Conformal Interfaces

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Conformal Interfaces

• based on CB, I. Brunner , hep-th/0712.0076
• CB, S. Monnier, to appear
• previous CB, M. Gaberdiel , hep-th/0411067
• CB, J. de Boer, R. Dijkgraaf, H. Ooguri ,
  hep-th/0111210

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Summary of talk

• What are they?
• Why are they interesting?
• Interfaces in c1 models
• Fusion of interfaces
• A black hole analogy
• Semiclassical theory
• Outlook

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1. What are they?
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t
CFT1
CFT2
x

• no energy trapped in interface

No scale
Txt must be continuous
Conformal maps that preserve the interface are
symmetries
5
Two special cases
D-branes

• Boundaries CFT2 trivial theory
• Defects CFT1 CFT2

Term used for general interfaces in some of the
recent literature
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Conformal interfaces can be mapped to boundary
states by the folding trick
Affleck, Oshikawa 1995 CB, de Boer, Dijkgraaf,
Ooguri 2001
CFT1
CFT1
CFT2
CFT2
Interface between CFT1 and CFT2
boundary of CFT1 CFT2
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Any conformal boundary in a (tensor product)
of two CFTs that do not interact in the bulk can
be unfolded to a Conformal Interface.
Much of the D-brane technology can be
carried over readily to study CIs.
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2. Why are they interesting?
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Critical behavior in Statistical Mechanics
critical q-state Potts
critical Ising
local spin-spin interaction
would flow in the infrared to a RG fixed point
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in Condensed Matter
junction of quatum wires
R2
R1
Luttinger liquids, i.e. massless-scalar theories
with radius discontinuity
Alternatively constrictions in q-Hall systems
Kane, Fisher 1992 . . . . CB, B. Doucot,
Froehlich, in progress
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Tomonaga-Luttinger liquid
bosonization
fermion excitations at two Fermi points
Charge and current density
Periodic identification
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Effective action
Coulomb interactions
Redefine the scalar field
so that
Compactification radius depends on q-wire
(but so does the speed of light )
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Introduce an impurity
reflected
transmitted
incident
At low T and energies two simplest fixed points
Dirichlet ?? vanishes, so current is zero
Neuman charge density continuous, perfect
transmission
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Quantum dots and different types of q-wire allow
for a much richer set of (experimentally
accessible?) critical behaviors.
I will not discuss this here.
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in AdS/CFT
Conformal interface in holographic field theory
AdSn branes in AdSn1 bulk
Karch, Randall 2001 CB, Petropoulos 2000 CB,
de Boer, Dijkgraaf, Ooguri 2001 DeWolfe,
Freedman, Ooguri 2001 Erdmenger, Guralnik,
Kirsch 2002 Yamaguchi 2002 . . .
Brane worlds without fine tuning
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lastly
CFT1
conformal boundary B1
CFT2
conformal boundary B2
Graham, Watts 2003 Fredenhagen 2003 CB,
Gaberdiel 2004 Fuchs, Gaberdiel, Runkel,
Schweigert 2006 CB, Monnier to appear
Universal boundary RG flows !
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in String Theory
worldsheet
closed string state
spectrum-generating symmetry of classical string
field theory, reminiscent but different from the
Harvey-Moore algebra.
CB, GGI talk 2007
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Conformal defects implement all perturbative
string symmetries, including T-dualities
Froehlich, Fuchs, Runkel, Schweigert 2004
They may, however, change the mass and charges of
D-branes, and are thus generalized
Ehlers-Geroch transformations
Interfaces transport furthermore the solution
over the closed-string moduli space.
Should understand them as symmetries of classical
String Field Theory ?
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3. Interfaces in c1 models
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f f 2pR1
f f2pR2
fold
Boundary of c2 theory
Simplest D0, D1, D2 brane (or any
superposition thereoff)
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e.g. a D1-brane can wind (k1,k2) times
around the two cycles of the torus, and has two
periodic moduli
2pR2
here (k1,k2) (3, 2)
)
2pR1
angle ?
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The useful device for our purposes here is the
associated conformal boundary state
Callan, Lovelace, Nappi, Yost 1987
t
boundary
. . .
V2
V1
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For example, for a periodic free scalar field
a Dirichlet boundary condition corresponds to
One checks easily that
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The overall normalization is fixed (up to
integer multiplicity) by Cardys conditions.
These are a consequence of locality.
The coefficient of the ground state is the
g-factor of the boundary. Its logarithm is the
boundary entropy.
Affleck, Ludwig 1991
One can endow the boundary with a
n-dimensional (Chan-Patton or quantum dot) space
of states. If these do not participate in the
dynamics
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momentum and winding in the directions 1 and 2
For the D1-brane on the torus
oscillator part
where
and
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momentum and winding in the directions 1 and 2
Likewise for the D0/D2 branes
oscillator part
where
and
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Unfolding hermitean conjugates the states of
CFT2, and exchanges its left- and right-movers
unfold
real
I is thus a formal operator from HCFT2 to HCFT1
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Two extreme situations
Totally-reflecting interfaces
, CFT1 and CFT2 decouple
Totally-transmitting interfaces
Ttt Tss also continuous, so full conformal
invariance is preserved.
Such interfaces are topological

Petkova, Zuber 2000
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Inspection of the oscillator part shows that the
reflectivity only depends on the angle ?
total reflection (factorizable)
? 0 or 90o
total transmission (topological)
? 45o
The topological condition fixes a combination of
the two radii
and minimizes (for fixed non-vanishing k1,k2) the
entropy
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The k1 k2 1 topological interfaces have zero
entropy. They are the U(1) ? Z2 2
automorphisms of the CFT.
separate translations and reflections of fL
and fR
The identity operator corresponds after folding
to a diagonal D1-brane.
The T-duality generator is a bound state of one
D0 and one D2-brane
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All other topological interfaces have positive
entropy. They correspond to isomorphisms between
projected out subsectors of the two CFTs.
For example, if R1 2R2
oscillator part trivial permutation
Projecting out subsectors can be also achieved by
periodic arrays of symmetry defects destructive
interference
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A conjecture

• topological interfaces have non-negative
  entropy,
• that only vanishes for CFT automorphisms.

valid in all examples known to us can the
conjecture be strengthened?
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4. Fusion of interfaces
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Two (or more) interfaces can be added
Chan-Patton or quantum-dot degrees of freedom
They can be also multiplied (or more properly
fused)
juxtaposition in space
This defines an interface  algebra  over ? ,
no additive and, in general, no multiplicative
inverse
different from the Verlinde algebra of local
operators.
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R1
R2
R3
e
Fusion shrinking of the middle region. The
result should be finite modulo a divergent energy
counterterm. On the cylinder inverse temperature
ß2p
Casimir energy
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The fusion is non-singular (d0) if one of
the two interfaces is topological.
locality
I I is sum with integer coefficients

strong consistency check
The algebra in the case at hand can be derived by
explicit computation, using coherent-state methods
. Let us enter a little in the details
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Deformed identity operator
Topological (un)dressing equations
Chosen to make topological
After separation of the topological dress,
the only singularities appear in the product of
deformed identity operators
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Claim product of deformed identities
proof
with
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and
can be treated as c-numbers.
Now use
?
and Baker-Haussdorf formula to get
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with
Now take the q

• limit and use simple trigonometry

to show that
angle of 13 interface
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The only non-trivial remaining part is the
overall constant
where
Use Euler-MacLaurin to get
subtracted Casimir energy
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and the trigonometric identity
to show that
qed
All remaining calculations are non-singular
operator products. The results can be summarizeas
follows
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• The continuity argument is correct the fusion
  product does
• not depend on R2 , i.e. memory of the
  collapsed region is
• lost only d depends on it .

We may thus choose R2 to make I or I
topological, so that the fusion algebra is
determined by the non-singular products of
topological operators.
(2) The fusion algebra is simpler to present for
equivalence classes of interfaces, mod CFT
automorphisms. The law of composition of
winding numbers is multiplicative
Fuchs, Gaberdiel,Runkel, Schweigert 2007
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(3) The inverse of fusion is the process of
dissociation. The entropy released is equal
to
It depends on the value of R2 , and can have
either sign unless I?I is topological in which
case its entropy cannot be lowered. This sign is
the same as that of the Casimir force
depend on R2
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5. A black hole analogy
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Conformal interfaces share features of black
holes in supersymmetric theories. Topological
interfaces in particular, exactly like BPS black
holes
(1) minimize the entropy for given charges
(2) fix, through an attractor mechanism a a
combination of the bulk moduli
(3) are marginally (un)stable against
dissociation to more elementary ones
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Topological interfaces may be protected by their
moduli space, which for a k1,k2 interface has
dimension 2?gcd(k1,k2). Thus for instance
3,3
3,1 o 1,3
?
dim 6
dim 4
NB this protects from decay supersymmetric BHs
Seiberg, Witten 1999
Is this a superficial analogy, or is there
some more profound reason behind?
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The Conformal Interface Algebra is reminiscent of
the Harvey/Moore proposal for a
BPS-vertex operator algebra in the heterotic
string
cf also Giveon, Porrati 1991
But there are important differences
? it is not an algebra of local
operators ? the conserved charges
are logarithms of natural numbers,
rather than taking values in a charge lattice

cf Julias arithmetic gas
? its elements transport one over both the
open and the closed-string
moduli space.
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? Many examples in interacting CFTs are known.
Quella, Schomerus 2002 . . Brunner,
Roggenkamp 2007
? Topological interfaces only exist if c1 c2
In unitary theories
Quella, Runkel, Watts 2006
Even in the simplest c1 model the full
algebra has not yet been derived
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The c1 theory has also topological
interfaces for which both bulk moduli
must be fixed. These have no geometric
semiclassical limit  quantum BHs 

S1/Z2
S1
circle/orbifold automorphism
SU(2)L SU(2)R automorphisms
all fusions with k1,k2()
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6. Semiclassical theory
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Consider a general sigma model with
(weakly-curved, large-V) target space manifold
M . The general renormalizable defect operator
is
where
matrix-valued 1-forms on M
These describe the perturbations of a
diagonally-embedded middle-dimensional D-brane.
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M
M
CB, Gaberdiel 2004
Fixed-point 1-forms (B , C) are solutions of
(non-abelian) DBI equations.
Pulling-back these forms on the tangent/normal
bundle of a D-brane of M effects the
semiclassical boundary flow from the UV to the IR
fixed point.
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Explicit computations can be done in the WZW
and coset GKO models, where defect operators and
their flows can be restricted by symmetry. Chiral
defect operators can be regularized and
renormalized consistent with symmetries, and
explicitly constructed at fixed points.
CB, Gaberdiel 2004 Alekseev, Monnier 2007 CB,
Monnier to appear Cf Bazhanov, Lukyanov,
Zamolodchikov 1994 Runkel 2007
No time to describe this here in any detail.
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7. Outlook

• Conformal interfaces and defects describe a rich
  variety of critical phenomena in 11 dimensional
  systems (ICFT). They can be added and multiplied
  (superposed and juxtaposed)
• They are special examples of extended operators
  (like Wilson loops and other extended sources)
  which are a still largely-unexplored chapter of
  QFT in 2 dimensions.
• They could play, in particular, a role in
  q-integrability Runkel 2007, Mikhailov,
  Schafer-Nameki 2007
• They define a large spectrum-generating algebra
  over the space
• of CFTs, that includes perturbative string
  symmetries. Its relation to other
• extended symmetries (E10 ?) must be
  explored.