# Lectures on K-theoretic computations in enumerative geometry

@article{Okounkov2015LecturesOK, title={Lectures on K-theoretic computations in enumerative geometry}, author={Andrei Okounkov}, journal={arXiv: Algebraic Geometry}, year={2015} }

These are notes from my lectures on quantum K-theory of Nakajima quiver varieties and K-theoretic Donaldson-Thomas theory of threefolds given at Columbia and Park City Mathematics Institute. They contain an introduction to the subject and a number of new results. In particular, we prove the main conjecture of arXiv:hep-th/0412021 and the conjecture of arXiv:1404.2323 in the simplest case of reduced smooth curves. We also prove the the absence of quantum corrections to the capped vertex with…

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## 132 Citations

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## References

SHOWING 1-10 OF 94 REFERENCES

Quantum Groups and Quantum Cohomology

- Mathematics
- 2012

In this paper, we study the classical and quantum equivariant cohomology of Nakajima quiver varieties for a general quiver Q. Using a geometric R-matrix formalism, we construct a Hopf algebra Y_Q,…

Etingof’s conjecture for quantized quiver varieties

- Mathematics
- 2013

We compute the number of finite dimensional irreducible modules for the algebras quantizing Nakajima quiver varieties. We get a lower bound for all quivers and vectors of framing and provide an exact…

On the WDVV equation in quantum K-theory.

- Mathematics
- 2000

0. Introduction. Quantum cohomology theory can be described in general words as intersection theory in spaces of holomorphic curves in a given Kahler or almost Kahler manifold X. By quantum K-theory…

Quantum cohomology of the Hilbert scheme of points on A_n-resolutions

- Mathematics
- 2008

We determine the two-point invariants of the equivariant quantum cohomology of the Hilbert scheme of points of surface resolutions associated to type A_n singularities. The operators encoding these…

Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras

- Mathematics
- 1994

To Professor Shoshichi Kobayashi on his 60th birthday 1. Introduction. In this paper we shall introduce a new family of varieties, which we call quiver varieties, and study their geometric…

Quantum difference equation for Nakajima varieties

- MathematicsInventiones mathematicae
- 2022

For an arbitrary Nakajima quiver variety $X$, we construct an analog of the quantum dynamical Weyl group acting in its equivariant K-theory. The correct generalization of the Weyl group here is the…

Wreath Macdonald polynomials and categorical McKay correspondence

- Mathematics
- 2012

Mark Haiman has reduced Macdonald positivity conjecture to a statement about geometry of the Hilbert scheme of points on the plane, and formulated a generalization of the conjectures where the…

The Hirzebruch--Riemann--Roch theorem in true genus-0 quantum K-theory

- Mathematics
- 2011

We completely characterize genus-0 K-theoretic Gromov-Witten invariants of a compact complex algebraic manifold in terms of cohomological Gromov-Witten invariants of this manifold. This is done by…

Random surfaces enumerating algebraic curves

- Mathematics
- 2004

The discovery that a relation exists between the two topics in the title was made by physicists who viewed them as two approaches to Feynman integral over all surfaces in string theory: one via…

OBSTRUCTIONS TO THE EXISTENCE OF S1-ACTIONS. BORDISM OF RAMIFIED COVERINGS

- Mathematics
- 1976

In this article the author proves that the values of the multiplicative genera Ak under discussion, where K = 2, 3,..., are obstructions to the existence of nontrivial S1-actions on a unitary…