# 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

physically relevant Yang-Mills theories. In addition, it is already known that

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.

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

physically relevant Yang-Mills theories. In addition, it is already known that

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.