Theory and applications of Boundary

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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.
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Boundary Sine-Gordon Theory
Approaches to solution i) semi-classical
techniques ii) conformal field theory iii)
Integrability Bethe ansatz iv) fermionization

References H. Saleur arXivcond-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.
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Applications i)Tachyons in string theory.
A.Sen, hep-th/0410103 ii)Quantum impurity
problems. H. Saleur, arXivcond-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
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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.
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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.

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V(ltTgt)
Open String Tachyon Potential
perturbative open string vacuum D25-brane.
rolling tachyon
TD25
closed string vacuum.
ltT(x)gt
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).
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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
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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).
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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.
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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
0
Free field theory in the bulk
Interactions on the boundaries
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a
0
We will compute the partition function
Field theory with Euclidean time which is
periodic
b
Space-time is a Euclidean cylinder.
a
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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.
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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
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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
Bgt and ltB are initial and final states and H is
the Hamiltonian of field theory on a circle.
b
Periodic space
initial boundary state
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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
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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.
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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.
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Consider the boson theory defined on the
cylinder, (now we have interchanged and
from the previous discussion)
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Normal ordering puts negatively moded oscillators
to the left of positively moded oscillators, zero
modes are left alone..
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Consider a compact boson with radius R
The total momentum is quantized in units, n/R.
The integers w are wrapping numbers.
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n is the number of momentum quanta and w is the
wrapping number.
w
n
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Fermionization
are co-cycles which make and
anti-commute.
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The same correlators are produced by fermion
fields with the mode expansion
and the zero momentum vacuum
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Quantum states of the fermion theory are obtained
by
These should correspond to states of the bosonic
theory. Which states?
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Quantization of the fermions
v.s. quantum theory of bosons
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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?
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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).
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Half-odd-integer fermion number occurs when there
are fermion zero modes in the R sector
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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)
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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
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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.
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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?
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Using the bosonic boundary conditions, we can
also deduce the Boundary conditions for the
fermions,
These conditions have the solutions
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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.
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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.
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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.
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T-duality
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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.
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We can now fermionize using
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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
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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

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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.
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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
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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
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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.
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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
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In Fermionic variables, the boundary conditions
are
A solution of these is given by a (nonunitary)
current algebra transformation
or, explicitly
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Application
Compute the partition function of the half-brane
Disc amplitude
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Conclusion
What about other radii?
Some progress can be made when is a
rational number less than 2. Work in progress.