Schedule for: 18w5190 - Geometry and Physics of F-theory

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

I will introduce a large ensemble of F-theory geometries, generated by crepant base-changes. I will discuss how understanding the construction algorithm allows us to read off universal behavior in this ensemble, and comment on what this behavior means for F-theory.

I'm going to present algorithms that explicitly generate a large number of topologically distinct smooth toric threefold bases used in 4D F-theory compactification. We found inherent patterns in these bases although they are seemingly random. I'll also talk about the characterization of these bases using local geometric data and non-Higgsable gauge group structure. Finally, I'm going to briefly mention our current understanding of non-toric base threefolds.

I will discuss joint work with Gabriele Di Cerbo on boundedness of Calabi-Yau pairs. Given an elliptically fibered Calabi-Yau manifold, the base of the fibration naturally carries the structure of a Calabi-Yau pair, that is, there exists an effective divisor D on the base, with nice singularities, such that K+D=0. Recent works in the minimal model program suggest that rationally connected Calabi-Yau pairs should satisfy some boundedness properties, that is, they should be parametrized by a finite type scheme. I will show that Calabi-Yau pairs which are not birational to a product are indeed log birationally bounded, if the dimension is less than four. In dimension three, we can actually obtain some more general results, by relaxing some of technical assumptions (joint work in progress with Chen, Di Cerbo, Han, Jiang).

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

If two $K3$ surfaces $X$ and $Y$ over $\mathbb{C}$ admit a rational map of finite degree $X\to Y$, Inose proved that their Picard numbers $\rho(X)$ and $\rho(Y)$ are equal. Suppose $X$ admits an elliptic fibration $\pi:X\to \mathbf{P}^{1}$. By a base change $b:\mathbf{P}^{1}\to \mathbf{P}^{1}$, we obtain another elliptic surface $\pi\times b:X':=X\times_{\mathbf{P}^{1}}\mathbf{P}^{1}\to \mathbf{P}^{1}$. If $X'$ is once again a $K3$ surface, we know $\rho(X')=\rho(X)$. However, it is difficult in general to find generators of the N\'eron-Severi goup of $X'$. Starting from various $K3$ surfaces $X$ having a Shioda-Inose structure, we construct $X'\to \mathbf{P}^{1}$ whose Mordell-Weil rank is large, and explore methods of finding generators of the Mordell-Weil group.

We outline several number-theoretical contexts where K3 surfaces and elliptic fibrations arise naturally: Diophantine equations, Euclidean and hyperbolic quadratic forms, elliptic and Shimura modular curves and higher-dimensional analogues, record ranks for elliptic curves and related tasks, and complex reflection groups and their invariants. Several of these contexts call for explicit formulas for surfaces are known to exist only by transcendental means (Torelli theorem for K3 surfaces). One of these formulas also yields a family of elliptically fibered Calabi-Yau threefolds with Mordell-Weil rank 10.

The global gauge group structure of F-theory compactifications is tied to the embedding of the Mordell—Weil lattice into the Neron—Severi group, also known as the Shioda map. While this has been previously explored in the case of purely non-abelian gauge algebras, I will extend it to the more general case by including u(1)s. In particular, I will discuss phenomenological implications, such as the natural realization of the full Standard Model gauge group or possible implications on the F-theory swampland. Furthermore, I will argue how similar structures are also present in genus-one fibrations without sections.

Together with Klaus Hulek we proved in 2011 that there is an effective algorithm which computes the Mordell-Weil group of X for ``most'' elliptic threefolds X with base P2. In the first part of the talk we explain what this statement means if one specializes to elliptic threefolds which are relevant for F-theory. Moreover, we explain several relations between singularity-theory invariants of the discriminant curve of an elliptic fibration and the Mordell-Weil rank of this fibration. In the second part we discuss extensions of these results to elliptic threefolds over arbitrary base surfaces and to certain classes of elliptic fourfolds.

While F-theory models readily admit relatively simple representations, it is difficult to construct models with “exotic” representations that go beyond these simple types. This talk will discuss ways of systematically constructing and understanding Weierstrass models with these exotic representations. For non-abelian groups, I will focus on “higher genus” representations, which involve 7-branes wrapping singular divisors. While models with higher genus representations involve intricate, complicated structures, they can be systematically constructed using techniques related to the normalization of singular varieties. I will also describe some results regarding non-abelian representations and matter spectra that cannot be realized in F-theory compactifications. I will then turn to the issue of matter in U(1) models with large charges. First, I will discuss a new strategy for constructing models with charge-3 matter that has interesting parallels with the techniques for constructing higher genus representations. I will also describe the construction of explicit models with charge-4 matter. The talk will conclude with some conjectures on matter with charges larger than 4.

In this talk, we give an explicit description for the relation between algebraic Kummer surfaces of Jacobians of genus-two curves with principal polarization and those associated to (1, 2)-polarized abelian surfaces from three different angles: the point of view of 1) the binational geometry of quartic surfaces in P^3 using even-eights, 2) elliptic fibrations on K3 surfaces of Picard-rank 17 over P^1 using Nikulin involutions, 3) theta-functions of genus-two using two-isogeny. Finally, we will explain how these (1,2)-polarized Kummer surfaces naturally allow for an identification of the complex gauge coupling in Seiberg-Witten gauge theory with the axion-dilaton modulus in string theory using an old idea of Sen. (This is joint work with Adrian Clingher.)

Certain classes of superconformal field theories in six dimensions (6D SCFTs) are in one-to-one correspondence with certain classes of group homomorphisms. In this talk, we will see that this correspondence allows us to classify homomorphisms that were previously unknown in the mathematics literature and understand aspects of 6DSCFTs that were previously unknown in the physics literature. We speculate on the implications for the future study of 6D SCFTs and these group homomorphisms.

In the past six years, several efforts have been directed towards developing a dictionary in between the physics of six-dimesional theories and the algebraic geometry of elliptically-fibered Calabi-Yau three-folds, building upon methods in string theory. This has lead to novel results in enumerative geometry as well, in particular, the Gopakumar-Vafa invariants of the topological string of such Calabi-Yau varieties translate to an operator counting in the six-dimensional theory. This gives an unorthodox perspective on topological strings, unveiling novel modular properties and a connection to Weyl invariant Jacobi forms. This connection follows from six-dimensional physics and can be ultimately traced back to the BPS strings subsector of these models. In this talk we are going to discuss certain aspects and applications of this correspondence, as well as its interplay with recent results about universal features of the BPS strings.

We explore 6-dimensional compactifications of F-theory exhibiting (2,0) superconformal theories coupled to gravity that include discretely charged superconformal matter. Those theories are constructed as smooth non-simply connected Calabi-Yau quotients with a†singular base whereas additional discrete charged singlets get introduced by the resolution of the base.

T-branes are bound states of ordinary D-branes, characterized by non-commuting expectation values for the worldvolume scalars. They defy usual geometric intuition and feature a variety of unconventional brane-model-building phenomena. In this talk I will discuss some general global properties of the vacuum solutions corresponding to T-branes, and show how supersymmetry puts strong constraints on the manifolds supporting them. Moreover, I will consider stability issues of such bound states at large volume, and give conditions to prevent their decay.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.