We study the effect of non-Gaussian average over the random couplings in a complex version of the celebrated Sachdev-Ye-Kitaev (SYK) model. Using a Polchinski-like equation and random tensor Gaussian universality, we show that the effect of this non-
Gaussian averaging leads to a modification of the variance of the Gaussian distribution of couplings at leading order in N. We then derive the form of the effective action to all orders. An explicit computation of the modification of the variance in the case of a quartic perturbation is performed for both the complex SYK model mentioned above and the SYK generalization proposed in D. Gross and V. Rosenhaus, JHEP 1702 (2017) 093.
We show that Wigner semi-circle law holds for Hermitian matrices with dependent entries, provided the deviation of the cumulants from the normalised Gaussian case obeys a simple power law bound in the size of the matrix. To establish this result, we
use replicas interpreted as a zero-dimensional quantum field theoretical model whose effective potential obey a renormalisation group equation.
In this paper we give a new proof of the universality of the Tutte polynomial for matroids. This proof uses appropriate characters of Hopf algebra of matroids, algebra introduced by Schmitt (1994). We show that these Hopf algebra characters are solut
ions of some differential equations which are of the same type as the differential equations used to describe the renormalization group flow in quantum field theory. This approach allows us to also prove, in a different way, a matroid Tutte polynomial convolution formula published by Kook, Reiner and Stanton (1999). This FPSAC contribution is an extended abstract.
Motivated by string theory on the orbifold ${cal M}/G$ in presence of a Kalb-Ramond field strength $H$, we define the operators that lift the group action to the twisted sectors. These operators turn out to generate the quasi-quantum group $D_{omega}
[G]$, introduced in the context of conformal field theory by R. Dijkgraaf, V. Pasquier and P. Roche, with $omega$ a 3-cocycle determined by a series of cohomological equations in a tricomplex combining de Rham, u{C}ech and group cohomologies. We further illustrate some properties of the quasi-quantum group from a string theoretical point of view.
We present the general form of the operators that lift the group action on the twisted sectors of a bosonic string on an orbifold ${cal M}/G$, in the presence of a Kalb-Ramond field strength $H$. These operators turn out to generate the quasi-quantum
group $D_{omega}[G]$, introduced in the context of orbifold conformal field theory by R. Dijkgraaf, V. Pasquier and P. Roche. The 3-cocycle $omega$ entering in the definition of $D_{omega}[G]$ is related to $H$ by a series of cohomological equations in a tricomplex combining de Rham, Cech and group coboundaries. We construct magnetic amplitudes for the twisted sectors and show that $omega=1$ arises as a consistency condition for the orbifold theory. Finally, we recover discrete torsion as an ambiguity in the lift of the group action to twisted sectors, in accordance with previous results presented by E. Sharpe.
We propose a general formulation of perturbative quantum field theory on (finitely generated) projective modules over noncommutative algebras. This is the analogue of scalar field theories with non-trivial topology in the noncommutative realm. We tre
at in detail the case of Heisenberg modules over noncommutative tori and show how these models can be understood as large rectangular pxq matrix models, in the limit p/q->theta, where theta is a possibly irrational number. We find out that the modele is highly sensitive to the number-theoretical aspect of theta and suffers from an UV/IR-mixing. We give a way to cure the entanglement and prove one-loop renormalizability.
This paper aims at presenting the first steps towards a formulation of the Exact Renormalization Group Equation in the Hopf algebra setting of Connes and Kreimer. It mostly deals with some algebraic preliminaries allowing to formulate perturbative re
normalization within the theory of differential equations. The relation between renormalization, formulated as a change of boundary condition for a differential equation, and an algebraic Birkhoff decomposition for rooted trees is explicited.
We construct exact solitons on noncommutative tori for the type of actions arising from open string field theory. Given any projector that describes an extremum of the tachyon potential, we interpret the remaining gauge degrees of freedom as a gauge
theory on the projective module determined by the tachyon. Whenever this module admits a constant curvature connection, it solves exactly the equations of motion of the effective string field theory. We describe in detail such a construction on the noncommutative tori. Whereas our exact solution relies on the coupling to a gauge theory, we comment on the construction of approximate solutions in the absence of gauge fields.
