24 research outputs found
Large Limits in Tensor Models: Towards More Universality Classes of Colored Triangulations in Dimension
We review an approach which aims at studying discrete (pseudo-)manifolds in
dimension and called random tensor models. More specifically, we
insist on generalizing the two-dimensional notion of -angulations to higher
dimensions. To do so, we consider families of triangulations built out of
simplices with colored faces. Those simplices can be glued to form new building
blocks, called bubbles which are pseudo-manifolds with boundaries. Bubbles can
in turn be glued together to form triangulations. The main challenge is to
classify the triangulations built from a given set of bubbles with respect to
their numbers of bubbles and simplices of codimension two. While the colored
triangulations which maximize the number of simplices of codimension two at
fixed number of simplices are series-parallel objects called melonic
triangulations, this is not always true anymore when restricting attention to
colored triangulations built from specific bubbles. This opens up the
possibility of new universality classes of colored triangulations. We present
three existing strategies to find those universality classes. The first two
strategies consist in building new bubbles from old ones for which the problem
can be solved. The third strategy is a bijection between those colored
triangulations and stuffed, edge-colored maps, which are some sort of hypermaps
whose hyperedges are replaced with edge-colored maps. We then show that the
present approach can lead to enumeration results and identification of
universality classes, by working out the example of quartic tensor models. They
feature a tree-like phase, a planar phase similar to two-dimensional quantum
gravity and a phase transition between them which is interpreted as a
proliferation of baby universes
Random Tensors and Quantum Gravity
We provide an informal introduction to tensor field theories and to their
associated renormalization group. We focus more on the general motivations
coming from quantum gravity than on the technical details. In particular we
discuss how asymptotic freedom of such tensor field theories gives a concrete
example of a natural "quantum relativity" postulate: physics in the deep
ultraviolet regime becomes asymptotically more and more independent of any
particular choice of Hilbert basis in the space of states of the universe.Comment: Section 6 is essentially reproduced from author's arXiv:1507.04190
for self-contained purpose of the revie
A New Large N Expansion for General Matrix-Tensor Models
We define a new large limit for general or
invariant tensor models, based on an enhanced large
scaling of the coupling constants. The resulting large expansion is
organized in terms of a half-integer associated with Feynman graphs that we
call the index. This index has a natural interpretation in terms of the many
matrix models embedded in the tensor model. Our new scaling can be shown to be
optimal for a wide class of non-melonic interactions, which includes all the
maximally single-trace terms. Our construction allows to define a new large
expansion of the sum over diagrams of fixed genus in matrix models with an
additional global symmetry. When the interaction is the
complete vertex of order , we identify in detail the leading order graphs
for a prime number. This slightly surprising condition is equivalent to the
complete interaction being maximally single-trace.Comment: 57 pages, 20 figures (additional discussion in Sec. 4.1.1. and
additional figure (Fig. 5)