Lonti Autumn 2021 Series: Lecture 4. Live Tutorial. Please register at https://lonti.weebly.com/registration.html to receive joining instructions for this live session which will be held via Zoom. Gauge theories are ubiquitous in theoretical physics, not to mention that the standard model is one. It is therefore of utmost importance to know what the observables of these theories are, quantities that can be calculated and measured. I start with a long discussion based on the most familiar gauge theory, Maxwell's electromagnetism, where a lot of computations can be done explicitly. I then take the lessons from that to non-abelian gauge theories. The observables covered are local, Wilson loops, and briefly 't Hooft loops and surface operators.

Lonti Autumn 2021 Series: Lecture 4. Release of Recorded Lecture. Available at https://youtu.be/JLbuSnt2OyA. Gauge theories are ubiquitous in theoretical physics, not to mention that the standard model is one. It is therefore of utmost importance to know what the observables of these theories are, quantities that can be calculated and measured. I start with a long discussion based on the most familiar gauge theory, Maxwell's electromagnetism, where a lot of computations can be done explicitly. I then take the lessons from that to non-abelian gauge theories. The observables covered are local, Wilson loops, and briefly 't Hooft loops and surface operators.

In this talk I will reexamine the classification of BPS Wilson loops in 3d super Chern-Simons-matter theories. Over the last several years a large class of increasingly intricate constructions of such operators have been found. They involve both discrete and continuous parameters chosen to satisfy varied conditions. In my talk I will explain that the discrete parameters are related to choosing a graded quiver diagram, which may be a subquiver or a cover of the one defining the theory. The continuous parameters are then a singular limit of the variety, a complex manifold, associated to that quiver.

The gauge-gravity correspondence identifies a field theory with a gravitational theory. The gravitational theory is weakly coupled when the field theory has large coupling and vice versa, which mostly prevents matching nontrivial results between the two descriptions. I will discuss cases when the field theory calculation can be reduced to a finite dimensional matrix integral, representing some counting problems. I will then evaluate the integral exactly and reexpand the exact result, which is valid for all coupling, at strong coupling. The resulting expression should match a weak coupling gravitational (or string theoretic) calculation and I’ll comment on what is known from that direction.

Matrix models are toy models for quantum field theories. They can be extremely complicated but can also be solved in a variety of ways. In my talk I will discuss general properties of matrix models and their solutions and focus on particular matrix models that arise in the study of 4d SUSY field theories. Those matrix models describe the index of the field theory, counting the number of states of the theory (with + sign for a boson and - for a fermion) and have been known for over 10 years. Though they look very complicated I will show how some simple tricks allow in certain cases to solve those matrix models exactly in terms of elementary functions. My talk will focus on the matrix model calculation and no specialized knowledge of SUSY field theories or indices would be required to follow it.

I will review the calculation of the partition function of 3d supersymmetric field theories on S^3 using the fermi-gas approach to solve the matrix integral. The resulting expression is an Airy function and is valid perturbatively to all orders in 1/N for a wide class of theories (including ABJM). This suggests that a similar formula can be derived by studying quantum gravity on AdS_4. I will explain several of the steps needed to implement this idea and some intriguing results.

Note: Sunil Mukhi's talk was cancelled!

I will review the matrix model which calculates the partition function of ABJM theory on S3 as well as the expectation value of Wilson loop operators. I will then explain how this matrix model is solved and present the results for these quantities at all values of the couplings. At strong coupling these calculations reproduce the results of supergravity on Ads4 x CP3 and in particular the N to the 3/2 scaling of the free energy of the theory.

I will present the 1/2 BPS Wilson loop operator of N=6 super Chern- Simons-matter (ABJM theory) which is dual to the simplest macroscopic open string in AdS4 x CP3. The Wilson loop couples, in addition to the gauge and scalar fields of the theory, also to the fermions in the bi-fundamental representation of the U(N) x U(M) gauge group. These ingredients are naturally combined into a superconnection whose holonomy gives the Wilson loop, which can be defined for any representation of the supergroup U(NlM). Using the localization calculation of Kapustin et al. I will then show that the circular loop is computed by a supermatrix model and discuss the connection to pure Chern-Simons theory with supergroup U(NlM).

A large family of interacting conformal field theories in four dimensions with N=2 supersymmetry was recently constructed by Gaiotto. Each gauge theory is associated to a Riemann surface with certain allowed singularities. In fact, it was proposed by Alday, Gaiotto and Tachikawa that the partition function of these theories (based on SU(2) gauge groups) is equal to correlation function in Liouville theory with central charge c=25. After reviewing these constructions I will turn to a detailed exploration of S-duality using loop operators: Wilson, 't Hooft and dyonic. I will explain the classification and evaluation of arbitrary loops in arbitrary theories and show how they transform into each-other under S-duality.

The standard prescription for calculating a Wilson loop in the AdS/CFT correspondence is by a string world-sheet ending along the loop at the boundary of AdS. For a multiply wrapped Wilson loop this leads to many coincident strings, which may interact among themselves. In such cases a better description of the system is in terms of a D3-brane carrying electric flux. We find such solutions for the single straight line and the circular loop. The action agrees with the string calculation at small coupling and in addition captures all the higher genus corrections at leading order in alpha'. The resulting expression is in remarkable agreement with that found from a zero dimensional Gaussian matrix model.