The Witten equation, mirror symmetry and quantum singularity theory

By Huijun Fan, Tyler Jarvis, Y. Ruan. 2007

377 citations recorded by the source corpus.

For any non-degenerate, quasi-homogeneous hypersurface singularity, we describe a family of moduli spaces, a virtual cycle, and a corresponding cohomological field theory associated to the singularity. This theory is analogous to Gromov-Witten theory and generalizes the theory of r-spin curves, which corresponds to the simple singularity A_{r-1}. We also resolve two outstanding conjectures of Witten. The first conjecture is that ADE-singularities are self-dual; and the second conjecture is that the total potential functions of ADE-singularities satisfy corresponding ADE-integrable hierarchies. Other cases of integrable hierarchies are also discussed.

Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/c2f09a76b45181589419b929c4d2ce0f0b3aaecd

How researchers see this paper

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

In Geometry and Topology

1. Kazushi Ueda
2. M. Futaki
3. L. Katzarkov
4. Matthew R. Ballard
5. David Favero
6. Y. Ruan
7. A. Chiodo
8. P. Seidel

In Mathematical Physics

1. Kazushi Ueda
2. M. Futaki
3. L. Katzarkov
4. Matthew R. Ballard
5. David Favero
6. Y. Ruan
7. A. Chiodo
8. P. Seidel

In Statistical and Nonlinear Physics

1. A. Chiodo
2. Kazushi Ueda
3. Matthew R. Ballard
4. David Favero
5. M. Futaki
6. L. Katzarkov
7. Y. Ruan
8. P. Seidel

Citations in the corpus

• A new look at the path integral of quantum mechanics (2010, neutral): “For mathematical studies of this equation, with an elucidation of some important points, see [21,22].”
• Relations on Mbar_{g,n} via 3-spin structures (2013, neutral): “Analytic constructions have been given by Mochizuki [12] and later by Fan, Jarvis, and Ruan [9].”
• Landau–Ginzburg/Calabi–Yau correspondence for quintic three-folds via symplectic transformations (2008, methodology): “mprehensive moduli theory of the above Witten equation has been established by Fan, Jarvis, and Ruan [FJR1, FJR2, FJR3]. Besides extending Witten’s 90’s conjecture (see [FSZ] for An-singularities and [FJR1] for all simple singularities), one outcome of Fan–Jarvis–Ruan–Witten theory is that it plays ”
• Homological mirror symmetry for Brieskorn–Pham singularities (2009, background): “The corresponding theory on the Landau-Ginzburg side is recently developed by Fan, Jarvis and Ruan [7, 6] who call it Fan-Jarvis-Ruan-Witten theory.”
• A category of kernels for equivariant factorizations and its implications for Hodge theory (2011, background): “orizations in this case. Despite the overlap in these foundational results, their inspiration and focus are ultimately distinct fromthe work here. They provide apurely algebraic version ofFJRW-theory [FJR07] KERNELS FOR EQUIVARIANT FACTORIZATIONS AND HODGE THEORY 5 by way of matrix factorizations. T”