221 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
Invitation to composition
In 1963 [Ann. of Math. {\bf 78}, 267-288], Gerstenhaber invented a
\emph{comp(osition)} calculus in the Hochschild complex of an associative
algebra. In this paper, the first steps of the Gerstenhaber theory are exposed
in an abstract (comp system) setting. In particular, as in the Hochschild
complex, a graded Lie algebra and a pre-coboundary operator can be associated
to every comp system. A derivation deviation of the pre-coboundary operator
over the total composition is calculated in two ways, (the long) one of which
is essentially new and can be seen as an example and elaboration of the
auxiliary variables method proposed by Gerstenhaber in the early days of the
comp calculus.Comment: 18 pages, AMSLaTe
The sh Lie structure of Poisson brackets in field theory
A general construction of an sh Lie algebra from a homological resolution of
a Lie algebra is given. It is applied to the space of local functionals
equipped with a Poisson bracket, induced by a bracket for local functions along
the lines suggested by Gel'fand, Dickey and Dorfman. In this way, higher order
maps are constructed which combine to form an sh Lie algebra on the graded
differential algebra of horizontal forms. The same construction applies for
graded brackets in field theory such as the Batalin-Fradkin-Vilkovisky bracket
of the Hamiltonian BRST theory or the Batalin-Vilkovisky antibracket.Comment: 24 pages Latex fil
The Maurer-Cartan structure of BRST differential
In this paper, we construct a new sequence of generators of the BRST complex
and reformulate the BRST differential so that it acts on elements of the
complex much like the Maurer-Cartan differential acts on left-invariant forms.
Thus our BRST differential is formally analogous to the differential defined on
the BRST formulation of the Chevalley-Eilenberg cochain complex of a Lie
algebra. Moreover, for an important class of physical theories, we show that in
fact the differential is a Chevalley-Eilenberg differential. As one of the
applications of our formalism, we show that the BRST differential provides a
mechanism which permits us to extend a nonintegrable system of vector fields on
a manifold to an integrable system on an extended manifold
BV-BFV approach to General Relativity, Einstein-Hilbert action
The present paper shows that general relativity in the Arnowitt-Deser-Misner
formalism admits a BV-BFV formulation. More precisely, for any
(pseudo-) Riemannian manifold M with space-like or time-like boundary
components, the BV data on the bulk induces compatible BFV data on the
boundary. As a byproduct, the usual canonical formulation of general relativity
is recovered in a straightforward way.Comment: 16 page
On the Strong Homotopy Lie-Rinehart Algebra of a Foliation
It is well known that a foliation F of a smooth manifold M gives rise to a
rich cohomological theory, its characteristic (i.e., leafwise) cohomology.
Characteristic cohomologies of F may be interpreted, to some extent, as
functions on the space P of integral manifolds (of any dimension) of the
characteristic distribution C of F. Similarly, characteristic cohomologies with
local coefficients in the normal bundle TM/C of F may be interpreted as vector
fields on P. In particular, they possess a (graded) Lie bracket and act on
characteristic cohomology H. In this paper, I discuss how both the Lie bracket
and the action on H come from a strong homotopy structure at the level of
cochains. Finally, I show that such a strong homotopy structure is canonical up
to isomorphisms.Comment: 41 pages, v2: almost completely rewritten, title changed; v3:
presentation partly changed after numerous suggestions by Jim Stasheff,
mathematical content unchanged; v4: minor revisions, references added. v5:
(hopefully) final versio
Irreducible antifield-BRST approach to reducible gauge theories
An irreducible antifield BRST quantization method for reducible gauge
theories is proposed. The general formalism is illustrated in the case of the
Freedman-Townsend model.Comment: 19 pages, LaTeX 2.0
A Contour Integral Representation for the Dual Five-Point Function and a Symmetry of the Genus Four Surface in R6
The invention of the "dual resonance model" N-point functions BN motivated
the development of current string theory. The simplest of these models, the
four-point function B4, is the classical Euler Beta function. Many standard
methods of complex analysis in a single variable have been applied to elucidate
the properties of the Euler Beta function, leading, for example, to analytic
continuation formulas such as the contour-integral representation obtained by
Pochhammer in 1890. Here we explore the geometry underlying the dual five-point
function B5, the simplest generalization of the Euler Beta function. Analyzing
the B5 integrand leads to a polyhedral structure for the five-crosscap surface,
embedded in RP5, that has 12 pentagonal faces and a symmetry group of order 120
in PGL(6). We find a Pochhammer-like representation for B5 that is a contour
integral along a surface of genus five. The symmetric embedding of the
five-crosscap surface in RP5 is doubly covered by a symmetric embedding of the
surface of genus four in R6 that has a polyhedral structure with 24 pentagonal
faces and a symmetry group of order 240 in O(6). The methods appear
generalizable to all N, and the resulting structures seem to be related to
associahedra in arbitrary dimensions.Comment: 43 pages and 44 figure
On the cohomological derivation of topological Yang-Mills theory
Topological Yang-Mills theory is derived in the framework of Lagrangian BRST
cohomology.Comment: LaTeX 2.09, 12 page
Non-commutative tachyon action and D-brane geometry
We analyse open string correlators in non-constant background fields,
including the metric , the antisymmetric -field, and the gauge field .
Working with a derivative expansion for the background fields, but exact in
their constant parts, we obtain a tachyonic on-shell condition for the inserted
functions and extract the kinetic term for the tachyon action. The 3-point
correlator yields a non-commutative tachyon potential. We also find a
remarkable feature of the differential structure on the D-brane: Although the
boundary metric plays an essential role in the action, the natural
connection on the D-brane is the same as in closed string theory, i.e. it is
compatible with the bulk metric and has torsion . This means, in
particular, that the parallel transport on the brane is independent of the
gauge field .Comment: 12 pages, no figure
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