By Paul S. Aspinwall et al.

Members: Paul S. Aspinwall, Tom Bridgeland, Alastair Craw, Michael R. Douglas, Mark Gross, Anton Kapustin, Gregory W. Moore, Graeme Segal, Balazs Szendroi, and P.M.H. Wilson

Research in string idea over the past a number of a long time has yielded a wealthy interplay with algebraic geometry. In 1985, the creation of Calabi-Yau manifolds into physics for you to compactify ten-dimensional space-time has ended in intriguing cross-fertilization among physics and arithmetic, specially with the invention of replicate symmetry in 1989. a brand new string revolution within the mid-1990s introduced the thought of branes to the vanguard. As foreseen via Kontsevich, those became out to have mathematical opposite numbers within the derived class of coherent sheaves on an algebraic type and the Fukaya classification of a symplectic manifold.

This has ended in interesting new paintings, together with the Strominger-Yau-Zaslow conjecture, which used the speculation of branes to suggest a geometrical foundation for replicate symmetry, the speculation of balance stipulations on triangulated different types, and a actual foundation for the McKay correspondence. those advancements have resulted in loads of new mathematical work.

One hassle in knowing all points of this paintings is that it calls for having the ability to communicate varied languages, the language of string thought and the language of algebraic geometry. The 2002 Clay institution on Geometry and String concept got down to bridge this hole, and this monograph builds at the expository lectures given there to supply an up to date dialogue together with next advancements. A ordinary sequel to the 1st Clay monograph on replicate Symmetry, it offers the recent principles popping out of the interactions of string thought and algebraic geometry in a coherent logical context. we are hoping it's going to let scholars and researchers who're accustomed to the language of 1 of the 2 fields to achieve acquaintance with the language of the other.

The booklet first introduces the inspiration of Dirichlet brane within the context of topological quantum box theories, after which experiences the fundamentals of string thought. After exhibiting how notions of branes arose in string idea, it turns to an creation to the algebraic geometry, sheaf idea, and homological algebra had to outline and paintings with derived different types. The actual lifestyles stipulations for branes are then mentioned and in comparison within the context of reflect symmetry, culminating in Bridgeland's definition of balance constructions, and its functions to the McKay correspondence and quantum geometry. The e-book maintains with designated remedies of the Strominger-Yau-Zaslow conjecture, Calabi-Yau metrics and homological replicate symmetry, and discusses newer actual developments.

Titles during this sequence are co-published with the Clay arithmetic Institute (Cambridge, MA).

Readership

Graduate scholars and learn mathematicians attracted to mathematical elements of quantum box idea, specifically string idea and replicate symmetry.

This e-book is acceptable for graduate scholars and researchers with both a physics or arithmetic history, who're attracted to the interface among string idea and algebraic geometry.

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**Extra resources for Dirichlet Branes and Mirror Symmetry**

**Example text**

Then Cr is the space of sections of a vector bundle E on X, and it follows from the condition χns = χr that the fibre at each point must have dimension 1. Thus the whole structure is determined by the Frobenius algebra Cns together with a binary choice at each point x ∈ X of the grading of the fibre Ex of the line bundle E at x. We can now see that if we had not used the graded symmetry in defining the tensor category we should have forced the grading of Cr to be purely even. For on the odd part the inner product would have had to be skew, and that is impossible on a 1-dimensional space.

We are now ready to explain the Strominger-Yau-Zaslow proposal [433]. Consider a pair of compact Calabi-Yau 3-folds X and Y related by mirror symmetry. By the above, the set of BPS A-branes on X is isomorphic to the set of BPS B-branes on Y , while the set of BPS B-branes on X is isomorphic to the set of BPS A-branes on Y . The simplest BPS B-branes on X are points. These exist for all complex structures on X, even nonalgebraic ones. Their moduli space is X itself. Let us try to determine which BPS A-branes on Y they correspond to.

The category of boundary conditions is equivalent to the category Vect(X) of finite-dimensional complex vector bundles on X. , one which does not depend on the particular D-brane). Conversely, every semisimple Frobenius category B is the category of boundary conditions for a canonical 2-dimensional TFT, whose corresponding commutative Frobenius algebra is the ring of endomorphisms of the identity functor of B. We shall explain in the next section the sense in which the boundary conditions form a category.