Communications in Mathematical Physics

, Volume 257, Issue 1, pp 1–28 | Cite as

A Spin Decomposition of the Verlinde Formulas for Type A Modular Categories

  • Christian Blanchet


A modular category is a braided category with some additional algebraic features. The interest of this concept is that it provides a Topological Quantum Field Theory in dimension 3. The Verlinde formulas associated with a modular category are the dimensions of the TQFT modules. We discuss reductions and refinements of these formulas for modular categories related with SU(N). Our main result is a splitting of the Verlinde formula, corresponding to a brick decomposition of the TQFT modules whose summands are indexed by spin structures modulo an even integer. We introduce here the notion of a spin modular category, and give the proof of the decomposition theorem in this general context.


Neural Network Statistical Physic Field Theory Complex System Quantum Field Theory 
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  1. 1.
    Andersen, J., Masbaum, G.: Involutions on moduli spaces and refinements of the Verlinde formula. Math. Ann. 314(2), 291–326 (1999)CrossRefGoogle Scholar
  2. 2.
    Andersen, H., Paradowski, J.: Fusion category arising from semisimple Lie algebras. Commun. Math. Phys. 169(3), 563–588 (1995)Google Scholar
  3. 3.
    Atiyah, M. F.: Riemann surfaces and spin structures. Ann. Ecole Norm. Sup. (4)4, 47–62 (1971)Google Scholar
  4. 4.
    Bakalov, B., Kirillov, A.: Lecture on tensor categories and modular functors. Univ. Lecture Series No.21, Providence, RI: AMS 2001Google Scholar
  5. 5.
    Beauville, A.: Conformal blocks, fusion rules and the Verlinde formula. Israel Math. Conf. Proceedings, Vol.9, 75–96 (1996)Google Scholar
  6. 6.
    Beauville, A., Laszlo, Y.: Conformal blocks and generalized theta functions. Commun. Math. Phys. 164, 385–419 (1994)Google Scholar
  7. 7.
    Beliakova, A., Blanchet, C.: Modular categories of types B,C and D. Comment. Math. Helv. 76, 467–500 (2001)Google Scholar
  8. 8.
    Bismut, J-M., Labourie, F.: Formules de Verlinde pour les groupes simplement connexes et géométrie sympleptique. CRAS, t. 325, Série I, 1009–1014 (1997)Google Scholar
  9. 9.
    Bismut, J-M., Labourie, F.: Sympleptic geometry and the Verlinde formulas. In: Surveys in differential geometry: differential geometry inspired by string theory, Boston, MA: Int. Press, 1999, pp. 97–311Google Scholar
  10. 10.
    Blanchet, C.: Refined quantum invariants for three-manifolds with structure. In: Knot Theory, Banach Center Pub. Vol. 42, Warsaw: Polish Acad. of Sci, 11–22 (1998)Google Scholar
  11. 11.
    Blanchet, C.: Hecke algebras, modular categories and 3-manifolds quantum invariants. Topology, 39, 193–223 (2000)Google Scholar
  12. 12.
    Blanchet, C., Habegger, N., Masbaum, G., Vogel, P.: Topological Quantum Field Theories derived from the Kauffman bracket. Topology 34(4), 883–927 (1995)CrossRefGoogle Scholar
  13. 13.
    Blanchet, C., Masbaum, G.: Topological quantum field theories for surfaces with spin structure. Duke Math. J 82, 229–267 (1996)CrossRefGoogle Scholar
  14. 14.
    Bruguières, A.: Catégories prémodulaires, modularisations et invariants des variétés de dimension 3. Math. Ann. 316(2), 215–236 (2000)Google Scholar
  15. 15.
    Faltings, G.: A proof of the Verlinde formula. J. Alg. Geometry 3, 347–374 (1994)Google Scholar
  16. 16.
    Johnson, D.: Spin structures and quadratic forms on surfaces. J. London Math Soc. (2) 22, 365–377 (1980)Google Scholar
  17. 17.
    Kohno, T., Takata, T.: Level-Rank Duality of Witten 3-manifolds invariants. Adv. Studies in Pure Math. 24, Progress in Algebraic Combinatorics, Orlando, FL: Acad.Press, 1996 pp. 243–264Google Scholar
  18. 18.
    Le, T.: Quantum invariants of 3-manifolds: integrality, splitting, and perturbative expansion., 2000
  19. 19.
    Le, T., Turaev, V.: Quantum groups and ribbon G-categories. J. Pure Appl. Algebra 178 (2), 169–185 (2003)Google Scholar
  20. 20.
    Lickorish, W.B.R.: An Introduction to Knot Theory. Grad. Texts in Math. 175, Berlin-Heidelberg-New York: Springer Verlag, 1997Google Scholar
  21. 21.
    Macdonald, I. G.: Symmetric functions and Hall polynomial. 2nd ed. , Oxford: Oxford Science Pub 1995Google Scholar
  22. 22.
    Masbaum, G., Wenzl, H.: Integral modular categories and integrality of quantum invariants at roots of unity of prime order. J. Reine Angew. Math. 505, 209–235 (1998)Google Scholar
  23. 23.
    Milnor, J.: Spin structures on manifolds. L’Enseignement Math. 9, 198–203 (1963)Google Scholar
  24. 24.
    Müger, M.: Galois theory for braided tensor categories and the modular closure. Adv. Math. 150(2), 151–201 (2000)Google Scholar
  25. 25.
    MuPAD: The Open Computer Algebra System. Sciface Software, Scholar
  26. 26.
    Oxbury, W. M., Wilson, S. M. J.: Reciprocity laws in the Verlinde formulae for the classical groups. Trans. AMS 348(7), 2689–2710 (1996)Google Scholar
  27. 27.
    Kassel, C., Rosso, M., Turaev, V.: Quantum groups and knots invariants. Panoramas et Synthèses No 5, Paris: Soc. Math. France, 1997Google Scholar
  28. 28.
    Sorger, C.: La formule de Verlinde. Séminaire Bourbaki 794, 1994Google Scholar
  29. 29.
    Sawin, S.: Quantum groups at roots of unity and modularity., 2003
  30. 30.
    Turaev, V.: Quantum invariants of knots and 3-manifolds. De Gruyter Studies in Math. 18, Berlin: De Gruyler, 1994Google Scholar
  31. 31.
    Turaev, V.: Homotopy field theory in dimension 2 and crossed groups-algebras., 1999
  32. 32.
    Turaev, V.: Homotopy field theory in dimension 3 and crossed groups-categories., 2000
  33. 33.
    Turaev, V., Wenzl, H.: Quantum invariants of 3-manifolds associated with classical simple Lie algebras. Int. J. of Math. 4(2), 323–358 (1993)Google Scholar
  34. 34.
    Turaev, V., Wenzl, H.: Semisimple and modular categories from link invariants. Math. Ann. 309, 411–461 (1997)CrossRefGoogle Scholar
  35. 35.
    Verlinde, E.: Fusion rules and modular transformations in 2d conformal field theory. Nucl. Phys. B 300(3), 360–376 (1988)Google Scholar
  36. 36.
    Yokota, Y.: Skeins and quantum SU(N) invariants of 3-manifolds. Math. Ann. 307, 109–138 (1997)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.L.M.A.M.Université de Bretagne-SudVannesFrance

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