91 research outputs found
Relations Among Universal Equations For Gromov-Witten Invariants
In this paper, we study relations among known universal equations for
Gromov-Witten invariants at genus 1 and 2.Comment: LaTex file, 13 page
Deformations of coisotropic submanifolds for fibrewise entire Poisson structures
We show that deformations of a coisotropic submanifold inside a fibrewise
entire Poisson manifold are controlled by the -algebra introduced by
Oh-Park (for symplectic manifolds) and Cattaneo-Felder. In the symplectic case,
we recover results previously obtained by Oh-Park. Moreover we consider the
extended deformation problem and prove its obstructedness
Hypercommutative operad as a homotopy quotient of BV
We give an explicit formula for a quasi-isomorphism between the operads
Hycomm (the homology of the moduli space of stable genus 0 curves) and
BV/ (the homotopy quotient of Batalin-Vilkovisky operad by the
BV-operator). In other words we derive an equivalence of Hycomm-algebras and
BV-algebras enhanced with a homotopy that trivializes the BV-operator.
These formulas are given in terms of the Givental graphs, and are proved in
two different ways. One proof uses the Givental group action, and the other
proof goes through a chain of explicit formulas on resolutions of Hycomm and
BV. The second approach gives, in particular, a homological explanation of the
Givental group action on Hycomm-algebras.Comment: minor corrections added, to appear in Comm.Math.Phy
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
Jacobi structures revisited
Jacobi algebroids, that is graded Lie brackets on the Grassmann algebra
associated with a vector bundle which satisfy a property similar to that of the
Jacobi brackets, are introduced. They turn out to be equivalent to generalized
Lie algebroids in the sense of Iglesias and Marrero and can be viewed also as
odd Jacobi brackets on the supermanifolds associated with the vector bundles.
Jacobi bialgebroids are defined in the same manner. A lifting procedure of
elements of this Grassmann algebra to multivector fields on the total space of
the vector bundle which preserves the corresponding brackets is developed. This
gives the possibility of associating canonically a Lie algebroid with any local
Lie algebra in the sense of Kirillov.Comment: 20 page
Spectral triples and the super-Virasoro algebra
We construct infinite dimensional spectral triples associated with
representations of the super-Virasoro algebra. In particular the irreducible,
unitary positive energy representation of the Ramond algebra with central
charge c and minimal lowest weight h=c/24 is graded and gives rise to a net of
even theta-summable spectral triples with non-zero Fredholm index. The
irreducible unitary positive energy representations of the Neveu-Schwarz
algebra give rise to nets of even theta-summable generalised spectral triples
where there is no Dirac operator but only a superderivation.Comment: 27 pages; v2: a comment concerning the difficulty in defining cyclic
cocycles in the NS case have been adde
Non-Commutative Batalin-Vilkovisky Algebras, Homotopy Lie Algebras and the Courant Bracket
We consider two different constructions of higher brackets. First, based on a
Grassmann-odd, nilpotent \Delta operator, we define a non-commutative
generalization of the higher Koszul brackets, which are used in a generalized
Batalin-Vilkovisky algebra, and we show that they form a homotopy Lie algebra.
Secondly, we investigate higher, so-called derived brackets built from
symmetrized, nested Lie brackets with a fixed nilpotent Lie algebra element Q.
We find the most general Jacobi-like identity that such a hierarchy satisfies.
The numerical coefficients in front of each term in these generalized Jacobi
identities are related to the Bernoulli numbers. We suggest that the definition
of a homotopy Lie algebra should be enlarged to accommodate this important
case. Finally, we consider the Courant bracket as an example of a derived
bracket. We extend it to the "big bracket" of exterior forms and multi-vectors,
and give closed formulas for the higher Courant brackets.Comment: 42 pages, LaTeX. v2: Added remarks in Section 5. v3: Added further
explanation. v4: Minor adjustments. v5: Section 5 completely rewritten to
include covariant construction. v6: Minor adjustments. v7: Added references
and explanation to Section
The Extended Bigraded Toda hierarchy
We generalize the Toda lattice hierarchy by considering N+M dependent
variables. We construct roots and logarithms of the Lax operator which are
uniquely defined operators with coefficients that are -series of
differential polynomials in the dependent variables, and we use them to provide
a Lax pair definition of the extended bigraded Toda hierarchy. Using R-matrix
theory we give the bihamiltonian formulation of this hierarchy and we prove the
existence of a tau function for its solutions. Finally we study the
dispersionless limit and its connection with a class of Frobenius manifolds on
the orbit space of the extended affine Weyl groups of the series.Comment: 32 pages, corrected typo
Classification of real three-dimensional Lie bialgebras and their Poisson-Lie groups
Classical r-matrices of the three-dimensional real Lie bialgebras are
obtained. In this way all three-dimensional real coboundary Lie bialgebras and
their types (triangular, quasitriangular or factorizable) are classified. Then,
by using the Sklyanin bracket, the Poisson structures on the related
Poisson-Lie groups are obtained.Comment: 17 page
The structure of 2D semi-simple field theories
I classify all cohomological 2D field theories based on a semi-simple complex
Frobenius algebra A. They are controlled by a linear combination of
kappa-classes and by an extension datum to the Deligne-Mumford boundary. Their
effect on the Gromov-Witten potential is described by Givental's Fock space
formulae. This leads to the reconstruction of Gromov-Witten invariants from the
quantum cup-product at a single semi-simple point and from the first Chern
class, confirming Givental's higher-genus reconstruction conjecture. The proof
uses the Mumford conjecture proved by Madsen and Weiss.Comment: Small errors corrected in v3. Agrees with published versio
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