2022-23 Series
- 17, 24 October and 1 November 2022: Alessandro Torrielli, Lectures on the duality between the Sine-Gordon and the Thirring model
Time and Location: Mondays (except Tuesday November 1st) at 10:30 at the London Institute for Mathematical Sciences, 21 Albemarle St.
We will describe the duality between two integrable systems: the 2D Sine-Gordon model and the 2D Thirring model. We will spend some time describing the classical and quantum Sine-Gordon model, in particular its spectrum, S-matrices and underlying quantum-group symmetry. We will then present the duality with the Thirring model as originally stated by Coleman and refined in subsequent literature. All the basic elements will be provided without relying on too many pre-requisites beyond standard graduate-level quantum field theory. The notes comprise a series of exercises.
Resources:
- 7, 14, and 21 November 2022: Sameer Murthy, Introduction to black hole micro state counting
Time and Location: Mondays at 10:30 at the London Institute for Mathematical Sciences, 21 Albemarle St.
The pioneering work of Bekenstein and Hawking in the 1970s showed that black holes have thermodynamic properties like temperature and entropy in the quantum theory, just like the air in this room. This leads to the question: can we account for the thermodynamic entropy of a black hole as a statistical entropy of an ensemble of microscopic states? One of the big successes of string theory is to answer this question in the affirmative for a large class of black holes. The aim of these lectures is to introduce these ideas to a beginning PhD student in high energy physics.
The lectures will cover the following topics in succession:
- A review of the basics of black hole thermodynamics
- The ideas underlying the counting of statistical entropy of black holes in string theory
- The appearance of a special class of black holes called BPS black holes for which the statistical entropy can be calculated
- The nature of the corresponding microscopic ensemble of BPS states
- An illustration of the calculation of their statistical entropy in some simple examples.
At the end of these lectures, the student is meant to have gained an orientation with respect to the basic ideas and be equipped with some basic techniques used in microstate counting.
Resources:
- January 30, February 6, 13 and 20 2023: Andreas Stergiou, Introduction to CFT and Conformal Bootstrap
Time and Location: Mondays at 10:30 at the London Institute for Mathematical Sciences, 21 Albemarle St.
These lectures aim to provide a self-contained introduction to the modern conformal bootstrap method. The study of conformal field theory (CFT) will first be motivated and the “old” way of studying CFTs as endpoints of RG flows will be explained. The set of ideas necessary to understand the conformal bootstrap method will then be introduced, and both analytic and numerical implementations of the conformal bootstrap method will be discussed.
Schedule:
- 30 January: General remarks, motivation for the study of CFTs, renormalisation group, conformal transformations, conformal group/algebra, operators.
- 6 February: Correlation functions, radial quantisation, unitarity, operator product expansion, conformal blocks.
- 13 February: Crossing symmetry, conformal bootstrap basics, analytic bootstrap.
- 20 February: Numerical bootstrap, global symmetries, mixed correlators.
Resources:
- Suggested reading
- David Tong’s lectures on Statistical Field Theory (http://www.damtp.cam.ac.uk/user/tong/sft/sft.pdf),
- Slava Rychkov’s EPFL Lectures on Conformal Field Theory in D ≥ 3 Dimensions (arXiv:1601.05000),
- David Simmons-Duffin’s TASI Lectures on the Conformal Bootstrap (arXiv:1602.07982).
- Review article
- David Poland, Slava Rychkov, Alessandro Vichi, “The Conformal Bootstrap: Theory, Numerical Techniques, and Applications” (arXiv: 1805.04405).
- March 13, 20, and 27 2023: Daniel Waldram, Geometry and fluxes
Time and Location: Mondays at 10:30 at the London Institute for Mathematical Sciences, 21 Albemarle St.
“Special” geometries, such as Calabi-Yau manifolds, play a central role in multiple areas of string theory, as well as gravitational theories more generally. The goal of these lectures is to introduce some of the formalism and tools useful for characterising such geometries, pitched at the level of a starting PhD student. We will start with purely geometrical backgrounds using the general notions of a G-structure and special holonomy and then will go on to describe backgrounds that also have non-trivial fluxes. We will be guided by applications to string phenomenology and the AdS/CFT correspondence.
Topics covered in rough sequence
- introduction: examples of special geometries, relations to supersymmetry and compactification, phenomenology, AdS/cft and consistent truncations
- unified description of geometrical backgrounds using G-structures: connections and intrinsic torsion, special holonomy, Calabi-Yau and Sasaki-Einstein geometries
- fluxes, generalised geometry and generalised G-structures and applications
We'll assume some knowledge of the basics of differential geometry at the level of for example, the introductory parts of Nakahara, "Geometry, Topology and Physics".
There is no one good reference for the lectures, but the following may be helpful:
- Nakahara contains is useful background on holonomy, complex and Kähler geometry
- short mathematical introduction to special holonomy in Joyce "Lectures on Calabi-Yau and special Lagrangian geometry"
- nice lectures on G-structures, generalised (complex) geometry and flux backgrounds by Koerber "Lectures on Generalized Complex Geometry for Physicists"
- April 24, 27 and May 2 2023: Elli Pomoni, Introduction to Seiberg-Witten Theory
Time and Location: 10:30 at the London Institute for Mathematical Sciences, 21 Albemarle St.
These three lectures will aim to provide a pedagogical introduction to the dynamics of N=2 supersymmetric gauge theory and the work of Seiberg and Witten. We will assume only basic knowledge of supersymmetry.
We will mostly follow:
- https://arxiv.org/abs/hep-th/9601007
- https://arxiv.org/abs/hep-th/9701069
- https://people.sissa.it/~bertmat/susycourse.pdf
- the book of Yuji Tachikawa: N=2 supersymmetric dynamics for pedestrians (https://arxiv.org/abs/1312.2684)
- the book of John Terning: Modern Supersymmetry: Dynamics and Duality
For students that need a quick introduction to supersymmetry I recommend
https://arxiv.org/abs/hep-th/0101055 (in my experience students can go through it in roughly a week (with an intense pace))
I also highly recommend https://homepages.uc.edu/~argyrepc/cu661-gr-SUSY/index.html even though for these lectures we will not cover these topics.
- May 15 and 22 2023: Ulf Lindström, Leverhulme Lectures: Supersymmetry, complex geometry and the Hyperkähler quotient
Time and Location: 10:30 at the London Institute for Mathematical Sciences, 21 Albemarle St.
Ulf Lindsröm, a Leverhulme visiting professor at Imperial College will discuss sigma models, which are maps from a domain to a target space T. The geometry of the target space is determined by the dimension of the domain and symmetries of the model. When it has isometries that can be gauged, the quotient space, i.e., the space of orbits under the isometries, supports a new sigma model. The target space geometry of the new model is the quotient of the T by the isometry group.
This is first described for a bosonic sigma model and it is pointed out that we need to understand supersymmetric sigma models, their isometries and gauging as well as the quotient in order to apply the scheme to models with extended supersymmetry. We then look at these issues. The final goal is to construct new hyperkähler geometries from hyperkähler geometries with isometries , so making sure that the quotient construction preserves the symmetries etc.
Lecture notes
References
- Hyperkahler Metrics and Supersymmetry
- Nonlinear σσ Models and Their Gauging in and Out of Superspace
- Supersymmetric Sigma Model geometry
- Geometry, topology and physics, book of Nakahara
- Gravitation, Gauge Theories and Differential Geometry
- Superspace Or One Thousand and One Lessons in Supersymmetry, book of Gates, Grisaru, Rocek and Siegel
- Twisted Multiplets and New Supersymmetric Nonlinear Sigma Models