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A. Tanasa

Publications and source records attributed to A. Tanasa.

At least 19 recordsLinked to original sources
arXiv2023

Duality of O(N) and Sp(N) random tensor models: tensors with symmetries

In a recent series of papers, a duality between orthogonal and symplectic random tensor models has been proven, first for quartic models and then for models with interactions of arbitrary order. However, the tensor models considered so far in the literature had no symmetry under permutation of the indices. In this paper, we generalize these results for tensors models with interactions of arbitrary order which further have non-trivial symmetry under the permutation of the indices. Totally symmetric and anti-symmetric tensors are thus treated as a particular case of our result.

math-ph↗
arXiv2023

Double scaling limit of the prismatic tensor model

In S. Giombi, I. Klebanov, F. Popov, S. Prakash and G. Tarnopolsky, {\it Phys. Rev.} {\bf D} 98 (2018) 10, 105005, a prismatic tensor model was introduced. We study here the diagrammatics and the double scaling limit of this model, using the intermediate field method. We explicitly exhibit the next-to-leading order Feynman graphs in the $1/N$ expansion. Using appropriate combinatorial tools, we further study the general term of the $1/N$ expansion and we compute the $2-$point function in the double scaling limit.

hep-th↗
arXiv2020

On the large N limit of the Schwinger-Dyson equation of tensor field theory

We analyse in this paper the large N limit of the Schwinger-Dyson equations in a rank-3 tensor quantum field theory, which are derived with the help of Ward-Takahashi identities. In order to have a well-defined large N limit, appropriate scalings in powers of N for the various terms present in the action are explicitly found. A perturbative check of our results is done, up to second order in the coupling constant.

math-ph↗
arXiv2019

Diagrammatics of the quartic $O(N)^3$-invariant Sachdev-Ye-Kitaev-like tensor model

Various tensor models have been recently shown to have the same properties as the celebrated Sachdev-Ye-Kitaev (SYK) model. In this paper we study in detail the diagrammatics of two such SYK-like tensor models: the multi-orientable (MO) model which has an $U(N) \times O(N) \times U(N)$ symmetry and a quartic $O(N)^3$-invariant model whose interaction has the tetrahedral pattern. We show that the Feynman graphs of the MO model can be seen as the Feynman graphs of the $O(N)^3$-invariant model which have an orientable jacket. We then present a diagrammatic toolbox to analyze the $O(N)^3$-invariant graphs. This toolbox allows for a simple strategy to identify all the graphs of a given order in the $1/N$ expansion. We apply it to the next-to-next-to-leading and next-to-next-to-next-to-leading orders which are the graphs of degree $1$ and $3/2$ respectively.

hep-th↗
arXiv2018

Non-Gaussian disorder average in the Sachdev-Ye-Kitaev model

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.

hep-th↗
arXiv2016

Wigner law for matrices with dependent entries - a perturbative approach

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.

math-ph↗
arXiv2015

The Jacobian Conjecture, a Reduction of the Degree to the Quadratic Case

The Jacobian Conjecture states that any locally invertible polynomial system in C^n is globally invertible with polynomial inverse. C. W. Bass et al. (1982) proved a reduction theorem stating that the conjecture is true for any degree of the polynomial system if it is true in degree three. This degree reduction is obtained with the price of increasing the dimension n. We prove here a theorem concerning partial elimination of variables, which implies a reduction of the generic case to the quadratic one. The price to pay is the introduction of a supplementary parameter 0<n'< n, parameter which represents the dimension of a linear subspace where some particular conditions on the system must hold. We first give a purely algebraic proof of this reduction result and we then expose a distinct proof, in a Quantum Field Theoretical formulation, using the intermediate field method.

math.AG↗
arXiv2014

Dendriform structures for restriction-deletion and restriction-contraction matroid Hopf algebras

We endow the set of isomorphic classes of matroids with a new Hopf algebra structure, in which the coproduct is implemented via the combinatorial operations of restriction and deletion. We also initiate the investigation of dendriform coalgebra structures on matroids and introduce a monomial invariant which satisfy a convolution identity with respect to restriction and deletion.

math.CO↗
arXiv2013

Renormalization group-like proof of the universality of the Tutte polynomial for matroids

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 solutions 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.

math.CO↗
arXiv2013

A selection-quotient process for packed word Hopf algebra

In this paper, we define a Hopf algebra structure on the vector space spanned by packed words using a selection-quotient coproduct. We show that this algebra is free on its irreducible packed words. Finally, we give some brief explanations on the Maple codes we have used.

math.CO↗
arXiv2013

A combinatorial non-commutative Hopf algebra of graphs

A non-commutative, planar, Hopf algebra of rooted trees was proposed in L. Foissy, Bull. Sci. Math. 126 (2002) 193-239. In this paper we propose such a non-commutative Hopf algebra for graphs. In order to define a non-commutative product we use a quantum field theoretical (QFT) idea, namely the one of introducing discrete scales on each edge of the graph (which, within the QFT framework, corresponds to energy scales of the associated propagators).

math.CO↗
arXiv2013

A word Hopf algebra based on the selection/quotient principle

In this paper, we define a Hopf algebra structure on the vector space spanned by packed words using a selection/quotient coproduct. We show that this algebra is free on its irreducible packed words. We also construct the Hilbert series of this Hopf algebra and we investigate its primitive elements.

math.CO↗
arXiv2013

The 1/N expansion of multi-orientable random tensor models

Multi-orientable group field theory (GFT) has been introduced in A. Tanasa, J. Phys. A 45 (2012) 165401, arXiv:1109.0694, as a quantum field theoretical simplification of GFT, which retains a larger class of tensor graphs than the colored one. In this paper we define the associated multi-orientable identically independent distributed multi-orientable tensor model and we derive its 1/N expansion. In order to obtain this result, a partial classification of general tensor graphs is performed and the combinatorial notion of jacket is extended to the multi-orientable graphs. We prove that the leading sector is given, as in the case of colored models, by the so-called melon graphs.

hep-th↗
arXiv2008

Topological Graph Polynomials and Quantum Field Theory, Part I: Heat Kernel Theories

We investigate the relationship between the universal topological polynomials for graphs in mathematics and the parametric representation of Feynman amplitudes in quantum field theory. In this first paper we consider translation invariant theories with the usual heat-kernel-based propagator. We show how the Symanzik polynomials of quantum field theory are particular multivariate versions of the Tutte polynomial, and how the new polynomials of noncommutative quantum field theory are particular versions of the Bollobás-Riordan polynomials.

math-ph↗
arXiv2007

Poincaré and sl(2) algebras of order 3

In this paper we initiate a general classification for Lie algebras of order 3 and we give all Lie algebras of order 3 based on $\mathfrak{sl}(2,\mathbb C)$ and $\mathfrak{iso}(1,3)$ the Poincaré algebra in four-dimensions. We then set the basis of the theory of the deformations (in the Gerstenhaber sense) and contractions for Lie algebras of order 3.

math-ph↗