Lectures on K-theoretic computations in enumerative geometry (1512.07363v2)

Abstract: 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 descendents for sufficiently large framing (and polarization), which is a property we call large framing vanishing. The shift operators for minuscule shift are shown to be given by qKZ operators, which is a K-theoretic analog of the result of arXiv:1211.1287.

• The paper introduces equivariant K-theory to compute Euler characteristics on moduli spaces, establishing a new framework in enumerative geometry.
• It details the use of stable envelopes and symmetrized virtual structure sheaves to rigorously derive K-theoretic Donaldson-Thomas invariants.
• The work integrates quasimaps, Nakajima varieties, and algebraic symplectic reduction, effectively bridging theoretical physics with advanced mathematical methods.

Overview of Advanced Concepts in Enumerative Geometry and Quantum Algebra

This paper explores the expansive and intricate domain of enumerative geometry and quantum algebra, focusing particularly on K-theoretic computations, stable envelopes, and their connections to moduli spaces. The work comprehensively explores topics such as Nakajima varieties, Hilbert schemes, and quasimaps, providing deep insights into their theoretical underpinnings and implications for both mathematics and physics.

Foundation of K-Theory in Enumerative Geometry

The lectures presented here elaborate on K-theoretic enumerative geometry, starting with the foundational concepts related to coherent sheaves and the application of K-theory, particularly equivariant K-theory. The discussion emphasizes the relationship between enumerative geometry and moduli spaces, exploring how the Euler characteristics of naturally defined coherent sheaves on moduli spaces can be computed using the principles of equivariant K-theory. This approach is beneficial in transcending traditional numerical counting to encapsulate deeper properties of geometric objects.

In this context, the paper provides significant detail on the symmetrized virtual structure sheaf, denoted _vir, which is utilized to achieve more rigid results due to its self-dual features—a pivotal aspect when dealing with problems in mathematical physics. This symmetrized sheaf is crucial for computing K-theoretic Donaldson-Thomas invariants via moduli spaces, offering a version of the virtual structure sheaf that facilitates the rigorous analysis found in the sections on quasimaps and stability conditions.

Integration with Algebraic Symplectic Reduction

The paper extends the discussion into algebraic symplectic reduction, surveying Nakajima's quiver varieties in depth. This section highlights how modern enumerative geometry has evolved due to influences from algebraic geometry and high energy physics, delineating the transposition of abstract computational theories into symplectic manifolds. By working through various examples, such as Hilbert schemes and quasimaps to these varieties, the authors convey the complexity and beauty of symplectic reductions, explaining how they relate to different areas like representation theory and gauge theories in physics.

Stable Envelopes and Quantum Shift Operators

A central contribution of the paper is its detailed analysis of stable envelopes and their role in building a quantum group structure. The approach leverages stable envelopes to impart finer control over the computation of equivariant characteristics, employing them to define K-theoretic stable maps and quantum dynamical Weyl groups, among other constructs. Stable envelopes are expertly structured to handle varying slopes within the support and degree arguments, thereby enabling fine algebraic control over quantum group actions.

Quantum Algebra and Applications

Building upon classical invariants like Yangians, the paper explicates the relationship between these objects and quantum loop algebras, particularly focusing on their K-theoretic interpretations. Detailed accounts of quantum groups include the presentation of R-matrices and their connection with stable envelopes through triangle and braid relations, showcasing how these constructions are pivotal for understanding deeper algebraic relationships and symmetries.

Conclusion and Speculation on Future Developments

In bringing these foundational topics together, the authors promote a coherent exploration of K-theoretic tools for understanding complex geometric phenomena. They suggest pathways for future research, notably the exploration of these mathematical tools in higher-dimensional spaces and in the context of artificial intelligence. As computational capacity and mathematical insight grow, the paper hints at significant advances in both pure and applied mathematics, extending into realms that bridge theoretical physics and computational models.

This extensive treatise offers a compelling synthesis of ideas in enumerative geometry, stable envelopes, and quantum algebra, cementing these concepts as vital pillars for current and future research in these fields.