Lecture Notes

The material below was provided to the students as class notes and associated readings.

1 Introduction to Branes

Branes Ending on Branes

5 Superstring Theories, 11d Supergravity

For today’s lecture, we have the following references:

- Open p-Branes (Strominger)

- String Theory Dynamics in Various Dimensions (Witten)

2-3 Central Charges in the Supersymmetry Algebra

Branes in Type II Theories

Dualities in String Theory

Tensions of Branes

Weinberg, Steven. The Quantum Theory of Fields. Vol. 3. New York, NY: Cambridge University Press, 1995-2000. ISBN: 9780521660006.
A good reference for Supersymmetry algebras in higher dimensions can be found in chapter 32 of Weinberg’s book, The Quantum Theory of Fields (Vol. 3). In particular see 32.2 for the massless supermultiplets.

The central extension to the supersymmetry algebra was first discussed in a paper by Witten Olive.
In this paper there is an explicit computation of the extension for a two dimensional theory with N=2 supersymmetry (4 supercharges). In this paper there is also a central extension for the N=2 theory in 4 dimensions (8 supercharges). The paper derives the BPS bound for the two algebras as a result of the existence of these central charges.

The discussion we had today is a (straightforward) generalization of this paper. One replaces particles by branes and generalizes the number of space-time dimensions. The central extension for a p-brane becomes a p-form in the supersymmetry algebra instead of a 0-form for a particle.

A treatment of the central charge for 11d supergravity can be found in M-Theory from its Super algebra (Townsend). (PDF)

A derivation of the BPS bound which is the generalization of the Witten Olive computation can be found.

A good source for SL(2,Z) duality of Type IIB is An SL(2,Z) Multiplet of Type IIB Superstrings (J. Schwarz).

And, Some Relationships Between Dualities in String Theory (Aspinwall).

4-5 Supergravity in Various Dimensions - 32 Supercharges

Scalar Manifolds in Various Dimensions

Cosmic Strings

For collected works on supergravity in various dimensions see:

Salam, A., and E. Sezgin, eds. Supergravity In Diverse Dimensions. North-Holland: World Scientific, 1989.

For a discussion on scalar manifolds in various dimensions see for example Hull and Townsend.
Tables 1 and 2 summarize the results.

For a recent revival of the topic of cosmic strings in string theory see Copeland, Myers, and Polchinski.

6-7 u-dualities and Representations of p-forms

Electric and Magnetic Excitations on Branes

SYM Actions in Various Dimensions

Higgs Mechanism - Adjoint Representation

A nice treatment of anomalies in various dimensions can be found in the review paper of Alvarez-Gaume and Ginsparg.

In this paper there is a topological interpretation of the anomalies in terms of some index theorems. The 2 dimensional extension of the manifold for which the anomaly in calculated is described in section 3 where this 2 dimensional extension turns out to be a 2-sphere. A summary of the ideas can be found in section 4. This reference answers in great detail all questions which were asked today in class about the derivation of the anomaly in even dimensions.

A good source for the properties of supergravity theories in various dimensions can be found in Introduction to Supergravities in Diverse Dimensions (Tanii).

8-9 Brane Realization of Classical Groups for Theories with 16 Supercharges

McKay Correspondence


Central Charge Formulas for W Bosons

Root Systems of Lie Algebras

A good review of string dualities can be found in the paper of lectures on Superstring and M Theory Dualities (Schwarz).

The next lecture, will be focused on the brane realization of the classical groups A, B, C, D for theories with 16 supercharges. We will learn for the first time in the course the deep connection between geometry and algebra as realized by the branes. We will cover orientifolds on their various types and learn about central charge formulas for W bosons and their relation to the root systems of the lie algebras.

10-11 BPS Objects Living on Branes



Brane Realization of E_n Gauge Theories

E_n Representation Theory, Root Systems

Supergravity Multiplets - 16 Supercharges, Vector Multiplets and Tensor Multiplets

Gravitational Anomalies

In Lec #10, we will go over the BPS objects on the world-volume of various branes. For gauge theories (which as is familiar by now live on D branes) these include the instanton and the ’t Hooft Polyakov monopole.

Next, we will discuss compact space-times with orientifolds and learn about the brane realization of E_n gauge theories. This will be the first non-perturbative dynamical result which can not be verified by the usual methods in perturbation theory.

These are two possible references for learning about monopoles and instantons:

Cheng, Ta-Pei, and Ling-Fong Li. Gauge Theory of Elementary Particle Physics. Oxford, NY [Oxfordshire]: Clarendon Press; New York, NY: Oxford University Press, 1984. ISBN: 9780198519560, 9780198519614. (Paperback.)
Cheng and Li (in their book, Gauge Theory of Elementary Particle Physics) discuss the ’t Hooft-Polyakov monopole in chapter 15 and the instanton in chapter 16. Their discussion is of course in 4 dimensions. The generalization to higher dimensions is straightforward and assumes that the extra directions are spectators. They also give the original references.

Weinberg, Steven. The Quantum Theory of Fields. Vol. 2. New York, NY: Cambridge University Press, 1996. ISBN: 9780521550024.
Weinberg’s volume two includes a very lucid treatment of instantons and monopoles (again in 4d) in chapter 23.

Another useful reference for the classical monopole solutions can be found in the paper by Gibbons and Manton.

Polchinski, Joseph Gerard. String Theory. Vol. 1. Cambridge, UK; New York, NY: Cambridge University Press, 1998, p. 278. ISBN: 9780521633031. (Hardcover.)
There is a brief discussion by Polchinski on the reflection and incident gravitational waves on an orientifold plane. This is a possible answer to a question which was discussed in class about the non-dynamical nature of the orientifold plane.

12-13 Enhanced Gauge Groups in Type I'

W Bosons and Root Systems

O8 Plane

Montonen Olive Duality and SL(2,Z), Brane Picture in Type IIB

Strong Coupling Phenomena in N=8 SYM in 3d and its Brane Realization

Orientifolds in Compact Backgrounds

5d SYM with 16 Supercharges and (0,2) Theories in 6d

  • Read Lecture Note 1 (PDF) (Courtesy of Megha Padi. Used with permission.)

    Read Lecture Note 2 (PDF) (Courtesy of Megha Padi. Used with permission.)

    In Lec #13, we will discuss the following topics:

    - Introduce the O8 plane which plays the role of the orientifold plane for E_n theories.

    - Discuss Montonen Olive Duality and its SL(2,Z) generalization and how it is implied by Type IIB SL(2,Z) duality using the brane picture.

    - Discuss strong coupling phenomena for N= 8 SYM in 3d and its brane realization.

14-15 Magnetic Monopoles in Type I'

SL(2,Z) Duality and N=4 SYM on a Circle

UV Completion of 5d SYM with 16 Supercharges - D4 Brane Story

Splitting of O3 Planes

A good exposition of the duality between Type I’ superstring and the Heterotic string in 9 dimensions can be found in the paper String Creation and Heterotic-Type I’ Duality (Bergman, Gaberdiel and Lifschytz).
16-17 Anomalies in 6d

3d Gauge Theories with 16 Supercharges

Theories with 8 Supercharges - Various Supermultiplets

A useful summary of supersymmetry multiplets and their properties can be found in Vadim Kaplunovsky’s notes. 
18 Theories with 8 Supercharges

Higgs Branch - Moduli Space of Instantons

Dual Coxeter Numbers and the Moduli Space of Instantons

Matter Content for D_p_ Branes on O(p+4) planes

Moduli Space of Instantons - ADHM Construction

In Lec #18, we continue to look at theories with 8 supercharges and will discuss the Higgs branch and its relation to the moduli space of instantons.
19-20 Higgs Branch and the Moduli Space of Instantons - Classical Groups A, B, C, D

E_n Instantons

String Coupling Equation and Beta Function of Supersymmetric Gauge Theories

Vacuum Structure of Supersymmetric Gauge Theories with 8 Supercharges - Brane Picture

D Branes near Orientifold Planes

D4 Branes and 5d BPS States

Read Lecture Note 3 (PDF) (Courtesy of Megha Padi. Used with permission.)

In Lec #19, we continue to look at the relation between the Higgs branch and the moduli space of instantons and extend the discussion to the classical groups: A_n, B_n, C_n, D_n. We will also learn about some aspects of the moduli space of E_n instantons. If time allows we will discuss the relation between the string coupling equation of motion and the beta function of supersymmetric gauge theories.

21 Brane Intervals - Hanany Witten Brane Configurations A source for the dual coxeter number of various groups can be found for example in Instantons and Magnetic Monopoles on _R_3 × _S_1 with Arbitrary Simple Gauge Groups (Lee).
22-23 The String Coupling and Vector Multiplets

Effective Gauge Coupling at One Loop - D_p_-D(p+4) System

Asymptotically Free Gauge Theories with 8 Supercharges and D Brane Realization

In Lec #22, we will start looking at brane configurations involving D branes bounded by NS5-branes - Brane Intervals.

A source for supermultiplets of (2,1) supersymmetry can be found in:

Strathdee, J. (ICTP, Trieste), IC-86/94, May 1986. “Extended Poincare Supersymmetry.” Int J Mod Phys A2, no. 273 (1987): 40.

24-25 Mirror Symmetry in 3 Dimensions

Five Dimensional Fixed Points and their Low Energy Gauge Theory Limits

Brane Creation - The Hanany Witten Effect

Continuation Past Infinite Coupling

For an M theory realization of Mirror Symmetry in 3 dimensions see Porrati and Zaffaroni.

For a Type IIB realization of Mirror Symmetry in 3 dimensions, Brane Creation, and Continuation Past Infinite Coupling see Hanany and Witten.

For (p,q) Webs and their application to five dimensional fixed points see Aharony, Hanany, and Kol.

Course Info

As Taught In
Fall 2004
Learning Resource Types
Lecture Notes
Problem Sets with Solutions