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Igor B. Frenkel

Publications and source records attributed to Igor B. Frenkel.

12 recordsLinked to original sources
arXiv2022

Three dimensional construction of the Virasoro-Bott group

We present a three-dimensional geometric construction of the Virasoro-Bott group, which is a central extension of the group of diffeomorphisms of the circle. Our approach is analogous to the well-known construction of a central extension of the loop group by means of the Wess-Zumino topological term. In particular, the Virasoro-Bott group is realized as a quotient group of diffeomorphisms of the disc with special boundary conditions. We identify the Lie algebra corresponding to our group with the Virasoro algebra. We also show that for generalized boundary conditions the Virasoro algebra is extended to a semidirect product with the Heisenberg algebra. We discuss the relation between our construction, the Chern-Simons theory, and the three-dimensional gravity.

math.GT↗
arXiv2014

On the continuous series for affine sl(2,R)

We construct the representations of affine sl(2,R) starting from the unitary representations of the loop ax+b-group. Our approach involves a combinatorial analysis of the correlation functions of the generators and renormalization of the appearing divergencies. We view our construction as a step towards a realization of the principal series representations of affine sl(2,R).

math.RT↗
arXiv2012

Rademacher sums, moonshine and gravity

In 1939 Rademacher derived a conditionally convergent series expression for the elliptic modular invariant, and used this expression- the first Rademacher sum - to verify its modular invariance. By generalizing Rademacher's approach we construct bases for the spaces of automorphic integrals of arbitrary even integer weight, for all groups commensurable with the modular group. We use these Rademacher sums to illuminate various aspects of the structure of the spaces of automorphic integrals, including the actions of Hecke operators. We obtain a new characterization of the discrete groups of monstrous moonshine in terms of Rademacher sums, and we develop connections between Rademacher sums and a family of monstrous Lie algebras recently introduced by Carnahan. Our constructions suggest conjectures relating monstrous moonshine to a distinguished family of chiral three dimensional quantum gravities, and relating monstrous Lie algebras and their Verma modules to the second quantization of this family of chiral three dimensional quantum gravities.

math.RT↗
arXiv2011

Positive representations of split real quantum groups and future perspectives

We construct a special principal series representation for the modular double $U_{q\tilde{q}}(g_R)$ of type $A_r$ representing the generators by positive essentially self-adjoint operators satisfying the transcendental relations that also relate $q$ and $\tilde{q}$. We use the cluster variables parametrization of the positive unipotent matrices to derive the formulas in the classical case. Then we quantize them after applying the Mellin transform. Our construction is inspired by the previous results for $g_R=sl(2,R)$ and is expected to have a generalization to other simply-laced types. We conjecture that our positive representations are closed under the tensor product and we discuss the future perspectives of the new representation theory following the parallel with the established developments of the finite-dimensional representation theory of quantum groups.

math.RT↗
arXiv2010

Quantum Teichmüller space from quantum plane

We derive the quantum Teichmüller space, previously constructed by Kashaev and by Fock and Chekhov, from tensor products of a single canonical representation of the modular double of the quantum plane. We show that the quantum dilogarithm function appears naturally in the decomposition of the tensor square, the quantum mutation operator arises from the tensor cube, the pentagon identity from the tensor fourth power of the canonical representation, and an operator of order three from isomorphisms between canonical representation and its left and right duals. We also show that the quantum universal Teichmüller space is realized in the infinite tensor power of the canonical representation naturally indexed by rational numbers including the infinity. This suggests a relation to the same index set in the classification of projective modules over the quantum torus, the unitary counterpart of the quantum plane, and points to a new quantization of the universal Teichmüller space.

math.RT↗
arXiv2010

Quantum Group as Semi-infinite Cohomology

We obtain the quantum group $SL_q(2)$ as semi-infinite cohomology of the Virasoro algebra with values in a tensor product of two braided vertex operator algebras with complementary central charges $c+\bar{c}=26$. Each braided VOA is constructed from the free Fock space realization of the Virasoro algebra with an additional q-deformed harmonic oscillator degree of freedom. The braided VOA structure arises from the theory of local systems over configuration spaces and it yields an associative algebra structure on the cohomology. We explicitly provide the four cohomology classes that serve as the generators of $SL_q(2)$ and verify their relations. We also discuss the possible extensions of our construction and its connection to the Liouville model and minimal string theory.

math.RT↗
arXiv2004

Modified regular representations of affine and Virasoro algebras, VOA structure and semi-infinite cohomology

We identify the algebra of matrix elements of big projective modules in category O with the regular functions on the big Bruhat cell of G. Analogous extensions of the regular representations of the affine Lie and Virasoro algebras yield vertex operator algebras, equipped with two commuting actions with special values of the total central charge. In the generic case, we identify the structure of these VOAs, compute their semi-infinite cohomology. Its superalgebra structure encodes the fusion rules for the associated tensor categories. Full details are provided for the affine sl(2) and Virasoro algebras, and various generalizations and extensions are discussed.

math.QA↗
arXiv2003

Bases of representations of type A affine Lie algebras via quiver varieties and statistical mechanics

We relate two apparently different bases in the representations of affine Lie algebras of type A: one arising from statistical mechanics, the other from gauge theory. We show that the two are governed by the same combinatorics and therefore can be viewed as identical. In particular, we are able to give an alternative and much simpler geometric proof of a result of E. Date, M. Jimbo, A. Kuniba, T. Miwa and M. Okado on the construction of bases of affine Lie algebra representations. At the same time, we give a simple parametrization of the irreducible components of Nakajima quiver varieties associated to infinite and cyclic quivers. We also define new varieties whose irreducible components are in one-to-one correspondence with bases of the highest weight representations of affine gl_{n+1}.

math.AG↗
arXiv1998

Annihilating ideals and tilting functors

We use Kazhdan-Lusztig tensoring to, first, describe annihilating ideals of highest weight modules over an affine Lie algebra in terms of the corresponding VOA and, second, to classify tilting functors, an affine analogue of projective functors known in the case of a simple Lie algebra. For the sake of completeness, the classification of annihilating ideals is borrowed from our previous work, q-alg/9711011; the part on tilting functors is new.

math.QA↗
arXiv1993

Central extensions of current groups in two dimensions

In this paper we generalize some of these results for loop algebras and groups as well as for the Virasoro algebra to the two-dimensional case. We define and study a class of infinite dimensional complex Lie groups which are central extensions of the group of smooth maps from a two dimensional orientable surface without boundary to a simple complex Lie group G. These extensions naturally correspond to complex curves. The kernel of such an extension is the Jacobian of the curve. The study of the coadjoint action shows that its orbits are labelled by moduli of holomorphic principal G-bundles over the curve and can be described in the language of partial differential equations. In genus one it is also possible to describe the orbits as conjugacy classes of the twisted loop group, which leads to consideration of difference equations for holomorphic functions. This gives rise to a hope that the described groups should possess a counterpart of the rich representation theory that has been developed for loop groups. We also define a two-dimensional analogue of the Virasoro algebra associated with a complex curve. In genus one, a study of a complex analogue of Hill's operator yields a description of invariants of the coadjoint action of this Lie algebra. The answer turns out to be the same as in dimension one: the invariants coincide with those for the extended algebra of currents in sl(2).

hep-th↗