The course on String/M theory is divided into two parts,
String/M theory (5p) and Advanced string/M theory (5p). The first one
is offered in the Master program while the second, more advanced, course is
suitable for students who aim at doing a Master thesis (20p or 40p)
on a related topic.
Teacher: Professor Bengt E.W. Nilsson.
Introductory meeting 2004: March 17 at 13.15 in Origo 6115
(the physics building called Origo, 6th floor, north wing).
String/M theory (spring 2004) is a course that is
more stream-lined and
less demanding course than the version taught in 2002 and 2003.
It begins with an introductory
discussion of the fundamental problems encountered when
trying to understand our universe in terms of standard (quantum)
field theory methods, and how
string theory may solve them. String theory is then introduced and some of
its properties studied. In particular we cover
the conformal field theory formulation,
low energy supergravity theories, scattering amplitudes,
extra dimensions and how their
compactifications can be related to elementary particles.
D-branes and non-perturbative aspects are discussed and the
connection to eleven dimensional M-theory briefly explained.
Content: see the individual lectures below
(will partly overlap with Chapters 1,2.1-2.2,5.1-5.3,6 of
String theory by Bengt EW Nilsson)
Literature: "D-branes", by Clifford V. Johnson (CUP 2003) or its
shorter version hep-th/0007170 on the net called
"D-Brane Primer".
For additional references, see the link to "Guide..." below.
Advanced string theory is a course introducing more
advanced mathematical tools and concepts needed to penetrate deeper
into the structure of string/M theory. Loop corrections, partition
functions and Riemann surfaces are discussed, together with some recent
results connected to gauge theories, AdS/CFT, p-, D-, and M-branes, dualities
and the non-perturbative properties of
string/M theory.
Content: not yet decided (will partly overlap with
Chapters 2,4,5,8,10,11,12 of
String theory
by Bengt EW Nilsson)
Literature: see link to "Guide..." below.
LITERATURE:
Non-technical string literature:
An excellent popular account of the fundamental questions and ideas
of modern string/M theory can be found in
"The elegent universe", by Brian Greene (Jonathan Cape 1999).
Additional string literature: (abbreviation in bracket)
1. M. Green, J. Schwarz and E. Witten (GSW), 'Superstring theory',
volume I and II
(Cambridge university press 1987).
2. J. Polchinski (JP), 'String theory', volume I and II
(Cambridge university press 1998).
3. D. Lüst and S. Theisen (LT), 'Lectures on string theory',
(346 Lecture Notes in Physics, Springer verlag 1989).
General high-energy physics:
A very nice overview of elementary particle physics, gravitation and cosmology, Kaluza-Klein,
supersymmetry and introductory string theory can be found in
"Particle physics and cosmology", by P.D.B. Collins, A.D. Martin and E.J. Squires (Wiley 1989).
This book will be referred to as (CMS) in this course.
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String/M theory, Master course, 5p, spring 2004, Lp IV
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Course codes: CTH: FFM 485 and GU: FY 4850
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The main text used in this course is Clifford Johnson's review article
hep-th/0007170 on the net called
"D-Brane Primer".
Another set of introductory lecture notes
to string theory is:
Lecture notes by Paolo Di Vecchia
and here you will find a collection of other review articles
Guide to the string/M theory literature
Some articles from Physics Today:
Witten, Physics Today, April 1996, p. 24-30
Kane, Physics Today, Febr 1997, p. 40-42
Collins, Physics Today, March 1997, p. 19-22
Witten, Physics Today, May 1997, p. 28-33
Schedule: Three lectures each week in
Origo 6115
Tentative schedule: Monday, Wednesday and Friday at 10.00-12.00
Examination:Home problems and, if you aim at top marks, an oral exam.
Lecturer:
Bengt EW Nilsson :bengt.nilsson@fy.chalmers.se
phone: 772 3160
Home page with more info on string theory: fy.chalmers.se/~tfebn/
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Lecture(date) and content
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1ST WEEK: Introduction: classical and quantum point particles (week 12)
Lecture 1: Wednesday March 17, 13.15-15.00
Reviews of important topics from classical mechanics and QM:
a) point particles, harmonic oscillator, quantization, see Sakurai's book
"Modern Quantum Mechanics"
b) the relativistic point particles, world line physics:
read section 2.1 in
C. Johnson hep-th/0007170,
Lecture 2: Friday March 19, 10.00-12.00
Basic classical and quantum field theory
Problems with the Standard Model of elementary particles and gravitation
The motivation for strings and a brief overview of string/M-theory:
have a look at the
review by M.J. Duff but read only Section 1.
See also section 2.1 of
Polchinski hep-th/9411028.
For a review of results and a comparison to the alternative approach known
as Loop quantum gravity, see
Lee Smolin hep-th/0303185
but do
keep in mind that
the author is not a string theory person, although he claims he is.
HOME PROBLEMS 1ST WEEK:
1. Prove that the symmetries given on page 9 in Johnsons review article
hep-th/0007170
leave the particle action with the auxiliary field
eta invariant. Explain also the form of the variation of eta.
2. Consider a real scalar field in four dimensions with a potential V
that is bounded from below, and has two degenerate
global minima and one local
maximum between them. Give a function for such a potential and
derive the expressions
for the mass of the field at the three
different extrema? Discuss the different situations
in terms of stability of the vacuum.
3.-6.:see the handed out problem sheet. Note that some formulae
are written in a time-like metric: Make the changes needed for them
to be valid in space-like metric before you solve the problems.
The problems 3 to 6 are:
3: Homework problem 1.1 and 1.2,
4: Homework problem 2,
5: Homework problem 3.1,
6: Homework problem 4.
Deadline for handing in the problems: Friday April 2, 2004
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2ND WEEK: The classical bosonic string (week 13)
Lecture 3: QFT, and quantum gravity (cont. from last week)
Dirac eq, brief history of string theory
(see lecture notes and Di Vecchia's lecture notes Chap 1)
Lecture 4: World sheet physics
the two different but classically equivalent actions, symmetries
Lecture 5: More world sheet physics
Reading material on world sheet physics:
At a first reading, read the following pages in the order suggested:
1. Johnson hep-th/0007170, section 2.2, page 9 to page 11, eq 15,
2. Di Vecchia (link above), Chap 2: page 14 from eq. 2.2.10 to page 17 eq. 2.2.36,
3. Johnson pages 12, 14, 15 and 16,
4. The material on p. 16 and 17 under the heading 'More terms' requires
some understanding of topology and is optional here, but will be
important later in the course!
HOME PROBLEMS 2ND WEEK:
1. Write down the two possible actions, like S and S'
for the string and particle cases, for a membrane, i.e. a two
dimensional surface propagating
in spacetime producing a three dimensional world sheet, and show that
they are classically equivalent. Hint: The action with an independent
world sheet metric must contain a cosmological term for this to work.
Are these membrane actions Weyl invariant?
NOTE: some more info about this problem will be provided Monday week 3!.
2. Homework problem 5.1 and 5.2.
3. Homework problem 6.1 and 6.2.
4. If you have taken the course Gravitation (Weinberg) do also
Homework 6.3.
Deadline for handing in the problems: Friday April 23, 2004
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3RD WEEK: World sheet techniques, quantization and CFT (week 14)
Read Johnson hep-th/0007170:
a) The rest of sect 2.2, starting from
"The stress tensor": page 17-23;
b) Sect 2.3 and Sect 2.4 observing misprints
in eq (70) and in connection with eq (83)
Lecture 6: World sheet stress tensor etc Johnson: the rest of section 2.2
Lecture 7: Quantization of the bosonic string: Johnson sect 2.3
the mass spectrum (lecture notes and handed out copies from Lust and Theisen).
Lecture 8: Conformal field theory and the Virasoro algebra (lecture notes)
Note: when the propagator between X(z) and X(w) is computed in the closed string,
as done in the lecture, the oscillator and zero mode contributions will only add correctly
if both left and right moving parts are included. The answer is then -1/2ln|(z-w)|.
HOME PROBLEMS 3RD WEEK:
1. Compute the propagator between X(z,zbar) and X(w,wbar) in the closed string. (See note above.)
2. Use the previous result to compute the OPE (the operator product expansion)
of two stress tensors T(z) and T(w) (no zbar or wbar) defined as in the lecture.
Deadline for handing in the problems: Friday April 30, 2004
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4TH WEEK: CFT and the effective action
Lecture 9: OPE's, the Virasoro algebra and vertex operators
Read: lecture notes.
Lecture 10: Correlation functions and effective actions
Read: lecture notes.
Lecture 11: The path integral and the double perturbative expansion:
World sheet vs. spacetime perturbation theory. Curved backgrounds.
Read: lecture notes and Johnson section 2.7 plus the oriented cases
in "Insert 5" on page 44 and "More terms" on pages 16 and 17.
And take a brief look at section 2.5 in Johnson.
HOME PROBLEMS 4TH WEEK:
Derive the OPE of the stress tensor T(z) and the tachyon vertex
operator V(w) for the closed bosonic string.
Deadline for handing in the problems: Friday May 7, 2004
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5TH WEEK: Superstrings
Lecture 12: The bosonic string effective action.
Lecture 13: NSR formalism, GSO and the superstring spectra.
Lecture 14: Compactification, vertex operator realization of the SU(2) Kac-Moody algebra,
self-dual lattices and the heterotic string.
The classification of string theories.
Read: lecture notes for details and the over-view in Johnson chapter 5:
sections 5.1 and 5.2 (page 85 - including Insert 9),
and section 5.4 (p.95-p.97).
HOME PROBLEMS 5TH WEEK:
1. Derive the Weyl rescaling formula given in eq 115 in Johnson's
hep-th review from eq 114, and then show that eq 116 follows from eq 111
as claimed by Johnson.
2. Derive the SU(2) Kac-Moody algebra from the vertex operators :exp(ipX(z)):, :exp(-ipX(z)):
and H(z) which is X(z) with a z-derivative acting on it. Note that the square of p is 2 and that
X(z) is the compact left moving X(z) with propagator -log(z-w) (i.e. with no factor of 1/4).
Deadline for handing in the problems: Friday May 14, 2004
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6TH WEEK: Dualities and branes
NOTE: lectures this week Monday 10.00-12.00, Wednesday 10.00-12.00, Thursday (in Delfinen) 9.00-12.00
Read: Johnson sections 3.1, 3.3, 3.7, 4.1, 4.2, 4.4, 4.5, 5.3, 5.9 and 5.10
Lecture 15: T-duality and D-branes
Lecture 16: Effective actions, S-duality and intro to non-perturbative aspects
Lecture 17: More on non-perturbative aspects: solitonic solutions
in supergravity. Fundamental vs solitonic degrees of freedom.
HOME PROBLEMS 6TH WEEK:
Show that the type IIB supergravity action is invariant under S-duality
by showing that the S-duality transformation is part of a bigger
symmetry given by projective transformations in the complex variable
\tau=\chi+i\exp{-\phi}. Type IIB is therefore said to be S-self-dual.
Why is this a remarkable property?
Deadline for handing in the problems: Wednesday May 17, 2004
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7TH WEEK: M theory and the return to four dimensions
Lecture 18: Solitonic p-brane, ADM mass,
charge and the Bogomolyi bound and BPS condition: see Argurio's thesis
sect 3.1 to 3.2.2)
"Brane physics in M-theory".
Lecture 19: Eleven dimensions, ADM mass,
charge and the Bogomolyi bound and BPS condition
(see Argurio's thesis
sections 3.2.3 and 3.2.5
"Brane physics in M-theory").
the connection between 11 and 10
dim's,
compactifications, the big picture, membranes see Chap I in
Taylor
"M(atrix) theory").
see also e.g.
Duff, Nilsson and Pope, "Kaluza-Klein supergravity" Phys Rep vol 130 (1986) p. 1-142,
Duff "M-theory on manifolds of G_2 holonomy" hep-th/0201062,
Nicolai and Helling "Supermembranes and M(atrix) theory" hep-th/9809103
Lecture 20: Comments on the relation
to physics in four dimensions, summary
read Cheung hep-ph/0305003 (NB ..-ph/..)
and the Introduction of Denef et al hep-th/0404257
HOME PROBLEMS 7TH WEEK: None
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8TH WEEK: Examination week (week 21)
Please contact me about the oral examination. Possible times are
May 17-19, from 09.00 to 18.00 and May 24 from 13.00 to 20.00
Master thesis work in string/M theory:
Send me an email as soon as possible if you are interested
(even if you have already discussed it with me or some other supervisor
in theoretical physics).
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