A Vertex Formalism for Local Ruled Surfaces

We develop a vertex formalism for topological string amplitudes on ruled surfaces with an arbitrary number of reducible fibers embedded in a Calabi-Yau threefold. Our construction is based on large N duality and localization with respect to a degenerate torus action. We also discuss potential generalizations of our formalism to a broader class of Calabi-Yau threefolds using the same underlying principles.     

On the Birational Invariance of the BCOV Torsion of Calabi-Yau Threefolds

Fang et al. (J. Diff. Geom. 80(2):175–259, 2008, Sect. 4, Conj. 4.17) conjecture that a certain spectral string-theoretic invariant of Calabi-Yau threefolds (the BCOV torsion) is a birational invariant. We prove a weak form of this conjecture. The proof combines the arithmetic Riemann-Roch theorem in Arakelov geometry with some inputs from motivic integration theory.     

Low-energy analysis of M- and F-theories on Calabi-Yau threefolds

We elucidate the interplay between gauge and supersymmetry anomalies in six-dimensional N = 1 supergravity with generalized couplings between tensor and vector multiplets. We derive the structure of the five-dimensional supergravity resulting from the S₁ reduction of these models and give the constraints on Chem-Simons couplings that follow from duality to M-theory compactified on a Calabi-Yau threefold. The duality is supported only on a restricted class of Calabi-Yau threefolds and requires a special type of intersection form. We derive five-dimensional central-charge formulas and briefly discuss the associated phase transitions. Finally, we exhibit connections with F-theory compactifications on Calabi-Yau manifolds that admit elliptic fibrations. This analysis suggests that F-theory unifies type-IIb three-branes and M-theory five-branes.     

Vertex Algebras,Mirror Symmetry,and D-Branes: The Case of Complex Tori

 A vertex algebra is an algebraic counterpart of a two-dimensional conformal field theory. We give a new definition of a vertex algebra which includes chiral algebras as a special case, but allows for fields which are neither meromorphic nor anti-meromorphic. To any complex torus equipped with a flat K?hler metric and a closed 2-form we associate an N=2 superconformal vertex algebra (N=2 SCVA) in the sense of our definition. We find a criterion for two different tori to produce isomorphic N=2 SCVA's. We show that for algebraic tori the isomorphism of N=2 SCVA's implies the equivalence of the derived categories of coherent sheaves corresponding to the tori or their noncommutative generalizations (Azumaya algebras over tori). We also find a criterion for two different tori to produce N=2 SCVA's related by a mirror morphism. If the 2-form is of type (1,1), this condition is identical to the one proposed by Golyshev, Lunts, and Orlov, who used an entirely different approach inspired by the Homological Mirror Symmetry Conjecture of Kontsevich. Our results suggest that Kontsevich's conjecture must be modified: coherent sheaves must be replaced with modules over Azumaya algebras, and the Fukaya category must be ``twisted' by a closed 2-form. We also describe the implications of our results for BPS D-branes on Calabi-Yau manifolds. Received: 3 May 2001 / Accepted: 17 August 2002 Published online: 8 January 2003     

All Loop Topological String Amplitudes from Chern-Simons Theory

We demonstrate the equivalence of all loop closed topological string amplitudes on toric local Calabi-Yau threefolds with computations of certain knot invariants for Chern-Simons theory. We use this equivalence to compute the topological string amplitudes in certain cases to very high degree and to all genera. In particular we explicitly compute the topological string amplitudes for ² up to degree 12 and ¹× ¹ up to total degree 10 to all genera. This also leads to certain novel large N dualities in the context of ordinary superstrings, involving duals of type II superstrings on local Calabi-Yau three-folds without any fluxes.This research is supported in part by NSF grants PHY-9802709 and DMS-0074329     

Dirac Oscillator in Noncommutative Phase Space and (Anti)-Jaynes-Cummings Models

We study planar Dirac oscillator in noncommutative phase space. The model is solved exactly. The relation between this model and Jaynes-Cummings (JC) or anti-Jaynes-Cummings (AJC) models is investigated. We find that the behaviors of this model depend qualitatively on the signs of a dimensionless parameter κ. For a negative κ, we find that there is a map from this model to a model which contains only AJC terms. However, for a positive κ, there is a map from this model to a model which contains both AJC and JC terms simultaneously. Our investigation may afford a new way to study the noncommutative Dirac oscillator by means of quantum optics method, and vice verse.     

Stability and Duality in {\mathcal{N}\,{=}\,2} Supergravity

The BPS-spectrum is known to change when moduli cross a wall of marginal stability. This paper tests the compatibility of wall-crossing with S-duality and electric-magnetic duality for N=2{\mathcal{N}=2} supergravity. To this end, the BPS-spectrum of D4-D2-D0 branes is analyzed in the large volume limit of Calabi-Yau moduli space. Partition functions are presented, which capture the stability of BPS-states corresponding to two constituents with primitive charges and supported on very ample divisors in a compact Calabi-Yau. These functions are “mock modular invariant” and therefore confirm S-duality. Furthermore, wall-crossing preserves electric-magnetic duality, but is shown to break the “spectral flow” symmetry of the N=(4,0){\mathcal{N}=(4,0)} CFT, which captures the degrees of freedom of a single constituent.     

Complexified Gravity in Noncommutative Spaces

The presence of a constant background antisymmetric tensor for open strings or D-branes forces the space-time coordinates to be noncommutative. This effect is equivalent to replacing ordinary products in the effective theory by the deformed star product. An immediate consequence of this is that all fields get complexified. The only possible noncommutative Yang–Mills theory is the one with U(N) gauge symmetry. By applying this idea to gravity one discovers that the metric becomes complex. We show in this article that this procedure is completely consistent and one can obtain complexified gravity by gauging the symmetry U(1,D−1) instead of the usual SO(1,D−1). The final theory depends on a Hermitian tensor containing both the symmetric metric and antisymmetric tensor. In contrast to other theories of nonsymmetric gravity the action is both unique and gauge invariant. The results are then generalized to noncommutative spaces. Received: 1 June 2000 / Accepted: 27 November 2000     

Compactifications of F-theory on Calabi-Yau threefolds (II)

We continue our study of compactifications of F-theory on Calabi-Yau threefolds. We gain more insight into F-theory duals of heterotic strings and provide a recipe for building F-theory duals for arbitrary heterotic compactifications on elliptically fibered manifolds. As a byproduct we find that string/string duality in six dimensions gets mapped to fiber/base exchange in F-theory. We also construct a number of new N = 1, d = 6 examples of F-theory vacua and study transitions among them. We find that some of these transition points correspond upon further compactification to 4 dimensions to transitions through analogues of Argyres-Douglas points of N = 2 moduli. A key idea in these transitions is the notion of classifying (0,4) fivebranes of heterotic strings.     

Noncommutative space-time models

The FRT quantum Euclidean spaces O _q ^N are formulated in terms of Cartesian generators. The quantum analogs of N-dimensional Cayley-Klein spaces are obtained by contractions and analytical continuations. Noncommutative constant-curvature spaces are introduced as spheres in the quantum Cayley-Klein spaces. For N = 5 part of them is interpreted as the noncommutative analogs of (1+3) space-time models. As a result the quantum (anti) de Sitter, Minkowski, Newton, Galilei kinematics with the fundamental length and the fundamental time are suggested. Presented at the International Colloquium “Integrable Systems and Quantum Symmetries”, Prague, 16–18 June 2005.     

Noncommutative Bianchi Type II Quantum Cosmology

In this paper we present the noncommutative Bianchi Class A cosmological models coupled to barotropic perfect fluid. The commutative and noncommutative quantum solution to the Wheeler–DeWitt equation for any factor ordering, to the anisotropic Bianchi type II cosmological model are found, using a stiff fluid (γ=1). In our toy model, we introduce noncommutative scale factors, is say, we consider that all minisuperspace variables q ⁱ does not commute, so the simplectic structure was modified.     

Noncommutative Instantons and Twistor Transform

Recently N. Nekrasov and A. Schwarz proposed a modification of the ADHM construction of instantons which produces instantons on a noncommutative deformation of ℝ⁴. In this paper we study the relation between their construction and algebraic bundles on noncommutative projective spaces. We exhibit one-to-one correspondences between three classes of objects: framed bundles on a noncommutative ℙ², certain complexes of sheaves on a noncommutative ℙ³, and the modified ADHM data. The modified ADHM construction itself is interpreted in terms of a noncommutative version of the twistor transform. We also prove that the moduli space of framed bundles on the noncommutative ℙ² has a natural hyperk?hler metric and is isomorphic as a hyperk?hler manifold to the moduli space of framed torsion free sheaves on the commutative ℙ². The natural complex structures on the two moduli spaces do not coincide but are related by an SO(3) rotation. Finally, we propose a construction of instantons on a more general noncommutative ℝ⁴ than the one considered by Nekrasov and Schwarz (a q-deformed ℝ⁴). Received: 3 May 2000 / Accepted: 3 April 2001     

Fermionic screenings and line bundle twisted chiral de Rham complex on CY manifolds

We present a generalization of Borisov’s construction of the chiral de Rham complex in the case of the line-bundle-twisted chiral de Rham complex on a Calabi-Yau hypersurface in a projective space. We generalize the differential associated with a polytope Δ of the projective space ℙ ^{d − 1} by allowing nonzero modes for the screening currents forming this differential. It is shown that the numbers of screening current modes define the support function of the toric divisor of a line bundle on ℙ ^{d − 1} that twists the chiral de Rham complex on the Calabi-Yau hypersurface.     

The Existence and Stability of Noncommutative Scalar Solitons

 We establish existence and stability results for solitons in noncommutative scalar field theories in even space dimension 2d. In particular, for any finite rank spectral projection P of the number operator 𝒩 of the d-dimensional harmonic oscillator and sufficiently large noncommutativity parameter θ we prove the existence of a rotationally invariant soliton which depends smoothly on θ and converges to a multiple of P as θ→∞. In the two-dimensional case we prove that these solitons are stable at large θ, if P=P _N , where P _N projects onto the space spanned by the N+1 lowest eigenstates of 𝒩, and otherwise they are unstable. We also discuss the generalisation of the stability results to higher dimensions. In particular, we prove stability of the soliton corresponding to P=P ₀ for all θ in its domain of existence. Finally, for arbitrary d and small values of θ, we prove without assuming rotational invariance that there do not exist any solitons depending smoothly on θ. Received: 13 July 2001 / Accepted: 9 July 2002 Published online: 10 January 2003     

Higher-dimensional perfect fluids and empty singular boundaries

In order to find out whether empty singular boundaries can arise in higher dimensional Gravity, we study the solution of Einstein’s equations consisting in a (N + 2)-dimensional static and hyperplane symmetric perfect fluid satisfying the equation of state ρ = ηp, being η an arbitrary constant and N ≥ 2. We show that this spacetime has some weird properties. In particular, in the case η > −1, it has an empty (without matter) repulsive singular boundary. We also study the behavior of geodesics and the Cauchy problem for the propagation of massless scalar field in this spacetime. For η > 1, we find that only vertical null geodesics touch the boundary and bounce, and all of them start and finish at z = ∞; whereas non-vertical null as well as all time-like ones are bounded between two planes determined by initial conditions. We obtain that the Cauchy problem for the propagation of a massless scalar field is well-posed and waves are completely reflected at the singularity, if we only demand the waves to have finite energy, although no boundary condition is required.     

Hitchin–Kobayashi Correspondence, Quivers, and Vortices

 A twisted quiver bundle is a set of holomorphic vector bundles over a complex manifold, labelled by the vertices of a quiver, linked by a set of morphisms twisted by a fixed collection of holomorphic vector bundles, labelled by the arrows. When the manifold is K?hler, quiver bundles admit natural gauge-theoretic equations, which unify many known equations for bundles with extra structure. In this paper we prove a Hitchin–Kobayashi correspondence for twisted quiver bundles over a compact K?hler manifold, relating the existence of solutions to the gauge equations to a stability criterion, and consider its application to a number of situations related to Higgs bundles and dimensional reductions of the Hermitian–Einstein equations. Received: 10 December 2001 / Accepted: 10 November 2002 Published online: 28 May 2003 RID="⋆" ID="⋆" Current address: Mathematical Sciences, University of Bath, Bath, BA2 7AY, UK. E-mail:L.Alvarez-Consul@maths.bath.ac.uk RID="⋆⋆" ID="⋆⋆" Current address: Instituto de Matemáticas y Física Fundamental, CSIC, Serrano 113 bis, 28006 Madrid, Spain. E-mail:oscar.garcia-prada@uam.es Communicated by R.H. Dijkgraaf     

Covariant formulation of BPS black holes and the scalar weak gravity conjecture

We find through a systematic analysis that all but 29,223 of the 473.8 million 4D reflexive polytopes found by Kreuzer and Skarke have a 2D reflexive subpolytope. Such a subpolytope is generally associated with the presence of an elliptic or genus one fibration in the corresponding birational equivalence class of Calabi-Yau threefolds. This extends the growing body of evidence that most Calabi-Yau threefolds have an elliptically fibered phase.