Matrix factorisations and D-branes

Matthias Gaberdiel

ETH Zürich

Cambridge, 3 April 2006

D-branes on Calabi-Yau’s

Phenomenological interesting string backgrounds involve often orbifolds ororientifolds of Calabi-Yau manifolds.

In many such constructions D-branes play an important role.

Conformal field theory

From a conformal field theory point of view,strings on Calabi-Yau manifolds can, forexample, be described by

Toroidal orbifolds, for example

Gepner models

Gepner models

Gepner models are (orbifolds of) tensorproducts of N=2 minimal models:

For A-type modular invariant, the Calabi-Yau manifold is then the hypersurface W=0 in (weighted) complex projective space where

The Quintic

The simplest example is the quintic that is definedby the equation

in complex projective space

Its Gepner model is the tensor product offive (A-type) N=2 minimal models with k=3.

D-branes in Gepner models

The simplest D-branes for the Gepner modelsare the Recknagel-Schomerus (RS) branes. Theyare characterised by the property that they preservethe 5 N=2 superconformal algebras separately:

[Here I have described B-type branes.]

For many Gepner models these D-braneshowever do not account for all the RR charges.

Even if they account for all charges (as for the case of the quintic), the lattice of chargesthey generate is often only a sublattice of finite index.

RR charges

For example, for the case of the quintic,they generate a sublattice of rank 25.

Also, it has been known for some timethat none of them describes the D0-braneon the quintic.

D0-brane

In order to make progress, use the new insight of Kontsevich about (B-type) D-branes for Landau-Ginzburg models.

[Brunner, et.al.]

How can one describe the remaining charges, and in particular the single D0-brane?

Landau-Ginzburg models

Via the linear sigma model, Calabi-Yau manifoldsare closely related to Landau-Ginzburg models.

The Landau-Ginzburg models of interest areN=(2,2) supersymmetric field theories involving chiral (and anti-chiral) superfields .

[Witten]

Their F-term superpotential is of the form

where W is the same function as before.

Matrix factorisations

Recently, Kontsevich has proposed that the superconformal B-type D-branes of the Landau-Ginzburg model with superpotential W can be characterised in terms of matrix factorisations of W as

Here E and J are polynomial matrices in the variables

Matrix factorisations

This condition can be understood from a physicspoint of view by analysing the supersymmetry variation of the Landau-Ginzburg model on a world-sheet with boundary (Warner problem).

[Kapustin Li][Brunner, et.al.]

The factorisation matrices E and J are graded, andmust in fact be off-diagonal. In the superconformal case their entries must furthermore be homogeneous.

[The matrices describe (world-sheet) fermionic degrees of freedom at the boundary --- tachyon condensation.]

Simple factorisations

If W is a sum of different such polynomials,these separate factorisations can be `tensored’together to give a factorisation of W.

For the simple case of a polynomial in one variable, , all factorisations are equivalent to directsums of the factorisations with

[Ashok et al]

[Herbst, et al]

Boundary states

The corresponding boundary states in the Gepner model are in fact precisely the RS (or tensor) branes with

[Recall, that the B-type RS branes are labelled by ; I am suppressing here the labels that comefrom the orbifold.]

[Recknagel, Schomerus][Moore, Maldacena, Seiberg]

[Kapustin Li][Brunner, et.al.]

Identification

This identification can be checked by comparingthe topological open string spectrum of these branes.

In conformal field theory: consider the chiralprimaries in open string spectrum.

From matrix factorisation point of view: the topological spectrum is the cohomology of anoperator that is associated to the factorisations.

Missing factorisations

Since we know that the RS branes often donot generate all the charges, it is then clearthat the same must be true for these simplefactorisations.

What are the factorisations that account for the remaining charges?

And what are the corresponding boundary states?

Permutation factorisations

At least some of the missing factorisations involvethe rank 1 factorisations that come from writing

where the product runs over the d’th roots of -1.[Ashok et al]

In the simplest case, E (or J) is just a linear factor

D0 matrix factorisations

Now tensor this factorisation with the usualone-variable factorisations for the other factors.Then one can argue that the correspondingD-brane is `located’ at

This should thus describe a D0-brane on the Calabi-Yau hypersurface!

[Ashok et al]

Permutation boundary states

The boundary state corresponding to the rank 1factorisation can be identified witha permutation brane. [Brunner, MRG]

cf. also [Enger, et. al.]

[Recknagel]cf. also [MRG, Schafer-Nameki]

Permutation branes are characterised by:

D0-brane on CY

Combining with the geometric intuition from thematrix factorisation point of view, we can thus identify the D0-brane in many Gepner modelswith a specific permutation brane.

[Brunner, MRG]

CY (W=0)

LG-model with W Gepner model

holomorphic D-branes

matrix factorisations B-type boundary states

D0-brane on CY

We have also checked this identification in some cases by comparing the charges with those that can be obtained from the geometric identification of the RS branes.

[Brunner, et. al.]

D-brane charges

Do these constructions account now for all RR charges?

[Caviezel, Fredenhagen, MRG]

For the A-type Gepner models (superpotentialsof the form ) we have found thatthey do, except in 31 cases.

(The simplest example where they do not is theCalabi-Yau surface described by the Fermat hypersurface

Missing factorisations

In terms of matrix factorisations, the missingconstructions can be very easily described: theycorrespond to generalised permutation factorisationsthat come from writing

where d is the greatest common factor of the twoexponents. [Caviezel, Fredenhagen, MRG]

CFT construction

In terms of conformal field theory, the correspondingconstruction is not yet known.

cf [Fredenhagen, Quella]

For one simple example, we have recently managedto construct the boundary state explicitly in conformal field theory. [Fredenhagen, MRG]

However, it is not yet clear how to generalise the construction to the general situation.

Matrix deformations

Conformal field theory description is only knownat the Gepner points, but matrix factorisationdescription also possible for deformations ofFermat polynomials.

For example, for the quintic, these (complexstructure) deformations can be described by adding to the superpotential W any term of the form

A necessary condition

Given a matrix factorisation of W, we can thenask whether this can be deformed into a matrix factorisation of .

[Brunner, MRG, Keller]

We have recently found a necessary conditionfor this to be possible: the factorisation mustnot be charged with respect to the RR fieldthat is associated to the deformation viaspectral flow.

[Hori, Walcher]

Global deformations

Matrix factorisations are therefore oftenobstructed against perturbations of the closed string theory.

[Brunner, MRG, Keller]

On the other hand, we have also shown (byexplicit construction) that the tensor (RS) and the D0-like factorisations can always be extended for an arbitrary deformation.

D-branes on K3

There are 14 different (A-type) Gepner modelsthat describe K3s. The simplest one is the quartic

In order to understand these phenomena in moredetail, we have applied these ideas to theexample of D-branes on K3.

Deforming K3

In all 14 cases, the tensor and D0-like factorisations(that exist generically) account for the genericB-type D-brane charges, as determined from geometry.

[At special points in moduli space (for examplethe Gepner points), the actual rank is howeverhigher.]

Deforming the quartic

[Wendland]

In particular, we have managed to understand all of the above features directly in the orbifold theory.

As a nice and explicit example we have furthermore considered the orbifold K3 that corresponds to the 2-parameter deformation of the quartic

Conclusions

Matrix factorisations: new approach to characterise N=2 B-type D-branes.

Interesting applications (so far):

CFT construction of D0-brane on CYs.

Systematic analysis of fundamental D-branes.

Behaviour of D-branes under deformationsof underlying Calabi-Yau.