Abstract D-branes on K3 are analysed from three different points of view. For deformations of hypersurfaces in weighted projected space we use geometrical methods as well as matrix factorisation techniques. Furthermore, we study the D-branes on the T 4 /Z 4 orbifold line in conformal field theory. The behaviour of the D-branes under deformations of the bulk theory are studied in detail, and good agreement between the different descriptions is found. * Introduction D-branes in models with N = (2, 2) worldsheet supersymmetry have been discussed in recent years from various points of view. Prominent examples for such theories include non-linear sigma-models whose targets are Kähler manifolds, Gepner models and LandauGinzburg models. Boundary conditions can be formulated in terms of the N = (2, 2) supersymmetry algebra, and one is usually interested in boundary conditions that preserve half of the supersymmetry. As is well known, there are two different classes of such supersymmetry preserving boundary conditions, which are related by mirror symmetry and are called A-type and B-type. In the non-linear sigma-model, A-type boundary conditions correspond to D-branes wrapping (special) Lagrangian cycles, whereas B-type boundary conditions describe holomorphic branes [1] . In the Gepner model, which provides a rational conformal field theory description in the small volume regime of certain Calabi-Yau compactifications, A-type and B-type D-branes can be constructed as explicit boundary states with appropriate gluing conditions for the generators of the symmetry algebra. Finally, in the Landau-Ginzburg model, A-type D-branes correspond to Lagrangian submanifolds that are mapped by the superpotential to straight lines [2] , whereas B-type Dbranes can be described in terms of matrix factorisations of the superpotential. The study of matrix factorisations was initiated in this context by Kontsevich, who proposed that there is a B-type D-brane for any factorisation of the superpotential W (x i ) = E(x i )J(x i ) in terms of matrices E and J with homogeneous polynomial entries. One then associates a BRST operator of the form Introduction to this factorisation and determines the (topological) open string spectrum as the BRST cohomology of this operator [3, 4, 5, 6, 7, 8] . Introduction In this paper, we shall analyse supersymmetric D-branes on K3 surfaces using matrix factorisation and conformal field theory techniques. K3 surfaces are special since they actually preserve N = (4, 4) supersymmetry, not just N = (2, 2) . The relation between the geometrical and conformal field theory description of closed strings on K3 has been analysed in some detail in [9] ; here we shall concentrate mainly on open strings. A-type and B-type boundary conditions can be formulated as usual once a particular N = 2 subalgebra is chosen. Geometrically, all supersymmetric D-branes are holomorphic with respect to some complex structure. For a given complex structure, on the other hand, not all of these D-branes are holomorphic and the spectrum of holomorphic D-branes thus depends on the actual point in moduli space. One expects to find a D0 and D4 branes at any point in moduli space, but the rank of the Picard lattice, that determines the number of holomorphic curves and hence D2 branes, varies. Introduction An explicit conformal field theory description of string theory on K3 is only available at rather specific points in the moduli space, but the matrix factorisation description is, in principle, available for a much larger subspace of the moduli space. Thus the latter approach is very well suited to study the spectrum of B-type D-branes for generic points in the moduli space. In this paper we shall study some aspects of matrix factorisations for K3s that can be described as hypersurfaces in weighted projective space, where we restrict to the case of Fermat type polynomials and their perturbations. At the maximally symmetric point, string theory on this surface has a conformal field theory description in terms of a Gepner model. Introduction At this Gepner point we can easily construct factorisations by tensoring the usual single factor and the permutation factorisations together. Introduction § We can then analyse how these B-type D-branes can be deformed as one perturbs the superpotential. For certain factorisations (in particular, the tensor product factorisation and single transposition factorisations) we will be able to show that they can be extended over the whole moduli space of deformations. This can be done very explicitly by writing down the corresponding factorisations for arbitrarily deformed superpotentials (see section 3) . This result has a very nice geometrical interpretation: these factorisations account precisely for those holomorphic D2-branes that come from the embedding space or from resolving singular points of the embedded manifold, and are generically present for K3s that arise as hypersurfaces in weighted projective space. Introduction On the other hand, we can also prove that certain factorisations cannot be deformed. This can be shown by studying the infinitesimal deformations following [11, 51] . In particular, we shall show that a necessary condition for a deformation not to be obstructed is that the brane is uncharged under the RR ground state corresponding to the deforming polynomial. The fact that factorisations are generically obstructed is also in good agreement with the geometrical results. Introduction In order to compare these results with what can be analysed in conformal field theory we can make use of the fact that there is an interesting subspace of the moduli space of such K3 surfaces for which we have an explicit conformal field theory description. Indeed, the deformations of the quartic surface in P 3 by two particular bulk fields describes [12] the 2-parameter space of toroidal Z 4 orbifold K3s. For this subspace of the moduli space one should thus be able to relate the different matrix factorisations with explicit conformal field theory constructions of D-branes; this will be done in section 4. In the orbifold description it is straightforward to see that the space of B-type RR charges is 22-dimensional for any choice of the orbifold parameters a and b, ¶ and it is in principle not difficult to construct the relevant D-branes in the orbifold conformal field theory. On the matrix factorisation side it is likewise not difficult to construct the relevant 22 factorisations that account for all of these charges at the Gepner point (where a = b = 0). However, we can show that not all of them can be extended to arbitrary a, b. In fact, given our general results about obstructions, it is clear that for the factorisations that are charged under the RR ground states corresponding to x this will not be possible. Furthermore, we can show that this is the only real obstruction: we have identified a set of factorisations that account for 20 RR charges and that can be extended for arbitrary a and b. Introduction This apparent obstruction has in fact a very nice interpretation in terms of the orbifold conformal field theory. At least some of the relevant D-branes that carry these charges stretch diagonally across the two T 2 s at 45 degrees. Varying the parameters a and b corresponds then to changing the radii of the two tori (as well as switching on a B-field). The structure of the 45 degree D-brane then depends crucially on the relative radii: if their ratio is rational, the brane will have finite length, but it will wind infinitely many times around the torus if the ratio is irrational [13] . Thus the boundary state depends in a very discontinuous manner on the parameters of the closed string theory. (This is also familiar from the analysis of the N = 0 and N = 1 D-branes on a single circle [14, 15, 16] .) This explains why the corresponding matrix factorisation description cannot depend on the deformation parameters in a simple analytic way. Introduction The paper is organised as follows. In section 2 we review some background material about D-branes on K3 surfaces from a geometric point of view. In particular, we describe how the rank of the Picard lattice is always at least 1 for K3 surfaces that are embedded in weighted projective space. In section 3 we briefly review the matrix factorisation approach to D-branes in Landau Ginzburg models. We then discuss the behaviour of D-branes under bulk perturbations in this language, and explain, in particular, how the generic rank of the Picard lattice can be understood from this perspective. We also study the quartic in P 3 and the above deformations in detail. Finally, in section 4 we discuss the orbifold theory T 4 /Z 4 and its D-branes. We explain some parts of the correspondence between the boundary states of the orbifold theory and the matrix factorisations of the Landau-Ginzburg description. Finally we study their deformations in the orbifold theory and explain why certain deformations are obstructed. We have included an appendix in which some of the more technical aspects of the orbifold description and its dictionary to the Gepner model are explained in detail. BPS D-branes on K3 surfaces Let us begin by explaining some generalities about D-branes on K3 surfaces [1, 17, 18] . On K3 we are in the special situation that there is extended N = (4, 4) supersymmetry. The N = 4 algebra is an extension of the usual N = 2 superconformal algebra, where the u(1) current of the N = 2 theory is enhanced to an su(2) 1 algebra; the additional generators are the spectral flow operators (by one unit), which have conformal weight 1 for c = 6. BPS D-branes on K3 surfaces From the point of view of the extended N = (4, 4) symmetry there is therefore some freedom in how to choose the u(1) generator of the N = 2 algebra inside the su(2) 1 algebra of the N = 4 algebra. This is precisely the freedom of choosing a Cartan torus for the SU(2) group. Each N = 2 subalgebra determines uniquely an u(1) subalgebra of the su(2) 1 , but the converse is not true [12, 20] . Once we have identified in addition a particular N = (2, 2) subalgebra, we can formulate A and B type boundary conditions as usual. However, it is clear that the distinction between A-type and B-type branes depends on the choice of the particular N = (2, 2). BPS D-branes on K3 surfaces Mirror symmetry corresponds algebraically to flipping the sign of the u(1) current of the left moving supersymmetry algebra. Obviously, this operation requires that a particular N = (2, 2) structure has been picked. The mirror operation can then be viewed as a rotation of the Cartan torus (for the left movers). BPS D-branes on K3 surfaces Geometrically, a K3 surface S is a hyperkaehler manifold with H 2 (S, Z) = 22. With respect to the usual intersection product, the resulting lattice is even and self-dual, and has signature (+) 3 , (−) 19 . A hyperkaehler structure is determined by the positive 3-plane spanned by the periods of the three hyperkaehler forms in that lattice. Once a compatible complex structure is chosen, this three-plane has an orthogonal decomposition into the line generated by ω (the Kaehler form), and the plane spanned by the real and imaginary components of the holomorphic 2-form Ω = x+iy. A change of complex structure amounts to rotating the 2-plane spanned by the vectors x, y. In the context of string theory, the moduli space contains in addition the B-field, and the full moduli space takes the form of a Grassmannian parametrising 4-planes in R 4, 20 . A decomposition of the positive 4-plane into two orthogonal 2-planes then amounts to fixing the complex structure, a Kaehler class and a B-field. In [19] , the four 2-forms (three Hyperkaehler forms and the B-field) have been combined into a single quaternionic 2-form. Mirror symmetry, which interchanges the complex structure with the complexified Kaehler structure, acts in this language as a quaternionic rotation of the positive 4-plane. BPS D-branes on K3 surfaces Comparing with the conformal field theory description, the choice of a decomposition of the positive 4-plane into two perpendicular 2-planes amounts to the choice of an N = (2, 2) subalgebra inside the N = (4, 4). The two SU(2) enhancing the N = (2, 2) to N = (4, 4) can be understood as the freedom to rotate the two 2-planes. BPS D-branes on K3 surfaces Geometrically, B-type D-branes correspond to holomorphic branes, whereas A-type branes wrap (special) Lagrangian submanifolds. In the case of K3, B-type branes can have dimension 0, 2, 4, whereas A-type branes are always 2-dimensional. Some of the 22 2-cycles will thus be wrapped by A-type branes, and some by B-type branes, but the decomposition into A-type and B-type branes depends, of course, on the chosen complex structure. For example, the quaternionic rotation that induces mirror symmetry exchanges holomorphic and Lagrangian cycles. The action of mirror symmetry on the D-branes can also be understood from the point of view of [23] , where mirror symmetry was formulated for elliptic fibrations with a section as T-duality on the fiber. In the K3 context, this point of view has been used to extend mirror symmetry to the open string sector in [24] . Homological mirror symmetry has been proven for the quartic surface in [25] . B-type branes and the Picard lattice As we have explained above, supersymmetric 2-cycles on K3 are holomorphic curves with respect to some complex structure. If a 2-cycle is holomorphic with respect to a given complex structure, it can be wrapped by a D-brane that is B-type with respect to the corresponding N = (2, 2) subalgebra. In the following we review some background material from [26] . B-type branes and the Picard lattice The 2-cycles are naturally elements of H 2 (S, Z), or, using duality, of H 2 (S, Z). Holomorphicity imposes that the dual 2-form is in fact in H 1,1 (S), and the Picard lattice is thus B-type branes and the Picard lattice 3) B-type branes and the Picard lattice The rank of the Picard lattice is usually denoted by ρ. Generically, K3 surfaces will have ρ = 0, meaning that no B-type 2-branes are compatible with the given holomorphic structure. However, in this paper we will always consider special geometric points at which the rank of the Picard lattice is enhanced or even maximal. We are particularly interested in the case where S is a hypersurface described by a Fermat polynomial in a weighted projective space. In such a case, there is a correspondence [27] between the non-linear sigma model on the hypersurface and the Landau Ginzburg model with a superpotential that formally equals the polynomial appearing in the hypersurface equation. To be more precise, the hypersurface in P w 1 ,w 2 ,w 3 ,w 4 [H], with H = w i B-type branes and the Picard lattice where k i + 2 = H/w i , H = lcm{k i + 2}, corresponds to the Landau Ginzburg orbifold model with superpotential B-type branes and the Picard lattice In this equation the x i denote chiral superfields of charge q L = q R = 1/(k i + 2); our notation will not distinguish between the chiral fields of the Landau-Ginzburg model and the coordinates of the projective space in the geometric description. The orbifold Z H acts by phase multiplication on the chiral superfields x i → e 2πi k i +2 B-type branes and the Picard lattice x i ; this orbifold projects onto integer U(1) charges of the theory. Altogether, there are 14 different examples corresponding to Fermat polynomials in weighted projective space, which we list in table 1. These models also have a description in terms of rational conformal field theory, namely as the tensor product of 4 N = 2 minimal models at levels k i , modulo an integer charge projection Z H . In terms of conformal field theory, it is straightforward to see that the space of B-type RR charges is 22-dimensional for each of these models. By standard conformal field theory arguments one should therefore expect that the corresponding Btype D-branes exist, and thus that the rank of the Picard lattice is maximal for all of these points. B-type branes and the Picard lattice The condition that S can be written as a hypersurface in weighted projective space imposes constraints on the allowed complex structure deformations, and thus increases the generic rank of the Picard lattice [26] . For example, in the case of the quartic in P 3 B-type branes and the Picard lattice there is at least one holomorphic curve at any point in the complex structure moduli space, namely the intersection of the quartic polynomial with any hyperplane. This phenomenon generalises immediately to all hypersurface equations in weighted projective space, where one can always consider the intersection with a hyperplane. In some examples the generic rank of the Picard lattice may be enhanced even further. Consider for example the model P (1,1,2,2) [6] . The embedding weighted projective space has a Z 2 orbifold singularity with fixed point (0, 0, x 3 , x 4 ). The Z 2 singularity is resolved by an exceptional P 1 . It intersects with the hypersurface in the 3 points that are defined by the equations x 1 = x 2 = 0 and x 3 3 + x 3 4 = 0. This enhances the rank of the generic Picard lattice by 3. Altogether, the rank of the Picard lattice is therefore 4 at generic points in the complex structure moduli space for this example. Note that there is one 2-cycle that is inherited from the embedding space: it corresponds to the combination of the three spheres, which is invariant under Projective space B-type branes and the Picard lattice Minimal model RS RS,ST P (1,1,1,1 x 3 → exp(2πi/3)x 3 , which permutes the 3 singular points on the hypersurface. There are therefore 4 different brane charges, two D2, the D0 and D4 that the hypersurface inherits directly from the embedding space. More generally, whenever two weights have a greatest common divisor m by which the other two weights are not divisible, the embedding projective space acquires an orbifold singularity which locally has the form C 2 /Z m . Its resolution requires m − 1 P 1 s whose intersection pattern is given by the A m−1 Dynkin diagram. This means that any such singularity contributes m − 1 2-brane charges to the charge lattice that can be obtained by pull back from the embedding projective space. To determine the contribution to the Picard lattice of the hypersurface, one has to take into account that, as in the example above, the hypersurface might intersect the exceptional set in several points. Each of them gives a contribution to the Picard lattice. We will interpret these general features of the Picard lattice from the matrix factorisation point of view in section 3. The orbifold line Generically, the points in moduli space where a conformal field theory description is known are isolated. For example, for the above theories we only have a conformal field theory description (namely a Gepner model) for the unperturbed superpotential. There is, however, one interesting exception to this: since the (2) 4 model is in fact equivalent to the Z 4 toroidal orbifold [9] , there is a two-parameter family of orbifold theories all of which describe K3. The corresponding subspace of the moduli space has recently been identified to be [12] The orbifold theory will be described in more detail in section 4; the detailed mapping between a and b and the relevant parameters of the orbifold theory was given in [28, 12] . For this subspace of the moduli space we therefore have a good understanding of both the conformal field theory and the matrix factorisation approach. We should thus be able to compare the results from both points of view. The matrix factorisation description will be given in the following section, where we will in particular show that certain D-branes are obstructed against modifying the bulk parameters a and b. In section 4 we will identify the corresponding boundary states in the orbifold conformal field theory and reproduce these obstructions also from that point of view. The matrix factorisation point of view In this section we shall analyse the above theories from the matrix factorisation perspective. This approach was proposed in unpublished form by Kontsevich, and the physical interpretation of it was given in [3, 4, 5, 6, 7, 8] , for a review see also [29] . We shall first collect very briefly some basic facts about matrix factorisations that we shall need later on. Fundamentals Kontsevich has proposed that D-branes in a Landau-Ginzburg models are given by matrix factorisations of the superpotential, Fundamentals where Q is a square matrix with polynomial entries that satisfies Fundamentals If we choose the grading operator σ to be diagonal, Q is of the form Fundamentals Two factorisations (E, J) and (E ′ , J ′ ) are considered equivalent if they are related by a similarity transformation with invertible matrices with polynomial entries, Fundamentals The spectrum of open strings between D-branes determined by factorisations (E, J) and (E ′ , J ′ ) is then given by the cohomology of the boundary BRST operator Q. From a physics point of view, the factorisation condition can be derived by varying the LandauGinzburg action and cancelling the boundary terms [3, 4] . This analysis also confirms that Q is the correct boundary BRST operator. The proposal got further support by relating the results from the matrix factorisation perspective with those obtained in conformal field theory; in particular, this was done for the N = 2 minimal models in [4, 6, 30] and for tensor products of minimal models in [31, 32] . Finally, the matrix factorsation results for toroidal theories were shown to be in agreement with geometrical expectations [33, 34, 35] . Fundamentals We are particularly interested in Landau-Ginzburg orbifolds of the form (2.5). In this situation, the orbifold group Z H gives an additional finer grading. This grading correspond to the choice of a representation γ M such that Q satisfies Fundamentals There are H different choices for γ M that are labelled by M. Given a matrix factorisation Q, the charge of the corresponding D-brane under the RR ground states can be calculated using the formulas derived in [5, 36, 37] . RR ground states in Landau-Ginzburg orbifolds arise both from the twisted and untwisted sector and can be counted using the techniques of [38] . The RR ground states from the untwisted sector correspond to polynomials in the Landau-Ginzburg fields and have the property that the U(1) charges of the left and right moving part are equal, q L = q R . In the simplest case, where the weights do not have divisors (such as for the quartic in P 3 ), there is exactly one RR ground states from each twisted sector. Fundamentals For the general case, let n = 0, . . . , H − 1 label the twisted sectors. In each sector, consider only the untwisted fields x i such that nw i /H ∈ Z, and set all twisted fields to zero. Let φ Fundamentals The RR ground states |n; α are obtained by acting with φ α n on the unique state |n; 0 . (For n = 0 this representation corresponds to (4.7) -see below.) Note that not all of the RR ground states obtained in this way survive the orbifold projection, which has to be imposed for all twists. The RR charge of Q with respect to a surviving RR ground state |n; α is given by [5, 36] Fundamentals Here r n is the number of untwisted fields, and W n and Q n are the superpotential and the factorisation with all twisted fields set to zero. The supertrace is the trace taken with the grading operator σ included, i.e. Fundamentals In the context of the correspondence between Landau-Ginzburg models and CalabiYau manifolds, one would expect that one can associate to any matrix factorisation an element of the derived category of coherent sheaves of the Calabi-Yau manifold. The derived category of coherent sheaves and the category of graded matrix factorisations have to be equivalent since both are believed to describe the topological category of Btype branes [39, 36] , which is supposed to decouple from the Kaehler moduli. One way to investigate this correspondence would be to analyse matrix factorisations in the context of the linear sigma-model. Since this has not yet been done to date, we will use a result of Orlov [40] , who established mathematically a correspondence between the 'category of singularities' D Sg , and the category of matrix factorisations (with the equivalence relations mentioned above). The category of singularities D Sg is a certain quotient of the derived category of coherent sheaves that depends only on the singularity and would be empty on a smooth manifold. Fundamentals Orlov's equivalence was formulated for the case of the un-orbifolded Landau-Ginzburg model. Given a matrix factorisation W = EJ one interprets the two factors as maps between projective modules over the polynomial ring C[x i ] -in our case these modules are simply C[ Fundamentals ⊕n for a factorisation in terms of n × n matrices E, J. One then associates to a factorisation the object coker J, which naturally lives on W = 0. This assignment associates to a single transposition brane in an unorbifolded two variable model the set x 1 − ηx 2 = 0. It has been shown [31] that the geometric intersection numbers can be matched with the intersection numbers derived from matrix factorisations (as well as with those obtained from permutation boundary states in conformal field theory). For the case of graded matrix factorisations in Landau-Ginzburg orbifolds, the idea is then that linear factorisations still describe the geometric object coker J for one choice of the representations γ M ; the D-branes corresponding to the other representations are images of that brane under the Landau-Ginzburg monodromy. For a number of examples this assignment has been verified for linear transposition and tensor product factorisations in [39, 31] . This was done by using alternative methods [41] to calculate the large volume charges corresponding to the branes at the Landau-Ginzburg point. In this paper, we will use these ideas to guess linear matrix factorisations corresponding to certain geometric D-branes. Basic factorisations The factorisations we shall mainly consider in this paper can be obtained as graded tensor products Q 1 ⊙ Q 2 [39, 11] of two simple classes of factorisations. The first construction involves a single factor theory of the form W = x h , for which we can construct a factorisation as Basic factorisations The branes that correspond [39] to the tensor product of four such factorisations are the RS D-branes with L = 0 [42] . It follows from (3.6) that these branes do not couple to RR charges in the untwisted sector. The other construction involves two factors of the form Basic factorisations Let η denote an h th root of −1, then we have the factorisation [39] Basic factorisations It was shown in [31] (see also [32] ) that the corresponding branes are permutation branes with L = 0 [43] . More generally, these factorisations can also be constructed for the case that h 1 and h 2 have a non-trivial common factor (but are not equal) [10] . The corresponding branes should then be generalised permutation branes similar to those considered in [44] . If we tensor this permutation factorisation to two tensor factorisations, we get a transposition brane, denoted for example by (34) . Once again, it only couples to charges in the twisted sectors. One can, of course, also consider tensoring with another permutation brane whenever the divisibility properties of the weights allow this. We will call the resulting branes double transposition branes and denote them by (12)(34), etc. Deformations For the following it will also be important to understand how the B-type D-branes behave under deformations of the complex structure. In particular, we will consider deformations of the Landau-Ginzburg superpotential by suitable quasihomogeneous polynomials of appropriate weight V , W → W (ψ) = W + ψV , where ψ denotes the parameter of the deformation. If Q is a factorisation of W , then we ask whether there is Q(ψ) = Q + f (ψ)δQ with f (0) = 0 such that Q is a factorisation of W . If such a Q(ψ) exists (at least in the neighbourhood of ψ = 0) we shall say that the D-brane can be extended for the deformation described by V . Global deformations There exist some classes of branes that can be extended for all possible deformations. In particular, this is the case for the tensor factorisations that correspond to RS branes. In order to see this we note that we can write any superpotential W as 9) where the F are suitable polynomials. In fact, we have Global deformations We can thus define factorisations that are 4-fold tensor products of the factorisations x i F i . These factorisations are the deformations of the standard tensor factorisations. Indeed, as we approach the Gepner point, we have Global deformations , and these factorisations reduce to the tensor branes. Global deformations In a similar way, we can extend single transposition branes. For definiteness we assume that w 3 = w 4 and define Global deformations At the Gepner point, the solutions for α are given by the (k 3 + 2)th roots of −1 and the factorisation reduces to the transposition brane, as required. Global deformations Of particular interest is the case where w 3 = w 4 = 1. In this case we are geometrically in the situation that the projective space acquires a singularity and the hypersurface intersects with it for generic complex structure deformations. The intersection points are then exactly given by L 1 = L 2 = L 3 = 0, where α solves (3.12). Global deformations We should note that in both cases, the factorisations that can be deformed do not couple to the charges in the untwisted sector (that are in turn in one-to-one correspondence to the polynomial deformation moduli). We shall see later on that this is indeed a necessary condition for the deformation to be possible. Enhancement of the Picard lattice As we have seen in section 2, the rank ρ of the Picard lattice is enhanced for hypersurfaces in weighted projective space. We would now like to understand this enhancement from the point of view of matrix factorisations. Enhancement of the Picard lattice Let us first discuss the part of the charge lattice that is inherited from the embedding space. For this, we consider the tensor product factorisations that correspond to the RS-branes. As we have just seen, these factorisations exist for arbitrary deformations of the superpotential. We expect on general grounds [45, 46, 47, 48] that these branes carry precisely all the charges that can be obtained as pullbacks from the embedding space. If this is so, then it follows from the discussion in section 2 that their rank should be Enhancement of the Picard lattice Here, the 3 represents the D0, D4 and generic D2 charge, and the other contribution comes from the resolution of the singularities of the embedding weighted projective space. We have verified that this relation is indeed true for all 14 examples; the relevant rank is given in the penultimate column of table 1. This gives good support to the assertion that the tensor factorisations account precisely for the charges that can be understood in terms of the embedding projective space. Enhancement of the Picard lattice As we have seen, a Z m singularity of the embedding space can lead to an enhancement of the rank of the Picard lattice of the hypersurface by a multiple of m − 1 if the hypersurface intersects the exceptional locus in more than one point. For example, the rank of the Picard lattice of the example P 1,1,2,2 [6] was shown to have 4 as a lower bound, where 3 holomorphic curves come from replacing the points z 1 = z 2 = 0, z 3 − ηz 4 = 0, η 3 = −1 by P 1 's. It is now natural to believe that these additional charges can be obtained as matrix factorisations of type (34) , where η appears as the parameter in the (34) part of the factorisation. Enhancement of the Picard lattice In order to check this claim we have verified that for each η, the rank of the charge lattice of the tensor and (34) factorisations is bigger by one than that of the tensor factorisations. Furthermore, if we consider two different (34) factorisations with different η, the rank is increased by 2, but considering all three different values does not increase the rank any further (since the symmetric combination of the three η values is already part of the tensor charges). Furthermore, as we have just seen, all of these factorisations can be defined for arbitrary complex structure deformations. This explains from a matrix factorisation point of view that for P 1,1,2,2 [6] ρ ≥ 4 at a generic point in the complex structure moduli space. Enhancement of the Picard lattice We have studied these phenomena also for the other examples. The rank of the charge lattice spanned by the (34) branes, where lcm(w 3 , w 4 ) = m is (for fixed value of η) always by m − 1 bigger than the rank of the tensor product lattice. Furthermore, including all values of η we obtain the generic part of the Picard lattice that arises because the K3 surface is embedded in the weighted projective space under consideration. The rank of the charge lattice that is generated by these factorisations is given in the last column of table 1; this agrees always with what is expected based on the geometric analysis of section 2. An infinitesimal analysis In section 3.3 we considered special factorisations that could be globally deformed. We would now like to investigate under which conditions a given factorisation can at least be infinitesimally deformed. Given Q 0 , we want to find a Q(ψ) with Q(ψ) → Q 0 for ψ → 0 such that Q(ψ) 2 = W + ψV (3.14) An infinitesimal analysis at least for small ψ. We make the analytic ansatz [11] 15) and obtain to first order An infinitesimal analysis As V · 1 is Q 0 -closed, this reduces to a cohomology problem: if V is not exact, then Q 0 is obstructed and cannot be continued. At higher order we obtain similar conditions: since An infinitesimal analysis the right hand side must be Q 0 -closed as well. In principle, obstructions may occur at higher orders too, but we have not found any examples where higher order obstructions were important. The above ansatz (3.15) implies that the deformation is analytic, but it is conceivable that non-analytic deformations could exist. In particular, holomorphicity implies that there is only one smooth family of brane deformations, but physically there are certainly situations where more than a single deformation could compensate for a given bulk perturbation. In such cases we would expect non-analytic behavior of Q(ψ). We can make a more general ansatz by making ψ an analytic function of a parameter φ: An infinitesimal analysis However, this new ansatz is in fact only more general than (3.15) if the cohomology of Q 0 is non-trivial. (Physically, this corresponds to Q 0 having fermions in its self-spectrum.) While we cannot solve the obstruction problem in general, we can at least give a necessary condition for the analytic deformation to be unobstructed. A necessary condition for unobstructed deformations We can make a general statement about the conditions that allow a brane to be continued: if Q can be continued analytically under the deformation V , then Q is not charged with respect to the corresponding RR ground state φ. A necessary condition for unobstructed deformations To prove this we start out with (3.6) in the untwisted sector. First of all, it is clear that if r is odd, the charge is always zero. We can thus assume that there is an even number of factors (as is always the case for the K3 examples). The charge is given by A necessary condition for unobstructed deformations where we have used that V must be exact if Q can be continued. Consider now terms of the form Q∂Q. Since Q∂Q = ∂(Q 2 ) − ∂QQ and Q 2 = W 1, A necessary condition for unobstructed deformations and ∂ i W cancels. At all Gepner points, W = x A necessary condition for unobstructed deformations , so x i only appears in the numerator. The residue integral dx i it thus zero. (This argument works also if W is not of the particular form given above, see [5] .) This calculation shows that ∂Q and Q anticommute in the supertrace. Pulling Q through all the factors and using (anti-)cyclicity of the supertrace, we see that QA cancels out with AQ and ch(Q) is thus zero. A necessary condition for unobstructed deformations This proof works for the twisted sector as well. In this case it suffices to realise that Q n commutes with γ n M . This follows from the fact that A necessary condition for unobstructed deformations But according to the definition of Q n , only those x i appear for which ω nw i = 1. It is also clear that if we insert any fermionic boundary operator F such that {Q, F } = 0, the charge remains zero. If there is an odd number of factors now, this is trivial, otherwise F just provides the additional sign that makes the trace disappear. A necessary condition for unobstructed deformations We note in passing that the globally deformed branes that we discussed in section 3.3.1 do indeed satisfy this condition. A counterexample In the previous subsection we have seen that a necessary condition for the brane Q not to be obstructed under the deformation V is that Q is not charged under V . One may wonder whether this condition is also sufficient, but this is not true. As an explicit counterexample consider the deformation of the (12)(34) A counterexample , and consider the superposition Q of these two factorisations A counterexample . Nevertheless it cannot be analytically deformed. If it could, we would have to find a matrix A counterexample consisting of polynomial block matrices A, B, . . . , H such that A counterexample This yields eight (matrix) equations. The first and the fifth one are A counterexample These are, however, the very equations we find if we want to deform Q A counterexample and thus not analytically deformable, so (3.25) has no solution. This shows that Q is not analytically deformable either. Quartics on the orbifold line As an interesting application of the above techniques we now want to study the line of 'very attractive' quartics (2.7) [12] from a matrix factorisation perspective. First we need to collect some information regarding the RR charges. Quartics on the orbifold line It is easy to see that there are 21 monomial deformations of the quartic superpotential that have integer U(1) charge in the closed string theory. These correspond to 21 RR ground states in the untwisted sector. 19 of these monomials are of charge 1 -the corresponding RR ground states have charge 0 and can couple to both A-type and B-type branes. The remaining two integer charge monomials 1 and x Our first task is to find a set of matrix factorisations that span the full B-type charge lattice. At the Gepner point, such a set is given by the double transposition branes (12)(34), (13)(24), (14)(23), which span a charge lattice of rank 22, accounting for the D0, D4 and 20 D2 branes. This in particular verifies that the Picard lattice at the Gepner point has maximal rank (namely 20). In the following, we want to analyse the deformations of these factorisations along the orbifold line. Deformations of the Gepner point It follows from our general discussion above that the D0-brane factorisation (34) and the tensor factorisation can be extended over the full complex structure moduli space, and therefore in particular, also along the orbifold line. These branes only couple to the three twisted RR charges, and thus account for the three generic RR-charges (that correspond to the D0, the D4, and the one D2-brane). Deformations of the Gepner point Next we observe that the (12)(34) factorisations can also be continued to arbitrary points on the orbifold line. To see this, we make the ansatz By the same argument we also see that the (13)(24) branes with Deformations of the Gepner point can be extended to those a and b that satisfy aη (13)(24) and (14)(23) are obstructed, as follows immediately from the fact that they are charged under the corresponding deformations. On the other hand, it is possible to construct a factorisation of the deformed superpotential by writing it as We can then consider the tensor product of the permutation factorisation of the first two and the last two terms. For a → 0, b → 0, these factorisations reduce to tensor products of permutation factorisations in x Thus we have found a set of factorisations that can be deformed along the whole orbifold line and whose charges generate a sublattice of rank 20. As follows from the analysis of section 3.4.1, all of these factorisations are uncharged under x On the other hand, it also follows from our analysis of section 3.4.1, that any factorisation that is charged with respect to x cannot be analytically deformed. Furthermore, the (13)(24) brane for example does not have a fermion in its self-spectrum, and thus also the non-analytic solution of the type (3.18) cannot exist. This seems to predict that the corresponding D-brane in the orbifold theory should also be obstructed. We shall explain in detail in the next section that this is indeed so. The conformal field theory description In order to understand in more detail how the geometric and the matrix factorisation point of view fit together it is useful to study the quartic K3 surface (and its orbifold line) directly in conformal field theory. The conformal field theory we are interested in has two equivalent descriptions: it can be described as the Gepner model corresponding to the four-fold tensor product of four N = 2 minimal models with k = 2; on the other hand, the theory is also equivalent to the Z 4 orbifold of a T 4 -torus. The equivalence involves in fact mirror symmetry. In the following we shall first explain briefly the relevant Gepner model construction, and then describe in more detail the torus orbifold realisation and the correspondence between the two descriptions. Finally we shall describe some of the D-branes from both points of view, and explain how they deform under the Kaehler deformations of the orbifold theory. The Gepner model The Gepner description is standard [49] , so we shall be fairly brief in the following. (A more comprehensive introduction to Gepner models can be found in [50] ; our conventions are explained in more detail for example in [42, 31] .) The Gepner model The Gepner model of interest is the Z 4 -orbifold of the four-fold tensor product of k = 2 minimal models (each having c = 3/2, so that the total central charge is c tot = 6). As usual we label the representations of the bosonic subalgebra of the N = 2 superconformal algebra by triples Representations with s even belong to the Neveu-Schwarz sector, while those with s odd belong to the Ramond sector. The space of states of the full theory is of the form The Gepner model where n = 0, 1, 2, 3 denotes the twisted sector, and s i (ands i ) are all either even (NS) or all odd (R). The labels m i are subject to the integrality condition The Gepner model Finally, we may impose the type 0B GSO-projection which requires that The Gepner model Of particular importance are the RR ground states of this theory. Ramond ground states are characterised by the property that their conformal weight h equals c/24. One can easily show that the ground state of the sector (l, m, s) is a Ramond ground state if it is of the form (l, l + 1, 1) or (l, −l − 1, −1). The above Gepner model possesses 24 RR ground states; in each of the three twisted sectors (n = 1, 2, 3) there is one RR ground state which is the state The Gepner model The remaining 21 RR ground states come from the untwisted n = 0 sector; if we associate to the R ground state representations The Gepner model where the index i refers to the ith factor, then we have the state 1, x The torus orbifold The torus in question is simply the orthogonal product of four circles, which initially all have the self-dual radius and vanishing B-field. For the following it is convenient to write this 4-torus as The torus orbifold The Z 4 orbifold acts by a counterclockwise rotation by 90 degrees in the first T 2 , and by a clockwise rotation by 90 degrees in the second. We denote the four real directions by y i with i = 1, 2, 3, 4, and introduce complex coordinates in the usual way: z 1 = y 1 + iy 2 and z 2 = y 3 + iy 4 . The Z 4 action is then The torus orbifold We denote the fermionic fields by χ i andχ i , where i = 1, 2, 3, 4. The corresponding complex fields are then The torus orbifold (4.9) with similar formulae for the right-moving fields,ψ 1 , etc. The momentum ground states of the torus are labelled by four momentum numbers n i , i = 1, 2, 3, 4, and four winding numbers w j , j = 1, 2, 3, 4. For the ith direction the left-and right-moving momenta are then The torus orbifold where initially all The torus orbifold in our conventions. On the ground states, the Z 4 action maps (n 1 , w 1 , n 2 , w 2 , n 3 , w 3 , n 4 , w 4 ) → (−n 2 , −w 2 , n 1 , w 1 , n 4 , w 4 , −n 3 , −w 3 ) . This symmetry requires only that R 1 = R 2 and R 3 = R 4 , but neither needs to take the self-dual value. Thus there is a two (real)-dimensional space of deformations that preserve the orbifold symmetry. One should expect on general grounds that this is only part of a two (complex)-dimensional space of deformations, and this is indeed so. One easily sees that one can also switch on an arbitrary B-field in either of the two T 2 : if we concentrate on the first T 2 , then the momenta are of the form The torus orbifold It is then easy to see that the spectrum is invariant under the The torus orbifold In fact, this action still corresponds precisely to the action (on the first two coordinates) of (4.11). The torus orbifold In the following we shall mainly concentrate on the theory where the B-field vanishes and all the radii take the self-dual value The torus orbifold ; this is the theory that corresponds precisely to the Gepner model (2) 4 . Unless mentioned otherwise this is what we shall call the torus orbifold in the following. A partial dictionary Before proceeding we shall match a few low-lying states in order to understand how the identification works. In the untwisted NS-NS sector of the torus orbifold, the lowest lying states is the vacuum with h =h = 0, as well as four states of h =h = 1/4. The latter are the Z 4 -orbits of the states for which the only non-vanishing momentum and winding number is n 1 = 1, or w 1 = 1 or n 3 = 1 or w 3 = 1. In the first (g) and third (g 3 ) twisted sector of the torus orbifold, there are 4 fixed points each, and each of them has (in the NS-NS sector) ground state energy h =h = 1/4. (There are 10 Z 4 -orbifold invariant Z 2 -fixed points in the second (g 2 ) twisted sector, but their ground state energy is higher.) In total the torus orbifold therefore has 12 NS-NS states with h =h = 1/4. A partial dictionary In the N = 2 orbifold (the Gepner model), all of these states appear in the untwisted (n = 0) sector; the vacuum obviously
{'paper_id': '16501335', 'title': 'Matrix factorisations and D-branes on K3', 's2_url': 'https://api.semanticscholar.org/CorpusID:16501335', 'source': 'S2ORC'}