1. Matrix factorisations and D-branes Matthias Gaberdiel ETH Zürich Cambridge, 3 April 2006

2. D-branes on Calabi-Yau’s Phenomenological interesting string backgrounds involve often orbifolds or orientifolds of Calabi-Yau manifolds. In many such constructions D-branes play an important role.

3. Conformal field theory From a conformal field theory point of view, strings on Calabi-Yau manifolds can, for example, be described by Toroidal orbifolds, for example Gepner models

4. Gepner models are (orbifolds of) tensor products 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

5. The Quintic The simplest example is the quintic that is defined by the equation in complex projective space Its Gepner model is the tensor product of five (A-type) N=2 minimal models with k=3.

6. D-branes in Gepner models The simplest D-branes for the Gepner models are the Recknagel-Schomerus (RS) branes. They are characterised by the property that they preserve the 5 N=2 superconformal algebras separately: [Here I have described B-type branes.]

7. For many Gepner models these D-branes however 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 charges they 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.

8. Also, it has been known for some time that none of them describes the D0-brane on 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?

9. Landau-Ginzburg models Via the linear sigma model, Calabi-Yau manifolds are closely related to Landau-Ginzburg models. The Landau-Ginzburg models of interest are N=(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.

10. 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

11. Matrix factorisations This condition can be understood from a physics point 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, and must 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.]

12. 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 direct sums of the factorisations with [Ashok et al] [Herbst, et al]

13. 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 come from the orbifold.] [Recknagel, Schomerus] [Moore, Maldacena, Seiberg] [Kapustin Li] [Brunner, et.al.]

14. Identification This identification can be checked by comparing the topological open string spectrum of these branes. In conformal field theory: consider the chiral primaries in open string spectrum. From matrix factorisation point of view: the topological spectrum is the cohomology of an operator that is associated to the factorisations.

15. Missing factorisations Since we know that the RS branes often do not generate all the charges, it is then clear that the same must be true for these simple factorisations. What are the factorisations that account for the remaining charges? And what are the corresponding boundary states?

16. Permutation factorisations At least some of the missing factorisations involve the 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

17. D0 matrix factorisations Now tensor this factorisation with the usual one-variable factorisations for the other factors. Then one can argue that the corresponding D-brane is `located’ at This should thus describe a D0-brane on the Calabi-Yau hypersurface! [Ashok et al]

18. Permutation boundary states The boundary state corresponding to the rank 1 factorisation can be identified with a permutation brane. [Brunner, MRG] cf. also [Enger, et. al.] [Recknagel] cf. also [MRG, Schafer-Nameki] Permutation branes are characterised by:

19. D0-brane on CY Combining with the geometric intuition from the matrix factorisation point of view, we can thus identify the D0-brane in many Gepner models with a specific permutation brane. [Brunner, MRG] CY (W=0) LG-model with W Gepner model holomorphic D-branes matrix factorisations B-type boundary states

20. 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.]

21. D-brane charges Do these constructions account now for all RR charges? [Caviezel, Fredenhagen, MRG] For the A-type Gepner models (superpotentials of the form ) we have found that they do, except in 31 cases. (The simplest example where they do not is the Calabi-Yau surface described by the Fermat hypersurface

22. Missing factorisations In terms of matrix factorisations, the missing constructions can be very easily described: they correspond to generalised permutation factorisations that come from writing where d is the greatest common factor of the two exponents. [Caviezel, Fredenhagen, MRG]

23. CFT construction In terms of conformal field theory, the corresponding construction is not yet known. cf [Fredenhagen, Quella] For one simple example, we have recently managed to 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.

24. Matrix deformations Conformal field theory description is only known at the Gepner points, but matrix factorisation description also possible for deformations of Fermat polynomials. For example, for the quintic, these (complex structure) deformations can be described by adding to the superpotential W any term of the form

25. A necessary condition Given a matrix factorisation of W, we can then ask whether this can be deformed into a matrix factorisation of. [Brunner, MRG, Keller] We have recently found a necessary condition for this to be possible: the factorisation must not be charged with respect to the RR field that is associated to the deformation via spectral flow. [Hori, Walcher]

26. Global deformations Matrix factorisations are therefore often obstructed against perturbations of the closed string theory. [Brunner, MRG, Keller] On the other hand, we have also shown (by explicit construction) that the tensor (RS) and the D0-like factorisations can always be extended for an arbitrary deformation.

27. D-branes on K3 There are 14 different (A-type) Gepner models that describe K3s. The simplest one is the quartic In order to understand these phenomena in more detail, we have applied these ideas to the example of D-branes on K3.

28. Deforming K3 In all 14 cases, the tensor and D0-like factorisations (that exist generically) account for the generic B-type D-brane charges, as determined from geometry. [At special points in moduli space (for example the Gepner points), the actual rank is however higher.]

29. 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

30. 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 deformations of underlying Calabi-Yau.