# Integrability, Gauge Fields and Strings (07frg130)

Arriving in Banff, Alberta Sunday, July 22 and departing Sunday July 29, 2007

## Organizers

Gordon Semenoff (University of British Columbia)

## Objectives

Work in this field will develop the conjectured duality between gauge field
theory and string theory which has been a tantalizing idea in theoretical
elementary particle physics dating back almost fourty years. The most important
explicit example of such a duality that is known thus far is the conjectured
AdS/CFT correspondence between maximally supersymmetric Yang-Mills theory in four
space-time dimensions and IIB superstring theory in a particular ten-dimensional
backgroud geometry. This conjecture is perhaps the single most important result
of the string duality revolution of the late 1990's. The Yang-Mills theory
involved is a close relative of those quantum field theories that are components
of the Standard Model'' which are used to describe elementary particle physics
at all presently accessible energy scales. This practical utility of
four-dimensional Yang-Mills theories means that they will always inevitably be an
important part of physics. Understanding the details of their dynamics in some
kinematical regimes has proved a difficult problem. In fact, the puzzle of
solving the confinement problem, that is, if demonstrating that quantized
Yang-Mills theory has a gapped spectrum, is one of the Clay Foundation Millenium
Prize problems, see

http://www.claymath.org/millennium/Yang-Mills_Theory/ .

One appoach to this problem is to seek a string theory which is dual to the
Yang-Mills theory. The hope is that the string theory can yield quantitative
results in regimes where it is difficult to extract the same from Yang-Mills
theory, one plausibly being where the spectrum and mass gap are formed. The
maximally supersymmetric Yang-Mills theory that we are proposing to work on is
not of the kind which will generate a mass gap, so we will not address this
problem directly. However, we do believe that a thorough and perhaps even
complete understanding of this theory will give significant insight into the
classic problem of solving the planar limit of other closely related and more
the same integrable structures that we propose to study do indeed appear in
realistic theories, at least in certain kinematical regimes.

As well, a better understanding of the AdS/CFT duality will advance the
knowledge of the behavior of string theory. String theory is a theory of
quantum gravity and, seen from that point of view, duality between a string
theory in ten dimensions and Yang-Mills theory in four dimensions is an explicit
realization of the idea of holography, that the dynamical data in a quantum
theory of gravity can be encoded in another dynmaical theory which lives in a
space of lower dimension. The insights derived from this explicit example of
duality are driving a revolution in current thinking about quantum mechanical
effects in gravitational environments, an example being an entirely new paradigm
for the origin and nature of the thermal radiation that Hawking discovered is
emitted by black holes.

Work in this field will also advance the science of integrable systems. This
is normally in the domain of classical nonlinear partial differential equations,
but has also been extended to some two-dimensional quantum field theories. The
study of quantum integrability that is being developed by the current program is
important in lower dimensional models used in condensed matter systems. The two
best known examples are the Hubbard model and the spin chain, but there are many
others with applications in a wide array of physical situations, quantum
impurities, edge states in the Quantum Hall effect, quantum wires, carbon
nanotubes and models of cold atoms on optical lattices being examples.

In concrete terms, the long-term goal of this project is to find a complete
solution of the planar sector of four dimensional maximally supersymmetric
Yang-Mills theory. A secondary goal is to find a similar solution to the IIB
superstring theory and to understand how these are related. It is believed that
these are united in the same one-parameter model, whose complete quantum
integrable structure is yet to be established. A lot of evidence has been
accumulated by many groups to substantiate this claim, but there is as yet no
proof of this assertion.
What we specifically propose to do in this two-week focused research group is
to:

i) further develop the approach of Beisert, Kazakov, Marshakov and Zarambo which
encodes integrability data in algebraic curves.

ii) Incorporate techniques for solving long-ranged spin chains which are relevant
to some problems in Yang-Mills theory and that have been developed in the context
of condensed matter systems.

iii) Pursue a connection of the above strategy with the Hubbard model, which has
recently been noticed by Rej, Serban and Staudacher.