Categorical mirror symmetry: The elliptic curve

Polishchuk, Alexander; Zaslow, Eric

doi:10.4310/atmp.1998.v2.n2.a9

Cited by 192 publications

(302 citation statements)

References 26 publications

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“…We remark that employing this construction is similar to the use of isogenies needed to define the categorical isomorphism which proves Kontsevich's conjecture in the case of the elliptic curve [13]. The point is that multisections transforming to higher-rank can be handled via single sections giving line bundles, by imposing functoriality after pushing forward under finite covers.…”

Section: Transformation Of a Multi-sectionmentioning

confidence: 84%

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From special Lagrangian to Hermitian–Yang–Mills via Fourier–Mukai transform

Leung¹,

Yau²,

Zaslow³

2000

Advances in Theoretical and Mathematical Physics

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141

172

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We exhibit a transformation taking special Lagrangian submanifolds of a Calabi-Yau together with local systems to vector bundles over the mirror manifold with connections obeying deformed HermitianYang-Mills equations. That is, the transformation relates supersymmetric A-and B-cycles. In this paper, we assume that the mirror pair are dual torus fibrations with flat tori and that the A-cycle is a section.We also show that this transformation preserves the (holomorphic) Chern-Simons functional for all connections. Furthermore, on corresponding moduli spaces of super symmetric cycles it identifies the graded tangent spaces and the holomorphic m-forms. In particular, we verify Vafa's mirror conjecture with bundles in this special case. e-print archive: http://xxx.lanl.gov/math.DG/0005118

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Section: Transformation Of a Multi-sectionmentioning

confidence: 84%

“…For one can write (f) 13 In fact we get a torus family of one-forms, since y (hence A) has x-(or x-) dependence. Namely, we obtain a U (1) connection on W,…”

Section: Remark 2 We Note the Symmetry Between G (Resp Uj) Andjj (Rmentioning

confidence: 99%

From special Lagrangian to Hermitian–Yang–Mills via Fourier–Mukai transform

Leung¹,

Yau²,

Zaslow³

2000

Advances in Theoretical and Mathematical Physics

Self Cite

141

172

View full text Add to dashboard Cite

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“…In general, agreement between type IIA and type IIB chiral field superpotentials arising from D-branes is an intense area of mathematical research (see e.g. [80]), and lies at the very heart of Kontsevich's homological mirror symmetry conjecture [81]. 8 This field theory approach also allows to extract important information regarding the Kähler metrics for charged matter.…”

Section: Type Iib Model Buildingmentioning

confidence: 99%

Progress in D‐brane model building

Marchesano¹

2007

Fortschritte der Physik

154

234

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The state of the art in D-brane model building is briefly reviewed, focusing on recent achievements in the construction of D = 4 N = 1 type II string vacua with semi-realistic gauge sectors. Such progress relies on a better understanding of the spectrum of BPS D-branes, the effective field theory obtained from them and the explicit construction of vacua. We first consider D-branes in standard Calabi-Yau compactifications, and then the more involved case of compactifications with fluxes. We discuss how the non-trivial interplay between D-branes and fluxes modifies the previous model-building rules, as well as provides new possibilities to connect string theory to particle physics. MotivationAs often emphasized in the literature, string theory gained its popularity among physicists in the mid-80's, as it was recognized as an outstanding candidate for a unified theory of all forces in nature. Not only would it incorporate quantum gravity, but also supersymmetry and non-Abelian gauge interactions. With such ingredients it was soon realized that, upon compactification, one could conceive D = 4 N = 1 effective theories remarkably close to natural extensions of the Standard Model [1].Further developments of the theory showed that the set of string theory vacua is quite complex even at a qualitative level. On the one hand we learned that seemingly different vacua could be dual to each other and, in fact, should be considered to be the same. On the other hand, it has been estimated that the set of semi-realistic vacua is enormous. This latter point has recently raised the question of whether string theory can reproduce almost any D = 4 effective theory that one may think of and, in particular, many possible physics beyond the Standard Model. If that was the case, one may drop the classical strategy to focus on a particular string theory model by means of some vacuum selection principle, and instead try to extract physical information out of the ensemble of realistic vacua via an statistical approach [2]. Now, while string theory has proven to be an excellent generator of phenomenologically interesting scenarios, a string vacuum reproducing the physics observed at particle accelerators is yet to be found. Individual vacua analyzed so far not only fail to describe fully realistic D = 4 physics because of some technical details like extra massless particles or unrealistic couplings, but also because of deep conceptual issues like gauge coupling unification, hierarchy problems and supersymmetry breaking. Moreover, recent cosmological observations have slightly changed our perspective on how a semi-realistic vacuum should be obtained and, in general, agreement with cosmological data puts even more constraints on realistic string constructions.Pointing out these facts is not intended to discourage those interested in the field of model building. On the contrary, there has recently been huge progress (some of which the present review is based on) in constructing vacua which are more and more realistic, and we expect much more p...

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“…The monodromies, however, are most easily determined on the mirror torus, where the B-branes with RR charges, (r, c 1 ), are mapped to A-branes realized as special Lagrangian submanifolds with winding numbers, (p, q) [49]. On the mirror side the monodromies are generated by encircling the corresponding singularities in the complex structure moduli space, which, for the torus, is identical to the Teichmüller space depicted in figure 3 (a).…”

Section: Jhep02(2007)006mentioning

confidence: 99%