Quantum Fields And Strings A Course For Mathematicians

Why Every Mathematician Who Tries This Book Eventually Curses IAS

The American Mathematical Society published Quantum Fields And Strings A Course For Mathematicians back in 1999, pulled from a special year at the Institute for Advanced Study. The contributors were established names in the field. The result is two massive volumes that are simultaneously one of the most important resources for bridging quantum field theory and mathematics and one of the most frustrating books you will ever sit down to read. You do not read this book cover to cover. Nobody does. I tried that on the first volume once, working through Deligne's lectures on supersymmetry in what I thought would be a focused three-week project. It took me eight months and I still had to go back and re-derive half the conventions from scratch because the notation shifted between chapters without warning. The practical approach is to treat it as a reference archive and pick the volume and chapter that match the gap in your current understanding. Volume 1 covers the foundational physics side—conformal field theory, topological quantum field theory, and the geometric applications. Volume 2 goes deeper into superstrings, the c = 1 model, and the more technical algebraic geometry connections. Most people who actually use this book end up living in Volume 1, Appendix, and the second half of Volume 2. The early chapters of Volume 1 are more lecture-note-like and comparatively readable if you have a solid background in differential geometry and basic topology.

The real problem is that this book assumes you already know what you are looking for. There is no gradual introduction to path integrals, no hand-holding through why physicists care about renormalization before they start using it. You are expected to either already be comfortable with the physics motivations or be willing to suffer through parallel reading from actual physics textbooks like Peskin and Schroeder or Weinberg's later volumes just to understand what the mathematicians are formalizing.

The Chapters That Actually Work and The Ones That Do Not

Deligne's contribution on supersymmetry is genuinely good. It is the most self-contained chapter in the entire set and the one I recommend people start with if they want to test whether this book is worth the effort. It reads like a real lecture series rather than a reference manual. If you can follow Deligne, you can handle the rest. If you cannot, you should probably build more background first. The Gotte-Möller chapter on symplectic reduction is useful but extremely terse. I ran into a specific issue when trying to apply their framework to a gauge theory problem involving a non-compact quotient space. Their treatment assumes compactness almost everywhere, and when I tried to extend it to a situation with an asymptotic symmetry group, the convergence arguments just broke down around equation 4.17. The workaround I ended up using was to go back to the original papers by Perelman on moment maps and reconstruct the reduction step by step for my particular case. It added about two weeks of work but saved me from making a false claim in a paper. Kontsevich's chapter on deformation quantization is brilliant but dense to the point of being almost unusable on a first pass. He assumes familiarity with Fedosov's technique and microlocal analysis without much preamble. I spent roughly three weeks on that chapter alone before it started to make sense. The payoff is worth it—his results are foundational—but the initial engagement is brutal. I found it helped to read him alongside a more pedagogical source like Cattaneo's lecture notes on the same topic.

The Dijgraaf-Witten section on topological field theory is one of the most cited parts of this book for good reason. It connects cleanly to later work in low-dimensional topology. But the passage from the physical partition function to the rigorous mathematical definition is glossed over in a way that will irritate anyone who cares about that boundary. You will need to fill in the analytic details yourself if that matters to you.

Common Pitfalls That Wasted My Time

The biggest trap is assuming the book is internally consistent in its notation and conventions. It is not. The authors use different sign conventions, different normalizations for the path integral measure, and different bracketing for anticommuting variables across different chapters. I discovered this the hard way when I tried to use results from two different sections in a single calculation and got contradictory signs on my final answer. I spent a full day tracing it back to a convention mismatch rather than an actual error. You need to pick one chapter's conventions and commit to them, then track every other section's deviations explicitly. Another issue is the treatment of infinite-dimensional spaces. The book routinely manipulates function spaces, loop spaces, and state spaces without always being precise about which topology is in play. If you are working on something where the choice of topology matters—which is to say, almost anything involving actual analysis rather than pure formalism—you will encounter gaps. The physics literature has the same problem but tends to be more explicit about when it is being formal. Here the formalism is often presented as if it were rigorous without the caveats. The book also does not include exercises or solutions. Some chapters hint at problems you might try but never state them clearly. If you learn by doing, which most mathematicians do, this is a significant deficiency. You will need to supplement with problem sets from courses that use this material, like the ones occasionally posted by the IAS or related institutions.

How I Actually Use This Book in Practice

My current workflow is to keep both volumes on my desk as lookup references while I work through a problem. I have sticky notes marking every chapter where the conventions diverge from what I am using. I cross-reference with the physics literature whenever a mathematical statement feels too clean to be true. When I need to understand a concept quickly, I go to the relevant chapter, read it once for the big picture, then read it a second time slowly filling in the gaps from other sources. For someone preparing to do research at the intersection of QFT and geometry, this book is still arguably the best single resource available, despite its flaws. It is not a textbook. It is a collection of expert lectures that assume expertise and sometimes forget that not every reader shares the same assumptions. Treat it that way and it will serve you well. Try to read it straight through like a novel and you will waste a lot of time and frustration. The appendix material in Volume 1, particularly the section on geometric applications of conformal field theory, is often overlooked but contains some of the most immediately usable results. I have cited material from that appendix more times than from any other part of the book.