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Charlie Beil

Publications and source records attributed to Charlie Beil.

24 records · Page 2Linked to original sources
arXiv2015

Morita equivalences and Azumaya loci from Higgsing dimer algebras

Let $ψ: A \to A'$ be a cyclic contraction of dimer algebras, with $A$ non-cancellative and $A'$ cancellative. $A'$ is then prime, noetherian, and a finitely generated module over its center. In contrast, $A$ is often not prime, nonnoetherian, and an infinitely generated module over its center. We present certain Morita equivalences that relate the representation theory of $A$ with that of $A'$. We then characterize the Azumaya locus of $A$ in terms of the Azumaya locus of $A'$, and give an explicit classification of the simple $A$-modules parameterized by the Azumaya locus. Furthermore, we show that if the smooth and Azumaya loci of $A'$ coincide, then the smooth and Azumaya loci of $A$ coincide. This provides the first known class of algebras that are nonnoetherian and infinitely generated modules over their centers, with the property that their smooth and Azumaya loci coincide.

math.RA↗
arXiv2014

The Bell states in noncommutative algebraic geometry

We introduce new mathematical aspects of the Bell states using matrix factorizations, nonnoetherian singularities, and noncommutative blowups. A matrix factorization of a polynomial $p$ consists of two matrices $ϕ_1,ϕ_2$ such that $ϕ_1ϕ_2 = ϕ_2ϕ_1 = p \operatorname{id}$. Using this notion, we show how the Bell states emerge from the separable product of two mixtures, by defining pure states over complex matrices rather than just the complex numbers. We then show in an idealized algebraic setting that pure states are supported on nonnoetherian singularities. Moreover, we find that the collapse of a Bell state is intimately related to the representation theory of the noncommutative blowup along its singular support. This presents an exchange in geometry: the nonlocal commutative spacetime of the entangled state emerges from an underlying local noncommutative spacetime.

quant-ph↗
arXiv2012

On the noncommutative geometry of square superpotential algebras

A superpotential algebra is square if its quiver admits an embedding into a two-torus such that the image of its underlying graph is a square grid, possibly with diagonal edges in the unit squares; examples are provided by dimer models in physics. Such an embedding reveals much of the algebras representation theory through a device we introduce called an impression. Let A be a square superpotential algebra, Z its center, and \mathfrak{m} the maximal ideal at the origin of Spec(Z). Using an impression, we give a classification of all simple A-modules up to isomorphism, and give algebraic and homological characterizations of the simple A-modules of maximal k-dimension; show that Z is a 3-dimensional normal toric domain and Z_{\mathfrak{m}} is Gorenstein, by determining transcendence bases and Z-regular sequences; and show that A_{\mathfrak{m}} is a noncommutative crepant resolution of Z_{\mathfrak{m}}, and thus a local Calabi-Yau algebra. A particular class of square superpotential algebras, the Y^{p,q} algebras, is considered in detail. We show that the Azumaya and smooth loci of the centers coincide, and propose that each ramified maximal ideal sitting over the singular locus is the exceptional locus of a blowup shrunk to zero size.

math.AG↗
arXiv2011

The Geometry of Noncommutative Singularity Resolutions

We introduce a geometric realization of noncommutative singularity resolutions. To do this, we first present a new conjectural method of obtaining conventional resolutions using coordinate rings of matrix-valued functions. We verify this conjecture for all cyclic quotient surface singularities, the Kleinian D_n and E_6 surface singularities, the conifold singularity, and a non-isolated singularity, using appropriate quiver algebras. This conjecture provides a possible new generalization of the classical McKay correspondence. Then, using symplectic reduction within these rings, we obtain new, non-conventional resolutions that are hidden if only commutative functions are considered. Geometrically, these non-conventional resolutions result from shrinking exceptional loci to ramified (non-Azumaya) point-like spheres.

math.AG↗
arXiv2008

Geometric aspects of dibaryon operators

The AdS/CFT correspondence for N=1 Super conformal field theories suggests that dibaryon operators are dual to D-brane states that are point like in AdS and that wrap various cycles in a Sasaki-Einstein manifold. It also suggests that the volume of the D-brane gives the R-charge of the corresponding operator. We elucidate various aspects of this correspondence, paying particular care to study the case of branes at the tip of three different Calabi Yau cones. We show that the arrows in the quiver diagram describing the conformal field theory can be thought of as global sections of a non-trivial holomorphic vector bundle over the Calabi-Yau geometry. We suggest that the zero locus of these sections gives the geometric map that lets us tie a particular dibaryon to a holomorphic cycle, by intersecting the corresponding cycle with the Sasaki-Einstein locus at fixed distance from the origin. We show that this can be compared with the corresponding volumes of the Sasaki-Einstein space and that one gets exact agreement between the volumes of the cycles identified with this procedure and the R-charges of the operators.

hep-th↗