About

Paul Stephen Aspinwall is a Professor of Mathematics at Duke University, holding appointments since 2006 and serving as Associate Chair of the Department of Mathematics since 2016. His research focuses on string theory, particularly the geometric and algebraic structures underlying the theory. He investigates how string theories, which naturally exist in multiple dimensions, can be connected to the real world through the process of compactification, emphasizing the importance of algebraic geometry in understanding supersymmetric physics. Aspinwall's work explores the concept of duality in string theory, where different compactifications yield equivalent physical theories. This approach leverages mathematical insights to deepen the understanding of string theory's nonperturbative aspects and its implications for fundamental physics. His contributions include analyzing moduli spaces, mirror symmetry, and the geometry of D-branes, with significant publications in high-energy physics journals. His research has been supported by multiple grants from the National Science Foundation, reflecting his active engagement in advancing the mathematical physics of string theory.

Research topics

• Pure mathematics
• Mathematics
• Theoretical physics
• Physics
• Quantum mechanics
• Mathematical analysis
• Mathematical physics
• Geometry

Selected publications

• String moduli spaces and parabolic factorizations

  Journal of High Energy Physics · 2025-03-04 · 1 citations

  articleOpen access1st authorCorresponding

  A bstract The symmetric spaces that appear as moduli spaces in string theory and supergravity can be decomposed with explicit metrics using parabolic subgroups. The resulting isometry between the original moduli space and this decomposition can be used to find parametrizations of the moduli. One application is to determine the volume parameter in conformal field moduli spaces for K3 surfaces. Other applications involve simple Dynkin diagram manipulations inducing “going up and down” between symmetric spaces by adding parameters and going to limits respectively. For supersymmetries such as N = 6, this involves combinatorics of less familiar “restricted” Dynkin diagrams.

  PublisherOA PDFDOI

• An Unreasonably Quick Introduction to String Theory, Conformal Field Theory and Geometry

  arXiv (Cornell University) · 2024

  1st authorCorresponding
  • Geometry
  • Theoretical physics
  • Physics

  A very quick introduction to the bosonic string, conformal field theory, the superstring and geometry. No background in quantum field theory is assumed and the omissions are vast. Based on four lectures at the 2024 Physical Mathematics of Quantum Field Theory Summer School.

  PublisherOA PDFDOI

• String Moduli Spaces and Parabolic Factorizations

  arXiv (Cornell University) · 2024 · 1 citations

  1st authorCorresponding
  • Pure mathematics
  • Mathematics
  • Mathematical analysis

  The symmetric spaces that appear as moduli spaces in string theory and supergravity can be decomposed with explicit metrics using parabolic subgroups. The resulting isometry between the original moduli space and this decomposition can be used to find parametrizations of the moduli. One application is to determine the volume parameter in conformal field moduli spaces for K3 surfaces. Other applications involve simple Dynkin diagram manipulations inducing "going up and down" between symmetric spaces by adding parameters and going to limits respectively. For supersymmetries such as N=6, this involves combinatorics of less familiar "restricted" Dynkin diagrams.

  PublisherOA PDFDOI

• Mirror Symmetry and Discriminants

  arXiv (Cornell University) · 2017-02-15 · 3 citations

  preprintOpen access1st authorCorresponding

  We 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.

  PublisherOA PDFDOI

• Quivers from matrix factorizations

  2016-08-16 · 36 citations

  article1st authorCorresponding

  We discuss how matrix factorizations offer a practical method of computing the quiver and associated superpotential for a hypersurface singularity. This method also yields explicit geometrical interpretations of D-branes (i.e., quiver representations) on a resolution given in terms of Grassmannians. As an example we analyze some non-toric singularities which are resolved by a single P1 but have “length ” greater than one. These examples have a much richer structure than conifolds. A picture is proposed that relates matrix factorizations in Landau–Ginzburg theories to the way that matrix factorizations are used in this paper to perform noncommutative resolutions. ar X iv

  Publisher

• General Mirror Pairs for Gauged Linear Sigma Models

  arXiv (Cornell University) · 2015-07-01 · 1 citations

  preprintOpen access1st authorCorresponding

  We carefully analyze the conditions for an abelian gauged linear sigma-model to exhibit nontrivial IR behavior described by a nonsingular superconformal field theory determining a superstring vacuum. This is done without reference to a geometric phase, by associating singular behavior to a noncompact space of (semi-)classical vacua. We find that models determined by reflexive combinatorial data are nonsingular for generic values of their parameters. This condition has the pleasant feature that the mirror of a nonsingular gauged linear sigma-model is another such model, but it is clearly too strong and we provide an example of a non-reflexive mirror pair. We discuss a weaker condition inspired by considering extremal transitions, which is also mirror symmetric and which we conjecture to be sufficient. We apply these ideas to extremal transitions and to understanding the way in which both Berglund-Hubsch mirror symmetry and the Vafa-Witten mirror orbifold with discrete torsion can be seen as special cases of the general combinatorial duality of GLSMs. In the former case we encounter an example showing that our weaker condition is still not necessary.

  PublisherOA PDFDOI

• General mirror pairs for gauged linear sigma models

  Journal of High Energy Physics · 2015-11-01 · 13 citations

  articleOpen access1st authorCorresponding

  We carefully analyze the conditions for an abelian gauged linear σ-model to exhibit nontrivial IR behavior described by a nonsingular superconformal field theory determining a superstring vacuum. This is done without reference to a geometric phase, by associating singular behavior to a noncompact space of (semi-)classical vacua. We find that models determined by reflexive combinatorial data are nonsingular for generic values of their parameters. This condition has the pleasant feature that the mirror of a nonsingular gauged linear σ-model is another such model, but it is clearly too strong and we provide an example of a non-reflexive mirror pair. We discuss a weaker condition inspired by considering extremal transitions, which is also mirror symmetric and which we conjecture to be sufficient. We apply these ideas to extremal transitions and to understanding the way in which both Berglund-Hübsch mirror symmetry and the Vafa-Witten mirror orbifold with discrete torsion can be seen as special cases of the general combinatorial duality of gauged linear σ-models. In the former case we encounter an example showing that our weaker condition is still not necessary.

  PublisherOA PDFDOI

• Exoops in two dimensions

  2015-01-01

  article1st authorCorresponding

  An exoop occurs in the gauged linear �-model by varying the Kahler form so that a subspace appears to shrink to a point and then reemerge \outside the original manifold. This occurs for K3 surfaces where a rational curve is \opped from inside to outside the K3 surface. We see that whether a rational curve contracts to an orbifold phase or an exoop depends on whether this curve is a line or conic. We study how the D-brane category of the smooth K3 surface is described by the exoop and, in particular, �nd the location of a massless D-brane in the exoop limit. We relate exoops to noncommutative resolutions.

  Publisher

• Exoflops in two dimensions

  Journal of High Energy Physics · 2015-07-01 · 5 citations

  articleOpen access1st authorCorresponding

  An exoflop occurs in the gauged linear σ-model by varying the Kähler form so that a subspace appears to shrink to a point and then reemerge “outside” the original manifold. This occurs for K3 surfaces where a rational curve is “flopped” from inside to outside the K3 surface. We see that whether a rational curve contracts to an orbifold phase or an exoflop depends on whether this curve is a line or conic. We study how the D-brane category of the smooth K3 surface is described by the exoflop and, in particular, find the location of a massless D-brane in the exoflop limit. We relate exoflops to noncommutative resolutions.

  PublisherDOI

• Exoflops in Two Dimensions

  arXiv (Cornell University) · 2014-12-01

  preprintOpen access1st authorCorresponding

  An exoflop occurs in the gauged linear $σ$-model by varying the Kahler form so that a subspace appears to shrink to a point and then reemerge "outside" the original manifold. This occurs for K3 surfaces where a rational curve is "flopped" from inside to outside the K3 surface. We see that whether a rational curve contracts to an orbifold phase or an exoflop depends on whether this curve is a line or conic. We study how the D-brane category of the smooth K3 surface is described by the exoflop and, in particular, find the location of a massless D-brane in the exoflop limit. We relate exoflops to noncommutative resolutions.

  PublisherOA PDFDOI

Recent grants

• Geometry and Mathematical Physics of D-Branes

  NSF · $348k · 2009–2014

• Algebraic Geometry and Quantum Field Theory of D-Branes

  NSF · $512k · 2006–2011

• Moduli Spaces and String Theory

  NSF · $318k · 2012–2017

Frequent coauthors

• David R. Morrison

  University of California, Santa Barbara

  17 shared

• Brian Greene

  Columbia University

  10 shared

• M. Ronen Plesser

  Duke University

  10 shared

• Mark Gross

  4 shared

• Sheldon Katz

  University of Illinois Urbana-Champaign

  3 shared

• C.A. Lütken

  University of Oslo

  3 shared

• Michael R. Douglas

  3 shared

• Ilarion V. Melnikov

  Carrier (United States)

  2 shared