Dirichlet branes, homological mirror symmetry, and stability

By Michael R Douglas. 2002

183 citations recorded by the source corpus.

We discuss some mathematical conjectures which have come out of the Dirichlet branes in superstring theory, focusing on the case of supersymmetric branes in Calabi-Yau compactification. This has led to the formulation of a notion of stability for objects in a derived category, contact with Kontsevich's homological mirror symmetry conjecture, and "physics proofs" for many of the subsequent conjectures based on it, such as the representation of Calabi-Yau monodromy by autoequivalences of the derived category.

Subfields: Geometry and Topology, Mathematical Physics, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/b45e237161a41efd282dd6296b664faf7c88cf02

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Citations in the corpus

Stability Manifold of P^1 (2004, unknown): “A G ] 1 9 M ar”
Stability conditions and Stokes factors (2008, methodology): “Stability conditions and Stokes data 26 References 29”
Lectures on Bridgeland Stability (2016, background): “tability conditions in derived categories provide a framework for the study of moduli spaces of complexes of sheaves. They have been introduced in [Bri07], with inspiration from work in string theory [Dou02]. It turns out that the theory has far further reach. A non exhaustive list of inﬂuenced area”
Stability conditions and extremal contractions (2012, background): “Now there is a notion of stability conditions on derived categories by Bridgeland [11], which provides a mathematical formulation of Douglas’s Π-stability [15].”
Derived categories of irreducible projective curves of arithmetic genus one (2005, background): “With the exception of § 2.7, throughout this section E denotes a smooth elliptic curve over k.”
The notion of dimension in geometry and algebra (2005, background): “tegories D(C), for example coherent sheaves on an elliptic curve, have families of hearts Cθ indexed by real numbers θ. This “ﬂow of charges” was ﬁrst discovered in the context of Mirror Symmetry (cf [Dou]). For a mathematical treatment due to T. Bridgeland, see [Br]; one can ﬁnd there some physics ”
Gluing Stability Conditions (2009, methodology): “A G ] 1 3 M ay 2 01 0 Introduction Stability conditions on triangulated categories were introduced by Bridgeland in [4] as a mathematical formalization of Douglas’ work on Π-stability in [7, 8].”
Stability conditions on curves (2007, unknown): “A G ] 2 5 M ay 2 00”
Dirichlet Branes and Mirror Symmetry (2009, background): “More subtle clues come from the comparison of different CFT’s which are supposed to lead to the same TFT, as discussed in [135].”
D-Branes on Calabi-Yau Manifolds (2004, background): “anes. These subjects have been reviewed by Douglas in [6] from a somewhat diﬀerent direction than we employ here. Douglas also has a shorter, more mathematicallyoriented review in the ICM proceedings [7]. In order to motivate and better understand our constructions, a good deal of our discussion wil”
An integral structure in quantum cohomology and mirror symmetry for toric orbifolds (2009, background): “u 3-folds, the author hopes that our Z(V) gives the physics central charge of the B-type D-brane in the class V. This plays an important role in the Douglas-Bridgeland stability on derived categories [31, 13]. 2.5. Givental’s symplectic space, ∞ 2 VHS and J-function. Givental’s symplectic space [39,”
Moduli stacks and invariants of semistable objects on K3 surfaces (2007, background): “The notion of stability conditions on triangulated categories was introduced in [7] to give the mathematical framework for the Douglas’s work on Π-stability [10], [11].”