Abstract
We study quantum geometry of Nakajima quiver varieties of two different types—framed A-type quivers and ADHM quivers. While these spaces look completely different we find a surprising connection between equivariant K-theories thereof with a nontrivial match between their equivariant parameters. In particular, we demonstrate that quantum equivariant K-theory of \(A_n\) quiver varieties in a certain \(n\rightarrow \infty \) limit reproduces equivariant K-theory of the Hilbert scheme of points on \(\mathbb {C}^2\). We analyze the correspondence from the point of view of enumerative geometry, representation theory and integrable systems. We also propose a conjecture which relates spectra of quantum multiplication operators in K-theory of the ADHM moduli spaces with the solution of the elliptic Ruijsenaars–Schneider model.
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Notes
To be more precise, \(\hbar \) is the class in the representation ring of the weight one representation.
\(\zeta _1\) is also redefined to absorb factor \((-q^{1/2}\hbar ^{-1/2})^{d_1}\), which arises from E function above.
I thank A. Okounkov and S. Katz for interesting discussions on these matters.
To actually get Roger polynomials of one variable x one needs to put \(\zeta _1=\xi \) and \(\zeta _1=\xi ^{-1}\) and multiply the resulting expression by \(\xi ^{\lambda _2}\).
This moduli space is a hyperKähler manifold and the quantization is given in complex structure J.
It is spherical DAHA which often appears in physics applications, albeit with some exceptions [KSNS].
Schiffmann and Vasserot used this asymptotic partition to show that there is a triangular decomposition in Heisenberg algebras inside \(\mathfrak {E}\). (see Proposition 4.8. in loc. cit.).
To get a stationary eigenvalue problem one needs to turn off one of the equivariant parameters on either complex line in \(\mathbb {C}^2\).
Non affine.
We abusing the notation by denoting by \(T_r\) both the tRS class and the operator. Hopefully this will not confuse the reader.
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Acknowledgements
I would like to thank I. Cherednik, E. Gorsky, S. Gukov, S. Katz, A. Negut, A. Oblomkov, A. Okounkov, P. Pushkar, A. Smirnov, Y. Soibelman, A. Zeitlin for valuable discussions and suggestions. I acknowledge support of IHÉS and funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (QUASIFT grant agreement 677368). I would also like to thank Simons Center for Geometry and Physics in Stony Brook, Mathematical Sciences Research Institute in Berkeley, Yau Mathematical Center at Tsinghua Univeristy, Beijing [K], where part of this work was done. This research was supported in part by the National Science Foundation under Grant No. NSF PHY-1748958 and in part by AMS-Simons grant.
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Appendix A. tRS Difference Equation for \(T^*\mathbb {P}^1\)
Appendix A. tRS Difference Equation for \(T^*\mathbb {P}^1\)
Let us illustrate (2.16) for \(X_2=T^*\mathbb {P}^1\). The vertex function is given by
For \(X_2\) there are two tRS operators
By acting with these operators on the above function we get
where the first tRS class is given byFootnote 10
and is a linear combination of the tautological bundle \(\mathscr {V}\) and \(\Lambda ^2\mathscr {W}\otimes \mathscr {V}^*\) over \(X_2\) with coefficients dependent on quantum parameter \(z=\frac{\zeta _1}{\zeta _2}\) and the equivariant parameters.
In order to prove that \(\mathrm {V}^{\left( T_1(s)\right) }=(a_1+a_2)\mathrm {V}^{(1)}\) we shall use the following integral (see also [NPZ], Appendix D)
where contour C is chosen in such a way that shift \(s\rightarrow q^{-1}s\) does not pick up any poles. This can be straightforwardly generalized to the n-particle tRS model. Thus integral \(I_2\) in the shifted variable is equal to itself which leads us to
from where the statement follows since the expression in the square brackets must vanish.
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Koroteev, P. A-type Quiver Varieties and ADHM Moduli Spaces. Commun. Math. Phys. 381, 175–207 (2021). https://doi.org/10.1007/s00220-020-03915-w
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DOI: https://doi.org/10.1007/s00220-020-03915-w