Nonsemisimple Fusion Algebras and the Verlinde Formula

We find a nonsemisimple fusion algebra associated with each (1, p) Virasoro model. We present a nonsemisimple generalization of the Verlinde formula which allows us to derive from modular transformations of characters.     

Fusion Rings of Loop Group Representations

We compute the fusion rings of positive energy representations of the loop groups of the simple, simply connected Lie groups.     

D-Branes,RR-Fields and Duality on Noncommutative Manifolds

We develop some of the ingredients needed for string theory on noncommutative spacetimes, proposing an axiomatic formulation of T-duality as well as establishing a very general formula for D-brane charges. This formula is closely related to a noncommutative Grothendieck-Riemann-Roch theorem that is proved here. Our approach relies on a very general form of Poincaré duality, which is studied here in detail. Among the technical tools employed are calculations with iterated products in bivariant -theory and cyclic theory, which are simplified using a novel diagram calculus reminiscent of Feynman diagrams.     

Hilbert Schemes, Separated Variables, and D-Branes

We explain Sklyanin's separation of variables in geometrical terms and construct it for Hitchin and Mukai integrable systems. We construct Hilbert schemes of points on T ^*Σ for Σ=C, C ^* or elliptic curve, and on C ²/Γ and show that their complex deformations are integrable systems of Calogero–Sutherland–Moser type. We present the hyperk?hler quotient constructions for Hilbert schemes of points on cotangent bundles to the higher genus curves, utilizing the results of Hurtubise, Kronheimer and Nakajima. Finally we discuss the connections to physics of D-branes and string duality. Received: 2 November 2000 / Accepted: 7 May 2001     

Computation of Superpotentials for D-Branes

We present a general method for the computation of tree-level superpotentials for the world-volume theory of B-type D-branes. This includes quiver gauge theories in the case that the D-brane is marginally stable. The technique involves analyzing the A_∞-structure inherent in the derived category of coherent sheaves. This effectively gives a practical method of computing correlation functions in holomorphic Chern–Simons theory. As an example, we give a more rigorous proof of previous results concerning 3-branes on certain singularities including conifolds. We also provide a new example.     

Geometric K-Homology of Flat D-Branes

We use the Baum-Douglas construction of K-homology to explicitly describe various aspects of D-branes in Type II superstring theory in the absence of background supergravity form fields. We rigorously derive various stability criteria for states of D-branes and show how standard bound state constructions are naturally realized directly in terms of topological K-cycles. We formulate the mechanism of flux stabilization in terms of the K-homology of non-trivial fibre bundles. Along the way we derive a number of new mathematical results in topological K-homology of independent interest.     

Massless D-Branes on Calabi–Yau Threefolds and Monodromy

We analyze the link between the occurrence of massless B-type D-branes for specific values of moduli and monodromy around such points in the moduli space. This allows us to propose a classification of all massless B-type D-branes at any point in the moduli space of Calabi–Yau’s. This classification then justifies a previous conjecture due to Horja for the general form of monodromy. Our analysis is based on using monodromies around points in moduli space where a single D-brane becomes massless to generate monodromies around points where an infinite number become massless. We discuss the various possibilities within the classification.     

Cylindric Versions of Specialised Macdonald Functions and a Deformed Verlinde Algebra

We define cylindric versions of skew Macdonald functions P _λ/μ (q, t) for the special cases q = 0 or t = 0. Fixing two integers n > 2 and k > 0 we shift the skew diagram λ/μ, viewed as a subset of the two-dimensional integer lattice, by the period vector (n, ?k). Imposing a periodicity condition one defines cylindric skew tableaux and associated weight functions. The resulting weighted sums over these cylindric tableaux are symmetric functions. They appear in the coproduct of a commutative Frobenius algebra which is a particular quotient of the spherical Hecke algebra. We realise this Frobenius algebra as a commutative subalgebra in the endomorphisms over a ${U_{q}\widehat{\mathfrak{sl}}(n)}$ Kirillov-Reshetikhin module. Acting with special elements of this subalgebra, which are noncommutative analogues of Macdonald polynomials, on a highest weight vector, one obtains Lusztig’s canonical basis. In the limit q = t = 0, this Frobenius algebra is isomorphic to the ${\widehat{\mathfrak{sl}}(n)}$ Verlinde algebra at level k, i.e. the structure constants become the ${\widehat{\mathfrak{sl}}(n)_{k}}$ Wess-Zumino-Novikov-Witten fusion coefficients. Further motivation comes from exactly solvable lattice models in statistical mechanics: the cylindric Macdonald functions discussed here arise as partition functions of so-called vertex models obtained from solutions to the Yang-Baxter equation. We show this by stating explicit bijections between cylindric tableaux and lattice configurations of non-intersecting paths. Using the algebraic Bethe ansatz the idempotents of the Frobenius algebra are computed.     

Intersection Numbers on Moduli Spaces and Symmetries of a Verlinde Formula

We investigate the geometry and topology of a standard moduli space of stable bundles on a Riemann surface, and use a generalization of the Verlinde formula to derive results on intersection pairings. Received: 5 April 1996 / Accepted: 6 February 1997     

Space-Time Fluctuations Induced by D-Branes and Their Effects on Neutrino Oscillations

Neutrino oscillations are analyzed in the Ellis-Mavromatos-Nanopoulos-Volkov (ENMV) model, where the quantum gravitational fluctuations of the space-time background are described by virtual D branes. Such fluctuations may induce neutrino oscillations if a violation of the equivalence principle or a tiny violation of the Lorentz invariance is imposed. In this framework, the oscillation length of neutrinos turns out to be proportional to E ^–2 M, where E is the neutrino energy and M is the energy which is the scale characterizing the topological fluctuations in the vacuum.     

The Formulae of Kontsevich and Verlinde¶from the Perspective of the Drinfeld Double

A two dimensional gauge theory is canonically associated to every Drinfeld double. For particular doubles, the theory turns out to be e.g. the ordinary Yang–Mills theory, the G/G gauged WZNW model or the Poisson σ-model that underlies the Kontsevich quantization formula. We calculate the arbitrary genus partition function of the latter. The result is the q-deformation of the ordinary Yang–Mills partition function in the sense that the series over the weights is replaced by the same series over the q-weights. For q equal to a root of unity the series acquires the affine Weyl symmetry and its truncation to the alcove coincides with the Verlinde formula. Received: 10 December 1999 / Accepted: 8 October 2000     

Bohr-sommerfeld orbits in the moduli space of flat connections and the Verlinde dimension formula

We show how the moduli space of flatSU(2) connections on a two-manifold can be quantized in the real polarization of [15], using the methods of [6]. The dimension of the quantization, given by the number of integral fibres of the polarization, matches the Verlinde formula, which is known to give the dimension of the quantization of this space in a Kähler polarization.Supported in part by MSRI under NSF Grant 85-05550Supported in part by an NSF Graduate fellowship, and by a grant-in-aid from the J. Seward Johnson Charitable TrustSupported in part by NSF Mathematical Sciences Postdoctoral Research Fellowship DMS 88-07291. Address as of January 1, 1993: Department of Mathematics, Columbia University, New York, NY 10027, USA     

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     

A Spin Decomposition of the Verlinde Formulas for Type A Modular Categories

A modular category is a braided category with some additional algebraic features. The interest of this concept is that it provides a Topological Quantum Field Theory in dimension 3. The Verlinde formulas associated with a modular category are the dimensions of the TQFT modules. We discuss reductions and refinements of these formulas for modular categories related with SU(N). Our main result is a splitting of the Verlinde formula, corresponding to a brick decomposition of the TQFT modules whose summands are indexed by spin structures modulo an even integer. We introduce here the notion of a spin modular category, and give the proof of the decomposition theorem in this general context.     

Entropy bounds and Cardy–Verlinde formula in Yang–Mills theory

Using gauge formulation of gravity the three-dimensional SU(2) YM theory equations of motion are presented in equivalent form as FRW cosmological equations. With the radiation, the particular (periodic, big bang – big crunch) three-dimensional universe is constructed. Cosmological entropy bounds (so-called Cardy–Verlinde formula) have the standard form in such universe. Mapping such universe back to YM formulation we got the thermal solution of YM theory. The corresponding holographic entropy bounds (Cardy–Verlinde formula) in YM theory are constructed. This indicates to universal character of holographic relations.     

Remarks on the structure constants of the Verlinde algebra associated to sl3

We present a new formula for the structure constants of the Verlinde algebra associated to sl₃. We show that after an affine change of variables the structure constants, considered as a function of highest weights, become the weight function of a suitable sl₃ highest weight representation.Supported in part by NFS grants DMS-9400841 and DMS-9203929.     

Symmetric and asymmetric scission properties

Existence of two kinds of scission configurations associated with the symmetric and asymmetric fission modes is pointed out in the fission of actinides: elongated and compact configurations. Each symmetric and asymmetric scission property is discussed in terms of shape elongation evaluated from fragment total kinetic energy (TKE). It is found that the shape elongation associated with the asymmetric fission mode is nearly constant for a wide range of fissioning nuclei, while that with the symmetric one is also constant except for low-energy induced fission and spontaneous fission (SF) of heavy nuclei. Prom the systematic study of the scission properties, the bimodal fission observed in SF of the heavy actinides is interpreted as the presence of the two fission paths of the ordinary asymmetric mode and a strongly shell-influenced symmetric one.