1,721,014 research outputs found
An introduction to the Batalin-Vilkovisky formalism
No full textThe aim of these notes is to introduce the quantum master equation {S, S}−2i \hbar \Delta S = 0, and to show its relations to the theory of Lie algebras representations and to perturbative expansions of Gaussian integrals. The relations of the classical master equation {S, S} = 0 with the BRST formalisms are also described. Being an introduction, only
finite-dimensional examples will be considered
Sums over graphs and integration over discrete groupoids
No full textWe show that sums over graphs such as appear in the theory of Feynman diagrams can be seen as integrals over discrete groupoids. From this point of view, basic combinatorial formulas of the theory of Feynman diagrams can be interpreted as pull-back or push-forward formulas for integrals over suitable groupoids
Feynman diagrams via graphical calculus
No full textWe use Reshetikhin-Turaev graphical calculus to define Feynman diagrams and prove that asymptotic expansions of Gaussian integrals can be written as a sum over a suitable family of graphs. We discuss how different kinds of interactions give rise to different families of graphs. In particular, we show how symmetric and cyclic interactions lead to "ordinary" and "ribbon" graphs respectively. As an example, the 't Hooft-Kontsevich model for 2D quantum gravity is treated in some detail
FORMALITY OF KOSZUL BRACKETS AND DEFORMATIONS OF HOLOMORPHIC POISSON MANIFOLDS
No full textWe show that if a generator of a differential Gerstenhaber algebra satisfies certain Cartan-type identities, then the corresponding Lie bracket is formal. Geometric examples include the shifted de Rham complex of a Poisson manifold and the subcomplex of differential forms on a symplectic manifold vanishing on a Lagrangian submanifold, endowed with the Koszul bracket. As a corollary we generalize a recent result by Hitchin on deformations of holomorphic Poisson manifolds
Matrix integrals and Feynman diagrams in the Kontsevich model
No full textWe review some relations occurring between the combinatorial intersection theory on the moduli spaces of stable curves and the asymptotic behavior of the 't Hooft-Kontsevich matrix integrals. In particular, we give an alternative proof of the Witten-Di Francesco-Itzykson-
Zuber theorem |which expresses derivatives of the partition function of intersection numbers as matrix integrals| using techniques based on diagrammatic calculus and combinatorial relations among intersection numbers. These techniques extend to a more general interaction potential
Integrals detecting degree 3 string cobordism classes
Full text linkThe third string bordism group is known
to be . Using Waldorf's notion of a geometric string
structure on a manifold, Bunke--Naumann and Redden have exhibited integral
formulas involving the Chern-Weil form representative of the first Pontryagin
class and the canonical 3-form of a geometric string structure that realize the
isomorphism . We
will show how these formulas naturally emerge when one considers certain
natural -valued and -valued 3d TQFT associated with
the classifying stacks of Spin bundles with connection and of String bundles
with geometric structure, respectively.Comment: 25 pages, exposition improved, references added. Final versio
A short note on infinity-groupoids and the period map for projective manifolds
No full textWe show how several classical results on the infinitesimal behaviour of the period map for smooth projective manifolds can be read in a natural and unified way within the framework of infinity-categories
Associative algebras, punctured disks and the quantization of Poisson manifolds
No full textThe aim of the note is to provide an introduction to the algebraic, geometric and quantum field theoretic ideas that lie behind the Kontsevich–Cattaneo–Felder formula for the quantization of Poisson structures. We show how the quantization formula itself naturally arises
when one imposes the following two requirements to a Feynman integral: on the one side it has to reproduce the given Poisson structure as the first order term of its perturbative expansion; on the other side its three-point functions should describe an associative algebra.
It is further shown how the Magri–Koszul brackets on 1-forms naturally fits into the theory of the Poisson sigma-model
L-infinity structures on mapping cones
No full textWe show that the mapping cone of a morphism of differential graded Lie algebras, chi : L -> M, can be canonically endowed with an L-infinity-algebra structure which at the same time lifts the Lie algebra structure on L and the usual differential on the mapping cone. Moreover, this structure is unique up to isomorphisms of L-infinity-algebras
Higher U (1)-gerbe connections in geometric prequantization
No full textWe promote geometric prequantization to higher geometry (higher stacks), where a prequantization is given by a higher principal connection (a higher gerbe with connection). We show fairly generally how there is canonically a tower of higher gauge groupoids and Courant groupoids assigned to a higher prequantization, and establish the corresponding Atiyah sequence as an integrated Kostant-Souriau infinity-group extension of higher Hamiltonian symplectomorphisms by higher quantomorphisms. We also exhibit the infinity-group cocycle which classifies this extension and discuss how its restrictions along Hamiltonian infinity-actions yield higher Heisenberg cocycles. In the special case of higher differential geometry over smooth manifolds, we find the L-infinity-algebra extension of Hamiltonian vector fields - which is the higher Poisson bracket of local observables - and show that it is equivalent to the construction proposed by the second author in n-plectic geometry. Finally, we indicate a list of examples of applications of higher prequantization in the extended geometric quantization of local quantum field theories and specifically in string geometry
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