Theory and applications of Boundary Sine-Gordon Theory Gordon W. Semenoff University of British Columbia http://www.nbi.dk/~semenoff Field Theories for Quantum Coherent Devices, JOSNET School, Villa Orlandi, Capri, June, 2005.

Boundary Sine-Gordon Theory Approaches to solution: i) semi-classical techniques ii) conformal field theory iii) Integrability – Bethe ansatz iv) fermionization References: H. Saleur: arXiv:cond-mat/9812110, cond-mat/0007309 C.Callan, I.Klebanov, A.Ludwig and J.Maldacena, hep-th/9402113 J.Polchinski and L.Thorlacius, hep-th/9404008 T.Lee and G.W.Semenoff, hep-th/0502236, in preparation.

i)Tachyons in string theory. A.Sen, hep-th/0410103 Applications: i)Tachyons in string theory. A.Sen, hep-th/0410103 ii)Quantum impurity problems. H. Saleur, arXiv:cond-mat/9812110, cond-mat/0007309 iii)Dissipative Hoffstaeder model. C.Callan and D.Freed, hep-th/9110046 C.Callan, A.Felce and D.Freed, hep-th/9202085 iv)Josephson junction arrays. D.Giuliano and P.Sodano, cond-mat/0501378

Tachyons in String Theory: Witten (1992): The space of classical configurations of open string theory = the set of all boundary field theories. Polyakov action of the string where world-sheet is a disc, D. Boundary terms with condensates of open string fields. tachyon, photon, massive tensor,… The equation of motion in string theory is equivalent to requiring conformal invariance. An exact classical solution of bosonic open string theory on Minkowski space is a boundary conformal field theory on a disc.

rolling tachyon, which describes the decay of an unstable D-brane. Open String Tachyon Both open and closed bosonic string theories have a tachyon in their spectrum. The existence of a tachyon in the spectrum is a signal of instability of the perturbative ground state . Some beautiful ideas about open string tachyons have been introduced by Ashoke Sen. A D-brane is an extended object occupying a sub-manifold of space-time where open strings are allowed to begin and end. The vacuum energy of the open string is equal to the D-brane tension. The open string has a tachyon because the D-brane is unstable and it can decay. The end-point of the decay is a lower dimensional (and still unstable) D-brane or the closed string vacuum. There is an exact solution of classical open string theory, called the rolling tachyon, which describes the decay of an unstable D-brane.

Open String Tachyon Potential V(<T>) TD25 <T(x)> perturbative open string vacuum = D25-brane. rolling tachyon TD25 closed string vacuum. <T(x)> Boundary conformal field theories: 2-dim. (disc) field theory of scalar embedding functions of the string with i)Neumann boundary condition for the string coordinate = coordinate extended in world-volume of a D brane. ii)Dirichlet boundary condition for the string coordinate = coordinate transverse to the world-volume of a D-brane, where only closed strings propagate. iii)The rolling tachyon describes the decay of the unstable brane where a coordinate goes from i) to ii).

Examples of conformal field theories of tachyons. i) Bosonic string theory with Neumann boundary conditions ii) Bosonic string theory with Dirichlet boundary conditions iii) Bosonic string theory with ``rolling tachyon’’ condensateg

If the boundary operator is not an exactly marginal the boundary field theory is not a solution of classical open string theory, but an off-shell field configuration: When a0 this is a Neumann boundary condition. When ainfinity it is a Dirichlet boundary condition. Renormalization group flow. In the UV limit, the boundary condition is Neumann. In the IR limit, it is Dirichlet. The IR stable boundary condition is the more stable string state (no open string tachyon).

Relation with dissipative quantum mechanics: Begin with action for open string on disc geometry, Integrate out the modes of the scalar field that live in the bulk of the disc to get a theory of the field on the boundary only. Recognize first tern as Caldeira-Leggett term coming from coupling particles whose coordinate is Xi to an external heat bath. Often, V(X) is taken to be a periodic potential and the gauge field Ai(X) a constant magnetic field  dissipative Hoffstaeder model.

Boundary Sine-Gordon Model Consider the bosonic field theory in two dimensions with action Space-time is a strip: When , for any value of , this is a conformal field theory. It is also an interesting field theory for other values of . a Free field theory in the ``bulk’’ Interactions on the boundaries

a We will compute the partition function Field theory with Euclidean time which is periodic b Space-time is a Euclidean cylinder. a

The path integral representation of the partition function, We use the observation that this path integral can be viewed as either - thermodynamic partition function in field theory on a spatial line segment with periodic time and with boundary interactions or - the transition function in field theory on a circle, with Euclidean time between initial state and final state with wave-functionals of initial and final boundary states.

Boundary states – a partition function for a system with a boundary can be presented in two ways Periodic time (thermal partition function) b The partition function for field theory at temperature 1/b on a line of length a has particular boundary conditions at the spatial end-points and periodic Euclidean time b. a spatial boundaries

Alternatively, it is the amplitude for time evolution between two boundary states during time a in a field theory on the circle with circumference b. final boundary state a |B> and <B| are initial and final states and H is the Hamiltonian of field theory on a circle. b Periodic space initial boundary state

We can get this path integral by first of all quantizing the theory with action Then computing the euclidean time evolution amplitude between boundary states. Now is the time and it runs over The canonical commutator is When we go back to Euclidean space it becomes

If we consider a wave-functional of the boundary state, The canonical commutator implies that canonical momentum operates like a functional derivative The boundary state wave-function then obeys the equation Note that this is just the same condition that is obtained as a boundary condition on the strip. There, it guarantees that equations of motion do not have boundary terms.

Operator techniques: -Quantized Euclidean field theory on the cylinder. -For each boundary interaction, there is a boundary state. -We can consider field theories with a variety of boundary interactions on each of the two boundaries by constructing the appropriate boundary states. -When we can find the explicit boundary states, this is an effective technique for computing the partition function.

Consider the boson theory defined on the cylinder, (now we have interchanged and from the previous discussion)

Normal ordering puts negatively moded oscillators to the left of positively moded oscillators, zero modes are left alone..

Consider a compact boson with radius R: The total momentum is quantized in units, n/R. The integers w are wrapping numbers.

n is the number of momentum quanta and w is the wrapping

Fermionization: are co-cycles which make and anti-commute.

The same correlators are produced by fermion fields with the mode expansion and the zero momentum vacuum

Quantum states of the fermion theory are obtained by These should correspond to states of the bosonic theory. Which states?

Quantization of the fermions v.s. quantum theory of bosons

Momentum of a state of the boson theory is equal to the fermion number in the fermion theory. Fermion number comes in quantized units, it is related to fermion number: We get only those states which have integer values of pR and pL Does this produce the states of a compact boson?

That momentum comes in quantized units is consistent with periodic identification of the boson field at two special radii any integer any even integer In case i), when pL and pR are both integers, since 2n is even, w must be even. How do we produce the states of the boson theory where w is odd? We need states where pL and pR are both half-odd-integers. This occurs when the fermions are periodic, instead of anti-periodic. Anti-periodic fermions are called Neveu-Schwarz (NS) Periodic fermions are called Ramond (R).

Half-odd-integer fermion number occurs when there are fermion zero modes – in the R sector:

To reproduce We need two sectors: where both pL and pR are integers – the NS-NS sector,. where both pL and pR are half-odd-integers – the R-R sector,. Consider all of the states in these two sectors. Then pL + pR and pL - pR run over all possible integers. Then project onto those combinations with even pL + pR by an analog of the “GSO projection” This produces all of the states of a compact boson. This is very similar to the GSO projection which produces the Type 0 string from the fermionic NSR string (an alternative one Produces the Type II superstring)

Partition function: To show that the boson and projected fermion theory have the same spectra, we can compute and compare the partition functions. The Hamiltonians are: The boson partition function is

The partition function of the fermions is Use the Jacobi triple product identity The spectra of the projected fermion theory and the compact boson theory are identical.

Boundary states: Let us construct a boundary state at time Either Neuman boundary condition or Dirichlet boundary condition Explicitly, What do these look like in the Fermionic variables?

Using the bosonic boundary conditions, we can also deduce the Boundary conditions for the fermions, These conditions have the solutions

Homework exercise i): Confirm that the fermionic boundary states work by computing partition functions: This is the partition function of the scalar field theory on a line segment with Neumann boundary conditions. We can also find the parition functions for the theory with Dirichlet or mixed Neumann-Dirichlet boundary conditions. For this, we could use either the boson or fermion representation of the boundary states and the Hamiltonian.

Homework exercise ii): We can now fermionize the boundary interaction with one specific period: The cosine is made from plane waves, The equation for the boundary state in the fermion variables is Solve this condition and compute the partition function of the boundary Sine-Gordon model with the above interaction.

Question: Are we stuck with the boson compactification radius or can we fermionize bosons with other radii? are only ever consistent with identifications where Further, we understand that two radii are related by T-duality.

T-duality

Polchinski and Thorlacius (1994): Doubling trick. Consider the periodic boundary potential with self-dual radius How can one use fermions when the natural fermion variable is ? and Introduce a second boson which has Dirichlet boundary conditions.

We can now fermionize using

The boson theory has two bosons which are gotten by a rotation by 45 degrees in the plane: The equivalent fermion theory is Use it to analyze the bosonic theory

If the bosons have the compactification radii We get the correct matching of fermionic and bosonic states if we consider the sectors where all of the fermions are NS Where all of the fermions are R and we GSO project onto states where the total fermion number is even

There are SU(2)XSU(2) current algebras These are related to the bosonic current algebra which appears in the bosonic theory when the boson is compact with the self-dual radius.

Simple boundary states of the doubled system are easy to find. In the bosonic representation they obey the boundary conditions which can be solved to get

The fermion boundary conditions for the simple boundary states are (The simple form of these relations relies on careful choice of cocycles.) We observe that, the simple boundary states are related to each other by global current algebra rotations:

The boundary condition can be solved to find the boundary state in terms of fermionic variables Other simple boundary states can be found by global rotations using current algebra.

The rolling tachyon boundary state Consider the “half-brane” rolling tachyon field Which is an analytic continuation of the marginal Liouville potential The boundary state obeys

In Fermionic variables, the boundary conditions are A solution of these is given by a (nonunitary) current algebra transformation or, explicitly

Application: Compute the partition function of the half-brane Disc amplitude

Conclusion: What about other radii? Some progress can be made when is a rational number less than 2. Work in progress.