The interplay between mathematics and physics that has so often yielded rich fruit over the centuries has of late been much in evidence, what with the work of such figures as Witten and Kontsevich, to name but two obvious luminaries. Arguably this ecumenical view of things underlay, and underlies, the work of many others who are perhaps seen as pure mathematicians exclusively. Certainly the Russians come to mind: Gel’fand and Manin, for example, and, going back a couple of generations, the titanic analytic number theorist, Carl Ludwig Siegel, certainly the purest of pure mathematicians, who also qualified as an expert in celestial mechanics.

Indeed, going on to consider the great Göttingen school Siegel stemmed from, with Klein and Hilbert at its center, it is worth noting that the alleged border separating *Mathematik* from *Physik* was disregarded and denied by almost all the indicated players (with Siegel’s advisor, Edmund Landau, an obvious *Gegenbeispiel*). It is entertaining to speculate about the reasons for the ensuing separation of the fields; perhaps the catastrophe of World War II and the various diasporas of European scholars are part of the reason, coupled with the increase in intrinsic difficulty in any number of areas of mathematics that precipitated a narrowing of specialties and a prevailing orientation to more compact problems and research programs. Hilbert’s campaign to capture quantum mechanics under the banner of mathematics properly so called is generally viewed as an anachronism these days, and as one of Hilbert’s less successful undertakings.

Adopting a more polemical point of view, perhaps, and borrowing the marvelous if somewhat controversial imagery of frogs and birds introduced by Freeman Dyson (in the February 2009 issue of the *Notices of the AMS*), one might posit that, with birds not only being rarer than frogs but having been scattered by the events of the middle to late twentieth century, frogs had their way in mathematics — *ergo, *narrower specialties. But birds will be birds, and in due course borders between mathematical areas were vaulted by a few trailblazers, with Grothendieck possibly the most obvious example. His algebraic geometry, pulling together homological algebra (building in turn on the work of Zariski whose flight path first straddled the Italians’ more or less intuitive and sometimes non-rigorous geometry and the burgeoning commutative algebra coming primarily from German algebraic geometry, to oversimplify matters a bit), representation theory, various other branches of algebra, and algebraic topology, can be regarded as be the seed ground for what the present synergy between mathematics and physics is all about. This state of affairs is indeed beautifully illustrated by, for instance, the 1996–1997 Special Year at the Institute for Advanced Study devoted to Quantum Field Theory, bringing together pure mathematicians, including Pierre Deligne, who famously stems from Grothendieck’s school of algebraic geometry, and QFT specialists, including Edward Witten. The enterprise produced the two-volume opus, *Quantum Fields and Strings* (AMS & IAS, 1999): it’s again fitting and proper that mathematicians have a go at physics … but all in proper *Hochmathematikalisch*, of course: hence the book.

The book under review, *Dirichlet Branes and Mirror Symmetry*, published by the Clay Institute (and, yes, the AMS), is equally ecumenical: mirror symmetry certainly brings to mind the aforementioned Maxim Kontsevich (beautiful mathematics coming from particle physics and beyond), and Dirichlet branes have gone from a physicist’s enterprise (e.g., Michael Douglas) to a context for (Tom) Bridgeland(‘s) stability conditions in algebraic geometry. (My apologies for my not mentioning others; I have some acquaintance with Bridgeland’s work, hence my focus — it’s just a question of my own ignorance. By the way, both Douglas and Bridgeland are counted among the present authors.)

The book is nothing if nor hefty, coming in at 681 pages, and evinces encyclopedic coverage. The trajectory of chapters takes one from D-branes, K-theory in topological field theory, and some string theory (viz. Ch’s. 2 and 3), through representation theory, homological algebra, geometry, and Dirichlet branes and stability conditions (Ch’s. 4, 5), to a consideration of mirror symmetry *à la* Strominger-Yau-Zaslow, Calabi-Yau manifolds, and finally homological mirror symmetry (Ch’s 6, 7, 8).

The authors themselves advertise their product as follows: “This is the second of two books that provide the record of the [2002 Clay School on Geometry and String Theory … held at the Isaac Newton Institute for Mathematical Sciences …] The first book, *Strings and Geometry*, … largely focused on the topics of manifolds of special holonomy and supergravity. The present volume … covers mirror symmetry from the homological and torus fibration points of view. We hope that [it] is a natural sequel to *Mirror Symmetry* [AMS 2003] … A familiarity with the foundational material of [*Mirror Symmetry*] can be viewed as a prerequisite for reading this volume …” *Caveat*.

They go on to say that their goal is “to explore the physical and mathematical aspects of Dirichlet branes … organized around … Kontsevich’s Homological Mirror Symmetry conjecture and the Strominger-Yau-Zaslow conjecture. While Kontsevich’s conjecture predates … D-branes [in] physics … [it] is really equivalent to the identification of two different categories of D-branes. [W]e examine how the physics leads us naturally to [e.g.] … derived categories and Fukaya categories.” And I cannot resist noting that derived categories are traced back to none other than Grothendieck and his pupil Verdier.

That’s it then: not for the timid, clearly, but very exciting stuff, at the frontier of both mathematics and physics. By the way, following Wikipedia, a D-brane is, by definition, “an extended object [in string theory] upon which open strings can end with Dirichlet boundary conditions,” while “mirror symmetry is a relation that can exist between two Calabi-Yau manifolds.” Furthermore (loc.cit.), “[i]t happens, usually for two such six-dimensional manifolds, that the shapes may look very different geometrically, but nevertheless they are equivalent if they are employed as hidden dimensions of string theory. The classical formulation of mirror symmetry relates two Calabi-Yau threefolds *M* and *W* whose Hodge numbers *h*^{1,1} and *h*^{1,2} are swapped; string theory compactified on these two manifolds lead to identical effective field theories.” Again: very exciting!

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