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

I had the good fortune to have as my undergraduate advisor (de facto, not de jure: in those days these things were done more informally) the marvelous and wise scholar, V. S. Varadarajan, whose vision and work encompassed themes reaching from quantum physics to number theory, all under the umbrella of the theory of representations of Lie groups and Lie algebras.  Professor Varadarajan gave me incomparable guidance, inspiring my work (and approach to mathematics) to this day. One of his particularly memorable phrases rings true more and more loudly than ever before: number theory and physics are two sides of the same coin. As an illustration of the pithiness of this observation, consider the book under review: its focus is the synergetic interplay between the mainstays of the extended theory of modular functions (including the themes of elliptic functions and integrals) and nothing less than quantum field theory.  These apparently disparate subjects have seen a phenomenal amount of activity over the last so-many decades, and now their inter-weaving is a marvel to behold.

On the number theory side, it is perhaps most apposite to mention the Langlands Program and Wiles’ stunning proof of the Shimura-Taniyama-Weil Conjecture (implying the Last Theorem of Fermat), which can arguably be credited with supplying the impetus for almost all the modern developments in this area.  On the quantum physics side, we encounter most prominently quantum field theory, first conceived in its prototypal format by Dirac as quantum electrodynamics; the main players in QED were Schwinger, Tomonaga, and Feynman, with the latter’s idiosyncratic approach (Feynman diagrams and path integrals) making the biggest splash of all.  Feynman’s methods are central to the sexiest version of QFT for us mathematicians, namely topological QFT (or TQFT, of course), developed in the 1980s by Witten. And there we have it in a nutshell --- speaking of which, one of the most popular texts on QFT is Anthony Zee’s Quantum Field Theory in a Nutshell.  For us mathematicians (for whom the way physicists think is unheimlich --- and I am one of these), consider Quantum Field Theory: A Tourist Guide for Mathematicians.  The book under review is a compendium of articles exploring the remarkable interplay between the indicated subjects.  Àpropos of QFT, the contribution to this volume by one of the editors, Johannes Blümlein, is titled, “Iterative Non-iterative Integrals in Quantum Field Theory”; to discover the meaning of the apparent contradiction in the title, see §3 of this article.

That said, here is a cross-section (in the prosaic sense, not that of the quantum physicists) of what the book offers.  Mathematicians properly so-called are already made to feel at home with the first article, “Eta Quotients and Rademacher Sums,” but the chafing physicists are pacified with the second article, “On a Class of Feynman Integrals Evaluating to Iterated Integrals of Modular Forms.”  The next article is the one by Blümlein referred to above. And on and on it goes: truly an ecumenical business. The article, “Expansions at cusps and Petersson products in Pari/GP,” reads like hard-core number theoretic stuff meant only for modular formers, but it ain’t necessarily so, as the song goes: there is, e.g., §3.2, “Eisenstein series of level two in the string context,” of the article, “One-Loop String Scattering Amplitudes as Iterated Eisenstein Integrals,” where we read that “the parameterization of [a] cylinder worldsheet … gives rise to teMZVs with twists … in the non-planar amplitudes.  Hence, the differential equation (34) [see the book] allows [one] to express the '-expansion in terms of iterated Eisenstein integrals …”  Earlier in the same article we encounter “five independent Mandelstam variables for five massless particles …” (p. 141): somewhere a physicist is smiling.  (By the way, MTZ stands for “multiple zeta value”; for the “te” see the book.)

So there it is.  As we can glean from the foregoing, the book is explicitly concerned with “the mathematics of modular forms [as it is] of central importance for the analytic solution of Feynman diagrams,” but modular formers should be equally interested in the material offered in this compendium.  It’s not for beginners, though.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.