String structure at finite temperature

Menika Sharma(Indian Institute of Science)

Based on:1) Strings at Finite Temperature: Wilson Lines, Free Energies, and the Thermal

Landscape, Dienes, Lennek and Sharma, Phys. Rev. D86(2012) 0660072) S-duality at Finite Temperature, Dienes and Sharma, Submitted to Phys. Rev. D

In collaboration with: Keith Dienes, Michael Lennek

Why study string thermodynamics?

• Cosmological applications.

• Simple lab for supersymmetry breaking in string theory.

– Finite temperature effects always break susy

• In recent years, not much effort has been expended on traditional string thermodynamics, yet many thorny issues remain in the field.

– The high-temperature behaviour of string theory should say something about its fundamental degrees of freedom.

The temperature-geometry correspondence in field theory

• Partition function

• Free-energy density

• Free energy density = Vacuum energy density of the quantum theory compactified on a circle of radius

with bosons having periodic b.cfermions having anti-periodic b.c

Finite-temperature formulation of string theory

• Compactified string theoryShould obey the symmetries of the torus – modular invarianceHas winding modes in addition to momentum modes

• The finite-temperature partition function of a string theory has no knowledge about either of these.

Finite-temperature formulation of string theory

• Compactified string theoryShould obey the symmetries of the torus – modular invarianceHas winding modes in addition to momentum modes

• The finite-temperature partition function of a string theory has no knowledge about either of these.

The temperature-geometry correspondence holds for string theory as well.

Polchinski,Mclain and Roth, O’Brien and Tan

Theories with gauge fields have an extra degree of freedom at finite temperature.

Theories with gauge fields

• Momentum along the compactified direction is quantized

• Switch on a background gauge field given by,

• Locally this is pure gauge as field strength vanishes

• Nonetheless such a Wilson line shifts the momentum of a state in the string spectrum charged under the gauge field

It means that at finite temperature, bosons do not necessarily have integer modings and fermions half-integer modings around the compactified circle.

X X X

X XX X

X X X

Theories with gauge fields

• What is the interpretation of the gauge field on the thermal side?

– The gauge field corresponds to a chemical potential.

Heterotic and Type I strings:Very different provenances same massless states…. Yet strikingly similar behaviour

This similarity does not extend itself to finite temperature.

• Hagedorn temperature of heterotic strings:

• Hagedorn temperature of Type I strings:

Heterotic and Type I strings:Very different provenances same massless states…. Yet strikingly similar behaviour

Hagedorn temperature: Temperature at which the free energy diverges.

The odd man out

• Temperature always breaks supersymmetry• The traditional heterotic finite-temperature model

• Although the heterotic geometric model starts out equal to the Boltzmann sum, this correspondence breaks at a certain temperature.

Heterotic SO(32) Heterotic SO(32)

Dienes and Lennek

Consistent heterotic models at finite temperature with Wilson lines

Consistent heterotic models at finite temperature with Wilson lines

SO(32) B

SO(32) A

How to choose the correct thermal theory?

• The correct thermal model should be determined dynamically

• Choose the model that has the least free energy

Dienes, Lennek and Sharma

General Wilson line in heterotic string theory• The thermal partition of string theory deforms when one

switches on a general Wilson line

Structure of heterotic theory at finite temperature

Summary of finite-temperature heterotic theory

• At finite temperature, heterotic theory can occupy two possible states. Both the A and B theories are equally likely as thermalheterotic theories.

• If for some reason the lowest state is unphysical the theory will automatically be in the next available minima, state B.

• If it occupies the state B, it has a Hagedorn temperature equal to Type II and Type I theory. At the same time, it exhibits normal thermodynamic behaviour at all temperatures.

• It is also possible that the heterotic string exhibits a double phase transition, first switching from phase A to phase B and then undergoing a “normal” Hagedorn transition.

The structure of Type I theory at finite T

Essentially same behavior as Heterotic but also one major difference

Why is there a difference between Heterotic and Type I thermal behavior?

• Difference can be traced to the presence of massive gauge group spinor states in the heterotic theory.

• These states respond differently to the two Wilson lines

Why is there a difference between Heterotic and Type I thermal behavior?

• Difference can be traced to the presence of massive gauge group spinor states in the heterotic theory.

• These states respond differently to the two Wilson lines

• Non-perturbative spinorial states in Type I

D-string D-particle

Masses go as 1/g

Polchinski and Witten Sen

Hints at an S-duality at finite temperature

A picture of S-duality at finite temperature

Dienes and Sharma

A freely acting orbifold that continuously deforms supersymmetric theories that are dual pairs will lead to theories that are also dual pairs.

Vafa and Witten

The adiabatic argument

The A and B theories in the Type I’ picture

Spectral flow in heterotic theory

String creationin Type I’ theory

All supersymmetric theories are alikebut each non-supersymmetric theory is messy

in its own way

All supersymmetric theories are alikebut each non-supersymmetric theory is messy

in its own way

• A non-zero free energy generates a runaway potential for the dilaton field so that…

All supersymmetric theories are alikebut each non-supersymmetric theory is messy

in its own way

• What sense does it make to talk about a finite-temperature theory as one increases the coupling?

All supersymmetric theories are alikebut each non-supersymmetric theory is messy

in its own way

• What sense does it make to talk about temperature as one increases the coupling?

• Jeans Instability

Conclusions

• There is a way we can evade the Jeans instability and apply the adiabatic argument.

• Further, using D-strings we can show that n=1 winding states match on both sides of the duality relation. (for both the A and B theories)

• This may look like a fluke, but it may also signal the presence of hidden symmetries in finite-temperature string theories.

Conclusions

To know more see:

S-duality at finite temperature, Dienes and Sharma