References
217
Although the I BL ∞ -algebras used there are very particular ones, the one describing
closed strings corresponds to a loop homotopy algebra and the one used to
incorporate the open strings is an ordinary IBL-algebra, the full open-closed string
field theory comprises their morphism in the IBL ∞ world. If we write the action
as the sum S c + S oc + S o , where S c and S o are the closed and open parts of
the action, respectively, this morphism corresponds to the vertices in the S oc part.
Complementary to our description of IBL ∞ in Part II, we reviewed them also in
Appendix A in a form directly used in our physics discussion of Part I.
References
1. Barannikov, S.: Modular operads and Batalin-Vilkovisky geometry. Int. Math. Res. Notices
2007(19), Art. ID rnm075, 31 (2007). http://dx.doi.org/10.1093/imrn/rnm075
2. Bashkirov, D., Voronov, A.A.: The BV formalism for L ∞ -algebras. ArXiv e-prints (2014)
3. Cieliebak, K., Latschev, J.: The Role of String Topology in Symplectic Field Theory. CRM
Proc. Lecture Notes, vol. 49, pp. 113–146. American Mathematical Society, Providence (2009)
4. Cieliebak, K., Fukaya, K., Latschev, J.: Homological algebra related to surfaces with boundary.
ArXiv e-prints (2015)
5. Kassel, C.: Quantum Groups, Graduate Texts in Mathematics, vol. 155. Springer, New York
(1995). http://dx.doi.org/10.1007/978-1-4612-0783-2
6. Kravchenko, O.: Deformations of Batalin-Vilkovisky algebras. In: Poisson Geometry (Warsaw,
1998), Banach Center Publ., vol. 51, pp. 131–139. Polish Academy of Sciences, Warsaw (2000)
7. Lada, T., Markl, M.: Strongly homotopy Lie algebras. Commun. Algebra 23(6), 2147–2161
(1995). http://dx.doi.org/10.1080/00927879508825335
8. Markl, M.: Models for operads. Commun. Algebra 24(4), 1471–1500 (1996). http://dx.doi.
org/10.1080/00927879608825647
9. Markl, M.: Loop homotopy algebras in closed string field theory. Commun. Math. Phys.
221(2), 367–384 (2001). http://dx.doi.org/10.1007/PL00005575
10. Markl, M.: Operads and PROPs. In: Handbook of Algebra, vol. 5, pp. 87–140. Elsevier/NorthHolland, Amsterdam (2008). http://dx.doi.org/10.1016/S1570-7954(07)05002-4
11. Markl, M.: On the origin of higher braces and higher-order derivations. J. Homotopy Relat.
Struct. (2015). https://doi.org/10.1007/s40062-014-0079-2
12. Markl, M., Voronov, A.: The MV formalism for IBL ∞ - and BV ∞ -algebras. Lett. Math. Phys.
107(8), 1515–1543 (2017)
13. Münster, K., Sachs, I.: Quantum open-closed homotopy algebra and string field
theory. Commun. Math. Phys. 321(3), 769–801 (2013). http://dx.doi.org/10.1007/s00220012-1654-1
217
Although the I BL ∞ -algebras used there are very particular ones, the one describing
closed strings corresponds to a loop homotopy algebra and the one used to
incorporate the open strings is an ordinary IBL-algebra, the full open-closed string
field theory comprises their morphism in the IBL ∞ world. If we write the action
as the sum S c + S oc + S o , where S c and S o are the closed and open parts of
the action, respectively, this morphism corresponds to the vertices in the S oc part.
Complementary to our description of IBL ∞ in Part II, we reviewed them also in
Appendix A in a form directly used in our physics discussion of Part I.
References
1. Barannikov, S.: Modular operads and Batalin-Vilkovisky geometry. Int. Math. Res. Notices
2007(19), Art. ID rnm075, 31 (2007). http://dx.doi.org/10.1093/imrn/rnm075
2. Bashkirov, D., Voronov, A.A.: The BV formalism for L ∞ -algebras. ArXiv e-prints (2014)
3. Cieliebak, K., Latschev, J.: The Role of String Topology in Symplectic Field Theory. CRM
Proc. Lecture Notes, vol. 49, pp. 113–146. American Mathematical Society, Providence (2009)
4. Cieliebak, K., Fukaya, K., Latschev, J.: Homological algebra related to surfaces with boundary.
ArXiv e-prints (2015)
5. Kassel, C.: Quantum Groups, Graduate Texts in Mathematics, vol. 155. Springer, New York
(1995). http://dx.doi.org/10.1007/978-1-4612-0783-2
6. Kravchenko, O.: Deformations of Batalin-Vilkovisky algebras. In: Poisson Geometry (Warsaw,
1998), Banach Center Publ., vol. 51, pp. 131–139. Polish Academy of Sciences, Warsaw (2000)
7. Lada, T., Markl, M.: Strongly homotopy Lie algebras. Commun. Algebra 23(6), 2147–2161
(1995). http://dx.doi.org/10.1080/00927879508825335
8. Markl, M.: Models for operads. Commun. Algebra 24(4), 1471–1500 (1996). http://dx.doi.
org/10.1080/00927879608825647
9. Markl, M.: Loop homotopy algebras in closed string field theory. Commun. Math. Phys.
221(2), 367–384 (2001). http://dx.doi.org/10.1007/PL00005575
10. Markl, M.: Operads and PROPs. In: Handbook of Algebra, vol. 5, pp. 87–140. Elsevier/NorthHolland, Amsterdam (2008). http://dx.doi.org/10.1016/S1570-7954(07)05002-4
11. Markl, M.: On the origin of higher braces and higher-order derivations. J. Homotopy Relat.
Struct. (2015). https://doi.org/10.1007/s40062-014-0079-2
12. Markl, M., Voronov, A.: The MV formalism for IBL ∞ - and BV ∞ -algebras. Lett. Math. Phys.
107(8), 1515–1543 (2017)
13. Münster, K., Sachs, I.: Quantum open-closed homotopy algebra and string field
theory. Commun. Math. Phys. 321(3), 769–801 (2013). http://dx.doi.org/10.1007/s00220012-1654-1