# Physics

## Derived Geometry in Twists of Gauge Theories 3 of 4

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

## Derived Geometry in Twists of Gauge Theories 2 of 4

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

## Derived Geometry in Twists of Gauge Theories 1 of 4

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

## Elliptic Fibrations and Singularities to Anomalies and Spectra 4 of 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.

## Elliptic Fibrations and Singularities to Anomalies and Spectra 3 of 4

Throughout my lectures I will explain the geometry of elliptic fibration which can gåve 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.

## Elliptic Fibrations and Singularities to Anomalies and Spectra 2 of 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.

## Elliptic Fibrations and Singularities to Anomalies and Spectra 1 of 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.

## Connes fusion of the free fermions on the circle

A conformal net on $S^1$ is an assignment $\mathcal{A}:\left\{\textrm{open subsets of } S^1\right\} \to \left\{\mbox{von Neumann algebras acting on } \mathcal{F}\right\}$, which satisfies a slew of axioms motivated by quantum field theory. In this talk, I will consider the free fermionic conformal net. In this case, the Hilbert space $\mathcal{F}$ is the Fock space generated by the positive energy modes of square-integrable spinors on the circle $?^2(?^1,\mathbb{S})$; and the von Neumann algebras are Clifford algebras generated by those elements of $?^2(?^1,\mathbb{S})$ whose support lies in $?\subset ?^1$. After going over this construction, I will argue that given an open interval $?\subset ?^1$, one can equip $\mathcal{F}$ with the structure of $\mathcal{A}(I)-\mathcal{A}(I)$-bimodule. I will then outline the construction of a canonical isomorphism of bimodules $\mathcal{F}\boxtimes_{\mathcal{A}(I_\_)}\mathcal{F}\to\mathcal{F}$ where $\boxtimes_{\mathcal{A}(I_\_)}$ stands for the Connes fusion product over the algebra assigned to the lower semi-circle $I_\_$. If time permits, I will discuss some (anticipated) applications of this isomorphism, for example in string geometry, or in the construction of the free fermion extended topological field theory.

## SU(2) hadrons on a quantum computer

Lattice gauge theories are relevant in many fields of physics, and simulations with quantum computers can become a powerful tool to study them, especially in regimes inaccessible to classical numerical methods. In particular, non-Abelian gauge theories, which among other things describe fundamental particles’ interactions, are of great interest. In this talk I will discuss the first quantum simulation of a non-Abelian lattice gauge theory that includes dynamical matter. I will show how the theory is formulated in order to include colour degrees of freedom, and how this allows for the existence of baryons in the model, which do not exist in Abelian theories. A quantum computation of the low-lying spectrum of the model is performed on an IBM superconducting platform using a variational quantum eigensolver. This proof-of-concept demonstration was made possible by a resource-efficient approach in the design of the quantum algorithm, and lays out the foundation for further development of the field. This talk is based on arXiv:2102.08920.

## Quantum applications of harmonic analysis on the group of positive rationals

Harmonic analysis on the multiplicative group of positive rational numbers (ℚ+) has not been part of the common quantum-theoretic toolkit. In this talk, I will discuss how it lends itself to the analysis of operators in ℓ2(ℕ), in some cases leading to spectacular new insights into their spectral properties. I will also discuss its application in a study of the Bose-Hubbard model, i.e. a model of an array of bosons with the nearest-neighbour interactions. The Fourier transform on ℚ+ uncovers the model's unobvious symmetries and surprising connections with other structures. In addition, I will report a rigorous, albeit computer-assisted, proof of the existence of quantum phase transitions in finite quantum systems of this type. The study of the Bose-Hubbard model has been carried out in collaboration with Prof. Jonas Fransson (Department of Physics and Astronomy, University of Uppsala).