**Spring/Summer 2018**

*YMSC Graduate Course*

**Brane Dynamics and Supersymmetric Gauge Theories**

This course is an introduction to branes in string theory and how they give rise to supersymmetric gauge theories in various dimensions. We will first cover the classification of different types of branes that arise in string theory and the bound states they form. We will then move on to the brane realizations of supersymmetric gauge theories in 2,3,4,5 and 6 dimensions. While doing so, we will explore how gauge theory phenomena such as dualities arise in terms of branes and how brane dynamics gives a powerful geometrical interpretation of such gauge theory phenomena. This course is intended to be a pedagogical introduction to the topic of branes in string theory with examples taken from modern developments in string theory, including topics like mirror symmetry and the Higgs and Coulomb branch of N=4 theories, branes at Calabi-Yau singularities and recent applications of Hanany-Witten brane setups.

*Prerequisite:*

Quantum Field Theory, Supersymmetry, Representation Theory

*Confirmed Schedule:*

Tuesdays and Thursdays, 17:00-18:40

March 6 – June 14, 2018

*Location:*

Yau Mathematical Sciences Center, Tsinghua University,

Conference Room 1, Jin Chun Yuan West Building

*Announcement:
http://ymsc.tsinghua.edu.cn/sjcontent.asp?id=1056*

*
Lecture 1 [March 6, 2018]:
*1.0 Why String Theory?

1.1 Strings & Branes

1.2 Spacetime Revisited

1.3 Symmetries & Spacetime

1.4 SO(8), D4

* Lecture 2 [March 8, 2018]:
*2.0 Recap Lecture 1

2.1 Representation Theory Revisited: SU(n) & SO(2n) Irreps, Characters

2.2 Supersymmetry & Chirality

2.3 10d Massless Supergravity Multiplets

2.4 10d SO(8) vs 11d SO(9)

*3.0 Recap Lecture 2*

**Lecture 3 [March 13, 2018]:**3.1 Representation Theory Revisited: SU(2n+1) Irreps, Characters

3.1 Dimensional Reduction & Branching Rules

3.2 What about d<10?

3.3 Maxwell’s Equations Revisited

*4.0 Recap Lecture 3*

**Lecture 4 [March 15, 2018]:**

4.1 Rewriting Maxwell’s Equations

4.2 Brane Spectroscopy (M, IIA, IIB)

* Lecture 5 [March 20, 2018]:*5.0 Recap Lecture 4

5.1 Electromagnetic Duality

5.2 The Vacuum and Moduli Spaces

5.3 Strings on Branes

5.4 Gauss’ Law

5.5 Towards Branes ending on Branes

* Lecture 6 [March 22, 2018]:*6.0 Recap Lecture 5

6.1 D-Branes and Magnetically Charged Objects

6.2 Vector Multiplets

6.3 Symmetries and Symmetry Breaking under a D-Brane

6.4 M2 ending on M5

* Lecture 7 [March 27, 2018]:*7.0 Recap Lecture 6

7.1 M2 on M5 and Electromagnetic Duality

* Lecture 8 [March 29, 2018]:*8.0 Recap Lecture 7

8.1 NS5-Branes

8.2 Multiplets in 5d

8.3 NS5- and D5-Branes

* Lecture 9 [April 3, 2018]:*9.0 Recap Lecture 8

9.1 Counting Supersymmetries

9.2 Multiplets and Supersymmetry

9.3 The Tensor Multiplet and M-Theory

9.4 Towards Actions

**Qingming Festival (Tomb-Sweeping Holiday) [April 5, 2018]**

* Lecture 10 [April 10, 2018]:*10.0 Recap Lecture 9

10.1 Comments about Basic Interaction Terms

10.2 Gauge Invariance

10.3 Towards the Worldsheet Action

10.4 Preview of new term in the action

* Lecture 11 [April 12, 2018]:*11.0 Recap Lecture 10

11.1 Two Branes approaching each other

11.2 The Chern-Simons Term

* Lecture 12 [April 17, 2018]:*12.0 Recap Lecture 11

12.1 Electromagnetic Duality on a Brane

12.2 Examples of Branes ending on Branes

* Lecture 13 [April 19, 2018]:*13.0 Recap Lecture 12

13.1 D2 on D4

13.2 F1 on D1

13.3 D5 on NS5

13.4 IIB and SL(2,Z)

* Lecture 14 [April 24, 2018]:*14.0 Recap Lecture 13

14.1 The Theory on a Dp-Brane

14.2 Tension of a String and Couplings

14.3 S-duality

* Lecture 15 [April 26, 2018]:*15.0 Recap Lecture 14

15.1 Dimensional Reduction Revisited

15.2 S1 Compactification: M to IIA

15.3 Momentum Modes and KK Monopoles

15.4 M-Theory on 2-Torus

**May 1 Week Holiday [May 1 & 3, 2018]**

* Lecture 16 [May 8, 2018]:*16.0 Recap Lecture 15

16.1 M-Theory on T2 vs IIA/IIB on S1

16.2 T-duality

16.3 S-duality, T-duality and U-duality

* Lecture 17 [May 10, 2018]:*17.0 Recap Lecture 16

17.1 Duality Chains and Brane Configurations

17.2 Revisiting the Worldvolume Theory on a D-Brane

17.3 Dp on Dp+2 and Dp on NS5

17.4 Vector and Hypermultiplets

* Lecture 18 [May 15, 2018]:*18.0 Recap Lecture 17

18.1 Chan-Paton Factors

18.2 A NS5, D5 and D3 configuration

18.3 The Field Theory on the D3-brane

* Lecture 19 [May 17, 2018]:*19.0 Recap Lecture 18

19.1 Vector and Hypermultiplets in 3d N=4

19.2 Brane Singularities

19.3 The Coulomb Branch and Magnetic Monopoles in 3d N=4

* Lecture 20 [May 18, 2018 – extra lecture]:*20.0 Recap Lecture 19

20.1 The Construction of the 3d N=4 Coulomb Branch

20.2 Special Examples of the Coulomb Branch

20.3 A Hypermultiplet Paradox?

* Lecture 21 [May 22, 2018]:*21.0 Recap Lecture 20

21.1 Hanany-Witten Transitions

21.2 Linking Numbers for 5-Branes

21.3 Linking Number Properties

* Lecture 22 [May 24, 2018]:*22.0 Recap Lecture 21

22.1 Multiple Branes and Hanany-Witten Transitions

22.2 Linking Numbers and Partitions

22.3 Gaiotto-Witten Theories

22.4 Mirror Symmetry and Partitions

* Lecture 23 [May 29, 2018]:*23.0 Recap Lecture 22

23.1 Conditions on Partitions and Non-Supersymmetric Theories

23.2 Higgs and Coulomb Branches of 3d N=4 Theories

23.3 Kraft-Procesi Transitions and Hasse Diagrams

* Lecture 24 [May 31, 2018]:*24.0 Recap Lecture 23

24.1 Examples of Hasse Diagrams

24.2 4d N=1 Theories and SQCD

24.3 F-and D-Terms and Vacuum Moduli Spaces

24.4 The Moduli Space for SQCD

* Lecture 25 [June 1, 2018]:*25.0 Recap Lecture 24

25.1 Different Moduli Spaces for SQCD

25.2 Seiberg Duality for 4d N=1 Theories

25.3 Abelian Orbifolds and Orbifold Theories

* Lecture 26 [June 5, 2018]:*26.0 Recap Lecture 25

26.1 4d Orbifold Theories

26.2 Classification of Abelian Orbifolds

26.3 The Theory for k U(N) Instantons and the ADHM Construction

26.4 The Theory for k U(N) Vortices

* Lecture 27 [June 7, 2018]:*27.0 Recap Lecture 26

27.1 Plethystics and Hilbert Series

27.2 Algebraic Varieties and Coordinate Rings

27.3 Counting Gauge Invariant Operators

* Lecture 28 [June 8, 2018]:*28.0 Recap Lecture 27

28.1 Quiver Gauge Theories and Toric Varieties

28.2 Klebanov-Witten Theory and the Conifold

28.3 Brane Tilings

**Fin**