39 research outputs found
Large limit of irreducible tensor models: rank- tensors with mixed permutation symmetry
It has recently been proven that in rank three tensor models, the
anti-symmetric and symmetric traceless sectors both support a large
expansion dominated by melon diagrams [arXiv:1712.00249 [hep-th]]. We show how
to extend these results to the last irreducible tensor representation
available in this context, which carries a two-dimensional representation of
the symmetric group . Along the way, we emphasize the role of the
irreducibility condition: it prevents the generation of vector modes which are
not compatible with the large scaling of the tensor interaction. This
example supports the conjecture that a melonic large limit should exist
more generally for higher rank tensor models, provided that they are
appropriately restricted to an irreducible subspace.Comment: 17 pages, 7 figure
Flowing in Group Field Theory Space: a Review
We provide a non-technical overview of recent extensions of renormalization
methods and techniques to Group Field Theories (GFTs), a class of
combinatorially non-local quantum field theories which generalize matrix models
to dimension . More precisely, we focus on GFTs with so-called
closure constraint, which are closely related to lattice gauge theories and
quantum gravity spin foam models. With the help of recent tensor model tools, a
rich landscape of renormalizable theories has been unravelled. We review our
current understanding of their renormalization group flows, at both
perturbative and non-perturbative levels
Asymptotic safety in three-dimensional SU(2) Group Field Theory: evidence in the local potential approximation
We study the functional renormalization group of a three-dimensional
tensorial Group Field Theory (GFT) with gauge group SU(2). This model generates
(generalized) lattice gauge theory amplitudes, and is known to be
perturbatively renormalizable up to order 6 melonic interactions. We consider a
series of truncations of the exact Wetterich--Morris equation, which retain
increasingly many perturbatively irrelevant melonic interactions. This
tensorial analogue of the ordinary local potential approximation allows to
investigate the existence of non-perturbative fixed points of the
renormalization group flow. Our main finding is a candidate ultraviolet fixed
point, whose qualitative features are reproduced in all the truncations we have
checked (with up to order 12 interactions). This may be taken as evidence for
an ultraviolet completion of this GFT in the sense of asymptotic safety.
Moreover, this fixed point has a single relevant direction, which suggests the
presence of two distinct infrared phases. Our results generally support the
existence of GFT phases of the condensate type, which have recently been
conjectured and applied to quantum cosmology and black holes.Comment: 43 pages, many figures; v2: minor correction
Using Grassmann calculus in combinatorics: Lindstr\"om-Gessel-Viennot lemma and Schur functions
Grassmann (or anti-commuting) variables are extensively used in theoretical
physics. In this paper we use Grassmann variable calculus to give new proofs of
celebrated combinatorial identities such as the Lindstr\"om-Gessel-Viennot
formula for graphs with cycles and the Jacobi-Trudi identity. Moreover, we
define a one parameter extension of Schur polynomials that obey a natural
convolution identity.Comment: 10 pages, contribution to GASCom 2016; v2: minor correction
Singular topologies in the Boulatov model
Through the question of singular topologies in the Boulatov model, we
illustrate and summarize some of the recent advances in Group Field Theory.Comment: 4 pages; proceedings of Loops'11 (May 2011, Madrid); v2: minor
modifications matching published versio
The expansion of the symmetric traceless and the antisymmetric tensor models in rank three
We prove rigorously that the symmetric traceless and the antisymmetric tensor
models in rank three with tetrahedral interaction admit a expansion, and
that at leading order they are dominated by melon diagrams. This proves the
recent conjecture of I. Klebanov and G. Tarnopolsky in JHEP 10 (2017) 037
[arXiv:1706.00839], which they checked numerically up to 8th order in the
coupling constant.Comment: 40 pages, many figure
Melonic phase transition in group field theory
Group field theories have recently been shown to admit a 1/N expansion
dominated by so-called `melonic graphs', dual to triangulated spheres. In this
note, we deepen the analysis of this melonic sector. We obtain a combinatorial
formula for the melonic amplitudes in terms of a graph polynomial related to a
higher dimensional generalization of the Kirchhoff tree-matrix theorem. Simple
bounds on these amplitudes show the existence of a phase transition driven by
melonic interaction processes. We restrict our study to the Boulatov-Ooguri
models, which describe topological BF theories and are the basis for the
construction of four dimensional models of quantum gravity.Comment: 8 pages, 4 figures; to appear in Letters in Mathematical Physic
Bounding bubbles: the vertex representation of 3d Group Field Theory and the suppression of pseudo-manifolds
Based on recent work on simplicial diffeomorphisms in colored group field
theories, we develop a representation of the colored Boulatov model, in which
the GFT fields depend on variables associated to vertices of the associated
simplicial complex, as opposed to edges. On top of simplifying the action of
diffeomorphisms, the main advantage of this representation is that the GFT
Feynman graphs have a different stranded structure, which allows a direct
identification of subgraphs associated to bubbles, and their evaluation is
simplified drastically. As a first important application of this formulation,
we derive new scaling bounds for the regularized amplitudes, organized in terms
of the genera of the bubbles, and show how the pseudo-manifolds configurations
appearing in the perturbative expansion are suppressed as compared to
manifolds. Moreover, these bounds are proved to be optimal.Comment: 28 pages, 17 figures. Few typos fixed. Minor corrections in figure 6
and theorem
Bubbles and jackets: new scaling bounds in topological group field theories
We use a reformulation of topological group field theories in 3 and 4
dimensions in terms of variables associated to vertices, in 3d, and edges, in
4d, to obtain new scaling bounds for their Feynman amplitudes. In both 3 and 4
dimensions, we obtain a bubble bound proving the suppression of singular
topologies with respect to the first terms in the perturbative expansion (in
the cut-off). We also prove a new, stronger jacket bound than the one currently
available in the literature. We expect these results to be relevant for other
tensorial field theories of this type, as well as for group field theory models
for 4d quantum gravity.Comment: v2: Minor modifications to match published versio