Advertisement

Communications in Mathematical Physics

, Volume 330, Issue 3, pp 1227–1262 | Cite as

Homotopy Classification of Bosonic String Field Theory

  • Korbinian MünsterEmail author
  • Ivo Sachs
Article

Abstract

We prove the decomposition theorem for the loop homotopy Lie algebra of quantum closed string field theory and use it to show that closed string field theory is unique up to gauge transformations on a given string background and given S-matrix. For the theory of open and closed strings we use results in open-closed homotopy algebra to show that the space of inequivalent open string field theories is isomorphic to the space of classical closed string backgrounds. As a further application of the open-closed homotopy algebra, we show that string field theory is background independent and locally unique in a very precise sense. Finally, we discuss topological string theory in the framework of homotopy algebras and find a generalized correspondence between closed strings and open string field theories.

Keywords

Modulus Space Open String Topological String Closed String Coder Cycl 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Witten E.: Noncommutative geometry and string field theory. Nucl. Phys. B 268, 253 (1986)ADSCrossRefMathSciNetGoogle Scholar
  2. 2.
    LeClair A., Peskin M.E., Preitschopf C.R.: String field theory on the conformal plane. 1. Kinematical principles. Nucl. Phys. B 317, 411 (1989)ADSCrossRefMathSciNetGoogle Scholar
  3. 3.
    Thorn C.B.: String field theory. Phys. Rep. 175, 1–101 (1989)ADSCrossRefMathSciNetGoogle Scholar
  4. 4.
    Gaberdiel, M.R., Zwiebach, B.: Tensor constructions of open string theories. 1: Foundations. Nucl. Phys. B505, 569–624 (1997). [hep-th/9705038]
  5. 5.
    Zwiebach, B.: Closed string field theory: quantum action and the B-V master equation. Nucl. Phys. B390, 33–152 (1993). [hep-th/9206084]
  6. 6.
    Zwiebach, B.: Oriented open-closed string theory revisited. Ann. Phys. 267, 193–248 (1998). [hep-th/9705241]
  7. 7.
    Kajiura, H.:Noncommutative homotopy algebras associated with open strings. Rev. Math. Phys. 19, 1–99 (2007). [arXiv:math/0306332v2]
  8. 8.
    Kontsevich, M.: Deformation quantization of Poisson manifolds, I. Lett. Math. Phys. 66(3), 157–216 (2003). [q-alg/9709040v1]Google Scholar
  9. 9.
    Kajiura, H.: Homotopy algebra morphism and geometry of classical string field theory. Nucl. Phys. B630, 361–432 (2002). [hep-th/0112228].
  10. 10.
    Sen, A., Zwiebach, B.: Background independent algebraic structures in closed string field theory. Commun. Math. Phys. 177, 305–326 (1996). [arXiv:hep-th/9408053v1]
  11. 11.
    Sen, A., Zwiebach, B.: Quantum background independence of closed string field theory. Nucl. Phys. B 423, 580–630 (1994). [arXiv:hep-th/9311009v1]
  12. 12.
    Costello, K.J.: Topological conformal field theories and Calabi–Yau categories. Adv. Math. 210(1), 165–214 (2007). (math/0412149v7 [math.QA])
  13. 13.
    Kajiura, H., Stasheff, J.: Homotopy algebras inspired by classical open-closed string field theory. Commun. Math. Phys. 263, 553–581 (2006). [math/0410291 [math-qa]]
  14. 14.
    Kajiura, H., Stasheff, J.: Open-closed homotopy algebra in mathematical physics. J. Math. Phys. 47, 023506 (2006). [hep-th/0510118]
  15. 15.
    Muenster, K., Sachs, I.: Quantum open-closed homotopy algebra and string field theory. Commun. Math. Phys. 321(3), 769–801 (2013). [1109.4101v2 [hep-th]]
  16. 16.
    Harrelson, E., Voronov, A.A., Zuniga, J.J.: Open-closed moduli spaces and related algebraic structures. Lett. Math. Phys. 94(1), 1–26 (2010). (0709.3874v2 [math.QA])
  17. 17.
    Schwarz, A.S.: Geometry of Batalin-Vilkovisky quantization. Commun. Math. Phys. 155, 249–260 (1993). [hep-th/9205088]
  18. 18.
    Chen, X.: Lie bialgebras and the cyclic homology of A structures in topology. 1002.2939v3 [math.AT]
  19. 19.
    Markl, M.: Loop homotopy Lie algebras in closed string field theory. Commun. Math. Phys. 221, 367–384 (2001). [hep-th/9711045]
  20. 20.
    DeWitt, B.: Supermanifolds, Cambridge monographs on mathematical physics. Cambridge University Press, Cambridge (1984)Google Scholar
  21. 21.
    Cieliebak, K., Fukaya, K., Latschev, J.: Homological algebra related to surfaces with boundaries (2014)Google Scholar
  22. 22.
    Liu, C.C.M.: Moduli of J-holomorphic curves with Lagrangian boundary conditions and open Gromov-Witten invariants for an S 1-equivariant pair. Math/0210257v2 [math.SG]
  23. 23.
    Harrelson, E.: On the homology of open-closed string field theory. 0412249v2 [math.AT]
  24. 24.
    Witten, E., Zwiebach, B.: Algebraic structures and differential geometry in 2D string theory. Nucl. Phys. B. 377, 55–112 (1992). (hep-th/9201056v1)
  25. 25.
    Fukaya, K.: Deformation theory, homological algebra, and mirror symmetry. Available at http://ftp.mat.uniroma1.it/people/manetti/DT2011/fukaya.pdf
  26. 26.
    Verlinde, E.: The master equation of 2D string theory. Nucl. Phys. B 381, 141–157 (1992). (hep-th/9202021v1)
  27. 27.
    Chuang J., Lazarev A.: Feynman diagrams and minimal models for operadic algebras. J. Lond. Math. Soc. (2) 81, 317337 (2010)CrossRefMathSciNetGoogle Scholar
  28. 28.
    Chuang, J., Lazarev, A.: Abstract Hodge decomposition and minimal models for cyclic algebras. Lett. Math. Phys. 89(1), 33–49 (2009). [arXiv:0810.2393v1 [math.QA]]
  29. 29.
    Moeller, N., Sachs, I.: Closed string cohomology in open string field theory. JHEP 1107, 022 (2011). [arXiv:1010.4125 [hep-th]]
  30. 30.
    Nakatsu, T.: Classical open-string field theory; A -algebra, renormalization group and boundary states. Nucl. Phys. B. 642, 13–90 (2002). [hep-th/0105272v4]
  31. 31.
    Kiermaier, M., Okawa, Y., Rastelli, L., Zwiebach, B.: Analytic solutions for marginal deformations in open string field theory. JHEP 0801, 028 (2008). [hep-th/0701249]
  32. 32.
    Kiermaier, M., Okawa, Y.: Exact marginality in open string field theory: A General framework. JHEP 0911, 041 (2009). [arXiv:0707.4472 [hep-th]]
  33. 33.
    Herbst, M.: Quantum A-infinity structures for open-closed topological strings. [hep-th/0602018v1]
  34. 34.
    Witten, E.: Mirror manifolds and topological field theory. In: Yau, S.T. (ed.): Mirror symmetry I, pp. 121–160. [hep-th/9112056]
  35. 35.
    Witten E.: Chern-Simons theory as a string theory. Prog. Math. 133, 637678 (1995)MathSciNetGoogle Scholar
  36. 36.
    Bershadsky M., Cecotti S., Ooguri H., Vafa C.: Kodaira–Spencer theory of gravity and exact results for quantum string amplitudes. Commun. Math. Phys. 165(2), 311–427 (1994)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  37. 37.
    Bershadsky M., Sadov V.: Theory of Kähler gravity. IJMPA 11(26), 4689–4730 (1996)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  38. 38.
    Hofman, C.: On the open-closed B-model. JHEP 2003, (2003)Google Scholar
  39. 39.
    Carqueville, N., Kay, M.M.: Bulk deformations of open topological string theory. Commun. Math. Phys. 315(3), 739–769 (2012). (1104.5438 [hep-th] hep-th/0602018v1)
  40. 40.
    Witten E.: Interacting field theory of open superstrings. Nucl. Phys. B 276, 291 (1986)ADSCrossRefMathSciNetGoogle Scholar
  41. 41.
    Berkovits, N.: Super Poincare covariant quantization of the superstring. JHEP 0004, 018 (2000). [hep-th/0001035]
  42. 42.
    Akman, F.: On some generalizations of Batalin-Vilkovsky algebras. J. Pure. Appl. Algebra 120(2), 105–141 (1997). [arXiv:q-alg/9506027]
  43. 43.
    Bering, K., Damgaard, P.H., Alfaro, J.: Algebra of higher antibrackets. Nucl. Phys. B478, 459–504 (1996). [hep-th/9604027]
  44. 44.
    Markl, M.: Operads in algebra, topology and physics. Mathematical Surveys and Monographs, vol. 96. American Mathematical Society, Providence, RI (2002). MR 1898414 (2003f:18011)Google Scholar
  45. 45.
    Hoefel, E.: OCHA and the swiss-cheese operad. J. Homotopy Relat Struct. 4, 123–151 (2009). [0710.3546v5 [math.QA]]
  46. 46.
    Kapustin, A., Rozansky, L.: On the relation between open and closed topological strings. Commun. Math. Phys. 252, 393–414 (2004). [hep-th/0405232]
  47. 47.
    Witten E.: Topological sigma models. Commun. Math. Phys. 118(3), 411–449 (1988)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  48. 48.
    Getzler, E.: Batalin-Vilkovisky algebras and two-dimensional topological field theories. Commun. Math. Phys. 159, 265–285 (1994). [hep-th/9212043]
  49. 49.
    Barannikov, S.: Modular operads and Batalin-Vilkovisky geometry. Int. Math. Res. Not. 2007 (2007). doi: 10.1093/imrn/rnm075
  50. 50.
    Stasheff J.: Homotopy associativity of H-spaces I. Am. Math. Soc. 108(2), 275–292 (1963)CrossRefzbMATHMathSciNetGoogle Scholar
  51. 51.
    Stasheff J.: Homotopy associativity of H-spaces II. Am. Math. Soc. 108(2), 293–312 (1963)MathSciNetGoogle Scholar
  52. 52.
    Getzler E., Jones J.D.S.: A -algebras and the cyclic bar complex. Ill. J. Math. 34(2), 256–283 (1990)zbMATHMathSciNetGoogle Scholar
  53. 53.
    Penkava, M., Schwarz, A.S.: A (infinity) algebras and the cohomology of moduli spaces. Am. Math. Soc. Transl. 169(2) (1995). [hep-th/9408064]
  54. 54.
    Lada, T., Markl, M.: Strongly homotopy Lie algebras. Commun. Algebra 23(6), 2147–2161 (1995). [hep-th/9406095]

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Arnold Sommerfeld Center for Theoretical PhysicsMunichGermany

Personalised recommendations