Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 7 (2011), 070, 15 pages      arXiv:0910.3813

Klein Topological Field Theories from Group Representations

Sergey A. Loktev a, b and Sergey M. Natanzon a, b, c
a) Department of Mathematics, Higher School of Economics, 7 Vavilova Str., Moscow 117312, Russia
b) Institute of Theoretical and Experimental Physics, 25 Bolshaya Cheremushkinskaya Str., Moscow 117218, Russia
c) A.N. Belozersky Institute, Moscow State University, Leninskie Gory 1, Bldg. 40, Moscow 119991, Russia

Received December 15, 2010, in final form July 04, 2011; Published online July 16, 2011

We show that any complex (respectively real) representation of finite group naturally generates a open-closed (respectively Klein) topological field theory over complex numbers. We relate the 1-point correlator for the projective plane in this theory with the Frobenius-Schur indicator on the representation. We relate any complex simple Klein TFT to a real division ring.

Key words: topological quantum field theory; group representation.

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  1. Alexeevski A., Natanzon S., Noncommutative two-dimensional topological field theories and Hurwitz numbers for real algebraic curves, Selecta Math. (N.S.) 12 (2006), 307-377, math.GT/0202164.
  2. Alexeevski A., Natanzon S., Algebra of Hurwitz numbers for seamed surfaces, Russian Math. Surveys 61 (2006), no. 4, 767-769.
  3. Alexeevski A., Natanzon S., Algebra of bipartite graphs and Hurwitz numbers of seamed surfaces, Izv. Math. 72 (2008), 627-646.
  4. Alexeevski A., Natanzon S., Hurwitz numbers for regular coverings of surfaces by seamed surfaces and Cardy-Frobenius algebras of finite groups, in Geometry, Topology, and Mathematical Physics, Amer. Math. Soc. Transl. Ser. 2, Vol. 224, Amer. Math. Soc., Providence, RI, 2008, 1-25, arXiv:0709.3601.
  5. Atiyah M., Topological quantum field theories, Inst. Hautes Études Sci. Publ. Math. (1988), no. 68, 175-186.
  6. Bantay P., The Frobenius-Schur indicator in conformal field theory, Phys. Lett. B 394 (1997), 87-88, hep-th/9610192.
  7. Cardy J.L., Operator content of two-dimensional conformal invariant theories, Nuclear Phys. B 270 (1986), 186-204.
  8. Cardy J.L., Effect of boundary conditions on the operator content of two-dimensional conformally invariant theories, Nuclear Phys. B 275 (1986), 200-218.
  9. Dijkgraaf R., Geometrical approach to two-dimensional conformal field theory, Ph.D. Thesis, Utrecht, 1989.
  10. Dijkgraaf R., Mirror symmetry and elliptic curves, in The Moduli Space of Curves (Texel Island, 1994), Progr. Math., Vol. 129, Birkhäuser Boston, Boston, MA, 1995, 149-163.
  11. Dubrovin B., Geometry of 2D topological field theories, in Integrable Systems and Quantum Groups (Montecatini Terme, 1993), Lecture Notes in Math., Vol. 1620, Springer, Berlin, 1996, 120-348, hep-th/9407018.
  12. Faith C., Algebra. II. Ring theory, Grundlehren der Mathematischen Wissenschaften, no. 191, Springer-Verlag, Berlin - New York, 1976.
  13. Felder G., Fröhlich J., Fuchs J., Schweigert C., Correlation functions and boundary conditions in rational conformal field theory and three-dimensional topology, Compositio Math. 131 (2002), 189-237, hep-th/9912239.
  14. Fuchs J., Runkel I., Schweigert C., TFT construction of RCFT correlators. II. Unoriented world sheets, Nuclear Phys. B 678 (2004), 511-637, hep-th/0306164.
  15. Fulton W., Harris J., Representation theory. A first course, Graduate Texts in Mathematics, Vol. 129, Springer-Verlag, New York, 1991.
  16. Kock J., Frobenius algebras and 2D topological quantum field theories, London Mathematical Society Student Texts, Vol. 59, Cambridge University Press, Cambridge, 2004.
  17. Kong L., Runkel I., Cardy algebras and sewing constraints. I, Comm. Math. Phys. 292 (2009), 871-912, arXiv:0807.3356.
  18. Lazaroiu C.I., On the structure of open-closed topological field theory in two-dimensions, Nuclear Phys. B 603 (2001), 497-530, hep-th/0010269.
  19. Lauda A.D., Pfeiffer H., Open-closed strings: two-dimensional extended TQFTs and Frobenius algebras, Topology Appl. 155 (2008), 623-666, math.AT/0510664.
  20. Moore G., Some comments on branes, G-flux, and K-theory, Internat. J. Modern Phys. A 16 (2001), 936-944, hep-th/0012007.
  21. Moore G., Segal G., D-branes and K-theory in 2D topological field theory, hep-th/0609042.
  22. Natanzon S.M., Exstended cohomological field theories and noncommutative Frobenius manifolds, J. Geom. Phys. 51 (2003), 387-403, math-ph/0206033.
  23. Natanzon S.M., Cyclic foam topological field theory, arXiv:0712.3557.
  24. Natanzon S.M., Brane topological field theory and Hurwitz numbers for CW-complexes, arXiv:0904.0239.
  25. Schwarz A.S., The partition function of degenerate quadratic functional and Ray-Singer invariants, Lett. Math. Phys. 2 (1977/78), 247-252.
  26. Segal G., Two-dimensional conformal field theories and modular functors, in IXth International Congress on Mathematical Physics (Swansea, 1988), Hilger, Bristol, 1989, 22-37.
  27. Turaev V., Turner P., Unoriented topological quantum field theory and link homology, Algebr. Geom. Topol. 6 (2006), 1069-1093, math.GT/0506229.
  28. Witten E., Topological quantum field theory, Comm. Math. Phys. 117 (1988), 353-386.

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