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Quantum Field Theory and Manifold Invariants

Daniel S. Freed, Sergei Gukov, Ciprian Manolescu, Constantin Teleman, and Ulrike Tillmann, eds.
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[Reviewed by
Ryan Grady
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Quantum Field Theory and Manifold Invariants is a text that continues the exceptionally high quality of published lectures associated with the Park City Mathematical Institute. This IAS/PCMI publication captures the content of the 2019 Graduate Summer School. The general level of exposition and the inclusion of exercises makes this text imminently useable by junior and senior researchers alike.
The text under review is a collection of eight series of lectures: Introduction to Gauge Theory (Haydys); Knots, Polynomials, and Categorification (Rasmussen); Lectures on Heegaard Floer Homology (Hom); Mathematics and Physics of Higgs Bundles (Schaposnik); Gauge Theory and a Few Applications to Knot Theory (Mrowka and Wang); Lectures on Invertible Field Theories (Galatius); TQFT, Knots and BPS States (Putrov); and Lectures on BPS States and Spectral Networks (Neitzke).  There is also a short introduction the collection by the editors: Freed, Gukov, Manolescu, Teleman, and Tillmann. This six-and-one-half page introduction is a beautiful overview of how quantum field theory has impacted topology and geometry over the past thirty years, and should be required reading for topologists young and old.
While all of the lectures are well organized and developed, this reviewer found the series by Haydys, Rasmussen, and Neitzke truly exceptional. Galatius' lectures on invertible TQFT's are also a welcome resource. His lectures use homotopy theory to relate the classification of invertible theories to the classifying spaces of diffeomorphism groups. These lectures are an excellent compliment to the recent work of Freed and Hopkins on the classification of invertible TQFT's, symmetry protected topological phases, and Fermionic systems.
Andriy Haydys' lectures, Introduction to Gauge Theory, assume no more than a first year course in topology and build to Seiberg--Witten Theory. The classical---and beautiful---results needed for construction of the Seiberg--Witten moduli spaces, e.g., Dirac operators, Fredholm theory, are recalled succinctly, palatably, and with style. This lecture series finishes by defining the Seiberg--Witten Invariant (via the compact, oriented perturbed moduli space) and some simple applications of these invariants.
The series of lectures by Jacob Rasmussen are a compelling yarn of knot homology theories. Beginning from consideration and example computations of the Alexander and Jones polynomials, Rasmussen introduces Khovanov Homology as the homological/categorification of the Jones polynomial. The HOMFLY--PT polynomial is a common generalization of the Alexander polynomial and the Jones polynomial and is introduced next. These knot polynomials are introduced via their skein relations and their categorifications by construction of TFT's with domain various cobordism/tangle categories. Rasmussen's lectures conclude with a discussion of TFT's due to Witten and Reshetikhin--Turaev which lead to a large class of knot, link, and 3-manifold invariants. Pavel Putrov's lectures are a good followup to Rasmussen's and discuss the connection to gauge theory in greater detail.
Rounding up the volume, Andy Neitzke's lectures provide a short, but illustrative introduction to several topics in supersymmetric quantum field theory that mathematicians often find confounding. Neitzke motivates and discusses BPS states/spectra, chiral rings, and spectral networks in a manner aimed towards mathematicians. Moreover, his choice to carry the guiding examples of cubic and quartic Landau--Ginzburg (LG) models throughout the lectures is incredibly helpful. These LG models depend on simple data: a polynomial on \( \mathbb{C} \) (or \( \mathbb{C}^{2} \) ), yet are surprisingly rich and much is gained by considering a cubic superpotential versus a quartic superpotential. Throughout Neitzke also highlights connections to current research frontiers, e.g., Donaldson--Thomas invariants, \( tt^{\ast} \) geometry, and maximally supersymmetric theories.
Ryan Grady is an Associate Professor of Mathematics at Montana State University.