Perverse coherent extensions on Calabi-Yau threefolds and representations of cohomological Hall algebras

2023/09/28 by Dylan Butson, Butson, Dylan, Miroslav Rapčák +1 · 2 citations
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph) #Quantum Algebra (math.QA) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2309.16582

openalex publication_date 2023/09/28 · openalex created_date 2023/10/01 · openalex updated_date 2026/07/28

Abstract

For Y→ X a toric Calabi-Yau threefold resolution and M∈ \DD^b\Coh(Y)^T satisfying some hypotheses, we define a stack \mf M(Y,M) parameterizing perverse coherent extensions of M, iterated extensions of M and the compactly supported perverse coherent sheaves of Bridgeland. We define framed variants \mf M^\f(Y,M), prove that they are equivalent to stacks of representations of framed quivers with potential (Q^\f,W^\f), and deduce natural monad presentations for these sheaves. Moreover, following Soibelman we prove that the homology H_\bullet(\mf M^\f,ζ(Y,M),φ_W^\f) of the space of ζ-stable, \f-framed perverse coherent extensions of M, with coefficients in the sheaf φ_W^\f of vanishing cycles for W^\f, is a representation of the Kontsevich-Soibelman cohomological Hall algebra of Y. For M=\mc O_Y[1], \mf M^\f(Y,M) is the stack of perverse coherent systems of Nagao-Nakajima, so \bb V_Y^ζ=H_\bullet(\mf M^\f,ζ(Y,M),φ_W^\f) is the DT/PT series of Y for ζ=ζ_\DT/\PT by Szendroi and loc. cit., and we conjecture that \V_Y^ζ_\NCDT is the vacuum module for the quiver Yangian of Li-Yamazaki. For M=\mc O_S[1] with S⊂ Y a divisor, \mf M^\f(Y,M) provides a definition in algebraic geometry for Nekrasov's spiked instanton variant of the ADHM construction, and analogous variants of the constructions of Kronheimer-Nakajima, Nakajima-Yoshioka, and Finkelberg-Rybnikov. We conjecture that H_\bullet(\mf M^\f,ζ(Y,M),φ_W^\f) is the vacuum module of the vertex algebra \V(Y,S) defined by the authors in a companion paper, generalizing the AGT conjecture to this setting. For Y→ X=\xy-z^mwⁿ\, this gives a geometric approach to the relationship between W-algebras and Yangians for affine \gl_m|n.

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