Quantum Field Theory: Batalin–Vilkovisky Formalism and Its Applications

In Quantum Field Theory: Batalin-Vilkovisky Formalism and Its Applications, Pavel Mnev has written a beautiful treatise propelling an advanced graduate student or researcher to the front lines of mathematical research on quantum field theory (QFT). Moreover, Mnev adroitly recalls the physical motivations and constructions underpinning the modern mathematical formalism of QFT and gauge theory.  He carries a handful of examples throughout the text: scalar field theory, e.g., \(\phi^3\)-theory; Background Field (BF) theory; Chern-Simons theory; Poisson sigma models; and more generally AKSZ (Alexandrov, Kontsevich, Schwarz, Zaboronsky) sigma models. The text contains many other small examples, and while it contains no exercises, the frequent examples provide a sufficient check of the reader's understanding. While not an easy ride, the diligent reader is handsomely rewarded for engaging this text.

 

Much of Mnev's text assumes a graduate course in differential topology, and some familiarity with graded mathematics (graded vector spaces and manifolds); the introductory chapter requires much less. Indeed, the first chapter provides an excellent overview of mathematical treatments of QFT, complete with the examples of quantum mechanics and finite gauge theory, and should be required reading for anyone with an interest in physical mathematics. Throughout this chapter (and the whole text), the author provides more than 100 references for the reader wanting to dig deeper into historical elements or see the most current research results.

 

The Batalin-Vilkovisky (BV) formalism was developed in the 1980's in order to compute path integrals in the presence of symmetry. The approach built on the work of Faddeev and Popov on resolving degeneracies in the Yang-Mills action. The BRST (Becchi, Rouet, and Stora; Tyutin) approach to gauge theory also preceded the more inclusive BV package. In the 1990's Albert Schwartz showed that the BV formalism could be interpreted through the lens of supermanifolds equipped with graded symplectic/Poisson structures (Manin also played a major role in this geometric viewpoint). A few years later, Jim Stasheff offered an interpretation of BV quantization in terms of homological algebra and Poisson reduction.  In the current text, Mnev does an excellent job of recounting this story while building up the mathematical machinery to discuss the modern formulation of mathematical BV theory.

 

As indicated above, path integrals and Feynman diagrammatics are central to both the physical and mathematical story of BV. These objects are notoriously subtle from a mathematical perspective and Mnev does a good job of providing the reader with a rigorous overview of the essentials. The chapter on Feynman Diagrams contains many salient results on Gaussian measures/integrals, Fresnel integrals, and their graphical expansions. When possible to provide an economical proof, it is given, with detailed references for the remaining results. Path integrals and their expansions are often only discussed at a superficial level or their mathematical results are black-boxed, but Mnev's treatment is appreciably different (of course a few other notable exceptions also exist, e.g., the tome of Glimm and Jaffe). This chapter provides a great service to the mathematician encountering these ideas for the first time.  

 

The text closes with discussion of mathematical applications of BV theory.  Mnev's overview of Kontsevich's deformation quantization a la Cattaneo and Felder is particularly coherent and a beautiful story.  

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Ryan Grady is an Assistant Professor of Mathematics and the Director of the Directed Reading Program in Mathematical Sciences at Montana State University.