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# Formal Quantum Field and String Theory

### Video Briefs

String theory has enjoyed tremendously fruitful interactions with modern mathematics. Some of the simplest and most interesting questions in number theory and geometry involve counting (for instance, determining numbers of integer solutions to certain equations).

Many mathematical concepts trace their origins to everyday experience, from astronomy to mechanics. Remarkably, ideas from quantum theory turn out to carry tremendous mathematical power too, even though we have little intuition dealing with elementary particles. The bizarre quantum world not only represents a more fundamental description of nature, it also inspires a new realm of mathematics that might be called “quantum mathematics” that turns out to be a powerful tool to solve deep outstanding mathematical problems.

This Strings 2014 talk by SITP Professor Shamit Kachru reviews recent developments in the study of moonshine.

This Strings 2013 talk explores a moonshine relating coefficients of the instanton expansions of gauge coupling functions of 4d N=2 string vacua, and the sporadic simple group M24.

### People

Liam McAllister is a professor at Cornell University. He received his Ph.D from Stanford University in 2005. He is interested in using string theory to understand the early universe, and in developing compactifications of string theory that lead to realistic four-dimensional physics.
Raphael Flauger is a professor at the University of California, San Diego. He received his Ph.D from the University of Texas at Austin in 2009. Raphael is a member of the Modern Inflationary Cosmology research team led by SITP faculty member Eva Silverstein.
### News Items

## Pages

The study of the formal and mathematical structure of quantum field theory and string theory has undergone a renaissance in recent decades. These subjects underlie our descriptions of phenomena across a range of energy scales, from condensed matter physics and particle physics at accessible energies, to more speculative thoughts about early-universe cosmology and physics of the Big Bang and Black Holes. Development of understanding of the mathematical structures underlying quantum field theories and string theories in their own right has thus often had fruitful applications. Theorists at Stanford have contributed to some fascinating developments in this area.

One direction has involved the exploration of exactly soluble models. Theorists presently at Stanford (Kachru, Shenker) were involved with the discovery of exactly solvable models of conformal field theory in two dimensions; soluble models of string theory in d<2 dimensions; and the first techniques to find exact solutions of 4d N=2 supersymmetric field theories using string dualities and the geometry of Calabi-Yau singularities.

Stanford theorists (Kachru, Silverstein) led the development of techniques to find concrete candidate "flux vacua" in the landscape of string theory, and explore their mathematical structure. These include foundational studies of the structure of Calabi-Yau flux vacua, as well solutions involving strings on spaces of negative curvature.

A major direction in recent years is the exploration of finiteness properties of supergravity. Kallosh was involved with this subject since its inception and has contributed many conjectures and results.

Mysterious dualities lie at the foundation of many new discoveries in string theory and its interaction with mathematical physics. The "D-duality" relating strings on negatively curved spaces to supercritical string theory was first described at Stanford. Explorations of new avatars of the mysterious "moonshine" relating (mock) modular forms, sporadic simple groups, algebraic geometry, and string vacua has been a subject of significant recent interest at Stanford.

Nov 1 2020

The Stanford Institute for Theoretical Physics (SITP) is searching for postdoctoral fellows across the full range of theoretical physics.

Dec 6 2018

Leonard Susskind has identified a possible quantum origin for the ever-growing volume of black holes.

Mar 31 2016

Two dimensional conformal field theories (CFTs) have a very powerful property called modular invariance, which relates the high and low temperature limits of the theory.

Mar 28 2016

Quantum gravity in anti de Sitter space in three spacetime dimensions is conjectured to be dual to two dimensional conformal field theories (CFTs) with sparse spectra.