American Mathematical Society. Clay Mathematics Institute. Clay Mathematics Monographs. Volume 4. **DIRICHLET** **BRANES**. AND **MIRROR** **SYMMETRY**. Paul S. Aspinwall. Tom Bridgeland. Alastair Craw. Michael R. Douglas. Mark Gross. Anton Kapustin. Gregory W. Moore. Graeme Segal. Balázs Szendrői. P.M.H. Wilson ...

arXiv:1612.09380v1 [math.SG] 30 Dec 2016. SYZ **Mirror** **Symmetry** for. **Dirichlet** **Branes**. CHUNG, Shun Wai. A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of. Master of Philosophy in. Mathematics. The Chinese University of Hong Kong. July 2016 ...

Keywords and Phrases: Superstring theory, **Dirichlet** **Branes**, Homological. **Mirror** **Symmetry**, stability. 1. Introduction. Let M be a complex manifold with Kähler metric, and E a vector bundle on. M. The hermitian Yang-Mills (HYM) equations are nonlinear partial differential equations for a connection

The concept of a D-brane (**Dirichlet** brane) in string theory had evolved considerably [?] in recent years: Original .... the Strominger-Yau-Zaslow [SYZ96] picture of **mirror** **symmetry** and suggests the following. Questions ... We will explain how these mathematical symmetries act on D-brane moduli

1.3 'Unifying string theories and p-**branes**. 2 General Formulation of **Dirichlet** p-**Branes**. 2.1 D-**branes** and boundary conformal field theory. 2.2 Supersymmetric cycles in Calabi-Yau manifolds. 2.3 Boundary states for D-**branes** ... 2.4 Exceptional supersymmetric cycles. 3 Applications to **Mirror** **Symmetry**.

a consequence of **mirror** **symmetry** acting on D-**branes**. The organization of this ... section 6 we derive, using the results of [2] the **mirror** of certain D-**branes** on Fano varieties in terms of D-**branes** in the ...... coordinate fields as (3.21), from pure Neumann to mixed **Dirichlet**-Neumann condition

G. Segal, B. Szendr˝oi, and P.M.H. Wilson, **Dirichlet** **branes** and **mirror** **symmetry**, Clay. Mathematics Monographs, vol. 4, American Mathematical Society, Providence, RI, 2009. 4. Serguei Barannikov, Semi-infinite Hodge structures and **mirror** **symmetry** for projective spaces, arXiv:math/0010157, 2000. 5

string theory, and give an intuitive description of string compactification,. **Dirichlet** **branes**, T-duality and the other physical concepts we will discuss in more depth, primarily in Chapters 2, 3 and 5. We then summarize the mathematics behind homological **mirror** **symmetry** and SYZ, which we

23. 1. The SYZ proposal. In 1996, drawing on the new idea of D-**branes** from string theory, Strominger,. Yau and Zaslow [77] made a ground-breaking proposal to explain **mirror** **symmetry** ...... B. Szendr˝oi, and P. M. H. Wilson, **Dirichlet** **branes** and **mirror** **symmetry**, Clay Mathemat- ics Monographs, vol.

D–**branes** and Calabi–Yaus. Topological Strings. **Mirror** **Symmetry**. Conclusions. D–**branes**. • D–**branes** are objects where open strings can end. (**Dirichlet** boundary conditions). • D–**branes** break translation invariance and supersymmetry. • The string gets additional degrees of freedom on the boundary.

Over the past year, **Dirichlet** **branes** (D-**branes**) which do not preserve supersymmetry, and therefore do not saturate the ... to make progress in analysing the possible dualities of theories without supersymmetry†, we have to develop ..... of the string that stretches between the D5-brane and its

the comprehensive **Mirror** **Symmetry** monograph [Clay03] and the more recent **Dirichlet**. **Branes** and **Mirror** **Symmetry** monograph [Clay09], the recent survey article by Mark Gross. [CDM] and the expository article [GS08] should both be reasonably accessible to workshop participants. (Chap. 6 through 8

Duality symmetries, D-brane categories, and extended field theory. 2. 3. ... The duality symmetries here are primarily **mirror** **symmetry** and S-duality. D-**branes** are connected to two central developments in this area of Mathematics: 1. ..... [22] J. Polchinski, “**Dirichlet** **Branes** and Ramond-Ramond

D-**Branes** on Calabi-Yau Manifolds. Michael R. Douglas. Abstract. We give an overview of recent work on **Dirichlet** **branes** on Calabi-. Yau threefolds which makes contact with Kontsevich's homological **mirror** **symmetry** proposal, proposes a new definition of stability which is appropriate in string theory

The homological **mirror** **symmetry** (HMS) conjecture, proposed by Kontsevich in his 1994 ICM address ... provides a geometric explanation for **mirror** **symmetry** – the celebrated SYZ conjec- ture asserts that a pair ...... MR 2914956. 33. P. Aspinwall et al., **Dirichlet** **branes** and **mirror** **symmetry**, Clay

(15) with P. Aspinwall et al, **Dirichlet** **Branes** and **Mirror** **Symmetry**, Clay. Mathematics Monographs, x+681 pp. Hardback 684 pp. (2009). (16) with David Stern, Helices on del Pezzo surfaces and tilting Calabi-Yau algebras,. Adv. Math. 224 1672–1716 (2010). (17) Hall algebras and curve-counting

markable conjecture (Conjecture 2.3), which builds a bridge between **mirror** **symmetry** for the ... by gluing the underlying affine base manifolds of X1 and X2 in SYZ **mirror** **symmetry**. This is based ...... Szendr˝oi B., Wilson P.M.H., **Dirichlet** **branes** and **mirror** **symmetry**, Clay Mathematics Monographs

D-**BRANES**. ANTON KAPUSTIN AND DMITRI ORLOV. Abstract. This paper is an introduction to Homological **Mirror** **Symmetry**, derived cat- egories, and topological D-**branes** aimed mainly at a mathematical audience. In the paper we explain ..... For example, one can impose **Dirichlet** boundary conditions (i.e

The present mathematical challenges of **mirror** **symmetry** include making the statements (1.3), (1.4) more precise, proving them, and finding relations between either of them and (1.2). All these conjectures are strongly backed by physical ideas from string theory, centrally involving D-**branes**. D

[1] P. S. Aspinwall, T. Bridgeland, A. Craw, M. R. Douglas, M. Gross, A. Kapustin, G. W.. Moore, G. Segal, B. Szendr˝oi, and P. M. H. Wilson, **Dirichlet** **branes** and **mirror** **symmetry**, Clay. Mathematics Monographs 4, American Mathematical Society, Providence, RI, 2009, ISBN 978-0-8218-. 3848-8.

mann **branes** and **Dirichlet** **branes** on these manifolds. ... For C-VCPs, there are two types of **branes** corresponding to the **Dirichlet** type ... Relations to **Mirror**. **Symmetry** will also be discussed there. 2. Instantons and **branes**. In this section we introduce and study instantons and **branes** on manifolds

I am interested in algebraic geometry with applications to **mirror** **symmetry**. I use methods of ... which both sides of **mirror** **symmetry** have been understood deeply and with explicit methods. The method in ..... G. Segal, B. Szendröi, P.M.H. Wilson: **Dirichlet** **branes** and **mirror** **symmetry**, Clay Math

on the homological **mirror** **symmetry** conjecture. This note can be read without ... end product of a conden- sation process is generally not a **Dirichlet** brane, since it typically cannot be described ... on **Dirichlet** **branes**, and – in a slightly less direct manner – also in the case of generalized. D

open string **mirror** **symmetry** for the simplest compact Calabi–Yau, the torus. ..... (**Dirichlet** conditions). **Dirichlet** boundary conditions have been ignored in string theory for a long time since there were no suitable objects in the theory for the open string to end ... Generic D–brane configurations

small resolution in IIA string theory. The image under **mirror** **symmetry** of the 3-**branes** wrapping the S. 3 providing additional non-perurbative massless states is the 2-**branes** of the IIA theory wrapping the S. 2. 's again giving additional massless states. On the “other side” of the conifold

Algebraic geometry— Seattle 2005. Part 1, 149–192, Proc. Sympos. Pure Math.,. 80, AMS, Providence, RI, 2009. [31] With Aspinwall et al, “**Dirichlet** **branes** and **mirror** **symmetry**,” Clay Mathe- matics Monographs, 4, American Mathematical Society, Providence, RI; Clay. Mathematics Institute, Cambridge, MA