BRST quantization ofN = 2, 3, and 4 superconformal theories on higher-genus Riemann surfaces

Complex contour integral techniques, developed in a previous paper for theN=0 and 1 superconformal theories on higher-genus Riemann surfaces, are applied to a Becchi-Rouet-Stora-Tyutin (BRST) quantization procedure of superconformal theories withN=2, 3, and 4 super-Krichever-Novikov (KN) constraint algebras on a genus-g Riemann surface. The BRST charges of the superconformai theories are constructed and the nilpotency of the BRST charges is checked. The critical spacetime dimension and the intercepts are found for theN=2 and 4 cases. Also calculated are the central charge and the intercept for theN=3 case.     

Extended super-Kač-Moody algebras and their super-derivation algebras

We study theN-extended super-Ka-Moody algebras, i.e. extensions of the Lie algebra of the loop group over the super-circleA _N . The extensions are characterized by 2-cocycles which are computed in terms of the cyclic cohomology of the Grassmann algebra withN generators. The graded algebra of super-derivations compatible with each extension is determined. The casesN=1,2,3 are examined in detail and their relation with the Ademollo et al. superconformal algebras is discussed. We examine the possibility of defining new superconformal algebras which, forN>1, generalize theN=1 Ramond-Neveu-Schwarz algebra.     

N=2 supergravity,type IIB superstrings,and algebraic geometry

The geometry ofN=2 supergravity is related to the variations of Hodge structure for formal Calabi-Yau spaces. All known results in this branch of algebraic geometry are easily recovered from supersymmetry arguments. This identification has a physical meaning for a type IIB superstring compactified on a Calabi-Yau 3-fold. We give exact (non-perturbative) results for the string effective lagrangian. Our geometrical framework suggests a re-formulation of the Gepner conjecture about (2,2) superconformal theories as the solution to theSchottky problem for algebraic complex manifolds having trivial canonical bundle.     

Homogeneous Kähler manifolds andT-algebras inN=2 supergravity and superstrings

Motivated by the problem of the moduli space of superconformal theories, we classify all the (normal) homogeneous Kähler spaces which are allowed in the coupling of vector multiplets toN=2 SUGRA. Such homogeneous spaces are in one-to-one correspondence with the homogeneous quaternionic spaces (H ⁿ ) found by Alekseevskii. There are two infinite families of homogeneous non-symmetric spaces, each labelled by two integers. We construct explicitly the corresponding supergravity models. They are described by acubic functionF, as in flat-potential models. They are Kähler-Einstein if and only if they are symmetric. We describe in detail the geometry of the relevant manifolds. They are Siegel (bounded) domains of the first type. We discuss the physical relevance of this class of bounded domains for string theory and the moduli geometry. Finally, we introduce theT-algebraic formalism of Vinberg to describe in an efficient way the geometry of these manifolds. The homogeneous spaces allowed inN=2 SUGRA are associated to rank 3T-algebras in exactly the same way as the symmetric spaces are related to Jordan algebras. We characterize theT-algebras allowed inN=2 supergravity. They are those for which theungraded determinant is a polynomial in the matrix entries. The Kähler potential is simply minus the logarithm of this naive determinant.     

Polynomial identities,indices, and duality for theN=1 superconformal modelSM(2, 4v)

We prove polynomial identities for theN=1 superconformal modelSM(2, 4v) which generalize and extend the known Fermi/Bose character identities. Our proof uses theq-trinomial coefficients of Andrews and Baxter on the bosonic side and a recently introduced very general method of producing recursion relations forq-series on the fermionic side. We use these polynomials to demonstrate a dual relation underqq ^–1 betweenSM(2, 4v) andM(2v–1, 4v). We also introduce a genralization of the Witten index which is expressible in terms of the Rogers false theta functions.Dedicated to the memory of Claude Itzykson.     

New identities between unitary minimal Virasoro characters

Two sets of identities between unitary minimal Virasoro characters at levelsm=3, 4, 5 are presented and proven. The first identity suggests a connection between the Ising and the tricritical Ising models since them=3 Virasoro characters are obtained as bilinears ofm=4 Virasoro characters. The second identity given the tricritical Ising model characters as bilinears in the Ising model characters and the six combinations ofm=5 Virasoro characters which do not appear in the spectrum of the three state Potts model. The implication of these identities on the study of the branching rules ofN=4 superconformal characters into characters is discussed.     

N=2 affine superalgebras and Hamiltonian reduction inN=2 superspace

We constructN=2 affine current algebras for the superalgebrassl(n/n-1)⁽¹⁾ in terms ofN=2 supercurrents subjected to nonlinear constraints and discuss the general procedure of the hamiltonian reduction inN=2 superspace at the classical level. We consider in detail the simplest case ofN=2sl(2/1)⁽¹⁾ and show howN=2 superconformal algebra inN=2 superspace follows via the hamiltonian reduction. Applying the hamiltonian reduction to the case ofN=2sl(3/2)⁽¹⁾, we find two new extendedN=2 superconformal algebras in a manifestly supersymmetricN=2 superfield form. Decoupling of four component currents of dimension 1/2 in them yields, respectively,u(2/1) andu(3) Knizhnik-Bershadsky superconformal algebras. We also discuss how theN=2 superfield formulations ofN=2W ₃ andN=2W ₃ ⁽²⁾ superconformal algebras come out in this framework, as well as some unusual extendedN=2 superconformal algebras containing constrainedN=2 stress tensor and/or spin 0 supercurrents.     

Exact generator coordinate wave functions for some simple hamiltonians

We obtain the superconformal transformation laws of theN=4 supersymmetric Yang-Mills theory and explicitly demonstrate the closure of the algebra.     

A newN= 6 superconformal algebra

In this paper we construct a newN = 6 superconformal algebra which extends the Virasoro algebra by theSO ₆ current algebra, by 6 odd primary fields of conformal weight 3/2 and by 10 odd primary fields of conformal weight 1/2. The commutation relations of this algebra, which we will refer to asCK ₆, are represented by short distance operator product expansions (OPE). We constructCK ₆, as a subalgebra of theSO(6) superconformal algebra K₆, thus giving it a natural representation as first order differential operators on the circle withN = 6 extended symmetry. We show thatCK ₆ has no nontrivial central extensions. Partially supported by NSC grant 85-2121-M-006-019 of the ROC. Partially supported by NSF grant DMS-9622870.     

Finite <Emphasis Type="Italic">N</Emphasis> index and angular momentum bound from gravity

We exactly compute the finite N index and BPS partition functions for SYM theory in a newly proposed maximal angular momentum limit. The new limit is not predicted from the superconformal algebra, but naturally arises from the supergravity dual. We show that the index does not receive any finite N corrections while the free BPS partition function does.     

Short Representations of SU(2,2/N) and Harmonic Superspace Analyticity

We consider the harmonic superspaces associated with SU(2,2/N) superconformal algebras. For arbitrary N, we show that massless representations, other than the chiral ones, correspond to [N/2] elementary ultrashort analytic superfields whose first component is a scalar in the k antisymmetric irrep of SU(N) (k=1...[N/2]) with top spin J _top=(N/2–k/2,0). For N=2n, we analyze UIR's obtained by tensoring the self-conjugate ultrashort multiplet J _top=(n/2,0) and show that N–1 different basic products give rise to all possible UIR's with residual shortening.     

Strings fromN=2 gauged Wess-Zumino-Witten models

We present an algebraic approach to string theory. An embedding ofsl(2|1) in a super Lie algebra together with a grading on the Lie algebra determines a nilpotent subalgebra of the super Lie algebra. Chirally gauging this subalgebra in the corresponding Wess-Zumino-Witten model, breaks the affine symmetry of the Wess-Zumino-Witten model to some extension of theN=2 superconformal algebra. The extension is completely determined by thesl(2|1) embedding. The realization of the superconformal algebra is determined by the grading. For a particular choice of grading, one obtains in this way, after twisting, the BRST structure of a string theory. We classify all embeddings ofsl(2|1) into Lie super algebras and give a detailed account of the branching of the adjoint representation. This provides an exhaustive classification and characterization of both all extendedN=2 superconformal algebras and all string theories which can be obtained in this way.     

Random matrix theory and equilibrium statistics for the quasienergies of classically chaotic systems

We calculate thek-level correlation function for the eigenphases of theN-dimensional Floquet operator of kicked dynamics whose classical counterpart is chaotic in the limitN by applying standard equilibrium statistical mechanics to a fictitiousN-particle system which is constructed from the eigenvalue equation of the Floquet operator and show that they become equal to those of random matrices from the circular ensemble associated with the appropriate universality class.     

New solvable lattice models in three dimensions

In this paper we establish a remarkable connection between two seemingly unrelated topics in the area of solvable lattice models. The first is the Zamolodchikov model, which is the only nontrivial model on a three-dimen-sional lattice so far solved. The second is the chiral Potts model on the square lattice and its generalization associated with theU _q(sl(n)) algebra, which is of current interest due to its connections with high-genus algebraic curves and with representations of quantum groups at roots of unity. We show that this last sl(n)-generalized chiral Potts model can be interpreted as a model on a threedimensional simple cubic lattice consisting ofn square-lattice layers with anN- valued (N2) spin at each site. Further, in theN=2 case this three-dimen-sional model reduces (after a modification of the boundary conditions) to the Zamolodchikov model we mentioned above.     

LargeN expansion of QED: asymptotic photon propagator and contributions to the muon anomaly,for any number of loops

We calculate analytical contributions to then-loop asymptotic photon propagator from diagrams withn–1 electron loops, i.e. theO(1/N) terms in the largeN limit. The corresponding contributions to the on-shell -function, ()=6 log / logm reduced to rational combinations of _s = _p p ^–s . For the -function of the MOM scheme (i.e. the Gell-Man-Low function) we obtain theO(1/N) terms of

Thermodynamics of theSU(N) Anderson impurity

The thermodynamic Bethe-ansatz equations of the degenerate Anderson model in theU limit with excluded multiple occupation of the localized level are solved numerically for the caseN=8. Thef-level occupation, the entropy, the spin and charge susceptibilities and the specific heat are obtained as a function of temperature for variousf-level energies. The results forN=6 andN=8 are compared with available data for CeTh and YbCuAl.     

Inverse backscattering in two dimensions

We interpretN=2 superconformal field theories (SCFTs) formulated by Kazama and Suzuki via Goddard-Kent-Olive (GKO) construction from a viewpoint of the Lie algebra cohomology theory for the affine Lie algebra. We determine the cohomology group completely in terms of a certain subset of the affine Weyl group. We find that this subset describing the cohomology group can be obtained from its classical counterpart by the action of the Dynkin diagram automorphisms. Some algebra automorphisms of theN=2 superconformal algebra are also formulated. Utilizing the algebra automorphisms, we study the field identification problem for the branching coefficient modules in the GKO-construction. Also the structure of the Poincaré polynomial defined for eachN=2 theory is revealed.Dedicated to Professor Noboru Tanaka on his sixtieth birthday     

The bi-Hamiltonian structure of fully supersymmetric Korteweg-de Vries systems

The bi-Hamiltonian structure of integrable supersymmetric extensions of the Korteweg-de Vries (KdV) equation related to theN=1 and theN=2 superconformal algebras is found. It turns out that some of these extensions admit inverse Hamiltonian formulations in terms of presymplectic operators rather than in terms of Poisson tensors. For one extension related to theN=2 case additional symmtries are found with bosonic parts that cannot be reduced to symmetries of the classical KdV. They can be explained by a factorization of the corresponding Lax operator. All the bi-Hamiltonian formulations are derived in a systematic way from the Lax operators.