I define and study TQFTs which describe the low-energy limit of
gapped phases of gauge theories where the microscopic gauge group is
partially Higgsed and partially confined. These TQFTs generalized the
Dijkgraaf-Witten TQFT and can be described on a lattice using discrete
1-form and 2-form gauge fields. The gauge group is replaced by a gauge
2-group (a 2-category with a single object and invertible 1-morphisms and
2-morphisms). It is proposed that 2-group TQFTs are associated with new
types of symmetry-protected gapped phases of matter.
Sen gave a description of the compactification of F-theory on a K3 surface in the orbifold limit where the surface appears as an orbifold of a four-torus. The corresponding picture in M-theory is a self-dual string from an intersecting M2 and M5-brane, and such a configuration corresponds to the lift of Seiberg-Witten theory to M-theory. This picture also provides an embedding of Seiberg-Witten theory into F-theory, by identifying the gauge coupling of gauge theory with the axion-dilaton modulus of string theory.
In my talk I will generalize Sen's procedure by constructing all 2-parameter families of lattice-polarized K3 surfaces that can be obtained from extremal rational elliptic surfaces through a quadratic twist. I will show that for all of these families the Picard-Fuchs system governing the K3-periods are obtained by an integral transform of a differential equation of hypergeometric or Heun type, and that in fact the K3-periods have an interpretation as modular forms. If time permits I will also explain how further generalization of this procedure naturally leads to K3 surfaces admitting double covers onto P2 branched along a plane sextic curve.
(This is joint work with Chuck Doran, University of Alberta)
Topological censorship is a basic principle of spacetime physics. It is a set of results that establishes the topological simplicity at the fundamental group level of the domain of outer communications (the region outside all black holes and white holes) under a variety of physically natural circumstances. An important precursor to the principle of topological censorship, which serves to motivate it, is the Gannon-Lee singularity theorem. All of these results are spacetime results, i.e., they involve conditions that are essentially global in time. From the evolutionary point of view, there is the difficult question of determining whether a given initial data set will give rise to a spacetime satisfying these conditions. In order to separate out the principle of topological censorship from these difficult questions of global evolution, it would be useful to have a pure initial data version of topological censorship. In this talk we give a brief review of topological censorship, and we formulate and present such an initial data version for 3-dimensional initial data sets. The approach taken here relies on recent developments in the existence theory for marginally outer trapped surfaces, and leads to a nontime-symmetric version of the purely Riemannian results of Meeks-Simon-Yau. Geometrization plays an essential role in the proofs. Results in higher dimensions will also be discussed. This talk is based on joint work with Michael Eichmair and Dan Pollack.
Singular elliptic fibrations play a critical role in describing certain physical theories; this includes Seiberg-Witten theory, for example, but also geometric compactifications of M-theory, F-theory, and string theory on elliptically fibered Calabi-Yau manifolds. In the latter cases the ADE structures associated to holomorphic curves is inherited by physical objects wrapped on those curves, giving multiplets in ADE representations. These are necessary ingredients for realistic theories of particle physics. I will comment on limitations on matter representations in the string landscape and potential importance for model building.
Mathematically, studying matter from geometry is usually done via standard blow-up methods in algebraic geometry. However, simple physical arguments show that there must exist another description when the singularities are deformed, rather than resolved. I will present recent work with A. Grassi and J. Shaneson on these ideas. As an appetizer, ADE root systems arise from two-cycles which can be described as vectors in ZN, where N is determined by Kodaira's classification, but is never the rank of the gauge group. Other aspects of representation theory arise naturally. As a concluding example, I will study monodromy induced by a codimension two singularity which reduces D4 to G2 in the deformation picture.
The Hirzebruch-Riemann-Roch theorem, and its cousin the Atiyah-Singer index theorem, relate differential geometry and elliptic operators on a manifold to topological invariants of that manifold -- the precise statements involve characteristic classes. I will review the theory of characteristic classes and Hirzebruch's generalization of the classic Riemann-Roch theorem. I will then explain some results of Mukai which suggest introducing some characteristic classes which had not explicitly considered by Hirzebruch. The latest version of Mukai's idea (proposed by Kontsevich among others) involves the gamma function.
These considerations have a surprising connection to the theory of two-dimensional sigma models with arbitrary Riemannian manifolds as their target spaces, and the behavior of target space metrics under the RG-flow of the physical theory. The perturbative corrections to the classical results on this RG-flow involve the gamma characteristic classes.
I will make this talk accessible to as broad an audience as possible.
It is based on joint work with Jim Halverson, Hans Jockers, and Josh Lapan.
This talk is a more in-depth look at knot contact homology (following the colloquium from the previous day, though we'll start from scratch). We'll examine how this knot invariant can be used to produce a three-variable knot polynomial, the augmentation polynomial, which specializes to a "stable" version of the A-polynomial (in work of Chris Cornwell) and is conjecturally related to colored HOMFLY polynomials. We'll then discuss a conjectural interpretation of this polynomial that uses string theory and mirror symmetry (in joint work with Mina Aganagic, Tobias Ekholm, and Cumrun Vafa).
Many quantum systems can be described by the tensor product of small
(e.g. two-dimensional) Hilbert spaces associated with atoms positioned in
Rn, usually called "sites". I will define a class of quantum states that can
be reconstructed from local data corresponding to balls of radius r. Such
states form a topological space, denoted by Bn for Bose systems and Fn
for Fermi systems. (In the latter case, the Hilbert spaces are Z2-graded.)
Only partial information is known about Bn and Fn, in particular, that they
form Ω-spectra. An analogous problem for so-called "free-fermion systems"
is completely solved, and the corresponding spaces
are given by the
KO spectrum. Interestingly, if we impose some symmetry described by an
action of a compact group G on each site, then finding the fixed points in Bn
and Fn is equivalent to finding the homotopy fixed points. That is not true
in the free case.