Cherednik algebras, W algebras and the equivariant cohomology of the moduli space of instantons on A^2 (1202.2756v2)

Abstract: We construct a representation of the affine W-algebra of gl_r on the equivariant homology space of the moduli space of U_r-instantons on A^2, and identify the corresponding module. As a corollary we give a proof of a version of the AGT conjecture concerning pure N=2 gauge theory for the group SU(r). Another proof has been announced by Maulik and Okounkov. Our approach uses a suitable deformation of the universal enveloping algebra of the Witt algebra W_{1+\infty}, which is shown to act on the above homology spaces (for any r) and which specializes to all W(gl_r). This deformation is in turn constructed from a limit, as n tends to infinity, of the spherical degenerate double affine Hecke algebra of GL_n.

• The paper demonstrates pivotal representations of affine W-algebras on the equivariant cohomology of U(r)-instanton moduli spaces.
• It introduces deformation techniques of universal enveloping algebras that yield a novel proof of the AGT conjecture in N=2 SU(r) gauge theory.
• The work provides deep structural insights linking algebraic geometry with two- and four-dimensional field theories.

Overview of "Cherednik Algebras, W-Algebras and the Equivariant Cohomology of the Moduli Space of Instantons on A"

The paper by O. Schiffmann and E. Vasserot explores pivotal connections between Cherednik algebras, W-algebras, and the equivariant cohomology of moduli spaces of instantons. Specifically, it investigates the representation of affine W-algebras on equivariant homology spaces, as tied to certain moduli spaces linked to U(r)-instantons. One key outcome is a novel proof of the AGT conjecture, particularly relevant for N = 2 gauge theory affiliated with the group SU(r), aligning with foundational symmetries postulated in physics.

Representation and Moduli Spaces

The authors construct representations of affine W-algebras of gl on the equivariant cohomology of moduli spaces of U(r)-instantons, pinpointing the corresponding modules. They utilize deformations of universal enveloping algebras of W, acting on these homology spaces, which specialize to representations for W(gl) for all r. The affine Hecke algebra emerges from a limit process, pushing the algebraic boundaries as dimensions tend toward infinity.

AGT Conjecture and Theoretical Foundations

Highlighting the connection with the AGT conjecture, the authors establish a framework in which conformal blocks in two-dimensional field theories correspond to instanton configurations in four-dimensional gauge theories. This association underpins a critical equivalency between mathematical formulations in algebraic geometry and the physical conceptions within conformal field theory.

Numerical Results and Structural Insights

The paper provides structured insights into the underlying algebraic structures, substantiated by robust numerical formulations and evidence within the mathematical framework. Deformations and degenerations of double affine Hecke algebras, together with spherical degenerates, form a recurring theme as one navigates the complexities of representations across dimensional scales.

Implications and Future Developments

An advancement in the mathematical articulation of these algebraic structures presents significant implications for both theoretical development and practical computations involving moduli spaces and instantons. The rigorous approach portends further exploration in quantum groups, specifically regarding the interplay between geometry and algebra, with potential extensions into higher-dimensional analogs and varied gauge symmetries.

Concluding Remarks

This work enriches the mathematical toolset available for addressing sophisticated problems in modern algebraic geometry and theoretical physics. The intersection of such rich algebraic structures with intricate physical theories offers a potential pathway for broadening the understanding of gauge theories and moduli in quantum field contexts. Looking forward, the methodologies and results outlined by Schiffmann and Vasserot lay the groundwork for future inquiries into the symmetries and correspondences that define this high-dimensional landscape.