Updates
March 30: New lecture notes on modular forms
March 17: New references for modular forms
March 4: New references for representation theory of sl(2) and the Weyl character formula
Feb 19: New literature suggestions on Lie algebras
Feb 3: More detailed goals and new literature suggestions below!
Project goals
Basic task: Study the connection between automorphic forms and statistical mechanical
models of crystals and ice.
Visionary goal: Use the above connection to study instanton effects in string theory, which are known to
be captured by automorphic forms.
Road map:
Step 1. Understand why partition functions of the 6-vertex model (“square ice”) can be written in terms
of products of Schur polynomials. Show this explicitly in low-dimensional examples and try to understand
the general proof. Literature help indicated below.
Step 2. Show that the same partition function appears as in the Fourier coefficients of certain modular
functions known as Eisenstein series. Daniel and Henrik will provide substantial help with this point.
Step 3*. Analyze the relations between the above results and instanton effects in string theory. NB: This is
a research level problem and we will probably only have time to discuss this at the very end!
Literature
Statistical physics
A nice summary of the connection we seek to understand is the paper
B. Brubaker, D. Bump, S. Friedberg, “Eisenstein Series, Crystals and Ice” (pdf)
Don’t attempt to read this from page one, but skip directly to section 7 and try to get an
idea of the statistical mechanical concepts that are involved. You will find the following
book very useful:
R. J. Baxter, “Exactly solved models in statistical mechanics” (pdf)
Again, don’t attempt to read this from cover to cover! Use the paper above for guidance. Models
of ice are analyzed from chapter 7 and onwards.
In order to complete Step 1 of the road map you will in addition find help from the following references.
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B.Brubaker, D. Bump, S. Friedberg, “Schur polynomials and the Yang-Baxter equation” (pdf)
First look in section 2 of this paper for the basic stuff and some examples. You will also find
section 2 of the following paper useful:
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B.Brubaker, D. Bump, G. Chinta, S. Friedberg, P. E. Gunnells, “Metaplectic Ice” (pdf)
For the general proof that the partition function can be written in terms of Schur polynomials,
see Theorem 9 on page 16 of the paper “Schur polynomials...”. There it is written in terms of the
(a, b, c) labelling of the Boltzmann weights. A much nicer form is obtained by going to the (t,z)-variables
which is done in section 5 of the same paper; see eq. (25) for the final result.
As a first step, start by rewriting the partition function for the example \lambda=(0,0) on page 11 in the general
form given in Theorem 9. Then change variables as in eq. (24) and rewrite it in the form of eq. (25).
Group theory and Lie algebras
* The following paper is a good starting point:
A. Kirillov, “Introduction to Lie Groups and Lie Algebras” (pdf)
Here I recommend especially section 3.10 and the first sections of chapter 5.
* Another good source is:
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B.C. Hall, “An Elementary Introduction to Groups and Representations” (pdf)
See in particular the first few sections of chapter 3 which contains useful info on how
to exponentiate matrices to go from the Lie algebra to the group.
* It might also be useful to take a look at the notes I wrote during the bachelor
project on group theory a few years ago. They can be found on this page:
http://www.danper.se/Daniels_homepage/GroupTheory.html
See in particular parts 1, 2, 4.
* An additional source that can be useful for the representation theory of sl(2) is
section 2 of the following paper:
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L.Banu, “Representation theory of the Lie algebra sl(n, C)” (pdf)
* For more details on the Weyl character formula you can take a look at:
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B.Reason, “Weyl’s Character Formula for Representations of Semisimple Lie Algebras” (pdf)
See section 4.1 for the specific example of sl(2).
Modular forms
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*A basic introduction to modular forms can be found in the following
notes I wrote for a series of 4 lectures at the Albert Einstein Institute:
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*For some basic stuff about holomorphic modular forms, see
sections 1.1, 1.2, 2.1 and 2.2 of
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D.Zagier, “Elliptic modular forms and their applications” (pdf)
(ignore the parts of these sections marked with the spades symbol)
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*For the detailed calculation of the Fourier expansion of the non-holomorphic
Eisenstein series (which is the most relevant object for you), see section 9.2 of
Henrik’s master thesis:
http://hgustafsson.se/files/publications/Eisenstein%20Series.pdf
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*If you are interested in a first glimpse of its physical interpretation in string theory you
can take a look at chapter 8 in my PhD thesis:
http://arxiv.org/pdf/1001.3154.pdf
Useful concepts to study closer
Mathematics:
Schur polynomials
Holomorphic functions
Fourier expansions of holomorphic functions
Integral forms of the Gamma function and modified Bessel function
Basics of group theory
The group SL(2,R) of 2x2 matrices with det = 1 and real entries
Physics:
Ising model
Boltzmann weight
Partition function in statistical mechanics
Basic group theory:
https://dl.dropboxusercontent.com/u/2728166/Papers%20and%20Notes/fufx02-12-04_final.pdf
For details about SL(2,R) you can also take a look at chapter 3 in Henrik’s master thesis: