An introduction to the Batalin-Vilkovisky formalism

The 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.Comment: 19 pages. Lecture given at the "Rencontres Mathematiques de Glanon", July 2003. Final version. Minor correction

Feynman Diagrams via Graphical Calculus

This paper is an introduction to the language of Feynman Diagrams. We 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 kind 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.Comment: 30 pages, AMS-LaTeX, 19 EPS figures + several in-text XY-Pic, PostScript \specials, corrected attributions, 'PROP's instead of 'operads

Boundary Conditions for Topological Quantum Field Theories, Anomalies and Projective Modular Functors

We study boundary conditions for extended topological quantum field theories (TQFTs) and their relation to topological anomalies. We introduce the notion of TQFTs with moduli level $m$, and describe extended anomalous theories as natural transformations of invertible field theories of this type. We show how in such a framework anomalous theories give rise naturally to homotopy fixed points for $n$-characters on $\infty$-groups. By using dimensional reduction on manifolds with boundaries, we show how boundary conditions for $n+1$-dimensional TQFTs produce $n$-dimensional anomalous field theories. Finally, we analyse the case of fully extended TQFTs, and show that any fully extended anomalous theory produces a suitable boundary condition for the anomaly field theory.Comment: 26 pages, 6 figures. Exposition improved, bibliography updated. Final version, to appear in Comm. Math. Phy

Associative algebras, punctured disks and the quantization of Poisson manifolds

The 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.Comment: LaTeX, 8 pages, uses XY-pic. Few typos corrected. Final versio

A short note on infinity-groupoids and the period map for projective manifolds

A common criticism of infinity-categories in algebraic geometry is that they are an extremely technical subject, so abstract to be useless in everyday mathematics. The aim of this note is to show in a classical example that quite the converse is true: even a naive intuition of what an infinity-groupoid should be clarifies several aspects of the infinitesimal behaviour of the periods map of a projective manifold. In particular, the notion of Cartan homotopy turns out to be completely natural from this perspective, and so classical results such as Griffiths' expression for the differential of the periods map, the Kodaira principle on obstructions to deformations of projective manifolds, the Bogomolov-Tian-Todorov theorem, and Goldman-Millson quasi-abelianity theorem are easily recovered.Comment: 13 pages; uses xy-pic; exposition improved and a few inaccuracies corrected; an hypertextual version of this article is available at http://ncatlab.org/publications/published/FiorenzaMartinengo201

t-structures are normal torsion theories

We characterize $t$-structures in stable $\infty$-categories as suitable quasicategorical factorization systems. More precisely we show that a $t$-structure $\mathfrak{t}$ on a stable $\infty$-category $\mathbf{C}$ is equivalent to a normal torsion theory $\mathbb{F}$ on $\mathbf{C}$, i.e. to a factorization system $\mathbb{F}=(\mathcal{E},\mathcal{M})$ where both classes satisfy the 3-for-2 cancellation property, and a certain compatibility with pullbacks/pushouts.Comment: Minor typographical corrections from v1; 25 pages; to appear in "Applied Categorical Structures

Matrix Integrals and Feynman Diagrams in the Kontsevich Model

We 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.Comment: 52 pages; final versio