String theory and holographic duality

Lectures: 2 sessions / week, 1.5 hours / session

8.323 Relativistic Quantum Field Theory I, 8.324 Relativistic Quantum Field Theory II

In holographic duality, a quantum gravity system in a (d+1)-dimensional spacetime is equivalent to a many-body system defined on its d-dimensional boundary. The boundary system is referred to as a "hologram" of the bulk system.

How can a d-dimensional system be equivalent to a (d+1)-dimensional system? An essential element is that quantum gravity is very special because it contains many fewer degrees of freedom than one might naively assume. Operationally, the radial direction z in the bulk can be interpreted to correspond to the length scale in the boundary system. For example, two cows identical in the bulk except for their radial coordinate z correspond, in the boundary system, to two differently sized cows, with larger cow on the boundary corresponding to the cow deeper in the bulk. In other words, short distance physics in the boundary maps to bulk gravity near the boundary, but long-distance physics in the boundry maps to bulk gravity in the deep interior.

During last ten years string theory has revealed a surprising and deep connection between gravity and many-body physics, under the name of holographic duality (or gauge / gravity duality or AdS / CFT). The duality brings together many previously seemingly unconnected subjects including quantum gravity / black holes, QCD at extreme conditions, exotic condensed matter systems, and quantum information in an extremely elegant yet still mysterious manner. It also opens up new powerful approaches for studying these subjects from completely different perspectives.

This course aims to bring students to the forefront of this exciting field. Prior knowledge in string theory is not required. Certain familiarity with quantum field theory and general relativity will be assumed.

There is no required textbook, but there are several suggested books and reviews listed in the readings section. Lecture notes are available for every class.

Problem sets are a very important part of this course. We believe that sitting down yourself and trying to reason your way through a problem not only helps you learn the material deeply, but also develops analytical tools fundamental to a successful career in science. We recognize that students also learn a great deal from talking to and working with each other. We therefore encourage each student to make his / her own attempt on every problem and then, having done so, to discuss the problems with one another and collaborate on understanding them more fully. The solutions you submit must reflect your own work. They must not be transcriptions or reproductions of other people's work. Plagiarism is a serious offense and is easy to recognize. Don't submit work which is not your own.

Problem sets are normally posted on the course website on Thursday and will be due on Thursday two weeks later. Late problem sets will only be counted for half credits.

Each student is required to write a final project paper on a topic related to holographic duality. The paper can expand on a topic or problems covered in the course. It can also be based on the student's own reading or research. It does not need be original, although original materials are certainly welcome.

The paper should be written in the style and format of a brief journal article, and should be at a level accessible to fellow graduate students not familiar with the topic. Recommended length: 8–10 pages.

There is no exam for this course. The course grade will be based on 5 Problem Sets which are due every two weeks and a final project. I may alter grades to reflect class participation, improvement, effort and other qualitative measures of performance.

Part 1. Hints for holography

Weinberg–Witten no–go theorem

Bekenstein bound

Part 2. Introducing the gauge / gravity duality

Part 3. A duality tool box

Part 4. Holographic renormalization group flows

Part 5. Insights into many-body systems

Part 6. Insights into quantum gravity

Part 7. Status of the duality

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