Transcript Intersection Between SFT and Condensed Matter

Martin Schnabl
Collaborators:
T. Erler, C. Maccaferri, M. Murata, M. Kudrna, Y. Okawa and M. Rapčák
Institute of Physics AS CR
53rd Schladming winter school, March 2nd, 2015
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There are several critical phenomena in
condensed matter with a common denominator:
- surface critical behavior of 2D systems
- defect lines in 2D statistical models
- quantum impurities (dots) in 1D quantum wires
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A perfect tool to obtain many interesting results
analytically is a 2D conformal field theory with a
boundary or a defect.
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There is not much of a difference between
theory with a boundary or a defect thanks
to the folding trick which maps defect line
into a boundary of the folded theory, e.g.
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The folding trick translates the difficult
defect problem into a boundary (easier)
problem in a more complicated (folded)
theory.
Two types of defects are easier to
understand:
- factorizing (the two sides are independent)
- topological or fully transmissive (trivial
defect, or the spin flip in the Ising model)
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All possible boundary conditions
preserving conformal symmetry have been
classified for minimal models with c<1.
Most of the information about a boundary
condition can be encoded in the so called
boundary state:
Closed string channel: closed string evolves
from the boundary state
Open string channel: open string channel makes
a loop with prescribed
boundary conditions at ends
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Describe possible boundary conditions from
the closed string channel point of view.
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Conformal boundary states obey:
1) the gluing condition
2) Cardy condition (modular invariance)
3) sewing relations (factorization constraints)
See e.g. reviews by Gaberdiel or by Cardy
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The gluing condition is easy to solve:
For any spin-less primary
we can define
where
is the inverse of the real symmetric
Gram matrix
where
,
(with possible null states projected out).
Ishibashi 1989
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By demanding that
and noting that RHS can be expressed as
Cardy derived integrality constraints on the boundary
states. Surprisingly, for certain class of rational CFT’s he
found an elegant solution (relying on Verlinde formula)
where
is the modular matrix.
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Ising model is the simplest of the unitary
minimally models with c = ½.
It has 3 primary operators
1 (0,0)
ε (½,½)
σ (1/16, 1/16)
The modular S-matrix takes the form
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And thus the Ising model conformal
boundary states are
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The first two boundary states describe fixed
(+/-) boundary condition, the last one free
boundary condition
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In this talk we will show how to use OSFT
(Open String Field Theory) to characterize
possible boundary conditions (and hence
defects as well) in any given 2D CFT.
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Field theoretic description of all excitations of
a string (open or closed) at once.
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Useful especially for physics of backgrounds:
tachyon condensation or instanton physics, etc.
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Single Lagrangian field theory which around
its various critical points should describe
physics of diverse D-brane backgrounds,
possibly also gravitational backgrounds.
Open string field theory uses the following data
Let all the string degrees of freedom be assembled in
Witten (1986) proposed the following action
This action has a huge gauge symmetry
provided that the star product is associative, QB acts as a
graded derivation and < . > has properties of integration.
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The elements of string field star algebra are states in the
BCFT, they can be identified with a piece of a worldsheet.
By performing the path integral on the glued surface in
two steps, one sees that in fact:
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The star algebra is formed by vertex operators and the
operator K. The simplest subalgebra relevant for tachyon
condensation is therefore spanned by K and c. Let us be
more generous and add an operator B such that QB=K.
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The building elements thus obey
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The derivative Q acts as
This new understanding lets us construct solutions to
OSFT equations of motion
easily.
More general solutions are of the form
Here F=F(K) is arbitrary
M.S. 2005, Okawa, Erler 2006
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What do these solutions correspond to?
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In 2011 with Murata we succeeded in computing their
energy
in terms of the function
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For simple choices of G, one can get perturbative vacuum,
tachyon vacuum, or exotic multibrane solution. At the
moment the multibrane solutions appear to be a bit
singular. (see also follow-up work by Hata and Kojita)
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So far the discussion concerned background
independent solutions and aspects of OSFT.
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The new theme of the past two years, is that OSFT
can be very efficient in describing BCFT
backgrounds and their interrelations. (See recent
paper 1406.3021 by Erler and Maccaferri.)
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Traditionally, this has been studied using the
boundary states. General construction not known!
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To construct new D-branes in a given BCFT
with central charge c using OSFT, we
consider strings ‘propagating’ in a
background BCFTc ⊗ BCFT26-c and look for
solutions which do not excite any primaries
in BCFT26-c .
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To get started with OSFT, we first have to specify the starting
BCFT, i.e. we need to know:
- spectrum of boundary operators
- their 2pt and 3pt functions
- bulk-boundary 2pt functions (to extract physics)
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The spectrum for the open string stretched between D-branes a
and b is given by boundary operators which carry labels of
operators which appear in the fusion rules
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In the case of Ising the boundary spectrum
is particularly simple
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The coefficients of the boundary state
can be computed from OSFT solution via
See: Kudrna,Maccaferri, M.S. (2012)
Alternative attempt:
Kiermaier, Okawa, Zwiebach (2008)
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Let us explore the implications of the linearity of the
formula for the boundary state:
Let us assume that the solution
describes the
boundary state
as seen from
. Assuming that
the Verma modules “turned on” are present also on
and the structure of boundary operators is identical, then:
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This formula is valid for rational theories, and in some
cases also for irrational theories (e.g. chiral marginal deformations)
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The computation proceeds along the similar
line as for (Moeller, Sen, Zwiebach 2000)
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For the string field truncated to level 2
the action is
Level ½ potential
The new nontrivial solutions are:
These can be interpreted as
the 1- and ε-branes
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We can systematically search for the solutions in the level
truncation scheme up to at least level 20. (At higher levels
we have to take care of the null states.)
By computing the gauge
invariants (the boundary
state) we confirm the
interpretation of the
solutions corresponding
to 1- and ε-branes !
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Big surprise: on the 1-brane with no
nontrivial operators, we find a nontrivial
real solution at level 14 starting with a
complex solution at level 2!
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Cubic extrapolations of energy and Ellwood invariant
(boundary entropy) to infinite level
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This and the full boundary state provides evidence that we
discovered the σ-brane with higher boundary entropy! This
would not be possible with RG techniques.
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The problem of characterizing conformal
boundary conditions (or conformal defects)
in general CFTs is very interesting,
important and interdisciplinary but still
unsolved.
Open String Field Theory offers a novel
approach to the problem. At present it can
be used quite efficiently as a numerical tool,
but hopefully it will provide crucial analytic
insights in the near future.