The length classification of threefold flops via noncommutative algebras

Abstract

Smooth threefold flops with irreducible centres are classified by the length invariant, which takes values 1, 2, 3, 4, 5 or 6. This classification by Katz and Morrison identifies 6 possible partial resolutions of Kleinian singularities that can occur as generic hyperplane sections, and the simultaneous resolutions associated to such a partial resolution produce the universal flop of length l.In this paper we translate these ideas into noncommutative algebra. We introduce the universal flopping algebra of length l from which the universal flop of length l can be recovered by a moduli construction, and we present each of these algebras as the path algebra of a quiver with relations. This explicit realisation can then be used to construct examples of NCCRs associated threefold flops of any length as quiver with relations defined by superpotentials, to recover the matrix factorisation description of the universal flop conjectured by Curto and Morrison, and to realise examples of contraction algebras.

High electric charges in M-theory from quiver varieties

“…Let us now summarize our findings. The construction by Karmazyn [38] of the socalled universal flopping algebra of length provides us a CY n-fold, 7 from which we can take any three-dimensional slice (or appropriate covering of such a slice). Any such slice or cover will contain a U(1) vector multiplet, and matter with charge .…”

“…This is the case that will ultimately produce threefolds with matter of charge two, starting from the D 4 surface. Here, we expose the technology recently developed in [38], where one constructs these families in terms of quivers. In section 4, we actually proceed to the construction of threefolds with charge-two matter.…”

“…In section 6, we open up the investigation for higher charge. We list the universal quivers of [38], and explain how to construct general threefolds admitting high-charge matter loci with the aid of simple computer algebra. The outputs of this calculation are written in the Mathematica notebook included with the arXiv submission, 10 where we present the most general n-folds in elliptic form admitting charge three, four, five, and six.…”

“…While this approach is completely correct (and is the primary approach of [35]), it appears to be impractical for Lie algebras beyond D 4 . Instead of directly generalizing (2.3) at the level of hypersurface equations, we will use the approach put forth in [38]. The idea is to reconstruct the geometry from a quiver with relations.…”

“…95-96], further developed in [35] and [37]. Concretely, we will use some technology that has recently been developed in [38], whereby one describes not the geometry directly, but a quiver that encompasses all relevant information. Such a quiver, known as the universal flopping algebra, has the property that its space of representations is literally a model for the geometry of interest (up to taking completions of local rings), with deformations and resolutions being regarded as Fayet-Illiopoulos terms.…”