Geometric, Algebraic and Topological Methods for Quantum Field Theory

The following sections are included:

• INTRODUCTION
• CONTENTS

Motivated by the sharp contrast between classical and quantum physics as probability theories, in these lecture notes I introduce the basic notions of operator algebras that are relevant for the algebraic approach to quantum physics. Aspects of the representation theory of C*-algebras will be motivated and illustrated in physical terms. Particular emphasis will be given to explicit examples from the theory of quantum phase transitions, where concepts coming from strands as diverse as quantum information theory, algebraic quantum physics and statistical mechanics agreeably converge, providing a more complete picture of the physical phenomena involved.

To pave the way for the journey from geometry to conformal field theory (CFT), these notes present the background for some basic CFT constructions from Calabi-Yau geometry. Topics include the complex and Kähler geometry of Calabi-Yau manifolds and their classification in low dimensions. I furthermore discuss CFT constructions for the simplest known examples that are based in Calabi-Yau geometry, namely for the toroidal superconformal field theories and their ℤ₂-orbifolds. En route from geometry to CFT, I offer a discussion of K3 surfaces as the simplest class of Calabi-Yau manifolds where non-linear sigma model constructions bear mysteries to the very day. The elliptic genus in CFT and in geometry is recalled as an instructional piece of evidence in favor of a deep connection between geometry and conformal field theory.

Black Holes in General Relativity and Supergravity are presented and their geometrical aspects discussed. The microcanonical description of the entropy of the black hole in the D1-D5 system in type IIB supergravity is sketched. These are the class notes of the 12-hour series of lectures presented during July 15 to 27, 2013 at Villa de Leyva, Colombia as part of the Summer School on Geometric, Algebraic and Topological Methods for Quantum Field Theory.

The theory of real relativistic fluids is in the rather unique situation that there is a natural relativistic extension of the nonrelativistic theory, but it is physically untenable [1]. On the other hand, mounting evidence that matter created in relativistic heavy ion collisions behaves as a relativistic fluid with small but finite viscosity has given the quest for an alternative a definite goal [2]. We shall review different approaches to relativistic real fluids, their link to relativistic kinetic theory, and their application to the analysis of heavy ion collisions [3].

The following sections are included:

• Foreword
• Background: Poisson structures, Lie algebroids, Lie groupoids
• Symplectic groupoids
• Midword
• Poisson groupoids
• The diagrammatic approach
• Endword
• References

The following sections are included:

• Introduction
• Lie algebroids
• Examples
• Modules and universal enveloping algebras
• Lie algebroid cohomology
• Classiffication of invertible modules
• Characteristic classes
• References

These lecture notes provide basic concepts for the understanding of index theory, in particular on manifolds with boundary. Topics include Fredholm theory, pseudodifferential calculus, Boutet de Monvel's algebra, K-theory for C*-algebras and applications to the index theory for boundary value problems.

This is a leisurely introductory account addressed to non-experts and based on previous work by the authors, on how methods borrowed from physics can be used to “count” an infinite number of points. We begin with the classical case of counting integer points on the non-negative real axis and the classical Euler-Maclaurin formula. As an intermediate stage, we count integer points on product cones where the roles played by the coalgebra and the algebraic Birkhoff factorization can be appreciated in a relatively simple setting. We then consider the general case of (lattice) cones for which we introduce a connected cograded coalgebra of cones, with applications to renormalization of conical zeta values. When evaluated at zero arguments conical zeta functions indeed “count” integer points on cones.

We construct a different Batalin-Vilkovisky algebra structure on the Hochschild cohomology of the algebra of truncated polynomials with coefficients in a field of p elements with p prime. We accomplish this by transfering the Frobenius form of the group ring of cyclic groups. We also explicitly calculate the Batalin-Vilkovisky algebra structure of the Hochschild cohomology of the group rings of cyclic groups of prime order. The algebra structure, even the Gesterhaber structure has been calculated before, but the BV structure that we calculate is a new one.