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We consider the parametric representation of the amplitudes of Abelian models in the so-called framework of rank d tensorial group field theory. These models are called Abelian because their fields live on copies ofU(1)D.We concentrate on the case when these models are endowed with particular kinetic terms involving a linear power in momenta. A new dimensional regularization is introduced for particular models in this class: a rank 3 tensor model, an infinite tower of matrix models φ2n overU(1), and a matrix model over U(1)². We prove that all divergent amplitudes are meromorphic functions in the complexified group dimension D ∈ C. From this point, a standard subtraction program yielding analytic renormalized integrals could be applied. Furthermore, we identify and study in depth the Symanzik polynomials provided by the parametric amplitudes of generic rank d Abelian models. We find that these polynomials do not satisfy the ordinary Tutte’s rules (contraction/deletion). By scrutinizing the “face”-structure of these polynomials, we find a generalized polynomial which turns out to be stable only under contraction. [ABSTRACT FROM AUTHOR]

We investigate the finite and large N behaviors of independent-value O(N)-invariant matrix models. These are models defined with matrix-type fields and with no gradient term in their action. They are generically nonrenormalizable but can be handled by nonperturbative techniques. We find that the functional integral of any O(N) matrix trace invariant may be expressed in terms of an O(N)-invariant measure. Based on this result, we prove that, in the limit that all interaction coupling constants go to zero, any interacting theory is continuously connected to a pseudo-free theory. This theory differs radically from the familiar free theory consisting in putting the coupling constants to zero in the initial action. The proof is given for generic, finite-size matrix models, whereas, in the limiting case N→∞, we succeed in showing this behavior for restricted types of actions using a particular scaling of the parameters. [ABSTRACT FROM AUTHOR]

We identify new families of renormalizable tensor models from anterior renormalizable tensor models via a mapping capable of reducing or increasing the rank of the theory without having an effect on the renormalizability property. Mainly, a version of the rank 3 tensor model as defined by Ben Geloun and Samary [Ann. Henri Poincare 14, 1599 (2013); e-print arXiv:1201.0176 [hep-th]] and the Grosse-Wulkenhaar model in 4D and 2D generate three different classes of renormalizable models. The proof of the renormalizability is fully performed for the first reduced model. The same procedure can be applied for the remaining cases. Interestingly, we find that, due to the peculiar behavior of anisotropic wave function renormalizations, the rank 3 tensor model reduced to a matrix model generates a simple super-renormalizable vector model. [ABSTRACT FROM AUTHOR]