Mathematical Physics

Mathematical Physics

My work in mathematical physics deals with geometric field theories. These are quantum field theory that provide mathematically relevant on geometrical objects.

The study of geometric field theories started in 1984 by the Belavin, Polyakov and Zamolodchikov paper on 2D conformal field theories and really took off when Witten realized in 1988 that the Jones polynomial could be obtained as an expectation value of Wilson lines in the Chern Simons field theory.

Famous examples include topological field theories in 3D which give knot and 3-manifold invariants. In 2D, conformal field theories are related to holomorphic finite dimensional vector bundles over moduli spaces. The properties of geometric field theories arise from an underlying symmetry: general covariance for topological field theories and conformal invariance for conformal field theories.

Geometric field theory was the object of my PhD thesis (december 1992) supervised by Bernard Julia from the theroretical physics lab of the Ecole Normale Supérieure in Paris.

My subsequent work on the subject mainly focused on the connexion between topological field theory and arithmetics in close relations with ideas developed by A. Grothendieck in the Esquisse d’un programme.

CONSTRUCTION OF TOPOLOGICAL FIELD THEORIES

Title: Moore and Seiberg equations and 3D TFTs

Author: Pascal Degiovanni

Journal reference: Comm. Math. Phys. 145, 459-505 (1992)

Euclid identifier: euclid.cmp:1104249810

Link: local copy (PDF)

In his 1988 paper on the field theoretical interpretation of the Jones polynomial, Edward Witten conjectured a deep relation between solutions to Moore and Seiberg's equations and 3D TFTs. I have worked on this problem between 1989 and 1992 obtaining several results.

First of all, I proved that from Moore and Seiberg equations, one can reconstruct a 3D projective topological field theory. This result is really a proof of the assumption made by Witten in his 1988 paper that any rational conformal field theory leads to a 3D topological field theory.

The main mathematical tool is a refinement of Kirby's calculus on surgery presentations for compact oriented three-manifolds developed by R. Kirby [Invent. Math. 45, 35-56 (1978)] and R. Fenn & C. Rourke [Topology 18, 1-15 (1979)].

Related unpublished works:

An alternative approach for the construction of 3D TFT uses Heegaard splittings. These are decompositions of closed 3D manifolds as gluings of handlebodies. I developed it in a LPTENS preprint (LPTENS 89-25) and never published it since the surgery approach is somehow simpler and enables treating the case of manifolds with boundaries. It also provides a nice treatment of cocycle problems (projectivity of the TFT).

Nevertheless, the use of Heegard splittings [see Trans. Am. Math. Soc. 35, 88-111 (1933) and Can. Journ. Maths. XXIX 11-124 (1977)] enables the definition of a numerical invariant for closed three-manifolds.

If you want to know more: local copy (PDF file).

Systematic constructions of 2D TFTs

Using solutions to Moore and Seiberg's equations, one can easily construct 2D TFTs. All constructions are done in the framework of category theory. The simplest construction uses no more than the 3D TFT deduced from a particular solution to Moore and Seiberg's equations.

The basic idea is to define a functor from the 2D geometrical category on which the 2D TFT will be defined, into the 2D TFT on which the 2D TFT is defined. It gives a solution to the gauged WZW model associated with a simply connected compact simple Lie group.

If you want to know more, this construction is described in my PhD (Chap. 7): local copy (PDF).

Title: Equations de Moore et Seiberg, théories topologiques et théorie de Galois

Author: Pascal Degiovanni

Journal reference: Helvetica Physica Acta 67, 799-883 (1994)

Link: local copy (PDF)</a>

The following work arose from the reading of an unpublished manuscript by A. Grothendieck called Esquisse d'un programme. In this text, written in 1984, Grothendieck describes several vision of the tower of all moduli spaces of algebraic curves (for all values of genus and number of punctures). He claims that, with suitable sets of basse points, the tower of algebraic fundamental groupoids (called the Teichmüller tower) should be studied, and that the natural action of the absolute Galois group on them preserves all morphisms between these groupoids.

The basic idea was then to connect this work with physics work on topological field theory. Moore and Seiberg data provide a kind of representation of a particular tower of fundamental groupoids which corresponds to the case of maximal degeneration base points. It is then very natural to look for the translation on three dimensional topological field theories of the natural Galois action on the Teichmüller tower. The result is surprisingly simple: on rational topological field theories, the action is simply the number theoretical action of the Galois group on numerical values of the invariants, which are always algebraic numbers.

Grothendieck's vision of the absolute Galois group as the automorphism group of the Teichmüller tower then suggests that all topological invariants that are related by this Galois action contain the same information about 3D topology.

Title: Moore and Seiberg equations, topological field theories and Galois theory

Author: Pascal Degiovanni

Reference: in The Grothendieck theory of dessins d’enfants (L. Schneps), Lecture Notes in Mathematics 200 (1994), p. 359-368.

Link: local copy (PDF)

This work is a summary of the connexion relating 2D conformal field and 3D topological field theory on the physics side to the ideas developed by Grothendieck in the Esquisse d’un programme. It is a very synthetic summary of the ideas expanded in my long paper “Equations de Moore et Seiberg, theories topologiques et théorie de Galois”.

Usually people cite this short paper but all the details and real proofs are indeed given in the long one...

Title: Precise study of some number fields and Galois actions occuring in

Conformal Field Theory

Authors: Eric Buffenoir, Antoine Coste, Jean Lascoux, Pascal Degiovanni and Arnaud Buhot.

Journal reference: Annales de l'Institut Henri Poincaré 63, 41-79 (1995)

Link: local copy (PDF)

The following paper contains explicit examples of the general ideas of my general paper on Galois actions. It was written in collaboration with A. Coste who also explored the use of Galois actions on RCFT in order to solve for modular invariants for RCFTs.

In this paper, we present a detailed study of some number fields and Galois groups occurring in two dimensional models built from Wess-Zumino-Novikov-Witten (WZNW) and Z/nZ theories. The observed structures must be relevant for the classification of rational conformal theories (RCFT) and for the understanding of links and three manifold invariants.

ARITHMETICS AND TOPOLOGICAL FIELD THEORY

CONFORMAL FIELD THEORY

Title: Z/nZ Conformal Field Theories

Author: Pascal Degiovanni.

Journal reference: Comm. Math. Phys. 127, 71-99 (1990)

Euclid identifier: euclid.cmp:1104180040

Link: local copy (PDF)

We compute the modular properties of the possible genus-one characters of some Rational Conformal Field Theories starting from their fusion rules. We show that possible choices of S matrices are indexed by some automorphisms of the fusion algebra. We also classify the modular invariant partition functions of these theories. This gives the complete list of modular invariant partition functions of Rational Conformal Field Theories with respect to the A_N^(1) level one algebra.

This work has been presented at the Les Houches Spring school on "Number Theory and Physics" (March 1989).

Related unpublished work

Using the naturality theorem proved by Moore and Seiberg, or the version proved by E. Verlinde and R. Dijkgraaf, one can also obtain the classification of modular invariant partition function.

This alternate proof (see local copy: PDF file) is part of my PhD (Chapter 4). It also contains a discussion of orbifold models of some holomorphic conformal field theory with respect to some cyclic group in the case they lead to a Z/nZ RCFT.