1. Intersections between Open String Field Theory 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

2. Condensed matter perspective
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
A perfect tool to obtain many interesting results analytically is a 2D conformal field theory with a boundary or a defect.

3. Folding trick
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.

4. Conformal defects
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)

5. Boundary CFT
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

6. Boundary states
Describe possible boundary conditions from the closed string channel point of view.
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

7. Boundary states 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

8. Boundary states – Cardy’s solution
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.

9. Ising model CFT
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

10. Boundary states – Cardy’s solution
And thus the Ising model conformal boundary states are
The first two boundary states describe fixed (+/-) boundary condition, the last one free boundary condition

11. Goal of this talk
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.

12. What is String Field Theory?
Field theoretic description of all excitations of a string (open or closed) at once.
Useful especially for physics of backgrounds: tachyon condensation or instanton physics, etc.
Single Lagrangian field theory which around its various critical points should describe physics of diverse D-brane backgrounds, possibly also gravitational backgrounds.

13. First look at bosonic OSFT
Open string field theory uses the following data Let all the string degrees of freedom be assembled in Witten (1986) proposed the following action

14. First look at OSFT
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.

15. Witten’s star product
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:

16. Simple subsector of the star algebra
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.
The building elements thus obey
The derivative Q acts as

17. Universal classical solutions
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

18. Universal classical solutions
What do these solutions correspond to?
In 2011 with Murata we succeeded in computing their energy in terms of the function
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)

19. OSFT = physics of backgrounds
So far the discussion concerned background independent solutions and aspects of OSFT.
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 by Erler and Maccaferri.)
Traditionally, this has been studied using the boundary states. General construction not known!

20. Numerical solutions in OSFT
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 .

21. Numerical solutions in OSFT
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)
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

22. Numerical solutions in OSFT
In the case of Ising the boundary spectrum is particularly simple

23. Boundary state from Ellwood invariants
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)

24. Pasquier algebra from OSFT
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:
This formula is valid for rational theories, and in some cases also for irrational theories (e.g. chiral marginal deformations)

25. Tachyon condensation on the σ-brane
The computation proceeds along the similar line as for (Moeller, Sen, Zwiebach 2000)
For the string field truncated to level 2 the action is

26. Tachyon condensation on the σ-brane
Level ½ potential
The new nontrivial solutions are:
These can be interpreted as
the 1- and ε-branes

27. Tachyon condensation on the σ-brane
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 !

28. Positive energy solutions on the 1-brane
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!

29. Positive energy solutions on the 1-brane
Cubic extrapolations of energy and Ellwood invariant (boundary entropy) to infinite level
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.

30. Summary
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.