Abstract
We consider recognizable evaluations for a suitable category of oriented two-dimensional cobordisms with corners between finite unions of intervals. We call such cobordisms thin flat surfaces. An evaluation is given by a power series in two variables. Recognizable evaluations correspond to series that are ratios of a two-variable polynomial by the product of two one-variable polynomials, one for each variable. They are also in a bijection with isomorphism classes of commutative Frobenius algebras on two generators with a nondegenerate trace fixed. The latter algebras of dimension n correspond to points on the dual tautological bundle on the Hilbert scheme of n points on the affine plane, with a certain divisor removed from the bundle. A recognizable evaluation gives rise to a functor from the above cobordism category of thin flat surfaces to the category of finite-dimensional vector spaces. These functors may be non-monoidal in interesting cases. To a recognizable evaluation we also assign an analogue of the Deligne category and of its quotient by the ideal of negligible morphisms.
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Acknowledgements
M.K. was partially supported by the NSF grant DMS-1807425 while working on this paper. Y. Q. was partially supported by the NSF grant DMS-1947532. L.R. was partially supported by the NSF grant DMS-1760578.
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Khovanov, M., Qi, Y. & Rozansky, L. Evaluating Thin Flat Surfaces. Commun. Math. Phys. 385, 1835–1870 (2021). https://doi.org/10.1007/s00220-021-04011-3
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DOI: https://doi.org/10.1007/s00220-021-04011-3