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A weighted topological quantum field theory for Quot schemes on curves

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Abstract

We study Quot schemes of vector bundles on algebraic curves. Marian and Oprea gave a description of a topological quantum field theory (TQFT) studied by Witten in terms of intersection numbers on Quot schemes of trivial bundles. Since these Quot schemes can have the wrong dimension, virtual classes are required. But Quot schemes of general vector bundles always have the right dimension. Using the degree of the general vector bundle as an additional parameter, we construct a weighted TQFT containing both Witten’s TQFT and the small quantum cohomology TQFT of the Grassmannian. This weighted TQFT is completely geometric (no virtual classes are needed), can be explicitly computed, and recovers known formulas enumerating the points of finite Quot schemes.

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Notes

  1. We use the word “stable” loosely to include balanced vector bundles in the case of \(\mathbb {P}^1\) and semistable vector bundles in genus 1. See Sect. 6.1.

  2. In (1), \(Q_{e,\mathbb {P}^1}\) is empty for all \(e < 0\), even though the expected dimension could be non-negative. There are similar cases on \(\mathbb {P}^1\) when V is balanced, so (3) is slightly imprecise.

  3. Actually, in case (4), we only show that \(W \cap U_{e,V}\) is dense in the intersection of W with the top-dimensional component of \(Q_{e,V}\).

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Acknowledgements

I would like to thank Aaron Bertram for providing the vision that led to the weighted TQFT as well as guidance during countless discussions. I am grateful to Nicolas Perrin and Daewoong Cheong for suggesting useful references and to the referee for catching many mistakes and helping to make the last part of the proof of Main Theorem (a) comprehensible. Most of the work was completed as part of my PhD thesis at the University of Utah. The writing was completed at the Korea Institute for Advanced Study.

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  1. School of Mathematics, Korea Institute for Advanced Study, 85 Hoegiro Dongdaemun-gu, Seoul, 02455, Republic of Korea

    Thomas Goller

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The author was partially supported by the National Science Foundation Research Training Grant DMS-1246989.

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Goller, T. A weighted topological quantum field theory for Quot schemes on curves. Math. Z. 293, 1085–1120 (2019). https://doi.org/10.1007/s00209-018-2221-z

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