Ward identities from recursion formulas for correlation functions in conformal field theory

Alexander Zuevsky

Archivum Mathematicum (2015)

  • Volume: 051, Issue: 5, page 347-356
  • ISSN: 0044-8753

Abstract

top
A conformal block formulation for the Zhu recursion procedure in conformal field theory which allows to find n -point functions in terms of the lower correlations functions is introduced. Then the Zhu reduction operators acting on a tensor product of VOA modules are defined. By means of these operators we show that the Zhu reduction procedure generates explicit forms of Ward identities for conformal blocks of vertex operator algebras. Explicit examples of Ward identities for the Heisenberg and free fermionic vertex operator algebras are supplied.

How to cite

top

Zuevsky, Alexander. "Ward identities from recursion formulas for correlation functions in conformal field theory." Archivum Mathematicum 051.5 (2015): 347-356. <http://eudml.org/doc/276207>.

@article{Zuevsky2015,
abstract = {A conformal block formulation for the Zhu recursion procedure in conformal field theory which allows to find $n$-point functions in terms of the lower correlations functions is introduced. Then the Zhu reduction operators acting on a tensor product of VOA modules are defined. By means of these operators we show that the Zhu reduction procedure generates explicit forms of Ward identities for conformal blocks of vertex operator algebras. Explicit examples of Ward identities for the Heisenberg and free fermionic vertex operator algebras are supplied.},
author = {Zuevsky, Alexander},
journal = {Archivum Mathematicum},
keywords = {conformal field theory; conformal blocks; recursion formulas; vertex algebras},
language = {eng},
number = {5},
pages = {347-356},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Ward identities from recursion formulas for correlation functions in conformal field theory},
url = {http://eudml.org/doc/276207},
volume = {051},
year = {2015},
}

TY - JOUR
AU - Zuevsky, Alexander
TI - Ward identities from recursion formulas for correlation functions in conformal field theory
JO - Archivum Mathematicum
PY - 2015
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 051
IS - 5
SP - 347
EP - 356
AB - A conformal block formulation for the Zhu recursion procedure in conformal field theory which allows to find $n$-point functions in terms of the lower correlations functions is introduced. Then the Zhu reduction operators acting on a tensor product of VOA modules are defined. By means of these operators we show that the Zhu reduction procedure generates explicit forms of Ward identities for conformal blocks of vertex operator algebras. Explicit examples of Ward identities for the Heisenberg and free fermionic vertex operator algebras are supplied.
LA - eng
KW - conformal field theory; conformal blocks; recursion formulas; vertex algebras
UR - http://eudml.org/doc/276207
ER -

References

top
  1. Belavin, A., Polyakov, A., Zamolodchikov, A., 10.1016/0550-3213(84)90052-X, Nuclear. Phys. B 241 (1984), 333–380. (1984) MR0757857DOI10.1016/0550-3213(84)90052-X
  2. Ben-Zvi, D., Frenkel, E., Vertex algebras and algebraic curves, Mathematical Surveys and Monographs, vol. 88, American Mathematical Society, Providence, RI, 2004, Second ed. (2004) Zbl1106.17035MR2082709
  3. Borcherds, R.E., 10.1073/pnas.83.10.3068, Proc. Nat. Acad. Sci. U.S.A. 83 (1986), 3068–3071. (1986) Zbl0613.17012MR0843307DOI10.1073/pnas.83.10.3068
  4. Dong, C., Lepowsky, J., Generalized Vertex Algebras and Relative Vertex Operators, Progr. Math., vol. 112, Birkhäuser (Boston, MA), 1993. (1993) Zbl0803.17009MR1233387
  5. Frenkel, I., Huang, Y., Lepowsky, J., On axiomatic approaches to vertex operator algebras and modules, Mem. Amer. Math. Soc. 104 (494) (1993), viii+64 pp. (1993) Zbl0789.17022MR1142494
  6. Frenkel, I. B., Lepowsky, J., Meurman, A., Vertex Operator Algebras and the Monster, Pure Appl. Math., vol. 134, Academic Press, Boston, 1988. (1988) Zbl0674.17001MR0996026
  7. Friedan, D., Shenker, S., The analytic geometry of two dimensional conformal field theory, Nuclear Phys. B 281 (1987), 509–545. (1987) MR0869564
  8. Gilroy, T., Genus Two Zhu Theory for Vertex Operator Algebras, Ph.D. thesis, NUI Galway, 2013, 1–89. (2013) 
  9. Gunning, R.C., Lectures on Riemann Surfaces, Princeton Univ. Press, Princeton, 1966. (1966) Zbl0175.36801MR0207977
  10. Hurley, D., Tuite, M.P., 10.1142/S0129167X12501066, Internat. J. Math. 23 (10) (2012), 17 pp., 1250106. (2012) Zbl1335.17016MR2999051DOI10.1142/S0129167X12501066
  11. Kac, V., Vertex Operator Algebras for Beginners, University Lecture Series, vol. 10, AMS, Providence, 1998. (1998) 
  12. Kawamoto, N., Namikawa, Y., Tsuchiya, A., Yamada, Y., 10.1007/BF01225258, Comm. Math. Phys. 116 (1988), 247–308. (1988) Zbl0648.35080MR0939049DOI10.1007/BF01225258
  13. Mason, G., Tuite, M.P., 10.1007/s00220-002-0772-6, Comm. Math. Phys. 235 (1) (2003), 47–68. (2003) Zbl1020.17020MR1969720DOI10.1007/s00220-002-0772-6
  14. Mason, G., Tuite, M.P., 10.1007/s00220-010-1126-4, Comm. Math. Phys. 300 (2010), 673–713. (2010) Zbl1226.17024MR2736959DOI10.1007/s00220-010-1126-4
  15. Mason, G., Tuite, M.P., Vertex operators and modular forms. A window into zeta and modular physics, Math. Sci. Res. Inst. Publ., vol. 57, Cambridge Univ. Press, Cambridge, 2010, pp. 183–278. (2010) MR2648364
  16. Mason, G., Tuite, M.P., Free bosonic vertex operator algebras on genus two Riemann surfaces II, Contributions in Mathematical and Computational Sciences 8, Springer-Verlag, Berlin-Heidelberg, 2014. (2014) Zbl1329.17027MR2736959
  17. Mason, G., Tuite, M.P., Zuevsky, A., 10.1007/s00220-008-0510-9, Comm. Math. Phys. 283 (2008), 305–342. (2008) MR2430636DOI10.1007/s00220-008-0510-9
  18. Matsuo, A., Nagatomo, K., Axioms for a vertex algebra and the locality of quantum fields, MSJ Memoirs 4 (1999), x+110 pp. (1999) Zbl0928.17025MR1715197
  19. Serre, J.-P., A course in Arithmetic, Springer-Verlag, Berlin, 1978. (1978) Zbl0432.10001MR0344216
  20. Tsuchiya, A., Ueno, K., Yamada, Y., Conformal field theory on universal family of stable curves with gauge symmetrie, Adv. Stud. Pure Math. 19 (1989), 459–566. (1989) MR1048605
  21. Tuite, M.P., Zuevsky, A., Genus two partition and correlation functions for fermionic vertex operator superalgebras II, arXiv:1308.2441v1, submitted. MR2824477
  22. Tuite, M.P., Zuevsky, A., 10.1007/s00220-011-1258-1, Comm. Math. Phys. 306 (2011), 419–447. (2011) Zbl1254.17024MR2824477DOI10.1007/s00220-011-1258-1
  23. Tuite, M.P., Zuevsky, A., 10.1007/s00220-011-1310-1, Comm. Math. Phys. 306 (2011), 617–645. (2011) Zbl1238.30029MR2825503DOI10.1007/s00220-011-1310-1
  24. Tuite, M.P., Zuevsky, A., 10.1016/j.jpaa.2011.10.025, J. Pure Appl. Algebra 216 (2012), 1442–1453. (2012) Zbl1287.17050MR2890514DOI10.1016/j.jpaa.2011.10.025
  25. Ueno, K., Introduction to conformal field theory with gauge symmetries, Geometry and Physics - Proceedings of the Conference at Aarhus Univeristy, Aarhus, Denmark, New York, Marcel Dekker, 1997. (1997) Zbl0873.32022MR1423195
  26. Zhu, Y., 10.1090/S0894-0347-96-00182-8, J. Amer. Math. Soc. 9 (1996), 237–302. (1996) Zbl0854.17034MR1317233DOI10.1090/S0894-0347-96-00182-8

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.