator by the variable and with . Using this theorem, we construct a Weyl representation of the CCR from the weak Weyl representation . This work is supported by the Grant-in-Aid No.17340032 for Scientific Research from Japan Society for the Promotion of Science (JSPS).