Multiple Zeta Values and Modular Forms in Quantum Field Theory

We establish a novel representation of arbitrary Euler-Zagier sums in terms of weighted vacuum graphs. This representation uses a toy quantum field theory with infinitely many propagators and interaction vertices. The propagators involve Bernoulli polynomials and Clausen functions to arbitrary orders. The Feynman integrals of this model can be decomposed in terms of a vertex algebra whose structure we investigate. We derive a large class of relations between multiple zeta values, of arbitrary lengths and weights, using only a certain set of graphical manipulations on Feynman diagrams. Further uses and possible generalisations of the model are pointed out.

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ARITHMETIC OF POTTS MODEL HYPERSURFACES

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887813500059 ◽

2013 ◽

Vol 10 (04) ◽

pp. 1350005 ◽

Cited By ~ 3

Author(s):

MATILDE MARCOLLI ◽

JESSICA SU

Keyword(s):

Field Theory ◽

Quantum Field Theory ◽

Potts Model ◽

Tutte Polynomial ◽

Polymer Chains ◽

Quantum Field ◽

Multiple Zeta ◽

Graph Hypersurfaces ◽

Zeta Values ◽

Perturbative Quantum

We consider Potts model hypersurfaces defined by the multivariate Tutte polynomial of graphs (Potts model partition function). We focus on the behavior of the number of points over finite fields for these hypersurfaces, in comparison with the graph hypersurfaces of perturbative quantum field theory defined by the Kirchhoff graph polynomial. We give a very simple example of the failure of the "fibration condition" in the dependence of the Grothendieck class on the number of spin states and of the polynomial countability condition for these Potts model hypersurfaces. We then show that a period computation, formally similar to the parametric Feynman integrals of quantum field theory, arises by considering certain thermodynamic averages. One can show that these evaluate to combinations of multiple zeta values for Potts models on polygon polymer chains, while silicate tetrahedral chains provide a candidate for a possible occurrence of non-mixed Tate periods.

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ZETA ELEMENTS IN DEPTH 3 AND THE FUNDAMENTAL LIE ALGEBRA OF THE INFINITESIMAL TATE CURVE

Forum of Mathematics Sigma ◽

10.1017/fms.2016.29 ◽

2017 ◽

Vol 5 ◽

Cited By ~ 6

Author(s):

FRANCIS BROWN

Keyword(s):

Lie Algebra ◽

Modular Forms ◽

Multiple Zeta Values ◽

Multiple Zeta ◽

Zeta Values ◽

Explicit Formulae ◽

Tate Curve ◽

Tate Motives

This paper draws connections between the double shuffle equations and structure of associators; Hain and Matsumoto’s universal mixed elliptic motives; and the Rankin–Selberg method for modular forms for $\text{SL}_{2}(\mathbb{Z})$. We write down explicit formulae for zeta elements $\unicode[STIX]{x1D70E}_{2n-1}$ (generators of the Tannaka Lie algebra of the category of mixed Tate motives over $\mathbb{Z}$) in depths up to four, give applications to the Broadhurst–Kreimer conjecture, and solve the double shuffle equations for multiple zeta values in depths two and three.

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Depth-graded motivic multiple zeta values

Compositio Mathematica ◽

10.1112/s0010437x20007654 ◽

2021 ◽

Vol 157 (3) ◽

pp. 529-572

Author(s):

Francis Brown

Keyword(s):

Galois Group ◽

Lie Algebra ◽

Modular Forms ◽

Cusp Forms ◽

Multiple Zeta Values ◽

Multiple Zeta ◽

Depth Filtration ◽

Zeta Values ◽

Lower Depth ◽

Tate Motives

We study the depth filtration on multiple zeta values, on the motivic Galois group of mixed Tate motives over $\mathbb {Z}$ and on the Grothendieck–Teichmüller group, and its relation to modular forms. Using period polynomials for cusp forms for $\mathrm {SL} _2(\mathbb {Z})$, we construct an explicit Lie algebra of solutions to the linearized double shuffle equations, which gives a conjectural description of all identities between multiple zeta values modulo $\zeta (2)$ and modulo lower depth. We formulate a single conjecture about the homology of this Lie algebra which implies conjectures due to Broadhurst and Kreimer, Racinet, Zagier, and Drinfeld on the structure of multiple zeta values and on the Grothendieck–Teichmüller Lie algebra.

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Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory

10.1007/978-3-030-04480-0 ◽

2019 ◽

Cited By ~ 2

Keyword(s):

Field Theory ◽

Quantum Field Theory ◽

Modular Forms ◽

Elliptic Functions ◽

Elliptic Integrals ◽

Quantum Field

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A CLASS OF NONHOLOMORPHIC MODULAR FORMS II: EQUIVARIANT ITERATED EISENSTEIN INTEGRALS

Forum of Mathematics Sigma ◽

10.1017/fms.2020.24 ◽

2020 ◽

Vol 8 ◽

Cited By ~ 2

Author(s):

FRANCIS BROWN

Keyword(s):

Eisenstein Series ◽

Modular Forms ◽

Multiple Zeta Values ◽

New Family ◽

Algebra Of Functions ◽

Multiple Zeta ◽

Zeta Values ◽

Upper Half Plane ◽

Real Analytic ◽

Holomorphic Modular Forms

We introduce a new family of real-analytic modular forms on the upper-half plane. They are arguably the simplest class of ‘mixed’ versions of modular forms of level one and are constructed out of real and imaginary parts of iterated integrals of holomorphic Eisenstein series. They form an algebra of functions satisfying many properties analogous to classical holomorphic modular forms. In particular, they admit expansions in $q,\overline{q}$ and $\log |q|$ involving only rational numbers and single-valued multiple zeta values. The first nontrivial functions in this class are real-analytic Eisenstein series.

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