Semi)simple exercises in quantum cohomology

By Arend Bayer, Yuri I. Manin. 2001

68 citations recorded by the source corpus.

The paper is dedicated to the study of algebraic manifolds whose quantum cohomology or a part of it is a semisimple Frobenius manifold. Theorem 1.8.1 says, roughly speaking, that the sum of $(p,p)$--cohomology spaces is a maximal Frobenius submanifold that has chances to be semisimple. Theorem 1.8.3 provides a version of the Reconstruction theorem, assuming semisimplicity but not $H^2$--generation. Theorem 3.6.1 establishes the semisimplicity for all del Pezzo surfaces, providing an evidence for the conjecture that semisimplicity is related to the existence of a full system of exceptional sheaves of the appropriate length. Finally, in \S 2 we calculate special coordinates for three families of Fano threefolds with minimal cohomology.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/f3c2972d42109ec635cb2ca81bae11f21b3221c7

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

1. V. Golyshev
2. H. Iritani
3. C. Teleman
4. Y.-P. Lee
5. C. Vafa
6. Andrew Neitzke
7. Sergei Gukov
8. R. Dijkgraaf

In Discrete Mathematics and Combinatorics

1. V. Golyshev
2. Sergei Gukov
3. R. Dijkgraaf
4. H. Iritani
5. Y.-P. Lee
6. C. Vafa
7. Andrew Neitzke
8. Arend Bayer

In Geometry and Topology

1. C. Vafa
2. R. Dijkgraaf
3. Sergei Gukov
4. Michael Entov
5. L. Polterovich
6. V. Golyshev
7. H. Iritani
8. Michael Usher

Citations in the corpus

• Topological M-theory as unification of form theories of gravity (2004, methodology): “We note that the A model can be defined on 3-folds which are not necessarily Calabi-Yau; it has been studied in the mathematical literature on Gromov-Witten theory [68,69,70,71].”
• An update on semisimple quantum cohomology and F-manifolds (2008, background): “1 of Bayer–Manin [5] can be strengthened in the following way: If the even quantum cohomology of a projective algebraic manifold V is generically semisimple, then V has no odd cohomology and is of Hodge–Tate type.”
• Semisimple Quantum Cohomology and Blow-ups (2004, background): “5. Further Questions. The first example where our theorem applies is the case of X = Pn. For n = 2, this yields the semisimplicity of quantum cohomology for all Del Pezzo surfaces as proven earlier in [BM01]. Further, semisimplicity has been established in [TX97] by Tian and Xu, using results of Beau”
• Classification problems and mirror duality (2005, methodology): “again agreed with the ones predicted by the conjecture. To our knowledge, quantum multiplication by the divisor class on V 5 (case (5,2) ) was first computed by Beauville [Bea]. We refer the reader to [BM] which makes use of Beauville’s and Kuznetsov’s results. To summarize, we have checked the conje”
• Symplectic quasi-states and semi-simplicity of quantum homology (2007, background): “Theorem 5.1 below states that this assumption is equivalent to semi-simplicity of QH2n(M) in the sense of the present paper.”
• Convergence of quantum cohomology by quantum Lefschetz (2005, background): “ule of rank 4 and isomorphic to the restriction of the big quantum D-module to some non-linear subvariety of H∗(Fn). On the other hand, MMell is of rank n+2 over C[~,q±]. (In fact, it is generated by [1],[~∂1],[(~∂1)2],[~∂2 −n~∂1],...,[(~∂2 − n~∂1)n−1].) The characteristic variety of MMell is given ”
• Quantum $K$-theory, I: Foundations (2001, background): “we show thatthe construction ofquantum K-ringproduces a (new) class of Frobenius manifolds, in the sense of [27]. This in particular gives a positive answer to the question raised by Bayer and Manin ([3] 1.1.1 Question). Note that in this case the identity element e0 is not flat. The data of K-theore”