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arXiv1992

Lie-algebraic approach to the theory of polynomial solutions. I. Ordinary differential equations and finite-difference equations in one variable

A classification of ordinary differential equations and finite-difference equations in one variable having polynomial solutions (the generalized Bochner problem) is given. The method used is based on the spectral problem for a polynomial element of the universal enveloping algebra of $sl_2({\bf R})$ (for differential equations) or $sl_2({\bf R})_q$ (for finite-difference equations) in the "projectivized" representation possessing an invariant subspace. Connection to the recently-discovered quasi-exactly-solvable problems is discussed.

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arXiv1992

Lie-algebraic approach to the theory of polynomial solutions. II. Differential equations in one real and one Grassmann variables and 2x2 matrix differential equations

A classification theorem for linear differential equations in two variables (one real and one Grassmann) having polynomial solutions(the generalized Bochner problem) is given. The main result is based on the consideration of the eigenvalue problem for a polynomial element of the universal enveloping algebra of the algebra $osp(2,2)$ in the "projectivized" representation (in differential operators of the first order) possessing an invariant subspace. A classification of 2 x 2 matrix differential equations in one real variable possessing polynomial solutions is described. Connection to the recently-discovered quasi-exactly-solvable problems is discussed.

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arXiv1992

Lie-algebraic approach to the theory of polynomial solutions. III. Differential equations in two real variables and general outlook

Classification theorems for linear differential equations in two real variables, possessing eigenfunctions in the form of the polynomials (the generalized Bochner problem) are given. The main result is based on the consideration of the eigenvalue problem for a polynomial elements of the universal enveloping algebras of the algebras $sl_3({\bf R})$, $sl_2({\bf R})\oplus sl_2({\bf R})$ and $gl_2 ({\bf R})\ \triangleright\!\!\!< {\bf R}^{r+1}\ , r>0$ taken in the "projectivized" representations (in differential operators of the first order in two real variables) possessing an invariant subspace. General insight to the problem of a description of linear differential operators possessing an invariant sub-space with a basis in polynomials is presented. Connection to the recently-discovered quasi-exactly-solvable problems is discussed.

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arXiv1993

Surgical invariants of four-manifolds

A new topological invariant of closed connected orientable four-dimensional manifolds is proposed. The invariant, constructed via surgery on a special link, is a four-dimensional counterpart of the celebrated SU(2) three-manifold invariant of Reshetikhin, Turaev and Witten.

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arXiv1993

On Solvable Lattice Models and Knot Invariants

Recently, a class of solvable interaction round the face lattice models (IRF) were constructed for an arbitrary rational conformal field theory (RCFT) and an arbitrary field in it. The Boltzmann weights of the lattice models are related in the extreme ultra violet limit to the braiding matrices of the rational conformal field theory. In this note we use these new lattice models to construct a link invariant for any such pair of an RCFT and a field in it. Using the properties of RCFT and the IRF lattice models, we prove that the invariants so constructed always obey the Markov properties, and thus are true link invariants. Further, all the known link invariants, such as the Jones, HOMFLY and Kauffman polynomials arise in this way, along with giving a host of new invariants, and thus also a unified approach to link polynomials. It is speculated that all link invariants arise from some RCFT, and thus the problem of classifying link and knot invariants is equivalent to that of classifying two dimensional conformal field theory.

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arXiv1993

Graph IRF Models and Fusion Rings

Recently, a class of interaction round the face (IRF) solvable lattice models were introduced, based on any rational conformal field theory (RCFT). We investigate here the connection between the general solvable IRF models and the fusion ones. To this end, we introduce an associative algebra associated to any graph, as the algebra of products of the eigenvalues of the incidence matrix. If a model is based on an RCFT, its associated graph algebra is the fusion ring of the RCFT. A number of examples are studied. The Gordon--generalized IRF models are studied, and are shown to come from RCFT, by the graph algebra construction. The IRF models based on the Dynkin diagrams of A-D-E are studied. While the $A$ case stems from an RCFT, it is shown that the $D-E$ cases do not. The graph algebras are constructed, and it is speculated that a natural isomorphism relating these to RCFT exists. The question whether all solvable IRF models stems from an RCFT remains open, though the $D-E$ cases shows that a mixing of the primary fields is needed.

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arXiv1993

Differential Operator Algebras on compact Riemann Surfaces

Invited talk at the International Symposium on Generalized Symmetries in Physics at the Arnold-Sommerfeld-Institute, Clausthal, Germany, July 26 -- July 29, 1993. This talk reviews results on the structure of algebras consisting of meromorphic differential operators which are holomorphic outside a finite set of points on compact Riemann surfaces. For each partition into two disjoint subsets of the set of points where poles are allowed, a grading of the algebra and of the modules of lambda - forms is introduced. With respect to this grading the Lie structure of the algebra and of the modules are almost graded ones. Central extensions and semi-infinite wedge representations are studied. If one considers only differential operators of degree 1 then these algebras are generalizations of the Virasoro algebra in genus zero, resp. of Krichever Novikov algebras in higher genus.

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arXiv1993

Where is the large radius limit?

By properly accounting for the invariance of a Calabi-Yau sigma-model under shifts of the $B$-field by integral amounts (analagous to the $θ$-angle in QCD), we show that the moduli spaces of such sigma-models can often be enlarged to include ``large radius limit'' points. In the simplest cases, there are holomorphic coordinates on the enlarged moduli space which vanish at the limit point, and which appear as multipliers in front of instanton contributions to Yukawa couplings. (Those instanton contributions are therefore suppressed at the limit point.) In more complicated cases, the instanton contributions are still suppressed but the enlarged space is singular at the limit point. This singularity may have interesting effects on the effective four-dimensional theory, when the Calabi-Yau is used to compactify the heterotic string.

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arXiv1994

Differential Geometry of Toda Systems

In the present paper we give a differential geometry formulation of the basic dynamical principle of the group--algebraic approach \cite{LeS92} --- the grading condition --- in terms of some holomorphic distributions on flag manifolds associated with the parabolic subgroups of a complex Lie group; and a derivation of the corresponding nonlinear integrable systems, and their general solutions. Moreover, the reality condition for these solutions is introduced. For the case of the simple Lie groups endowed with the canonical gradation, when the systems in question are reduced to the abelian Toda equations, we obtain the generalised Plücker representation for the pseudo--metrics specified by the Kähler metrics on the flag manifolds related to the maximal nonsemisimple parabolic subgroups; and the generalised infinitesimal Plücker formulas for the Ricci curvature tensors of these pseudo--metrics. In accordance with these formulas, the fundamental forms of the pseudo--metrics and the Ricci curvature tensors are expressed directly in terms of the abelian Toda fields, which have here the sense of Kähler potentials.

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arXiv1993

Spacetime Topology Change: The Physics of Calabi-Yau Moduli Space

We review recent work which has significantly sharpened our geometric understanding and interpretation of the moduli space of certain $N$=2 superconformal field theories. This has resolved some important issues in mirror symmetry and has also established that string theory admits physically smooth processes which can result in a change in topology of the spatial universe.

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arXiv1994

Infinite Dimensional Geometry and Quantum Field Theory of Strings. I. Infinite Dimensional Geometry of Second Quantized Free String

There are investigated several objects of an INFINITE DIMENSIONAL GEOMETRY appearing from the second quantization of a free string. The paper contains 2 chapters: 1st is devoted to the infinite dimensional geometry of flag, fundamental and $Π$-spaces for Virasoro-Bott group and its nonassociative deformation defined by Gelfand-Fuchs 3-cocycle (Gelfand-Fuchs loop) as well as of infinite-dimensional non-Euclidean symplectic grassmannian, to the constructions of Verma modules, their models and skladens over Virasoro algebra; an infinite dimensional geometry of the configuration space for the second quantized free string in flat and curved backgrounds as well as author version of Bowick- Rajeev formalism of the separation of internal and external degrees of freedom of a closed string are described in 2nd chapter. In the 1st chapter the main objects are infinite dimensional Lie algebras, groups and loops, homogeneous, Kähler, Finsler, contact and symmetric spaces, complex, real and CR-manifolds, determinant sheaves, manifolds with subsymmetries, polarizations and Fock spaces, bibundles and objects of integral geometry, nonholonomic spaces, deformations of geometric structures and moduli spaces. In the 2nd chapter they are gauge fields, Faddeev-Popov ghosts, Gauss-Manin connections, Kostant-Blattner-Sternberg pairings, BRST-operators.

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arXiv1994

String Theory on K3 Surfaces

The moduli space of N=(4,4) string theories with a K3 target space is determined, establishing in particular that the discrete symmetry group is the full integral orthogonal group of an even unimodular lattice of signature (4,20). The method combines an analysis of the classical theory of K3 moduli spaces with mirror symmetry. A description of the moduli space is also presented from the viewpoint of quantum geometry, and consequences are drawn concerning mirror symmetry for algebraic K3 surfaces.

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arXiv1994

The Role of Solvable Groups in Quantization of Lie Algebras

The elements of the wide class of quantum universal enveloping algebras are prooved to be Hopf algebras $H$ with spectrum $Q(H)$ in the category of groups. Such quantum algebras are quantum groups for simply connected solvable Lie groups $P(H)$. This provides utilities for a new algorithm of constructing quantum algebras especially useful for nonsemisimple ones. The quantization procedure can be carried out over an arbitrary field. The properties of the algorithm are demonstrated on examples.

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arXiv1994

Group-like Structures in Quantum Lie Algebras and the Process of Quantization

For a certain class of Lie bialgebras $(A,A^*)$ the corresponding quantum universal enveloping algebras $U_q(A)$ are prooved to be equivalent to quantum groups Fun$_q(F^*)$, $F^*$ being the factor group for the dual group $G^*$. This property can be used to simplify the process of quantization. The described class appears to be wide enough to contain all the standard quantizations of infinite series. The properties of the groups $F^*$ are explicitly demonstrated for the standard deformations $U_q(SL(n))$. It is shown that for different $A^*$ (remaining in the described class of Lie bialgebras) the same algorithm leads to the nonstandard quantizations.

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arXiv1994

Topics in Hidden Symmetries

These three topics are an attempt to explicate some curiosities of the inverse problem of representation theory (i.e. having a set of operators to describe the "correct" algebraic object, which is represented by them) on simple examples related to the Lie algebra sl(2,C). Such consideration maybe regarded as toy one for analogous infinite dimensional problems in the modern quantum field theory (conformal field theory, integrable models, field theory in non-trivial backgrounds). Some constructions inspired by these topics and related to infinite dimensional hidden symmetries, which are produced by vertex operator fields, will be discussed in the forthcoming article.

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arXiv1995

Lagrangian Formalism Over Graded Algebras

This paper provides a description of an algebraic setting for the Lagrangian formalism over graded algebras and is intended as the necessary first step towards the noncommutative C-spectral sequence (variational bicomplex). A noncommutative version of integration procedure, the notion of adjoint operator, Green's formula, the relation between integral and differential forms, conservation laws, Euler operator, Noether's theorem is considered.

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arXiv1994

Localization of $\frak{u}$-modules. I. Intersection cohomology of real arrangements

This paper is the first in a series. The main goal of the series is to present a geometric construction of certain remarkable tensor categories arising from quantum groups coresponding to the value of deformation parameter $q$ equal to a root of unity. In the present paper we study perverse sheaves over a complex affine space which are smooth along the stratification determined by a finite arrangement of complex affine hyperplanes defined by real equations. In particular, we construct explicitely (in terms of combinatorial data) complexes computing cohomology of Goresky-MacPherson extensions of one-dimensional local systems over the complement of hyperplanes.

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arXiv1994

Local systems over complements of hyperplanes and the Kac-Kazhdan conditions for singular vectors

In this note we strenghten a theorem by Esnault-Schechtman-Viehweg which states that one can compute the cohomology of a complement of hyperplanes in a complex affine space with coefficients in a local system using only logarithmic global differential forms, provided certain "Aomoto non-resonance conditions" for monodromies are fulfilled at some "edges" (intersections of hyperplanes). We prove that it is enough to check these conditions on a smaller subset of edges. We show that for certain known one dimensional local systems over configuration spaces of points in a projective line defined by a root system and a finite set of affine weights (these local systems arise in the geometric study of Knizhnik-Zamolodchikov differential equations), the Aomoto resonance conditions at non-diagonal edges coincide with Kac-Kazhdan conditions of reducibility of Verma modules over affine Lie algebras.

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