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Dirichlet branes and mirror symmetry. (English) Zbl 1188.14026
Clay Mathematics Monographs 4. Providence, RI: American Mathematical Society (AMS); Cambridge, MA: Clay Mathematics Institute (ISBN 978-0-8218-3848-8/hbk). ix, 681 p. (2009).
The book under review grew out of the second school on mirror symmetry organized by the Clay Mathematics Institute at Cambridge, UK in spring 2002. It can be viewed as a sequel to Mirror Symmetry [Clay Mathematics Monographs 1. Providence, RI: American Mathematical Society (AMS) (2003; Zbl 1044.14018)], an account of the first school in 2000, despite the empty intersection between the lists of authors. Major differences include a narrower focus and conscious effort to integrate mathematical and physical points of view. This makes for a more unified presentation that will be helpful in bridging the gap between mathematical and physical languages for the subject. As the title suggests, the book is strongly influenced by the second superstring revolution of 1990’s with its focus on branes and open strings.
The book relies somewhat on foundational material from Mirror Symmetry, but the parts specific to the book’s focus are re-reviewed in Chapters 1-3. New developments of 2002-2007 are also well referenced and incorporated to a reasonable degree. The book will be useful to graduate students and young researchers in physics and mathematics wishing to work on mirror symmetry; it will also serve experts in the field as an encyclopedia and a dictionary.
Although more general framework is briefly surveyed, the book (like most of the past and current work) focuses on mirror symmetry in the context of topological string theory. Within this context, mirror symmetry refers to duality between two types of string theories, IIA and IIB, defined geometrically on Calabi-Yau threefolds. The type IIA theory depends only on Kähler moduli of the threefold, while the type IIB depends only on its complex structure. For closed strings, the IIA theory can be interpreted in terms of enumerative and Gromov-Witten invariants of algebraic geometry. Computations in the IIB theory (like solving the Picard-Fuchs equations) can be performed classically and make duality predictions for these invariants. This explains the deep impact mirror symmetry had on algebraic geometry.
Dirichlet Branes and Mirror Symmetry goes beyond this relationship and into the realm of open strings. Their ends require boundary conditions to be specified, geometric manifestations of these boundary conditions are the Dirichlet branes or D-branes of the title (brane is a shorthand for membrane). In topological string theory they can be roughly identified with submanifolds equipped with a vector bundle and a connection on it.
The book further narrows it down to BPS branes (after Bogomol’nyi and Prasad-Sommerfield), which preserve supersymmetry. Among other things, BPS brane submanifolds must be calibrated, e.g. be special Lagrangian in the type IIA, or holomorphic in the type IIB. The book’s approach to mirror symmetry is to compare D-branes in dual manifolds rather than to deal with open strings directly. To this end, two major frameworks are pursued: Kontsevich’s Homological Mirror Symmetry conjecture and the SYZ conjecture of Strominger-Yau-Zaslow.
The content of the book is organized as follows. Chapter 1 is a lucid introduction into physical intuitions of string theory and their partial mathematical counterparts. It also gives a bird view of the other chapters providing a gateway into the book. Chapter 2 introduces D-branes in the simplest setting of two-dimensional topological field theory. Closed strings in two dimensions can be encoded into a commutative Frobenius algebra using sewing relations, the D-branes correspond to modules over it. This establishes an instance of a general relationship between D-branes and K-theory. Chapter 3 outlines relevant elements of superconformal field theory, including topological \(\sigma\)-models and geometric definition of BPS branes.
Chapters 4, 5 and 8 tell the story of homological mirror symmetry. In this rather abstract approach branes are formalized into coherent sheaves and their complexes. Before introducing the derived category of coherent sheaves Chapters 4 briefly reviews the categorical language involved. The discussion then turns to Fourier-Mukai transform and its application to studying derived categories on \(\mathbb{P}^n\) (Beilinson’s trick), and to McKay correspondence between holomorphic group quotients and representation theory. Chapter 5 goes into more technical aspects of BPS B-branes, such as stability conditions required for them to carry a Hermitian Yang-Mills connection, another product of supersymmetry. This eventually leads to the notion of Pi-stability of Aspinwall-Douglas and Bridgeland. The discussion is motivated by physical world-volume considerations and comparisons to A-brane stability due to Joyce.
While Chapters 4 and 5 are centered on B-branes, Chapter 8 focuses on A-branes and the Fukaya category. Unlike the derived category of coherent sheaves, the Fukaya category is not really a category since the composition of morphisms is only associative ‘up to homotopy’. Accordingly, the chapter starts by introducing \(A_\infty\) structures that replace associativity. Conversely, the Fukaya category is triangulated, something not obvious for the derived category. After addressing these complications a precise mathematical statement of the Homological Mirror Symmetry conjecture is given, and a proof is outlined for the elliptic curves.
The SYZ conjecture discussed in Chapters 6, 7 follows geometric intuitions more closely than the homological mirror symmetry. The prototypical example is T-duality that relates string theories on tori with mutually inverse metrics. A generalization leads to fibering of mirror Calabi-Yau’s by dual families of special Lagrangian tori. Chapter 6 starts by describing McLean’s theory of special Lagrangian moduli, and proceeds with mirror symmetry for toric fibrations with flat metrics on each fiber. Following Hitchin, affine Kähler structures on the bases of fibrations are explored next, establishing a link with the homological mirror symmetry. The chapter concludes with construction of purely topological toric fibrations for some compact threefolds like the quintic in \(\mathbb{P}^4\).
Chapter 7 is the only one in the book that deals with metric issues, albeit in a survey-like manner. It begins by surveying non-compact examples of Ricci-flat metrics followed by ’pathological’ examples of special Lagrangian fibrations due to Joyce. These lead to doubts about the original SYZ conjecture, and it is replaced with a version that holds in the large complex structure limit of Calabi-Yau degenerations.

14J32 Calabi-Yau manifolds (algebro-geometric aspects)
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
32Q25 Calabi-Yau theory (complex-analytic aspects)
18E30 Derived categories, triangulated categories (MSC2010)
32G81 Applications of deformations of analytic structures to the sciences
83E30 String and superstring theories in gravitational theory
81T45 Topological field theories in quantum mechanics
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
81T75 Noncommutative geometry methods in quantum field theory