Abstract
We describe a formalism allowing a completely mathematical rigorous approach to closed and open conformal field theories with general anomaly. We also propose a way of formalizing modular functors with positive and negative parts, and outline some connections with other topics, in particular elliptic cohomology.
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Ando, M.: Power operations in elliptic cohomology and representations of loop groups. Trans. AMS 352, 5619–5666 (2000)
Baas, N.A., Dundas, B.I., Rognes, J.: Two-vector bundles and forms of elliptic cohomology. To appear in Segal Proceedings, Cambridge University Press
Borceux, F.: Handbook of categorical algebra 1–2, Encyclopedia of Mathematics and its Applications, Cambridge: Cambridge University Press, pp.50–52
Borcherds, R.E.: Monstrous moonshine and monstrous Lie superalgebra. Invent. Math. 109, 405–444 (1992)
d’Hoker, E.: String theory, In: Quantum fields and strings: a course for mathematicians, Vol. 2, Providence RI: AMS and IAS, 1999, pp. 807–1012
Deligne, P., Freed, D.: Notes on Supersymmetry (following J. Bernstein) In: Quantum fields and strings, a course for mathematicians, Vol. 1, Providence RI: AMS, 1999, pp. 41–98
Diaconescu, D.E.: Enhanced D-brane categories from string field theory. JHEP 0106, 16 (2001)
Douglas, M.R.: D-branes, categories and N=1 SUSY. J. Math. Phys. 42, 2818–2843 (2001)
Fiore, T.: Lax limits, lax adjoints and lax algebras: the categorical foundations of conformal field theory. To appear
Frenkel, I.: Vertex algebras and algebraic curves. Seminaire Bourbaki 1999–2000, Asterisque 276, 299–339 (2002)
Frenkel, I., Lepowsky, J., Meurman, A.: Vertex operator algebras and the monster. Pure and applied Mathematics, Vol. 134, London–NewYork: Academic Press, 1999
Fuchs, J., Runkel, I., Schweigert, C.: TFT construction of RCFT correlators I: Partition functions. Nucl. Phys. B 646, 353 (2002)
Fuchs, J., Runkel, I., Schweigert, C.: TFT construction of RCFT correlators II: Unoriented world sheets. Nucl. Phys. B 678, 511 (2004)
Green, M.B., Schwartz, J.H., Witten, E.: Superstring theory. Vol. 1,2, Cambridge: Cambridge University Press, 1988
Horava, P.: Equivariant Topological Sigma Models. Nucl. Phys. B 418, 571–602 (1994)
Hu, P., Kriz, I.: Conformal field theory and elliptic cohomology. Advances in Mathematics 189(2), 325–412 (2004)
Hu, P., Kriz, I., Voronov, A.A.: On Kontsevich’s Hochschild cohomology conjecture. http://arxiv.org/abs/amth.AT/0309369, 2003
Huang, Y.Z., Kong, L.: Open-string vertex algebras, tensor categories and operads. Comm. Math. Phys. 250, 433–471 (2004)
Kriz, I.: On spin and modularity in conformal field theory. Ann. Sci. de ENS 36, 57–112 (2003)
Lawvere, W.F.: Functorial semantics of algebraic theories. Proc. Nat. Acad. Sci. U.S.A. 50, 869–87 (1963)
Lazaroiu, C.I.: On the structure of open-closed topological field theory in two-dimensions. Nucl. Phys. B 603, 497–530 (2001)
Lazaroiu, C.I.: Generalized complexes and string field theory. JHEP 06, 52 (2001)
Lazaroiu, C.I.: Unitarity, D-brane dynamics and D-brane categories. JHEP 12, 31 (2001)
Lewellen, D.: Sewing constraints for conformal field theories on surfaces with boundaries. Nucl. Phys. B 372, 654 (1992)
Moore, G.: Some Comments on Branes, G-flux, and K-theory. Int. J. Mod. Phys. A 16, 936–944 (2001)
Moore, G.: Lectures on branes, K-theory and RR-charges. http://www.physics.rutgers.edu/∼gmoore/day1/12.html
Moore, G., Seiberg, N.: Classical and Quantum Conformal Field Theory. Commun. Math. Phys. 123, 177–254 (1989)
Moore, G., Seiberg, N.: Taming the conformal ZOO. Phys. Lett. B 220, 422–430 (1989)
Moore, G. Seiberg, N.: Lectures on RCFT. In: H.C. Lee (ed.), Physics, Geometry and Topology, RiverEdge, World Scientific, 1990, pp. 263–361
Ostrik, V.: Module categories, weak Hopf algebras and modular invariants. Transform. Groups 8, 177 (2003)
Polchinski, J.: String theory. Vols. 1,2, Cambridge: Cambridge Univ. Press, 1999
Pradisi, G., Sagnotti, A., Stanev, Y.A.: Planar duality in SU(2) WZW models. Phys. Lett. B 354, 279 (1995)
Pressley, A., Segal, G.: Loop groups. Oxford: Oxford University Press, 1986
Segal, G.: Elliptic cohomology. Seminaire Bourbaki 1987/88, Asterisque 161–162, Exp. No, 695, (1988) 4, 187–201 (1989)
Segal, G.: The definition of conformal field theory. Preprint, 1987
Segal, G.: ITP lectures. http://doug-pc.itp.ucsb.edu/online/geom99/, 1999
Segal, G.: Categories and cohomology theories. Topology 13, 293–312 (1974)
Stolz, S., Teichner, P.: What is an elliptic object?, In: U. Tillmann (ed.), Proc. of 2002 Oxford Symp. in Honour of G.Segal, Cambridge: Cambridge Univ. Press, 2004
Thomason, R.W.: Beware the phony multiplication on Quillen’s Proc. AMS 80(4), 569–573 (1980)
Verlinde, E.: Fusion rules and modular transformations in 2D conformal field theory. Nucl. Phys. B 300, 360–376 (1988)
Witten, E.: Overview of K-theory applied to strings. Int. J. Mod. Phys. A 16, 693–706 (2001)
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Communicated by M.R. Douglas
The authors were supported by the NSF.
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Hu, P., Kriz, I. Closed and Open Conformal Field Theories and Their Anomalies. Commun. Math. Phys. 254, 221–253 (2005). https://doi.org/10.1007/s00220-004-1202-8
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DOI: https://doi.org/10.1007/s00220-004-1202-8