Skip to main content
SpringerLink
Log in

Homotopy Classification of Bosonic String Field Theory

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Witten E.: Noncommutative geometry and string field theory. Nucl. Phys. B 268, 253 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  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)

    Article  ADS  MathSciNet  Google Scholar 

  3. Thorn C.B.: String field theory. Phys. Rep. 175, 1–101 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  4. Gaberdiel, M.R., Zwiebach, B.: Tensor constructions of open string theories. 1: Foundations. Nucl. Phys. B505, 569–624 (1997). [hep-th/9705038]

  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. Zwiebach, B.: Oriented open-closed string theory revisited. Ann. Phys. 267, 193–248 (1998). [hep-th/9705241]

  7. Kajiura, H.:Noncommutative homotopy algebras associated with open strings. Rev. Math. Phys. 19, 1–99 (2007). [arXiv:math/0306332v2]

  8. Kontsevich, M.: Deformation quantization of Poisson manifolds, I. Lett. Math. Phys. 66(3), 157–216 (2003). [q-alg/9709040v1]

  9. Kajiura, H.: Homotopy algebra morphism and geometry of classical string field theory. Nucl. Phys. B630, 361–432 (2002). [hep-th/0112228].

  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. Sen, A., Zwiebach, B.: Quantum background independence of closed string field theory. Nucl. Phys. B 423, 580–630 (1994). [arXiv:hep-th/9311009v1]

  12. Costello, K.J.: Topological conformal field theories and Calabi–Yau categories. Adv. Math. 210(1), 165–214 (2007). (math/0412149v7 [math.QA])

  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. Kajiura, H., Stasheff, J.: Open-closed homotopy algebra in mathematical physics. J. Math. Phys. 47, 023506 (2006). [hep-th/0510118]

  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. 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. Schwarz, A.S.: Geometry of Batalin-Vilkovisky quantization. Commun. Math. Phys. 155, 249–260 (1993). [hep-th/9205088]

  18. Chen, X.: Lie bialgebras and the cyclic homology of A ∞ structures in topology. 1002.2939v3 [math.AT]

  19. Markl, M.: Loop homotopy Lie algebras in closed string field theory. Commun. Math. Phys. 221, 367–384 (2001). [hep-th/9711045]

  20. DeWitt, B.: Supermanifolds, Cambridge monographs on mathematical physics. Cambridge University Press, Cambridge (1984)

    Google Scholar 

  21. Cieliebak, K., Fukaya, K., Latschev, J.: Homological algebra related to surfaces with boundaries (2014)

  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. Harrelson, E.: On the homology of open-closed string field theory. 0412249v2 [math.AT]

  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. Fukaya, K.: Deformation theory, homological algebra, and mirror symmetry. Available at http://ftp.mat.uniroma1.it/people/manetti/DT2011/fukaya.pdf

  26. Verlinde, E.: The master equation of 2D string theory. Nucl. Phys. B 381, 141–157 (1992). (hep-th/9202021v1)

  27. Chuang J., Lazarev A.: Feynman diagrams and minimal models for operadic algebras. J. Lond. Math. Soc. (2) 81, 317337 (2010)

    Article  MathSciNet  Google Scholar 

  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. Moeller, N., Sachs, I.: Closed string cohomology in open string field theory. JHEP 1107, 022 (2011). [arXiv:1010.4125 [hep-th]]

  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. 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. Kiermaier, M., Okawa, Y.: Exact marginality in open string field theory: A General framework. JHEP 0911, 041 (2009). [arXiv:0707.4472 [hep-th]]

  33. Herbst, M.: Quantum A-infinity structures for open-closed topological strings. [hep-th/0602018v1]

  34. Witten, E.: Mirror manifolds and topological field theory. In: Yau, S.T. (ed.): Mirror symmetry I, pp. 121–160. [hep-th/9112056]

  35. Witten E.: Chern-Simons theory as a string theory. Prog. Math. 133, 637678 (1995)

    MathSciNet  Google Scholar 

  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)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  37. Bershadsky M., Sadov V.: Theory of Kähler gravity. IJMPA 11(26), 4689–4730 (1996)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  38. Hofman, C.: On the open-closed B-model. JHEP 2003, (2003)

  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. Witten E.: Interacting field theory of open superstrings. Nucl. Phys. B 276, 291 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  41. Berkovits, N.: Super Poincare covariant quantization of the superstring. JHEP 0004, 018 (2000). [hep-th/0001035]

  42. Akman, F.: On some generalizations of Batalin-Vilkovsky algebras. J. Pure. Appl. Algebra 120(2), 105–141 (1997). [arXiv:q-alg/9506027]

  43. Bering, K., Damgaard, P.H., Alfaro, J.: Algebra of higher antibrackets. Nucl. Phys. B478, 459–504 (1996). [hep-th/9604027]

  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)

  45. Hoefel, E.: OCHA and the swiss-cheese operad. J. Homotopy Relat Struct. 4, 123–151 (2009). [0710.3546v5 [math.QA]]

  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. Witten E.: Topological sigma models. Commun. Math. Phys. 118(3), 411–449 (1988)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  48. Getzler, E.: Batalin-Vilkovisky algebras and two-dimensional topological field theories. Commun. Math. Phys. 159, 265–285 (1994). [hep-th/9212043]

  49. Barannikov, S.: Modular operads and Batalin-Vilkovisky geometry. Int. Math. Res. Not. 2007 (2007). doi:10.1093/imrn/rnm075

  50. Stasheff J.: Homotopy associativity of H-spaces I. Am. Math. Soc. 108(2), 275–292 (1963)

    Article  MATH  MathSciNet  Google Scholar 

  51. Stasheff J.: Homotopy associativity of H-spaces II. Am. Math. Soc. 108(2), 293–312 (1963)

    MathSciNet  Google Scholar 

  52. Getzler E., Jones J.D.S.: A ∞-algebras and the cyclic bar complex. Ill. J. Math. 34(2), 256–283 (1990)

    MATH  MathSciNet  Google Scholar 

  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. Lada, T., Markl, M.: Strongly homotopy Lie algebras. Commun. Algebra 23(6), 2147–2161 (1995). [hep-th/9406095]

Download references

Author information

Authors and Affiliations

  1. Arnold Sommerfeld Center for Theoretical Physics, Theresienstrasse 37, 80333, Munich, Germany

    Korbinian Münster & Ivo Sachs

Authors
  1. Korbinian Münster
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ivo Sachs
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Korbinian Münster.

Additional information

Communicated by N. A. Nekrasov

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Münster, K., Sachs, I. Homotopy Classification of Bosonic String Field Theory. Commun. Math. Phys. 330, 1227–1262 (2014). https://doi.org/10.1007/s00220-014-2027-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-014-2027-8

Keywords

Search

Navigation