Course Identification

BSc in math or physics

We will start with some background on Lie algebras, their representations and highest weight representation.

We will learn what a CFT in D dimensions is, and what is special in the case D=2 dimensions.

We will define the Virasoro algebra, study its representation and use them to define minimal models.

We will also define Vertex operator algebras and study their basic properties.

We will meet the conformal and modular bootstrap techniques and applications in physics.

We will overview the sphere packing problem and learn the Cohn-Elkies method. We will then study Viazovska's solution for Sphere packing in 8 dimensions and its relation with modular bootstrap, following  Hartman-Mazac-Rastelli  .

We will then study recent applications to spectra of surfaces.

-The students will get to know the concepts of representations of Lie algebras, Virasoro algebra and Vertex operator algebras.

-The students will learn the basics of CFT in dimension 2 and general dimensions.

-The students will be exposed to linear and semi definite programming tools in math and physics.

-The students will learn the recent breakthrough of Viazovska in sphere packing in dimension 8.

-"Bombay Lectures on highest weight representations of infinite dimensional Lie algebras" by A. Raina and V. Kac (Virasoro algebra and its representations).

-"Infinite conformal symmetry in 2 dimensional quantum field theories" by Belavin, Polyakov, Zamolodchikov.

-https://link.springer.com/book/10.1007/978-3-642-00450-6 (CFT in general dimension and dimension 2).

-https://www.worldscientific.com/doi/10.1142/S0217751X92000946 (Vertex operator algebras and CFT)

-https://webhomes.maths.ed.ac.uk/~lhenneca/Skye2023-Talk2.pdf (Examples of Vertex algebras).

-https://arxiv.org/abs/1805.04405 (Conformal bootstrap in D>2)

-https://arxiv.org/abs/1603.04246 (Viazovska sphere packing in dimension 8)

-https://www.pnas.org/doi/10.1073/pnas.2304891120 (Romik's proof of Viazovska's inequalities)

-https://arxiv.org/pdf/math/0110009 (Cohn-Elkies)

-https://arxiv.org/pdf/1905.01319 (Hartman-Mazac-Rastelli)

https://arxiv.org/pdf/2111.12716 (spectra of surfaces)

Optional:

-"Vertex algebras for beginners" by V. Kac 

https://arxiv.org/pdf/2006.02560 (towards high dim sphere packing)

https://arxiv.org/pdf/2308.11174 (applications for hyperbolic 3-manifolds)

https://arxiv.org/abs/2206.09876 (bounds for Cohn-Elkies)