Entropy of Operator-valued Random Variables: A Variational Principle for Large N Matrix Models

We show that, in 't Hooft's large N limit, matrix models can be formulated as a classical theory whose equations of motion are the factorized Schwinger--Dyson equations. We discover an action principle for this classical theory. This action contains a universal term describing the entropy of the non-commutative probability distributions. We show that this entropy is a nontrivial 1-cocycle of the non-commutative analogue of the diffeomorphism group and derive an explicit formula for it. The action principle allows us to solve matrix models using novel variational approximation methods; in the simple cases where comparisons with other methods are possible, we get reasonable agreement.Comment: 45 pages with 1 figure, added reference

Computations in formal symplectic geometry and characteristic classes of moduli spaces

We make explicit computations in the formal symplectic geometry of Kontsevich and determine the Euler characteristics of the three cases, namely commutative, Lie and associative ones, up to certain weights.From these, we obtain some non-triviality results in each case. In particular, we determine the integral Euler characteristics of the outer automorphism groups Out F_n of free groups for all n <= 10 and prove the existence of plenty of rational cohomology classes of odd degrees. We also clarify the relationship of the commutative graph homology with finite type invariants of homology 3-spheres as well as the leaf cohomology classes for transversely symplectic foliations. Furthermore we prove the existence of several new non-trivalent graph homology classes of odd degrees. Based on these computations, we propose a few conjectures and problems on the graph homology and the characteristic classes of the moduli spaces of graphs as well as curves.Comment: 33 pages, final version, to appear in Quantum Topolog

Feynman diagrams and minimal models for operadic algebras

We construct an explicit minimal model for an algebra over the cobar-construction of a differential graded operad. The structure maps of this minimal model are expressed in terms of sums over decorated trees. We introduce the appropriate notion of a homotopy equivalence of operadic algebras and show that our minimal model is homotopy equivalent to the original algebra. All this generalizes and gives a conceptual explanation of well-known results for A∞-algebras. Furthermore, we show that these results carry over to the case of algebras over modular operads; the sums over trees get replaced by sums over general Feynman graphs. As a by-product of our work we prove gauge-independence of Kontsevich's ‘dual construction’ producing graph cohomology classes from contractible differential graded Frobenius algebras

Cohomology theories for homotopy algebras and noncommutative geometry

This paper builds a general framework in which to study cohomology theories of strongly homotopy algebras, namely $A_\infty, C_\infty$ and $L_\infty$-algebras. This framework is based on noncommutative geometry as expounded by Connes and Kontsevich. The developed machinery is then used to establish a general form of Hodge decomposition of Hochschild and cyclic cohomology of $C_\infty$-algebras. This generalizes and puts in a conceptual framework previous work by Loday and Gerstenhaber-Schack.Comment: This 54 pages paper is a substantial revision of the part of math.QA/0410621 dealing with algebraic Hodge decompositions of Hochschild and cyclic cohomology theories. The main addition is the treatment of cohomology theories corresponding to unital infinity-structure

The calculus of multivectors on noncommutative jet spaces

The Leibniz rule for derivations is invariant under cyclic permutations of co-multiples within the arguments of derivations. We explore the implications of this principle: in effect, we construct a class of noncommutative bundles in which the sheaves of algebras of walks along a tesselated affine manifold form the base, whereas the fibres are free associative algebras or, at a later stage, such algebras quotients over the linear relation of equivalence under cyclic shifts. The calculus of variations is developed on the infinite jet spaces over such noncommutative bundles. In the frames of such field-theoretic extension of the Kontsevich formal noncommutative symplectic (super)geometry, we prove the main properties of the Batalin--Vilkovisky Laplacian and Schouten bracket. We show as by-product that the structures which arise in the classical variational Poisson geometry of infinite-dimensional integrable systems do actually not refer to the graded commutativity assumption.Comment: Talks given at Mathematics seminar (IHES, 25.11.2016) and Oberseminar (MPIM Bonn, 2.02.2017), 23 figures, 60 page

A new look at loop quantum gravity

I describe a possible perspective on the current state of loop quantum gravity, at the light of the developments of the last years. I point out that a theory is now available, having a well-defined background-independent kinematics and a dynamics allowing transition amplitudes to be computed explicitly in different regimes. I underline the fact that the dynamics can be given in terms of a simple vertex function, largely determined by locality, diffeomorphism invariance and local Lorentz invariance. I emphasize the importance of approximations. I list open problems.Comment: 15 pages, 5 figure

Planar homotopy algebras and open-string field theory

This thesis is concerned about the existence of a open-string field theory that is consistent at the quantum level without coupling to the closed string. We want to achieve this via a restriction to planar Feynman graphs. Our aim is to formulate this theory in the mathematical language of homotopy algebras. We further ask whether such a formulation is applies also to general gauge theories, in particular in the limit of large gauge groups. Finally, we will discuss the problems of such a formulation, as well as how these can be solved by lifting the restriction to planar diagrams only. This work should also serve as an extensive introduction to the Batalin-Vilkovisky formalism. We look at different mathematical aspects of this formalism and its relation to homotopy algebras