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

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.

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.

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)

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 stateOpen string channel: open string channel makes a loop with prescribed

boundary conditions at ends

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

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

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.

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

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

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.

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.

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

First look at OSFT

This action has a huge gauge symmetry

provided that the star product is associative, QB acts as agraded derivation and < . > has properties of integration.

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:

Witten’s star product

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

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

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)

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 1406.3021 by Erler and Maccaferri.)

Traditionally, this has been studied using the boundary states. General construction not known!

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 BCFT⊗ 26-c and look for solutions which do not excite any primaries in BCFT26-c .

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

Numerical solutions in OSFT

In the case of Ising the boundary spectrum is particularly simple

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)

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)

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

Tachyon condensation on the σ-brane

Level ½ potentialThe new nontrivial solutions are:

These can be interpreted as the 1- and ε-branes

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 !

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!

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.

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.