Feynman Motives by Matilde Marcolli (PDF)

Description

This book presents recent and ongoing research work aimed at understanding the mysterious relation between the computations of Feynman integrals in perturbative quantum field theory and the theory of motives of algebraic varieties and their periods. One of the main questions in the field is understanding when the residues of Feynman integrals in perturbative quantum field theory evaluate to periods of mixed Tate motives. The question originates from the occurrence of multiple zeta values in Feynman integrals calculations observed by Broadhurst and Kreimer.Two different approaches to the subject are described. The first, a “bottom-up” approach, constructs explicit algebraic varieties and periods from Feynman graphs and parametric Feynman integrals. This approach, which grew out of work of Bloch-Esnault-Kreimer and was more recently developed in joint work of Paolo Aluffi and the author, leads to algebro-geometric and motivic versions of the Feynman rules of quantum field theory and concentrates on explicit constructions of motives and classes in the Grothendieck ring of varieties associated to Feynman integrals. While the varieties obtained in this way can be arbitrarily complicated as motives, the part of the cohomology that is involved in the Feynman integral computation might still be of the special mixed Tate kind. A second, “top-down” approach to the problem, developed in the work of Alain Connes and the author, consists of comparing a Tannakian category constructed out of the data of renormalization of perturbative scalar field theories, obtained in the form of a Riemann-Hilbert correspondence, with Tannakian categories of mixed Tate motives. The book draws connections between these two approaches and gives an overview of other ongoing directions of research in the field, outlining the many connections of perturbative quantum field theory and renormalization to motives, singularity theory, Hodge structures, arithmetic geometry, supermanifolds, algebraic and non-commutative geometry.The text is aimed at researchers in mathematical physics, high energy physics, number theory and algebraic geometry. Partly based on lecture notes for a graduate course given by the author at Caltech in the fall of 2008, it can also be used by graduate students interested in working in this area.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐In the preface the author asserts that she is not an expert in the theory of motives. But by perusing the discussion on motives and periods that takes place in Chapter 2 of this book serves as another piece of evidence that it is the non-expert that is better at explaining difficult abstract mathematical concepts. In the opinion of the reviewer that even though it is cursory, this discussion is one of the best, if not the best, discussion on pure and mixed motives that exists in print, and readers who want to understand these objects, even if they are not interested in their connection to quantum field theory, will gain a lot by reading not only this chapter but the rest of the book. For readers who want to understand the connections between perturbative quantum field theory and the theory of motives that are outlined in the work of Alain Connes et al, this book would be a good start. The reviewer found the study of some of the publications of the author also of great assistance in understanding the real intuition behind pure and mixed motives.The quantum field theory in the book is strictly perturbative, and so readers who are interested in any possible connection of non-perturbative effects in quantum field theory and motives will probably be disappointed. An interesting avenue of future investigation may be to uncover this connection. Also, the algebraic geometry involved is primarily concentrated on understanding determinant hypersurfaces, since these are the varieties that arise from the calculation of Feynman integrals in quantum field theory. Concrete examples are rarely given in expositions on the theory of motives, and so the discussion on how to associate a motive to a determinant hypersurface helps readers to appreciate the theory in a way that simply studying the axioms of the theory does not.Of fundamental importance to the understanding of the book is the notion of a period, which arise in the computation of Feynman amplitudes in quantum field theory, in algebraic geometry arise when one considers the isomorphism between between the complexification of rational algebraic de Rham cohomology and rational Betti cohomology, and in the theory of motives where every rational motive is associated with a period matrix. More specifically, the author tries to establish to what degree the residues of Feynman diagrams are periods of mixed Tate motives.Mixed Tate motives are introduced in the book as a full triangulated thick subcategory of the Voevodsky triangulated category DMT(K) of mixed motives over a field K which is generated by the Tate objects Q(n). If K is a number field, then there is an abelian category MT(K) but the author does not prove this in book (references are given however). The connection of mixed Tate motives with Feynman diagrams is discussed by answering the question as to which mixed motives are mixed Tate motives. The answer to this question involves viewing DMT(K) as a triangulated subcategory of the triangulated category DM(K) of mixed motives, the latter of which is not explicitly constructed in the book (but references are again given). The author makes up for this omission by giving the reader a fine motivation as to the methodology involved in determining whether a motive of a particular variety in fact is a mixed Tate motive. This involves breaking up the variety into “strata” that allow the construction of the motive from pieces that are mixed Tate. As she explains, this strategy is dependent on properties of the (triangulated) category DM(K), namely the presence of a distinguished triangle, and the fact that identical motives are homotopic. Determinant varieties, which play the central role in illustrating the connection of Feynman graphs with the theory of motives, are shown to be mixed Tate motives using solely these two properties. This discussion, and other ones throughout the book, illustrate nicely that the algebraic variety that represents a Feynman graph is in general singular, thus requiring mixed motives as a consequence.This book therefore illustrates the value of perturbative quantum field theory in producing non-trivial examples of motives through the calculation of periods of Feynman integrals. When studying the book, the reviewer wondered whether motives, as objects of a particular category, can perhaps be manipulated mathematically as “quantum” objects of some sort. The Grothendieck ring, which plays a central role in this book, allows one to do algebra on motives that gives a sensible meaning to the operations of joining them together, and so on. “Factoring through” the Grothendieck ring is a requirement for showing the motivic nature of the resulting objects constructed using Feynman graphs. But can these constructions and algebraic manipulations be placed in a context that is “truly” quantum in the sense that one could meaningfully speak of transition amplitudes between motives or the entanglement of motives? A truly quantum formulation of the theory of motives might then reveal more transparently the underlying structure of a universal cohomology theory. An algebraic variety might then have a “classical” as well as a “quantum” piece, and the cohomologies of these two pieces may be necessary to obtain the “universal” cohomology of interest. This would have the effect of reversing the logic of this book, in that instead of using Feynman graphs to obtain varieties that are motivic, one studies Feynman graphs of motivic objects. Entanglement of motives might then be formulated in a similar fashion as what is done in quantum information theory. Two motives could be “equivalent” if one can find a transition amplitude, or “quantum morphism” between them. Equivalence under this conception would generalize the numerical or rational equivalence that one has in the “classical” theory of motives.

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