Cohomology of graph hypersurfaces associated to certain Feynman graphs

We consider the number $\bar N(q)$ of points in the projective complement of graph hypersurfaces over $\mathbb{F}_q$ and show that the smallest graphs with non-polynomial $\bar N(q)$ have 14 edges. We give six examples which fall into two classes. One class has an exceptional prime 2 whereas in the other class $\bar N(q)$ depends on the number of cube roots of unity in $\mathbb{F}_q$. At graphs with 16 edges we find examples where $\bar N(q)$ is given by a polynomial in $q$ plus $q^2$ times the number of points in the projective complement of a singular K3 in $\mathbb{P}^3$. In the second part of the paper we show that applying momentum space Feynman-rules over $\mathbb{F}_q$ lets the perturbation series terminate for renormalizable and non-renormalizable bosonic quantum field theories.

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Algebraic combinatorics and QFT

Combinatorial Physics ◽

10.1093/oso/9780192895493.003.0008 ◽

2021 ◽

pp. 76-94

Author(s):

Adrian Tanasa

Keyword(s):

Hopf Algebra ◽

Hochschild Cohomology ◽

Algebraic Combinatorics ◽

Fourth Section ◽

Analytic Combinatorics ◽

Multi Scale ◽

Starting Point ◽

Feynman Graphs ◽

Perturbative Renormalization

We have seen in the previous chapter some of the non-trivial interplay between analytic combinatorics and QFT. In this chapter, we illustrate how yet another branch of combinatorics, algebraic combinatorics, interferes with QFT. In this chapter, after a brief algebraic reminder in the first section, we introduce in the second section the Connes–Kreimer Hopf algebra of Feynman graphs and we show its relation with the combinatorics of QFT perturbative renormalization. We then study the algebra's Hochschild cohomology in relation with the combinatorial Dyson–Schwinger equation in QFT. In the fourth section we present a Hopf algebraic description of the so-called multi-scale renormalization (the multi-scale approach to the perturbative renormalization being the starting point for the constructive renormalization programme).

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From random matrices to random tensors

Combinatorial Physics ◽

10.1093/oso/9780192895493.003.0011 ◽

2021 ◽

pp. 166-177

Author(s):

Adrian Tanasa

Keyword(s):

Matrix Models ◽

Random Matrices ◽

Mathematical Physics ◽

Natural Generalization ◽

Random Surface ◽

The Third ◽

Planar Surfaces ◽

Tensor Models ◽

Feynman Graphs ◽

The Matrix

After a brief presentation of random matrices as a random surface QFT approach to 2D quantum gravity, we focus on two crucial mathematical physics results: the implementation of the large N limit (N being here the size of the matrix) and of the double-scaling mechanism for matrix models. It is worth emphasizing that, in the large N limit, it is the planar surfaces which dominate. In the third section of the chapter we introduce tensor models, seen as a natural generalization, in dimension higher than two, of matrix models. The last section of the chapter presents a potential generalisation of the Bollobás–Riordan polynomial for tensor graphs (which are the Feynman graphs of the perturbative expansion of QFT tensor models).

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