Algebraic Geometry

These lectures will review and develop methods in algebraic geometry (in particular, derived algebraic geometry) to describe topological and holomorphic sectors of quantum field theories. A recurring theme will be the interaction of local and extended operators, and of QFT's in different dimensions. The main examples will come from twists of supersymmetric gauge theories, and will connect to a large body of recent and ongoing work on 3d Coulomb branches, 3d mirror symmetry (and geometric Langlands), logarithmic VOA's and non-semisimple TQFT's, and categorified cluster algebras.

The basic plan for the lectures is:

• Lecture 1 (2d warmup): categories of boundary conditions, interfaces, and Koszul duality
• Lectures 2 and 3 (3d): twists of 3d N=2 and N=4 gauge theories; vertex algebras, chiral categories, and braided tensor categories; d mirror symmetry; quantum groups at roots of unity and derived non-semisimple 3d TQFT's (compared and contrasted with Chern-Simons theory)
• Lecture 4 (4d): line and surface operators in 4d N=2 gauge theory, the coherent Satake category, and relations to Schur indices and 4d N=2 vertex algebras

1. Lecture 1
2. Lecture 2
3. Lecture 3
4. Lecture 4

Throughout my lectures I will explain the geometry of elliptic fibration which can gave rise to understanding the spectra and anomalies in lower-dimensional theories from the Calabi-Yau compactifications of F-theory. I will first explain what elliptic fibration is and explain Kodaira types, which gives rise an ADE classification. Utilizing Weierstrass model of elliptic fibrations, I will discuss Tate’s algorithm and Mordell-Weil group. By considering codimension one and two singularities and studying the geometry of crepant resolutions, we can define G-models that are geometrically-engineered models from F-theory. I will discuss the dictionary between the gauge theory and the elliptic fibrations and how to incorporate this to learn about topological invariants of the compactified Calabi-Yau that is one of the ingredient to understand spectra in the compactified theories. I will explain the more refined connection to understand the Coulomb branch of the 5d N=1 theories and 6d (1,0) theories and their anomalies from this perspective.

1. Lecture 1
2. Lecture 2
3. Lecture 3
4. Lecture 4

Throughout my lectures I will explain the geometry of elliptic fibration which can give rise to understanding the spectra and anomalies in lower-dimensional theories from the Calabi-Yau compactifications of F-theory. I will first explain what elliptic fibration is and explain Kodaira types, which gives rise an ADE classification. Utilizing Weierstrass model of elliptic fibrations, I will discuss Tate’s algorithm and Mordell-Weil group. By considering codimension one and two singularities and studying the geometry of crepant resolutions, we can define G-models that are geometrically-engineered models from F-theory. I will discuss the dictionary between the gauge theory and the elliptic fibrations and how to incorporate this to learn about topological invariants of the compactified Calabi-Yau that is one of the ingredient to understand spectra in the compactified theories. I will explain the more refined connection to understand the Coulomb branch of the 5d N=1 theories and 6d (1,0) theories and their anomalies from this perspective.

1. Lecture 1
2. Lecture 2
3. Lecture 3
4. Lecture 4

A fundamental problem of algebraic geometry is to determine which algebraic varieties are rational, that is, isomorphic to projective space after removing lower-dimensional subvarieties from both sides. We discuss the history of the problem. Some dramatic progress in the past 5 years uses a new tool in this context: the Chow group of algebraic cycles.

The sign is a fundamental invariant of an abelian variety defined over a local (archimedian or p-adic) or global (number or function) field. The sign of an abelian varieties over a global field has arithmetic significance: it is the parity of Mordell-Weil group of the abelian variety. The sign also appears in the functional equation of the L-function of abelian variety, determining the parity of its order of vanishing at s=1. The modularity conjecture says that this L-function coincides with the L-function of an automorphic representation, and the sign can be expressed in terms of this representation. Although we know how to compute this sign using representation theory, this computation does not really shed any light on the representation theoretic significance of the sign. This representation theoretic significance was articulated first by Dipendra Prasad (in his thesis), where he relates the sign of a representation to branching laws — laws that govern how an irreducible group representation decomposes when restricted to a subgroup. The globalization of Prasad’s theory culminates in the conjectures of Gan, Gross and Prasad. These conjectures suggest non-torsion elements in Mordell-Weil groups of abelian varieties can be obstructions to the existence of branching laws. By exploiting p-adic variation, though, one can hope to actually produce the Mordell-Weil elements giving rise to these obstructions. Aspects of this last point are joint work with Marco Seveso.