Homotopy algebras inspired by classical open-closed string field theory

We define a homotopy algebra associated to classical open-closed strings. We call it an open-closed homotopy algebra (OCHA). It is inspired by Zwiebach's open-closed string field theory and also is related to the situation of Kontsevich's deformation quantization. We show that it is actually a homotopy invariant notion; for instance, the minimal model theorem holds. Also, we show that our open-closed homotopy algebra gives us a general scheme for deformation of open string structures (A(infinity)-algebras) by closed strings (L(infinity)-algebras).Comment: 30 pages, 14 figures; v2: added an appendix by M.Markl, ambiguous terminology fixed, minor corrections; v3: published versio

Homotopy algebra of open-closed strings

This paper is a survey of our previous works on open-closed homotopy algebras, together with geometrical background, especially in terms of compactifications of configuration spaces (one of Fred's specialities) of Riemann surfaces, structures on loop spaces, etc. We newly present Merkulov's geometric A_infty-structure [Internat. Math. Res. Notices (1999) 153--164, arxiv:math/0001007] as a special example of an OCHA. We also recall the relation of open-closed homotopy algebras to various aspects of deformation theory.Comment: This is the version published by Geometry & Topology Monographs on 22 February 200

Noether's variational theorem II and the BV formalism

We review the basics of the Lagrangian approach to field theory and recast Noether's Second Theorem formulated in her language of dependencies using a slight modernization of terminology and notation. We then present the Cattaneo-Felder sigma model and work out the Noether identities or dependencies for this model. We review the description of the Batalin-Vilkovisky formalism and show explicitly how the anti-ghosts encode the Noether identities in this example.Comment: 15 pages, submitted to the Proceedings of the 2002 Winter School ``Geometry and Physics'', Srni, Czech Republi

Sh-Lie algebras Induced by Gauge Transformations

The physics of ``particles of spin $\leq 2$'' leads to representations of a Lie algebra $\Xi$ of gauge parameters on a vector space $\Phi$ of fields. Attempts to develop an analogous theory for spin $>2$ have failed; in fact, there are claims that such a theory is impossible (though we have been unable to determine the hypotheses for such a `no-go' theorem). This led BBvD [burgers:diss,BBvd:three,BBvD:probs] to generalize to `field dependent parameters' in a setting where some analysis in terms of smooth functions is possible. Having recognized the resulting structure as that of an sh-lie algebra ($L_\infty$-algebra), we have now reproduced their structure entirely algebraically, hopefully shedding some light on what is going on.Comment: Now 24 pages, LaTeX, no figures Extensively revised in terms of the applications and on shell aspects. In particular, a new section 8 analyzes Ikeda's 2D example from our perspective. His bracket is revealed as a generalized Kirillov-Kostant bracket. Additional reference