40 research outputs found
-Algebras of Classical Field Theories and the Batalin-Vilkovisky Formalism
We review in detail the Batalin-Vilkovisky formalism for Lagrangian field
theories and its mathematical foundations with an emphasis on higher algebraic
structures and classical field theories. In particular, we show how a field
theory gives rise to an -algebra and how quasi-isomorphisms between
-algebras correspond to classical equivalences of field theories. A
few experts may be familiar with parts of our discussion, however, the material
is presented from the perspective of a very general notion of a gauge theory.
We also make a number of new observations and present some new results. Most
importantly, we discuss in great detail higher (categorified) Chern-Simons
theories and give some useful shortcuts in usually rather involved
computations.Comment: v3: 131 pages, minor improvements, published versio
Noncommutative homotopy algebras associated with open strings
We discuss general properties of -algebras and their applications
to the theory of open strings. The properties of cyclicity for
-algebras are examined in detail. We prove the decomposition theorem,
which is a stronger version of the minimal model theorem, for
-algebras and cyclic -algebras and discuss various
consequences of it. In particular it is applied to classical open string field
theories and it is shown that all classical open string field theories on a
fixed conformal background are cyclic -isomorphic to each other. The
same results hold for classical closed string field theories, whose algebraic
structure is governed by cyclic -algebras.Comment: 92 pages, 16 figuers; based on Ph.D thesis submitted to Graduate
School of Mathematical Sciences, Univ. of Tokyo on January, 2003; v2:
explanation improved, references added, published versio
-Algebras, the BV Formalism, and Classical Fields
We summarise some of our recent works on -algebras and quasi-groups
with regard to higher principal bundles and their applications in twistor
theory and gauge theory. In particular, after a lightning review of
-algebras, we discuss their Maurer-Cartan theory and explain that any
classical field theory admitting an action can be reformulated in this context
with the help of the Batalin-Vilkovisky formalism. As examples, we explore
higher Chern-Simons theory and Yang-Mills theory. We also explain how these
ideas can be combined with those of twistor theory to formulate maximally
superconformal gauge theories in four and six dimensions by means of
-quasi-isomorphisms, and we propose a twistor space action.Comment: 19 pages, Contribution to Proceedings of LMS/EPSRC Durham Symposium
Higher Structures in M-Theory, August 201
Open-closed homotopy algebra in mathematical physics
In this paper we discuss various aspects of open-closed homotopy algebras
(OCHAs) presented in our previous paper, inspired by Zwiebach's open-closed
string field theory, but that first paper concentrated on the mathematical
aspects. Here we show how an OCHA is obtained by extracting the tree part of
Zwiebach's quantum open-closed string field theory. We clarify the explicit
relation of an OCHA with Kontsevich's deformation quantization and with the
B-models of homological mirror symmetry. An explicit form of the minimal model
for an OCHA is given as well as its relation to the perturbative expansion of
open-closed string field theory. We show that our open-closed homotopy algebra
gives us a general scheme for deformation of open string structures
(-algebras) by closed strings (-algebras).Comment: 38 pages, 4 figures; v2: published versio
DG-algebras and derived A-infinity algebras
A differential graded algebra can be viewed as an A-infinity algebra. By a
theorem of Kadeishvili, a dga over a field admits a quasi-isomorphism from a
minimal A-infinity algebra. We introduce the notion of a derived A-infinity
algebra and show that any dga A over an arbitrary commutative ground ring k is
equivalent to a minimal derived A-infinity algebra. Such a minimal derived
A-infinity algebra model for A is a k-projective resolution of the homology
algebra of A together with a family of maps satisfying appropriate relations.
As in the case of A-infinity algebras, it is possible to recover the dga up
to quasi-isomorphism from a minimal derived A-infinity algebra model. Hence the
structure we are describing provides a complete description of the
quasi-isomorphism type of the dga.Comment: v3: 27 pages. Minor corrections, to appear in Crelle's Journa
An algebraic proof of Bogomolov-Tian-Todorov theorem
We give a completely algebraic proof of the Bogomolov-Tian-Todorov theorem.
More precisely, we shall prove that if X is a smooth projective variety with
trivial canonical bundle defined over an algebraically closed field of
characteristic 0, then the L-infinity algebra governing infinitesimal
deformations of X is quasi-isomorphic to an abelian differential graded Lie
algebra.Comment: 20 pages, amspro
D-brane categories
This is an exposition of recent progress in the categorical approach to
D-brane physics. I discuss the physical underpinnings of the appearance of
homotopy categories and triangulated categories of D-branes from a string field
theoretic perspective, and with a focus on applications to homological mirror
symmetry.Comment: 37 pages, IJMPA styl
Manin products, Koszul duality, Loday algebras and Deligne conjecture
In this article we give a conceptual definition of Manin products in any
category endowed with two coherent monoidal products. This construction can be
applied to associative algebras, non-symmetric operads, operads, colored
operads, and properads presented by generators and relations. These two
products, called black and white, are dual to each other under Koszul duality
functor. We study their properties and compute several examples of black and
white products for operads. These products allow us to define natural
operations on the chain complex defining cohomology theories. With these
operations, we are able to prove that Deligne's conjecture holds for a general
class of operads and is not specific to the case of associative algebras.
Finally, we prove generalized versions of a few conjectures raised by M. Aguiar
and J.-L. Loday related to the Koszul property of operads defined by black
products. These operads provide infinitely many examples for this generalized
Deligne's conjecture.Comment: Final version, a few references adde