# Browsing by Author "Aspinwall, Paul S"

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Item Open Access Aspects of the (0,2)-McKay Correspondence(2015) Gaines, Benjamin C.We study first order deformations of the tangent sheaf of resolutions of Calabi-Yau threefolds that are of the form $\CC^3/\ZZ_r$, focusing

on the cases where the orbifold has an isolated singularity. We prove a lower bound on the number

of deformations of the tangent bundle for any crepant resolution of this orbifold. We show that this lower bound is achieved when the resolution used is the

G-Hilbert scheme, and note that this lower bound can be found using a combinatorial count of (0,2)-deformation moduli fields for

N=(2,2) conformal field theories on the orbifold. We also find that in general this minimum is not achieved, and expect the discrepancy

to be explained by worldsheet instanton corrections coming from rational curves in the orbifold resolution. We show that

irreducible toric rational curves will account for some of the discrepancy, but also prove that there must be additional

worldsheet instanton corrections beyond those from smooth isolated rational curves.

Item Open Access Complete Mirror Pairs and Their Naive Stringy Hodge Numbers(2017) Fitzpatrick, Brian DavidThe Batyrev-Borisov construction associates a to dual pair of nef-partitions

$\Delta=\Delta_1+\dotsb+\Delta_c$ and $\nabla=\nabla_1+\dotsb+\nabla_c$ a pair

of Calabi-Yau complete intersections

$(Y_{\Delta_1,\dotsc,\Delta_c},Y_{\nabla_1,\dotsc,\nabla_c})$ in Gorenstein Fano

toric varieties $(X_\Delta,X_\nabla)$. These Calabi-Yau varieties are singular

in general. Batyrev and Nill have developed a generating function $\Est$ for the

stringy Hodge numbers of Batyrev-Borisov mirror pairs. This function depends

solely on the combinatorics of the nef-partitions and, under this framework,

Batyrev-Borisov mirror pairs pass the stringy topological mirror symmetry test

$\hst^{p,q}(Y_{\Delta_1,\dotsc,\Delta_c})=\hst^{d-p,q}(Y_{\nabla_1,\dotsc,\nabla_c})$.

Recently, Aspinwall and Plesser have defined the notion of a complete

non-reflexive mirror pair $(\scrA,\scrB)$ and used this notion to study

Calabi-Yau complete intersections in non-Gorenstein toric varieties. Complete

mirror pairs generalize the notion of a dual pair of almost reflexive Gorenstein

cones $(\sigma,\sigma^\bullet)$ developed by Mavlyutov to propose a

generalization of the Batyrev-Borisov mirror construction. The only known

example of either of these two notions is the complete intersection of a quintic

and a quadric in $\PP_{211111}^5$. We construct $2152$ distinct examples of

complete mirror pairs and $1077$ distinct examples of dual pairs of almost

reflexive Gorenstein cones. Additionally, we propose a generalization of Batyrev

and Nill's stringy $E$-function, called the na\"{i}ve stringy $E$-function

$\gEst$, that is well-defined for complete mirror pairs.

Item Open Access Elusive Worldsheet Instantons in Heterotic String Compactifications(STRING-MATH 2011, 2012-01-01) Aspinwall, Paul S; Plesser, M RonenWe compute the spectrum of massless gauge singlets in some heterotic string compactifications using Landau-Ginzburg, orbifold and non-linear sigma-model methods. This probes the worldsheet instanton corrections to the quadratic terms in the spacetime superpotential. Previous results predict that some of these states remain massless when instanton effects are included. We find vanishing masses in many cases not covered by these predictions. However, we discover that in the case of the Z-manifold the corrections do not vanish. Despite this, in all the examples studied, we find that the massless spectrum in the orbifold limit agrees with the nonlinear sigma-model computation.Item Open Access Mirror Symmetry and DiscriminantsAspinwall, Paul S; Plesser, M Ronen; Wang, KangkangWe analyze the locus, together with multiplicities, of "bad" conformal field theories in the compactified moduli space of N=(2,2) superconformal field theories in the context of the generalization of the Batyrev mirror construction using the gauged linear sigma-model. We find this discriminant of singular theories is described beautifully by the GKZ "A-determinant" but only if we use a noncompact toric Calabi-Yau variety on the A-model side and logarithmic coordinates on the B-model side. The two are related by "local" mirror symmetry. The corresponding statement for the compact case requires changing multiplicities in the GKZ determinant. We then describe a natural structure for monodromies around components of this discriminant in terms of spherical functors. This can be considered a categorification of the GKZ A-determinant. Each component of the discriminant is naturally associated with a category of massless D-branes.Item Open Access Towards A Stability Condition on the Quintic Threefold(2010) Roy, AryaIn this thesis we try to construct a stability condition on the quintic threefold. We have not succeeded in proving the existence of such a stability condition. However we have constructed a stability condition on a quotient category of projective space that approximates the quintic. We conjecture the existence of a stability condition on the quintic threefold generated by spherical objects and explore some consequences.