2014 Fiscal Year Research-status Report
W-constraints and the Eynard-Orantin topological recursion
| Project/Area Number |
26800003
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| Research Institution | The University of Tokyo |
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Principal Investigator |
MILANOV Todor 東京大学, カブリ数物連携宇宙研究機構, 准教授 (80596841)
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| Project Period (FY) |
2014-04-01 – 2017-03-31
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| Keywords | Gromov-Witten invariants / Eynard―Orantin recursion / period integrals / Frobenius structures / vertex operators |
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| Outline of Annual Research Achievements |
In my previous work I have obtained a local Eynard―Orantin (EO) recursion for the total ancestor potential of an isolated singularity. The term local here refers to the fact that the spectral curve of the EO recursion is a disjoint union of several discs. The 1st goal of my proposal is to find a global Eynard―Orantin recursion, namely to extend the disjoint union of discs to a Riemann surface and to identify the recursion kernel with the Bergman kernel of the Riemann surface. I managed to find the global EO recursion for simple singularities. The spectral curve is easy to describe: it is a complete intersection in the Cartan subalgebra defined by the invariant polynomials of non-maximal degree, while the polynomial of maximal degree provides a branched covering of P1. I managed to express the recursion kernel of the local EO recursion in terms of the Bergmann kernel. The former is a quadratic expression in period integrals known as the phase factor and my formula says that it is an average of the Bergmann kernel over the Weyl group. The 2nd goal of my proposal is to relate the EO recursion with the so-called W-constraints. I managed to prove a very important technical result about the compatibility of the phase factors with the monodromy representation. This was a question asked by Givental more than 10 years ago and the result was known only for simple singularities.
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| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
I solved one of the main problems in my proposal in the case of a simple singularity.
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| Strategy for Future Research Activity |
The ancestor potential and respectively the spectral curve depend on the deformation parameters of the singularity. My result applies to generic parameters, which are given by the complement of the caustic. The next step is to find the limit, i.e., to let a generic parameter approach a non-generic value. The main goal is to find the limit of the EO recursion at the most singular point on the caustic. My expectation is that the limit can be expressed in terms of the W-algebra. There are several difficult results about the structure of the W-algebra due to Feigin and Frenkel. If my approach is successful I might be able to reprove Feigin and Frenkel’s results.
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| Causes of Carryover |
My original plan was to visit US in the winter, but my collaborator Yongbin Ruan took a sabbatical and now he is in China. In addition I have made progress on a problem that I have been trying to solve for a very long time. That is why I preferred to cancel all travels and complete my work. In particular, there was no urgency to buy a laptop and iPad. On the other hand, in FY2015 my travel plans increased more than expected, so I am planning to use the amount left from FY2014.
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| Expenditure Plan for Carryover Budget |
I am planning to use the amount carried over from FY2014 for: Laptop MacAir (300,000), iPad (50,000), books (50,000), travel (426,736).
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