37 research outputs found
Renormalization of the Orientable Non-commutative Gross-Neveu Model
We prove that the non-commutative Gross-Neveu model on the two-dimensional
Moyal plane is renormalizable to all orders. Despite a remaining UV/IR mixing,
renormalizability can be achieved. However, in the massive case, this forces us
to introduce an additional counterterm of the form "psibar i gamma^{0}
gamma^{1} psi". The massless case is renormalizable without such an addition.Comment: 45 pages, 5 figure
The multivariate signed Bollobas-Riordan polynomial
We generalise the signed Bollobas-Riordan polynomial of S. Chmutov and I. Pak
[Moscow Math. J. 7 (2007), no. 3, 409-418] to a multivariate signed polynomial
Z and study its properties. We prove the invariance of Z under the recently
defined partial duality of S. Chmutov [J. Combinatorial Theory, Ser. B, 99 (3):
617-638, 2009] and show that the duality transformation of the multivariate
Tutte polynomial is a direct consequence of it.Comment: 17 pages, 2 figures. Published version: a section added about the
quasi-tree expansion of the multivariate Bollobas-Riordan polynomia
Partial duality of hypermaps
We introduce a collection of new operations on hypermaps, partial duality,
which include the classical Euler-Poincar\'e dualities as particular cases.
These operations generalize the partial duality for maps, or ribbon graphs,
recently discovered in a connection with knot theory. Partial duality is
different from previous studied operations of S. Wilson, G. Jones, L. James,
and A. Vince. Combinatorially hypermaps may be described in one of three ways:
as three involutions on the set of flags (-model), or as three
permutations on the set of half-edges (-model in orientable case), or
as edge 3-colored graphs. We express partial duality in each of these models.Comment: 19 pages, 16 figure
Quantum field theory on the degenerate Moyal space
We prove that the self-interacting scalar field on the four-dimensional
degenerate Moyal plane is renormalisable to all orders when adding a suitable
counterterm to the Lagrangean. Despite the apparent simplicity of the model, it
raises several non trivial questions. Our result is a first step towards the
definition of renormalisable quantum field theories on a non-commutative
Minkowski space.Comment: 21 pages, 4 figures. We use the techniques of the previous version in
a conclusive manner on the degenerate Moyal spac
Hopf algebra of non-commutative field theory
We contruct here the Hopf algebra structure underlying the process of
renormalization of non-commutative quantum field theory.Comment: 14 pages, 4 figure
Just Renormalizable TGFT's on U(1)^d with Gauge Invariance
We study the polynomial Abelian or U(1)^d Tensorial Group Field Theories
equipped with a gauge invariance condition in any dimension d. From our
analysis, we prove the just renormalizability at all orders of perturbation of
the phi^4_6 and phi^6_5 random tensor models. We also deduce that the phi^4_5
tensor model is super-renormalizable.Comment: 33 pages, 22 figures. One added paragraph on the different notions of
connectedness, preciser formulation of the proof of the power counting
theorem, more general statements about traciality of tensor graph
Topological graph polynomials and quantum field theory, Part II: Mehler kernel theories
We define a new topological polynomial extending the Bollobas-Riordan one,
which obeys a four-term reduction relation of the deletion/contraction type and
has a natural behavior under partial duality. This allows to write down a
completely explicit combinatorial evaluation of the polynomials, occurring in
the parametric representation of the non-commutative Grosse-Wulkenhaar quantum
field theory. An explicit solution of the parametric representation for
commutative field theories based on the Mehler kernel is also provided.Comment: 58 pages, 23 figures, correction in the references and addition of
preprint number
Renormalisation des theories de champs non commutatives
Very high energy physics needs a coherent description of the four fundamental
forces. Non-commutative geometry is a promising mathematical framework which
already allowed to unify the general relativity and the standard model, at the
classical level, thanks to the spectral action principle. Quantum field
theories on non-commutative spaces is a first step towards the quantification
of such a model. These theories can't be obtained simply by writing usual field
theory on non-commutative spaces. Such attempts exhibit indeed a new type of
divergencies, called ultraviolet/infrared mixing, which prevents
renormalisability. H. Grosse and R. Wulkenhaar showed, with an example, that a
modification of the propagator may restore renormalisability. This thesis aims
at studying the generalization of such a method. We studied two different
models which allowed to specify certain aspects of non-commutative field
theory. In x space, the major technical difficulty is due to oscillations in
the interaction part. We generalized the results of T. Filk in order to exploit
such oscillations at best. We were then able to distinguish between two
mixings, renormalizable or not. We also bring the notion of orientability to
light : the orientable non-commutative Gross-Neveu model is renormalizable
without any modification of its propagator. The adaptation of multi-scale
analysis to the matrix basis emphasized the importance of dual graphs and
represents a first step towards a formulation of field theory independent of
the underlying space.Comment: PhD thesis, 164 pages. In French. Also available at
http://tel.archives-ouvertes.fr/tel-0011804
Non-orientable quasi-trees for the Bollobas-Riordan polynomial
We extend the quasi-tree expansion of A. Champanerkar, I. Kofman, and N.
Stoltzfus to not necessarily orientable ribbon graphs. We study the duality
properties of the Bollobas-Riordan polynomial in terms of this expansion. As a
corollary, we get a "connected state" expansion of the Kauffman bracket of
virtual link diagrams. Our proofs use extensively the partial duality of S.
Chmutov.Comment: 31 pages, 11 figure