Katrina D. Barron  Associate Professor Office: 276C Hurley. Phone: 574-631-3981 E-mail: kbarron@nd.edu Mailing Address: 255 Hurley Hall Department of Mathematics University of Notre Dame Notre Dame, IN 46556

i miei bambini

Spring 2012 Courses:

Math 60220 Graduate Algebra II

Research: My research focuses on vertex operator superalgebras and the algebraic and geometric foundations of superconformal field theory.

Conformal field theory (CFT), or more specifically, string theory, and related superconformal field theories (SCFTs) are an attempt at developing a physical theory that combines all fundamental interactions of particles, including gravity.  The ``super" refers to certain assumed symmetries between bosons (integral spin particles with symmetric wave functions) and fermions (half integral spin particles with anti-symmetric wave functions).

The geometry of CFT and SCFT extends the use of Feynman diagrams, describing the interactions of point particles whose propagation in time sweeps out a line in space-time, to one-dimensional strings or superstrings whose propagation in time sweeps out a two-dimensional surface or supersurface called a ``worldsheet".

Much of my research involves the study of relationships between the worldsheet geometry of CFT and SCFT and properties of the algebras of correlation functions of the particle interactions.  For genus-zero holomorphic CFT these algebras are called vertex operator algebras. For genus-zero holomorphic SCFT these algebras involve supersymmetric vertex operator superalgebras and certain twisted and untwisted modules, where the ``supersymmetry" means there are certain symmetries present between fermions and bosons.

Selected Publications:

K. Barron, Twisted modules for N=2 supersymmetric vertex operator superalgebras arising from finite automorphisms of the N=2 Neveu-Schwarz algebra, submitted.

K. Barron, Uniformization of genus-zero and certain genus-one N=2 superconformal super-Riemann surfaces, submitted.

K. Barron, Automorphism groups of N=2 superconformal super-Riemann spheres, J. Pure Appl. Algebra, vol. 214 (2010), 1973-1987.

K. Barron, Axiomatic aspects of N=2 vertex superalgebras with odd formal variables, Commun. in Alg., vol. 38 (2010), 1199-1268.

K. Barron, Alternate notions of N=1 superconformality and deformations of N=1 vertex superalgebras, in ``Vertex Operator Algebras and Related Areas", Commun. in Math., Amer. Math. Soc., Vol. 497, (2009), 33-51.

K. Barron, The moduli space of N=2 super-Riemann spheres with tubes, Commun. in Contemp. Math., vol. 9 (2007), 857-940.

K. Barron, Y.-Z. Huang, J. Lepowsky, An equivalence of two constructions of permutation-twisted modules for lattice vertex operator algebras, J. Pure and Appl. Algebra, vol. 210 (2007), 797-826.

K. Barron, Superconformal change of variables for N=1 Neveu-Schwarz vertex operator superalgebras, J. of Algebra, vol. 277 (2004), 717-764.

K. Barron, The notion of N=1 supergeometric vertex operator superalgebra and the isomorphism theorem, Commun. in Contemp. Math., vol. 5 (2003), 481-567.

K. Barron, The moduli space of N=1 superspheres with tubes and the sewing operation, Memoirs of the AMS, vol. 162, no. 772, (2003).

K. Barron, C. Dong, G. Mason, Twisted sectors for tensor product vertex operator algebras associated to permutation groups, Commun. in Math. Phys., vol. 227 (2002), 349-384.

K. Barron, Y.-Z. Huang, J. Lepowsky, Factorization of formal exponentials and uniformization, J. of Algebra, vol. 228 (2000), 551-579.

K. Barron, N=1 Neveu-Schwarz vertex operator superalgebras over Grassmann algebras and with odd formal variables, in ``Representations and Quantizations: Proceedings of the International  Conference on Representation Theory, 1998", ed. by J. Wang and Z. Lin, China Higher Education Press & Springer-Verlag, Beijing, 2000, 9-36.

K. Barron, A supergeometric interpretation of vertex operator superalgebras, Internat. Math. Res. Notices 1996, no. 9, 409--430.