Scientific Training

The gauge/string integrability is a multidisciplinary field that interrelates quantum field theory, string theory and integrable systems alike. Integrable quantum field theories are beautiful examples of exactly solvable quantum systems in which the effects of strong quantum fluctuations can be analyzed exactly, without making any approximations. Mathematical apparatus developed for solving integrable models, although deeply rooted in conventional quantum field theory, employs very special tools and methods, such as bootstrap, exact S-matrices, various types of Bethe ansatz, inverse scattering method and the like, that are not covered in standard courses of quantum field theory.

Applications of integrability in gauge theories were boosted by gauge-string duality, which remains one of its major motivations. The key example of the gauge-string duality is the AdS/CFT correspondence, which in its turn arose from the study of string-theory solitons called D-branes and of subtle interplay between gravity and gauge interactions in string theory.

String theory and AdS/CFT duality rely on powerful symmetry principles, such as supersymmetry and conformal invariance. Applications of integrable systems to gauge-string duality incorporate conformal symmetry and supersymmetry in an essential way.

Mathematical background necessary for gauge-string integrability includes group theory, basic differential geometry, and complex analysis. Below we list a number of monographs and review articles on the quantum field theory and string theory:

- Quantum field theory: there are many introductory, as well as more advanced textbooks on QFT. The majority cover gauge fields and Feynman diagram technique of perturbation theory, for instance M.E. Peskin and D.V. Schroeder (1995). Some also contain an introduction to supersymmetry. Lecture notes on QFT can be found line, for instance

- Supersymmetry: The standard introduction is J. Wess and J. Bagger (1992)

- String theory: Two standard introductory texts are M.B. Green, J.H. Schwarz and E. Witten (1987) and J. Polchinski (1998). Some introductory reviews of string theory can be found online D. Tong: String Theory.

- Conformal field theory: the majority of introductory texts on CFT concentrate on two-dimensional theories. Some (for instance P. Ginsparg: Applied Conformal Field Theory) introduce higher-dimensional CFT as well.

Essential reading on integrable systems includes an introduction on Bethe ansatz, coordinate or algebraic, such as lectures L.D. Faddeev, as well as classical integrable systems, exact S-matrices and thermodynamic Bethe ansatz. Some of the monographs and reviews of the subject are listed below.

There are a number of reviews of the AdS/CFT correspondence. The standard reference is O. Aharony, S.S. Gubser, J. Maldacena, H. Ooguri and Y. Oz (2000): Large N Field Theories, String Theory and Gravity.

Various aspects of the AdS/CFT integrability are covered by a collection of review articles N. Beisert et al.: Review of AdS/CFT Integrability: An Overview.

Quantum field theory

• Textbooks; Monographs:

S. Pokorski (1987): Gauge Field Theories; Cambridge, UK; Univ. Pr. (Cambridge Monographs On Mathematical Physics).

J. Zinn-Justin (1993): Quantum field theory and critical phenomena; Int. Ser. Monogr. Phys. 85, 1 (1993).

M.E. Peskin and D.V. Schroeder (1995): An Introduction to quantum field theory; Reading, USA: Addison-Wesley.

M. Srednicki (2007): Quantum field theory; Cambridge, UK: Univ. Pr.; 641 p.

• Lecture Notes; Online lectures:

David Tong: Lectures on Quantum Field Theory

Program in QFT at IAS (1996-1997)

J. Zinn-Justin: Semiclassical methods: From quantum mechanics to quantum field theory; Saclay lectures, 2011

Prof. Niklas Beisert: Quantum Field Theory I (HS12)

Supersymmetry and supersymmetric gauge theory

• Textbooks; Monographs:

D. Bailin and A. Love (1994): Supersymmetric gauge field theory and string theory; Bristol, UK: IOP (Graduate student series in physics) 322 p.

J. Wess and J. Bagger (1992): Supersymmetry and supergravity; Princeton, USA: Univ. Pr. 259 p.

• Lecture Notes; Reports:

D.S. Berman and E. Rabinovici (2002): Supersymmetric gauge theories; Les Houches Lectures 2002, [hep-th/0210044].
Extensive introduction to supersymmetry, with emphasis on supersymmetric quantum mechanics, superspace variables, susy breaking, etc.

N. Seiberg (2001): The Power of duality: Exact results in 4-D SUSY field theory; Int. J. Mod. Phys. A 16, 4365 (2001), [hep-th/9506077].
Short introduction to supersymmetric gauge theory and dualities.

K.A. Intriligator and N. Seiberg (1996): Lectures on supersymmetric gauge theories and electric - magnetic duality; Nucl. Phys. Proc. Suppl. 45BC (1996) 1 [hep-th/9509066].

String theory

• Textbooks; Monographs:

B. Zwiebach (2009): A first course in string theory; Cambridge, UK: Univ. Pr.
Excellent introduction to string theory that requires only minimal background.

R. Blumenhagen, D. Lüst and S. Theisen (2013): Basic concepts of string theory.
Good introduction to basic string theory.

M.B. Green, J.H. Schwarz and E. Witten (1987):
Superstring Theory. Vol. 1: Introduction; Cambridge, UK: Univ. Pr.
Superstring Theory. Vol. 2: Loop Amplitudes, Anomalies And Phenomenology; Cambridge, UK: Univ. Pr.
The classic textbook on string theory.

J. Polchinski (1998):
String theory. Vol. 1: An introduction to the bosonic string, Cambridge, UK: Univ. Pr. (1998)
String theory. Vol. 2: Superstring theory and beyond, Cambridge, UK: Univ. Pr. (1998)

• Lecture Notes; Reports:

D. Tong (2012): String Theory; [arXiv:0908.0333/hep-th].
Nice modern introduction to string theory.

G. Arutyunov, S. Frolov (2009): Foundations of the AdS5 x S5 Superstring Part I; [arXiv:0901.4937].
Good introduction to the Green-Schwarz approach to string theory in AdS_5 \times S^5.

Freddy Cachazo's course on String theory at the PI

Barton Zwiebach's course on String theory at the PI

Conformal field theory

• Textbooks; Monographs:

P. Di Francesco, P. Mathieu, D. Senechal (1999): Conformal Field Theory; Springer;
Comprehensive textbook on CFT, focusing mostly on the bulk theory.

R. Blumenhagen and E. Plauschinn (2009): Introduction to conformal field theory;
Lect. Notes Phys.779 (2009) 1.
Textbook covering basic aspects of conformal field theory and its applications to perturbative string theory.

A. Recknagel, V. Schomerus (2013): Boundary conformal field theory and the world-sheet approach to D-branes; Cambridge University Press.
Comprehensive monograph on boundary conformal field theory.

• Lecture Notes; Reports:

P. Ginsparg (1988): Applied Conformal Field Theory;
A classic introduction to early developments in conformal field theory.

M. Gaberdiel (2000): An Introduction to Conformal Field Theory; Rept. Prog. Phys. 63 (2000) 607
A short introduction to bulk conformal field theory that focuses on an algebraic approach.

K. Gawedzki: Lectures on conformal field theory;
IHES-P-97-2; in Edward Witten, Robbert Dijkgraaf, Pierre Deligne u.a. Quantum field theory and strings: a course for mathematicians (IAS/Park City Lectures 1996/97), American Mathematical Society 1999
A short introduction to bulk conformal field theory that focuses on a path integral approach.

V. Schomerus (2002): Lectures on branes in curved backgrounds;
Class. Quant. Grav. 19 (2002) 5781 [hep-th/0209241].
A short introduction to boundary conformal field theory and applications to D-branes.

V. Schomerus (2006), Non-compact string backgrounds and non-rational CFT,
Phys. Rept. 431 (2006) 39, [hep-th/0509155].
A short introduction to non-rational conformal field theory and in particular to Liouville field theory.

Jaume Gomis' course on conformal field theory at PI
An introduction to the key ideas and techniques of CFT.


• Textbooks; Monographs:

R. Baxter (1982): Exactly solved models in statistical mechanics;

M. Gaudin (1983): La fonction d'onde de Bethe; Masson.

O. Babelon, D. Bernard and M. Talon (2007): Introduction to classical integrable systems;
Cambridge University Press.

V. E. Korepin, N.M. Bogoliubov and A.G. Izergin (1997): Quantum Inverse Scattering Method and Correlation Functions; Cambridge University Press.

• Lecture Notes; Reports:

L.D. Faddeev (1996): How Algebraic Bethe Ansatz works for integrable model; [hep-th9605187].
Self-contained review on the Algebraic Bethe Ansatz for the su(2) XXX spin chains.

O. Babelon (2007): A short introduction to classical and quantum integrable systems; Saclay lectures;

N. A. Slavnov (2007): The algebraic Bethe ansatz and quantum integrable systems; Russian Math. Surveys 62:4 727–766.

N. Reshetikhin (2010): Lectures on the integrability of the 6-vertex model;

H. Saleur (2011): Super Spin Chains and Super Sigma Models: a Short Introduction; Lectures Les Houches 2008, Oxford University Press.

R. Nepomechie (1999): A spin chain primer;

M. Karbach and G. Muller (1998): Introduction to the Bethe ansatz I;

M. Karbach, K. Hu and G. Muller: Introduction to the Bethe ansatz II, III;

• Exact S-matrices:

G. Mussardo (1992): Off critical statistical models: Factorized scattering theories and bootstrap program; Phys. Rept. 218 (1992) 215

P. Dorey (1998): Exact S-matrices

H. Saleur (1998): Lectures on non perturbative field theory and quantum impurity problems; proceedings of the 1998 Les Houches Summer School;

H. Saleur (2000): Lectures on non perturbative field theory and quantum impurity problems. Part II; [preprint cond-mat/0007309].


O. Aharony, S.S. Gubser, J. Maldacena, H. Ooguri and Y. Oz (2000): Large N Field Theories, String Theory and Gravity; Phys. Rept. 323 (2000) 183.
Background on N=4 SYM and aspects of the AdS/CFT correspondence.

E. D’Hoker and D. Z. Freedman (2002): Supersymmetric gauge theories and the AdS/CFT correspondence; [hep-th/0201253].
Background on N=4 SYM and aspects of the AdS/CFT correspondence.

AdS/CFT and Integrability

• Lecture Notes; Reports:

J. Plefka (2005): Spinning strings and integrable spin chains in the AdS/CFT correspondence; [hep-th/0507136].
Rather short review of basic aspects of the duality between spinning strings and spin chains.

N. Beisert (2005): The dilatation operator of N = 4 super Yang-Mills theory and Integrability; Phys. Rept. 405 (2005) 1, [hep-th/0407277].
N. Beisert's PhD thesis.

N. Beisert et al. (2010): Review of AdS/CFT Integrability: An Overview;
23 contributions on different aspects related to the AdS/CFT integrability.

V. Kazakov (2007): Integrability and its applications to AdS/CFT; Graduate lectures at Sapporo Winter School.

V. Kazakov (2007): Solving exactly the planar N=4 SYM theory; Third International School on Symmetry in Integrable Systems and Nuclear Physics; Zakhgadzor, Armenia.

• Special issues of J. Phys. A on integrability and the AdS/CFT correspondence:

Recent advances in low-dimensional quantum field theories; J. Phys. A 39 (2006) number 41

Gauge–string duality and integrability: progress and outlook; J. Phys. A 42 (2009) number 25

Quantum integrable models and gauge-string duality; J. Phys. A 44 (2011) number 12


R. Roiban, M. Spradlin and A. Volovich (2011): Scattering amplitudes in gauge theories: progress and outlook; J. Phys. A: Math. Theor. 44 450301 [doi:10.1088/1751-8113/44/45/450301].

R. Britto (2011): Introduction to Scattering amplitudes; Saclay lecture notes;

Matrix models

• Textbooks; Monographs:

M.L. Mehta (2004): Random Matrices; Academic Press, (3rd edition).

P.J. Forrester (2010): Log-gases and Random Matrices; Princeton University Press.

• Lecture Notes; Reports:

P. Di Francesco, P. H. Ginsparg, and J. Zinn-Justin (1995): 2D Gravity and Random Matrices; Phys. Rept. 254, 1 (1995); [hep-th/9306153].

I. R. Klebanov (1991): String Theory in Two Dimensions; [hep-th/9108019].

M. Marino (2014): Lectures on non-perturbative effects in large N gauge theories, matrix models and strings;

N=2 Integrability

• Textbooks; Monographs:
A. Marshakov (1999): Seiberg-Witten Theory and the Integrable Systems‬; ‪World Scientific‬.

Online lectures

Link for conferences and lectures held in IHES:

Link for lectures held in Perimeter Institute:


Link for lectures note for lectures held in Saclay: