Categorical Donaldson-Thomas Theory for Local Surfaces
Author | : Yukinobu Toda |
Publisher | : Springer Nature |
Total Pages | : 318 |
Release | : |
Genre | : |
ISBN | : 3031617053 |
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Author | : Yukinobu Toda |
Publisher | : Springer Nature |
Total Pages | : 318 |
Release | : |
Genre | : |
ISBN | : 3031617053 |
Author | : Yukinobu Toda |
Publisher | : Springer Nature |
Total Pages | : 110 |
Release | : 2021-12-15 |
Genre | : Science |
ISBN | : 9811678383 |
This book is an exposition of recent progress on the Donaldson–Thomas (DT) theory. The DT invariant was introduced by R. Thomas in 1998 as a virtual counting of stable coherent sheaves on Calabi–Yau 3-folds. Later, it turned out that the DT invariants have many interesting properties and appear in several contexts such as the Gromov–Witten/Donaldson–Thomas conjecture on curve-counting theories, wall-crossing in derived categories with respect to Bridgeland stability conditions, BPS state counting in string theory, and others. Recently, a deeper structure of the moduli spaces of coherent sheaves on Calabi–Yau 3-folds was found through derived algebraic geometry. These moduli spaces admit shifted symplectic structures and the associated d-critical structures, which lead to refined versions of DT invariants such as cohomological DT invariants. The idea of cohomological DT invariants led to a mathematical definition of the Gopakumar–Vafa invariant, which was first proposed by Gopakumar–Vafa in 1998, but its precise mathematical definition has not been available until recently. This book surveys the recent progress on DT invariants and related topics, with a focus on applications to curve-counting theories.
Author | : Yukinobu Toda |
Publisher | : |
Total Pages | : 0 |
Release | : 2021 |
Genre | : |
ISBN | : 9789811678394 |
This book is an exposition of recent progress on the Donaldson-Thomas (DT) theory. The DT invariant was introduced by R. Thomas in 1998 as a virtual counting of stable coherent sheaves on Calabi-Yau 3-folds. Later, it turned out that the DT invariants have many interesting properties and appear in several contexts such as the Gromov-Witten/Donaldson-Thomas conjecture on curve-counting theories, wall-crossing in derived categories with respect to Bridgeland stability conditions, BPS state counting in string theory, and others. Recently, a deeper structure of the moduli spaces of coherent sheaves on Calabi-Yau 3-folds was found through derived algebraic geometry. These moduli spaces admit shifted symplectic structures and the associated d-critical structures, which lead to refined versions of DT invariants such as cohomological DT invariants. The idea of cohomological DT invariants led to a mathematical definition of the Gopakumar-Vafa invariant, which was first proposed by Gopakumar-Vafa in 1998, but its precise mathematical definition has not been available until recently. This book surveys the recent progress on DT invariants and related topics, with a focus on applications to curve-counting theories.
Author | : Dominic D. Joyce |
Publisher | : |
Total Pages | : 199 |
Release | : 2011 |
Genre | : Calabi-Yau manifolds |
ISBN | : 9780821887523 |
Donaldson-Thomas invariants DTα(τ) are integers which `count' τ-stable coherent sheaves with Chern character α α on a Calabi-Yau 3-fold X, where τ denotes Gieseker stability for some ample line bundle on X. They are unchanged under deformations of X. The conventional definition works only for classes α containing no strictly τ-semistable sheaves. Behrend showed that DTα(τ) can be written as a weighted Euler characteristic χ(Mstα(τ),νMstα(τ)) of the stable moduli scheme Mstα(τ) by a constructible function νMstα(τ) we call the `Behrend function'. This book studies generalized Donaldson-Thomas invariants DT ̄α(τ). They are rational numbers which `count' both τ-stable and τ-semistable coherent sheaves with Chern character α on X; strictly τ-semistable sheaves must be counted with complicated rational weights. The DT ̄α(τ) are defined for all classes α α, and are equal to DTα(τ) when it is defined. They are unchanged under deformations of X, and transform by a wall-crossing formula under change of stability condition τ. To prove all this we study the local structure of the moduli stack M of coherent sheaves on X. We show that an atlas for M may be written locally as Crit(f) for f:U→C holomorphic and U smooth, and use this to deduce identities on the Behrend function νM. We compute our invariants DT ̄α(τ) in examples, and make a conjecture about their integrality properties. We also extend the theory to abelian categories Q with relations I I coming from a superpotential W on Q, and connect our ideas with Szendrői's noncommutative Donaldson-Thomas invariants, and work by Reineke and others on invariants counting quiver representations. Our book is closely related to Kontsevich and Soibelman's independent paper Stability structures, motivic Donaldson-Thomas invariants and cluster transformations.
Author | : Clay Mathematics Institute. Summer School |
Publisher | : American Mathematical Soc. |
Total Pages | : 396 |
Release | : 2004 |
Genre | : Mathematics |
ISBN | : 9780821837153 |
Contains selection of expository and research article by lecturers at the school. Highlights current interests of researchers working at the interface between string theory and algebraic supergravity, supersymmetry, D-branes, the McKay correspondence andFourer-Mukai transform.
Author | : Ricardo Castano-Bernard |
Publisher | : Springer |
Total Pages | : 445 |
Release | : 2014-10-07 |
Genre | : Mathematics |
ISBN | : 3319065149 |
The relationship between Tropical Geometry and Mirror Symmetry goes back to the work of Kontsevich and Y. Soibelman (2000), who applied methods of non-archimedean geometry (in particular, tropical curves) to Homological Mirror Symmetry. In combination with the subsequent work of Mikhalkin on the “tropical” approach to Gromov-Witten theory and the work of Gross and Siebert, Tropical Geometry has now become a powerful tool. Homological Mirror Symmetry is the area of mathematics concentrated around several categorical equivalences connecting symplectic and holomorphic (or algebraic) geometry. The central ideas first appeared in the work of Maxim Kontsevich (1993). Roughly speaking, the subject can be approached in two ways: either one uses Lagrangian torus fibrations of Calabi-Yau manifolds (the so-called Strominger-Yau-Zaslow picture, further developed by Kontsevich and Soibelman) or one uses Lefschetz fibrations of symplectic manifolds (suggested by Kontsevich and further developed by Seidel). Tropical Geometry studies piecewise-linear objects which appear as “degenerations” of the corresponding algebro-geometric objects.
Author | : Ana Cannas da Silva |
Publisher | : Springer |
Total Pages | : 240 |
Release | : 2004-10-27 |
Genre | : Mathematics |
ISBN | : 354045330X |
The goal of these notes is to provide a fast introduction to symplectic geometry for graduate students with some knowledge of differential geometry, de Rham theory and classical Lie groups. This text addresses symplectomorphisms, local forms, contact manifolds, compatible almost complex structures, Kaehler manifolds, hamiltonian mechanics, moment maps, symplectic reduction and symplectic toric manifolds. It contains guided problems, called homework, designed to complement the exposition or extend the reader's understanding. There are by now excellent references on symplectic geometry, a subset of which is in the bibliography of this book. However, the most efficient introduction to a subject is often a short elementary treatment, and these notes attempt to serve that purpose. This text provides a taste of areas of current research and will prepare the reader to explore recent papers and extensive books on symplectic geometry where the pace is much faster. For this reprint numerous corrections and clarifications have been made, and the layout has been improved.
Author | : |
Publisher | : American Mathematical Soc. |
Total Pages | : 698 |
Release | : 2009 |
Genre | : Mathematics |
ISBN | : 0821838482 |
Research in string theory has generated a rich interaction with algebraic geometry, with exciting work that includes the Strominger-Yau-Zaslow conjecture. This monograph builds on lectures at the 2002 Clay School on Geometry and String Theory that sought to bridge the gap between the languages of string theory and algebraic geometry.
Author | : Denis Auroux |
Publisher | : Birkhäuser |
Total Pages | : 368 |
Release | : 2017-07-27 |
Genre | : Mathematics |
ISBN | : 3319599399 |
This volume is a tribute to Maxim Kontsevich, one of the most original and influential mathematicians of our time. Maxim’s vision has inspired major developments in many areas of mathematics, ranging all the way from probability theory to motives over finite fields, and has brought forth a paradigm shift at the interface of modern geometry and mathematical physics. Many of his papers have opened completely new directions of research and led to the solutions of many classical problems. This book collects papers by leading experts currently engaged in research on topics close to Maxim’s heart. Contributors: S. Donaldson A. Goncharov D. Kaledin M. Kapranov A. Kapustin L. Katzarkov A. Noll P. Pandit S. Pimenov J. Ren P. Seidel C. Simpson Y. Soibelman R. Thorngren
Author | : David A. Cox |
Publisher | : American Mathematical Soc. |
Total Pages | : 498 |
Release | : 1999 |
Genre | : Mathematics |
ISBN | : 082182127X |
Mirror symmetry began when theoretical physicists made some astonishing predictions about rational curves on quintic hypersurfaces in four-dimensional projective space. Understanding the mathematics behind these predictions has been a substantial challenge. This book is the first completely comprehensive monograph on mirror symmetry, covering the original observations by the physicists through the most recent progress made to date. Subjects discussed include toric varieties, Hodge theory, Kahler geometry, moduli of stable maps, Calabi-Yau manifolds, quantum cohomology, Gromov-Witten invariants, and the mirror theorem. This title features: numerous examples worked out in detail; an appendix on mathematical physics; an exposition of the algebraic theory of Gromov-Witten invariants and quantum cohomology; and, a proof of the mirror theorem for the quintic threefold.