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

TL;DR

This paper constructs and studies stacks of perverse coherent extensions on Calabi-Yau threefolds, linking them to quiver representations, Hall algebras, and conjectural vertex algebra modules, with applications to DT/PT invariants and algebraic geometry.

Contribution

It introduces a new stack of perverse coherent extensions, establishes their equivalence to quiver representations, and connects their homology to cohomological Hall algebras and vertex algebras, extending geometric representation theory.

Findings

Homology of the stacks forms representations of cohomological Hall algebras.

Special cases recover DT/PT series and Nekrasov's spiked instantons.

Conjectures relate these structures to vacuum modules of vertex algebras.

Abstract

For $Y\to X$ a toric Calabi-Yau threefold resolution and $M\in \DD^b\Coh(Y)^T$ satisfying some hypotheses, we define a stack $\mf M(Y,M)$ parameterizing \emph{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 $H_\bullet(\mf M^{\f,\zeta}(Y,M),\varphi_{W^\f})$ of the space of $\zeta$-stable, $\f$-framed perverse coherent extensions of $M$, with coefficients in the sheaf $\varphi_{W^\f}$ 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 $\bb V_Y^\zeta=H_\bullet(\mf M^{\f,\zeta}(Y,M),\varphi_{W^\f})$ is the DT/PT series of $Y$ for $\zeta=\zeta_{\DT/\PT}$ by Szendroi and \emph{loc. cit.}, and we conjecture that $\V_Y^{\zeta_\NCDT}$ is the vacuum module for the quiver Yangian of Li-Yamazaki. For $M=\mc O_S[1]$ with $S\subset Y$ 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 $H_\bullet(\mf M^{\f,\zeta}(Y,M),\varphi_{W^{\f}})$ is the vacuum module of the vertex algebra $\V(Y,S)$ defined by the \mbox{authors} in a companion paper, generalizing the AGT conjecture to this setting. For $Y\to X=\{xy-z^mw^n\}$, this gives a geometric approach to the relationship between $W$-algebras and Yangians for affine $\gl_{m|n}$.