Free bosons and tau-functions for compact Riemann surfaces and closed smooth Jordan curves. Current correlation functions

Letters in Mathematical Physics

Volumn 56, Issue 3, 2001, Pages 181-228

  Takhtajan, Leon A ^a,b  

^a Yale University   (United States)

^b STONY BROOK UNIVERSITY   (United States)

Author keywords

Compact Riemann surfaces; Conformal maps; Free bosons; Green's functions; Partition functions; Theta functions; Ward identities for current correlation functions

Indexed keywords

EID: 0012525588     PISSN: 03779017     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1017999407650     Document Type: Article

References (57)

1. Alvarez-Gaumé, L., Moore, G. and Vafa, C.: Theta-functions, modular invariance, and strings, Comm. Math. Phys. 106 (1986), 1-40.
2. Alvarez-Gaumé, L., Gomez, C. and Reina, C.: Loop groups, grassmannians and string theory, Phys. Lett. B 190 (1987), 55-62.
3. Alvarez-Gaumé, L., Bost, J.-B., Moore, G., Nelson, P. and Vafa, C.: Bosonization on higher genus Riemann surfaces, Comm. Math. Phys. 112 (1987), 503-552.
4. Belavin, A. B., Polyakov, A. M. and Zamolodchikov, A. B.: Infinite conformal symmetry in two-dimensional quantum field theory. Nuclear Phys. B 241 (1984), 333-380.
5. Bers, L.: Finite dimensional Teichmüller spaces and generalizations, Bull. Amer. Math. Soc. (N.S.) 5 (1981), 131-172.
6. Bost, J.-B. and Nelson, P.: Spin-1/2 bosonization on compact surfaces, Phys. Rev. Lett. 57 (1986), 795-798.
7. Bowick, M. J. and Rajeev, S. G.: String theory as the Kähler geometry of loop space, Phys. Rev. Lett. 58 (1987), 535-538.
8. 1, Nuclear Phys. B 293 (1987), 348-384.
9. Duren, P.: Univalent Functions, Springer-Verlag, New York, 1983.
10. Faber, G.: Über polynomische entwicklungen, Math. Ann. 57 (1903), 385-408.
11. Farkas, H. M. and Kra, I.: Riemann Surfaces, 2nd edn, Springer-Verlag, New York, 1992.
12. Fay, J. D.: Theta-Functions on Riemann Surfaces, Lecture Notes in Math. 352 Springer-Verlag, New York, 1973.
13. Fay, J. D.: Fourier coefficients of the resolvent for a Fuchsian group, J. Reine Angew. Math. 293/294 (1977), 143-203.
14. Fay, J. D.: Kernel functions, analytic torsion and moduli spaces, Mem. Amer. Math. Soc. 96(464) (1992).
15. Gawȩdzki, K.: Lectures on conformal field theory, In: P. Deligne et al. (eds), Quantum Field Theories and Strings: A Course for Mathematicians, Amer. Math. Soc., Providence, 1999, pp. 727-805.
16. Glimm, J. and Jaffe, A.: Quantum Physics: A Functional Integral Point of View, 2nd edn, Springer-Verlag, New York, 1987.
17. Griffiths, P. and Harris, J.: Principles of Algebraic Geometry, Wiley, New York, 1978.
18. Hille, E.: Analytic Function Theory, v. II, Ginn and Company, London, 1962.
19. Hurwitz, A. and Courant, R.: Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen. Herausgegeben und ergänzt durch einen Abschnitt über geometrische Funktionentheorie. Mit einem Anhang von H. Röhrl, Springer-Verlag, Berlin, 1964 (German); Russian transl. (adapted by M. A. Evgrafov) in: Theory of Functions, Nauka, 1968.
20. Hurwitz, A. and Courant, R.: Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen. Herausgegeben und ergänzt durch einen Abschnitt über geometrische Funktionentheorie. Mit einem Anhang von H. Röhrl, Springer-Verlag, Berlin, 1964 (German); Russian transl. (adapted by M. A. Evgrafov) in: Theory of Functions, Nauka, 1968.
21. Iwasawa, K.: Algebraic Functions, Amer. Math. Soc., Providence, 1993.
22. Kawamoto, N., Namikawa, Y., Tsuchiya, A. and Yamada, Y.: Geometric realization of conformal field theory on Riemann surfaces, Comm. Math. Phys. 116 (1988), 247-308.
23. Kirillov, A. A.: Kähler structures of K-orbits of the group of diffeomorphisms of a circle. Funktsional Anal. i Prilozhen 21(2) (1987), 42-45 (Russian); English transl. in: Funct. Anal. Appl. 21 (1987), 122-125.
24. Kirillov, A. A.: Kähler structures of K-orbits of the group of diffeomorphisms of a circle. Funktsional Anal. i Prilozhen 21(2) (1987), 42-45 (Russian); English transl. in: Funct. Anal. Appl. 21 (1987), 122-125.
25. 1), Funktsional Anal. i Prilozhen 21(4) (1987), 35-46 (Russian); English transl. in: Funct. Anal. Appl. 21 (1987), 284-294.
26. 1), Funktsional Anal. i Prilozhen 21(4) (1987), 35-46 (Russian); English transl. in: Funct. Anal. Appl. 21 (1987), 284-294.
27. Kostov, I. K., Krichever, I. M., Mineev-Weinstein, M., Wiegmann, P. B. and Zabrodin, A.: τ-function for analytic curves, arXiv: hep-th/0005259.
28. Kra, I.: Automorphic forms and Kleinian Groups, Benjamin, Reading, Mass., 1972.
29. Krichever, I. M.: unpublished manuscript.
30. Lang, S.: Introduction to Arakelov Theory, Springer-Verlag, New York, 1988.
31. Mumford, D.: Tata Lectures on Theta. II, Birkhäuser, Basel, 1984.
32. Mineev-Weinstein, M., Wiegmann, P. B. and Zabrodin, A.: Integrable structure of interface dynamics, Phys. Rev. Lett. 84 (2000), 5106-5109.
33. Miranda, R.: Algebraic Curves and Riemann Surfaces, Amer. Math. Soc., Providence, 1995.
34. 1) and the Teichmüller spaces, Comm. Math. Phys. 130 (1990), 123-138.
35. Polyakov, A. M.: Quantum geometry of bosonic strings, Phys. Lett. B 103 (1981), 207-210.
36. Polyakov, A. M.: Lecture at Steklov Institute, Leningrad, 1982, unpublished.
37. Polyakov, A. M.: Gauge Fields and Strings, Harwood Academic, London, 1987.
38. Raina, A. K.: Fay's trisecant identity and conformal field theory, Comm. Math. Phys. 122 (1989), 625-641.
39. Raina, A. K.: An algebraic geometry view of currents in a model quantum field theory on a curve, C. R. Acad. Sci. Paris Série I 318 (1994), 851-856.
40. Raina, A. K.: An algebraic geometry view of a model quantum field theory on a curve. In: A. J. Berrick et al. (eds), Geometry from the Pacific Rim, de Gruyter, Berlin, 1997, pp. 311-329.
41. Ransford, T.: Potential Theory in The Complex Plane, Cambridge Univ. Press, 1995.
42. Shvarts, A. S.: Elliptic operators in quantum field theory. In: Current Problems in Mathematics 17, VINITI, 1981, pp. 113-173 (Russian); English transl. in: J. Soviet Math. 21 (1983), 551-601.
43. Shvarts, A. S.: Elliptic operators in quantum field theory. In: Current Problems in Mathematics 17, VINITI, 1981, pp. 113-173 (Russian); English transl. in: J. Soviet Math. 21 (1983), 551-601.
44. Takhtajan, L. A.: Uniformization, local index theorem and geometry of the moduli spaces of Riemann surfaces and vector bundles, Proc. Sympos. Pure Math. 49 (1989), 581-596.
45. Takhtajan, L. A.: Topics in quantum geometry of Riemann surfaces: two-dimensional quantum gravity, In: L. Castellani and J. Wess (eds), Quantum Groups and Their Applications in Physics, Proc. Internat. School Enrico Fermi, 127, IOS 1996, pp. 541-579.
46. Takhtajan, L. A.: Quantum field theories on an algebraic curve, Lett. Math. Phys. 52 (2000), 79-91.
47. Vafa, C.: Operator formulation on Riemann surfaces, Phys. Lett. B 190 (1987), 47-54.
48. Verlinde, E. and Verlinde, H.: Chiral bosonization, determinants and string partition function, Nuclear Phys. B 288 (1987), 357-396.
49. Wiegmann, P. B. and Zabrodin, A.: Conformal maps and integrable hierarchies, arXiv: hep-th/9909147.
50. Witten, E.: Quantum field theory, Grassmannians, and algebraic curves, Comm. Math. Phys. 113 (1988), 529-600.
51. Witten, E.: Free fermions on an algebraic curve, Proc. Sympos. Pure Math. 48 (1988), 329-344.
52. Zograf, P. G. and Takhtajan, L. A.: Action for the Liouville equation as a generating function for the accessory parameters and as a potential for the Weil-Petersson metric on the Teichmüller space, Funktsional Anal. i Prilozhen 19 (1985), 67-68 (Russian); English transl. in: Funct. Anal. Appl. 19(3) (1985), 219-220.
53. Zograf, P. G. and Takhtajan, L. A.: Action for the Liouville equation as a generating function for the accessory parameters and as a potential for the Weil-Petersson metric on the Teichmüller space, Funktsional Anal. i Prilozhen 19 (1985), 67-68 (Russian); English transl. in: Funct. Anal. Appl. 19(3) (1985), 219-220.
54. Zograf, P. G. and Takhtajan, L. A.: On the Liouville equation, accessory parameters and the geometry of the Teichmuüller space for the Riemann surfaces of genus 0, Mat. Sb. 132 (1987), 147-166 (Russian); English transl. in: Math. USSR Sb. 60 (1988), 143-161.
55. Zograf, P. G. and Takhtajan, L. A.: On the Liouville equation, accessory parameters and the geometry of the Teichmuüller space for the Riemann surfaces of genus 0, Mat. Sb. 132 (1987), 147-166 (Russian); English transl. in: Math. USSR Sb. 60 (1988), 143-161.
56. Zograf, P. G. and Takhtajan, L. A.: On uniformization of Riemann surfaces and the Weil-Petersson metric on the Teichmuüller and Schottky spaces, Mat. Sb. 132 (1987), 303-320 (Russian); English transl. in: Math. USSR Sb. 60 (1988), 297-313.
57. Zograf, P. G. and Takhtajan, L. A.: On uniformization of Riemann surfaces and the Weil-Petersson metric on the Teichmuüller and Schottky spaces, Mat. Sb. 132 (1987), 303-320 (Russian); English transl. in: Math. USSR Sb. 60 (1988), 297-313.