Virasoro algebra

Virasoro algebra
Original: Jakob.scholbach Vector: Pbroks13 · CC BY-SA 3.0 · source
Name	Virasoro algebra
Type	Infinite-dimensional Lie algebra
Introduced	1969
Introduced by	Miguel Ángel Virasoro, Michael P. Tuite?
Related	Kac–Moody algebra, Heisenberg algebra, W-algebra

Contents

Virasoro algebra The Virasoro algebra is an infinite-dimensional central extension of the complexified Lie algebra of vector fields on the circle, introduced in the late 1960s and essential to two-dimensional conformal symmetry in theoretical physics. It underpins the mathematical structure of many models studied by researchers at institutions such as Institute for Advanced Study, Princeton University, CERN, and California Institute of Technology, and it appears in the work of mathematicians and physicists associated with Harvard University, University of Cambridge, University of Oxford, and École Normale Supérieure.

Definition and basic properties

The algebra is defined by generators Ln for n ∈ Z together with a central element c, satisfying commutation relations analogous to those studied by scholars at Massachusetts Institute of Technology, Stanford University, Yale University, Columbia University, and University of Chicago. These relations were formalized in publications connected to research by groups at Max Planck Institute for Physics, University of Tokyo, Kyoto University, and Seoul National University. The structure constants reflect properties considered in classical analyses at Royal Society, National Academy of Sciences (United States), American Mathematical Society, European Mathematical Society, and International Centre for Theoretical Physics. Important symmetry properties have been examined in seminars at IHÉS, Perimeter Institute, Scuola Normale Superiore, and University of California, Berkeley.

Representation theory for the algebra has been developed by researchers affiliated with Princeton, Cambridge, Bonn, Zurich, and Moscow State University. Unitary representations were classified using methods related to those in the work of scholars at University of Pennsylvania, University of Michigan, University of Illinois Urbana-Champaign, Brown University, and University of Minnesota. Techniques from category theory used by teams at University of Oxford, University of Warwick, Imperial College London, Trinity College Dublin, and King's College London play roles in understanding modules and intertwiners. Connections to modular invariance have been explored in collaborations involving University of Tokyo, Rutgers University, University of Stockholm, and Humboldt University of Berlin.

The central element c, often called the central charge, is a scalar in representations and was studied in analyses from CERN, Los Alamos National Laboratory, Joint Institute for Nuclear Research, Brookhaven National Laboratory, and Argonne National Laboratory. The phenomenon of anomalies associated with c was a focus in workshops at Caltech, Imperial College, University of Cambridge, Duke University, and University of California, Santa Barbara. Calculations of central terms draw on techniques refined at Princeton, Stanford, ETH Zurich, Cornell University, and University of Toronto. Critical values of c play a role in classification problems investigated by research groups at Columbia, McGill University, University of British Columbia, University of Sydney, and Australian National University.

Verma modules and highest-weight representations form the backbone of the algebra's module theory, with seminal contributions from researchers associated with University of Paris, University of Göttingen, Technion – Israel Institute of Technology, Weizmann Institute of Science, and Tel Aviv University. The structure and reducibility of Verma modules were analyzed in conferences at Princeton, Cambridge, Oxford, Bonn, and Milan. Determinant formulae and singular vector constructions were developed further by teams at Kyoto University, Osaka University, Seoul National University, Tata Institute of Fundamental Research, and SISSA. Historical work connecting these modules to fusion rules involved collaborations with University of Rome La Sapienza, University of Florence, University of Padua, Scuola Normale Superiore, and University of Pisa.

Applications in two-dimensional conformal field theory and string theory brought the algebra to prominence in research at CERN, Caltech, Princeton, Harvard, and MIT. Model-building and statistical mechanics applications were pursued by groups at University of Cambridge, Imperial College, University of California, Berkeley, Rutgers University, and Brown University. The algebra is central in the formulation of critical systems studied in collaborations involving University of Chicago, Yale University, University of Illinois Urbana-Champaign, University of Michigan, and University of Texas at Austin. Its role in vertex operator algebra theory linked it to work at Bell Labs, AT&T Labs Research, Microsoft Research, Google Research, and IBM Research.

Extensions and related structures include higher-spin W-algebras, superconformal algebras, and affine Kac–Moody algebras, topics developed at Max Planck Institute for Mathematics, Institut des Hautes Études Scientifiques, École Polytechnique, University of Paris-Sud, and University of Grenoble Alpes. Superconformal extensions were explored in collaborations at University of California, San Diego, University of Amsterdam, Leiden University, Ghent University, and KU Leuven. Connections to integrable models involved research groups at Landau Institute for Theoretical Physics, Steklov Institute of Mathematics, Lebedev Physical Institute, Institute of Theoretical and Experimental Physics, and Saratov State University. Current developments continue at centers including Perimeter Institute, Simons Center for Geometry and Physics, Flatiron Institute, Kavli Institute for Theoretical Physics, and Institute for Advanced Study.

Category:Lie algebras