Katrina D. Barron Associate Professor Office: 276C Hurley. Phone: 574-631-3981 E-mail: kbarron@nd.edu Mailing Address: 255 Hurley Hall |
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i miei bambini |

Lie Algebras, Vertex Operator Algebras and Related Topics, in honor of James Lepowsky and Robert Wilson. University of Notre Dame, August 14–18, 2015.

Special Session on Representation Theory, Vertex Operator Algebras, and Related Topics , AMS Special Session, November 14–15, 2015, Rutgers University.

Vertex Algebras and Quantum Groups, February 7–12, 2016, Banff International Research Station.

Special Session on Vertex Algebra and Related Algebraic and Geometric Structures , AMS Special Session, March 19–20, 2016, SUNY-Stonybrook.

**Upcoming Conferences (needs to be updated): **

Special Session on Representation Theory and Algebraic Mathematical Physics, AMS Special Session, March 11–12, 2017, Charleston, SC.

Representation Theory XV, Inter University Center, Dubrovnik, Croatia, 18–25 June 2017.

INdAM Roma, Italy, December 2017.

Sacramento, California, June 2018.

Oaxaca, Mexico, September 2018.

University of Delaware, September 2018.

Utrecht, Netherlands, May 2019.

Chern Institute, Nankai, China, June 2019.

Dubrovnik, Croatia, June 2019.

Banff, Canada, September 2020.

**Math 30710 ** — Algebra. This course introduces and studies the concepts of groups, rings, and fields.

**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.

In addition to being a physical model for particle interactions, vertex operator superalgebras have important and deep links to the theory of finite simple groups, number theory, topology, etc. My research often also touches upon or has applications to such other branches of mathematics.

**Selected Publications (updated 2014):**

K. Barron and N. Vander Werf, Permutation-twisted modules for even order cycles acting on tensor product vertex operator superalgebras, Internat. Jour. of Math, 25 (2014), no. 2, 1450018 (35 pages).

K. Barron, On the correspondence between mirror-twisted sectors for N=2 supersymmetric vertex operator superalgebras of the form (V tensor V) and N=1 Ramond sectors of V , in: ``Proceedings of the Xth International Workshop on Lie Theory and Its Applications in Physics", June 2013, Varna, Bulgaria. ed. V. Dobrev. Springer Proceedings in Mathematics & Statistics, Vol. 111, 2014.

K. Barron, Twisted modules for tensor product vertex operator superalgebras and permutation automorphisms of odd order, in: "Proceedings of the Southeastern Lie Theory Workshop Series 2012-2014", ed. by K. Misra, D. Nakano and B. Parshall, Conf. Proc. Series of the AMS, 2015.

K. Barron and N. Vander Werf, On permutation-twisted free fermions and two conjectures, in: ``Proceedings of the XXIst International Conference on Integrable Systems and Quantum Symmetries", June 2013, Prague, Czech Republic; ed. C. Burdik, O. Navratil and S. Posta; Jour. of Physics: Conference Series, vol. 474 (2013), 012009; 35 pages.

K. Barron, On twisted modules for N=2 supersymmetric vertex operator superalgebras, in: ``Proceedings of the IXth International Workshop on Lie Theory and Its Applications in Physics", June 2011, Varna, Bulgaria; ed. V. Dobrev, Springer 2013, 411--420. Longer preprint: Twisted modules for N=2 supersymmetric vertex operator superalgebras arising from finite automorphisms of the N=2 Neveu-Schwarz algebra.

K. Barron, On uniformization of N=2 superconformal and N=1 superanalytic DeWitt 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.

Last updated January 2019.