One-dimensional super Calabi-Yau manifolds and their mirrors

Journal of High Energy Physics, Apr 2017

We apply a definition of generalised super Calabi-Yau variety (SCY) to supermanifolds of complex dimension one. One of our results is that there are two SCY’s having reduced manifold equal to \( {\mathrm{\mathbb{P}}}^1 \), namely the projective super space \( {\mathrm{\mathbb{P}}}^{\left.1\right|2} \) and the weighted projective super space \( \mathbb{W}{\mathrm{\mathbb{P}}}_{(2)}^{\left.1\right|1} \). Then we compute the corresponding sheaf cohomology of superforms, showing that the cohomology with picture number one is infinite dimensional, while the de Rham cohomology, which is what matters from a physical point of view, remains finite dimensional. Moreover, we provide the complete real and holomorphic de Rham cohomology for generic projective super spaces \( {\mathrm{\mathbb{P}}}^{\left.n\right|m} \). We also determine the automorphism groups: these always match the dimension of the projective super group with the only exception of \( {\mathrm{\mathbb{P}}}^{\left.1\right|2} \), whose automorphism group turns out to be larger than the projective super group. By considering the cohomology of the super tangent sheaf, we compute the deformations of \( {\mathrm{\mathbb{P}}}^{\left.1\right|m} \), discovering that the presence of a fermionic structure allows for deformations even if the reduced manifold is rigid. Finally, we show that \( {\mathrm{\mathbb{P}}}^{\left.1\right|2} \) is self-mirror, whereas \( \mathbb{W}{\mathrm{\mathbb{P}}}_{(2)}^{\left.1\right|1} \) has a zero dimensional mirror. Also, the mirror map for \( {\mathrm{\mathbb{P}}}^{\left.1\right|2} \) naturally endows it with a structure of N = 2 super Riemann surface.

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One-dimensional super Calabi-Yau manifolds and their mirrors

Received: September One-dimensional super Calabi-Yau manifolds and their mirrors S. Noja 0 1 3 5 6 7 8 9 10 S.L. Cacciatori 0 1 3 4 6 7 8 9 10 F. Dalla Piazza 0 1 3 6 7 8 9 10 A. Marrani 0 1 2 3 6 7 8 9 10 R. Re 0 1 3 6 7 8 9 10 0 Via Celoria 16, I-20133 Milano , Italy 1 Via Valleggio 11, I-22100 Como , Italy 2 Centro Studi e Ricerche `Enrico Fermi' 3 Via Saldini 50, I-20133 Milano , Italy 4 INFN , Sezione di Milano 5 Dipartimento di Matematica, Universita degli Studi di Milano 6 Open Access , c The Authors 7 Viale Andrea Doria 6 , 95125 Catania , Italy 8 Via Marzolo 8, I-35131 Padova , Italy 9 and INFN , Sezione di Padova 10 Via Panisperna 89A, I-00184 Roma , Italy We apply a de nition of generalised super Calabi-Yau variety (SCY) to supermanifolds of complex dimension one. One of our results is that there are two SCY's having reduced manifold equal to P1, namely the projective super space P1j2 and the weighted projective super space WP(12j1). Then we compute the corresponding sheaf cohomology of superforms, showing that the cohomology with picture number one is in nite dimensional, while the de Rham cohomology, which is what matters from a physical point of view, remains nite dimensional. Moreover, we provide the complete real and holomorphic de Rham cohomology for generic projective super spaces Pnjm. We also determine the automorphism groups: these always match the dimension of the projective super group with the only exception of P1j2, whose automorphism group turns out to be larger than the projective super group. By considering the cohomology of the super tangent sheaf, we compute the deformations of P1jm, discovering that the presence of a fermionic structure allows for deformations even if the reduced manifold is rigid. Finally, we show that P1j2 is self-mirror, whereas WP(12j1) has a zero dimensional mirror. Also, the mirror map for P1j2 naturally endows it with a structure of N = 2 super Riemann surface. ArXiv ePrint: 1609.03801 mirrors; Di erential and Algebraic Geometry; Superspaces; String Duality; Supersym- - metry and Duality 1 Introduction Supermanifolds and projective super spaces De nitions and notions in supergeometry Projective super spaces and weighted projective super spaces Vector bundles over P1, Grothendieck's theorem and cohomology of OPn (k)-bundles Super Calabi-Yau varieties The sheaf cohomology of di erential and integral forms De Rham cohomology of WP(12j1) and P1j2 The complete de Rham cohomology of Pnjm Automorphisms and deformations of P1jm A super mirror map for SCY in reduced dimension 1 Mirror construction for P1j2 Mirror construction for WP(2j) P1j2 as a N = 2 super Riemann surface A Super Fubini-Study metric and Ricci atness of P1j2 Introduction \Super-mathematics" has quite a long history, starting from the pioneering papers by Martin, [1, 2] and Berezin, [3, 4], before the discovery of supersymmetry in physics.1 After its appearance in physics in the 70s, however, supergeometry has caught more attention in the mathematical community, and corresponding developments appeared not only in numerous articles but also in devoted books, see e.g. [6]{[15]. In most of the concrete applications of supersymmetry, like in quantum eld theory or in supergravity, algebraic properties play a key role, whereas geometry has almost always a marginal role (apart from the geometric formulation of superspace techniques; see further below). This is perhaps the reason why some subtle questions in supergeometry (see for example [12]) have not attracted too much the attention of physicists and, as a consequence, the necessity of further developments has not been stimulated. 1Even though anticommutation was proposed yet previously by Schwinger and other physicists, see [5] for a more detailed account. String theory makes exception. Perturbative super string theory is expected to be described in terms of the moduli space of super Riemann surfaces, which results to be itself a supermanifold. However, some ambiguities in de ning super string amplitudes at genus higher than one suggested, already in the 80s, that the geometry of such super moduli space may not be trivially obtained from the geometry of the bosonic underlying space [16]. More than twenty years of e orts have been necessary in order to be able to unambiguously compute genus two amplitudes; cfr. e.g. the papers by D'Hoker and Phong [17]{[27], which also include attempts in de ning genus three amplitudes, without success but renewing the interest of the physical community in looking for a solution to the problem of constructing higher genus amplitudes. Through the years, various proposals have been put forward, see e.g. [28]{[48]. However, most of such constructions were based on the assumption that the supermoduli space is projected (see below for an explanation), but a careful analysis of perturbative string theory and of the corresponding role of supergeometry [49]{[53] suggested that this could not be the case. Indeed, it has been proved in [54] (see also [55]) that the supermoduli space is not split and not projected at least for genus g 5. Obviously, such result gave rise to new interest in understanding the peculiarities of supergeometry with respect to the usual geometry, in particular from the viewpoint of algebraic geometry. A second framework in which supergeometry plays a prominent role is the geometric approach to the superspace formalism, centred on integral forms (discussed e.g. in [56{58]; see below), whose application in physics can be traced back to [49, 59]. Superspace techniques are well understood and used in quantum eld theory, supergravity as well as in string theory (see e.g. [60, 61]). They provide a very powerful method to deal with supersymmetric multiplets and to determine supersymmetric quantities, such as actions, currents, operators, vertex operators, correlators, and so on. However, even when the superspace formulation exists, it is often di cult to extract the component action. This occurs often in supergravity, in which the superdeterminant of the supervielbein is needed for the construction of the action, making the computation pretty cumbersome in a number of cases. On the other hand, the so-called \Ectoplasmic Integration Theorem" (EIT) [62]{[65] can be used in order the extract the component action from the superspace formulation. Generally, supermanifolds are endowed with a tangent bundle (generated by commuting and anticommuting vector elds) and with an exterior bundle; thus, one would navely expect the geometric theory of integration on manifolds to be exported tout court in supersymmetric context. Unfortunately, such an extension is not straightforward at all, because top superforms do not exist, due to the fact that the wedge products of the di erentials d ( being the anticommuting coordinates) are commuting, and therefore there is no upper bound on the length of the usual exterior d-complex. In order to solve this problem, distribution-like quantities (d ) are introduced, for which a complete Cartan calculus can be developed. Such distributions (d ) then enter the very de nition of the integral forms [66]{[71], which are a new type of di erential forms requiring the enlargement of the conventional space spanned by the fundamental 1-forms, admitting distribution-like expressions (essentially, Dirac delta functions and Heaviside step functions). Within such an extension of the d di erential, a complex with an upper bound arises, and this latter can be used to de ne a meaningful geometric integration theory for forms on supermanifolds. In recent years, this led to the development of a complete formalism (integral-, pseudoand super- forms, their complexes and related integration theory) in a number of papers by Castellani, Catenacci and Grassi [58, 73, 74]. In [73], the exploitation of integral forms naturally yielded the de nition of the Hodge dual operator F for supermanifolds, by means of the Grassmannian Fourier transform of superforms, which in turn gave rise to new supersymmetric actions with higher derivative terms (these latter being required by the invertibility of the Hodge operator itself). Such a de nition of F was then converted into a Fourier-Berezin integral representation in [75], exploiting the Berezin convolution. It should also be recalled that integral forms were as an integral on a supermanifold [76]. Furthermore, in [74], the cohomology of superforms and integral forms was discussed, within a new perspective based on the Hodge dual operator introduced in [73]. Therein, it was also shown how the superspace constraints (i.e., the rheonomic parametrisation) are translated from the space of superforms (pj0) to the space of integral forms (pjm) where 0 6 p 6 n, with n and m respectively denoting the bosonic and fermionic dimensions of the supermanifold; this naturally let to the introduction of the so-called Lowering and Picture Raising Operators (namely, the Picture Changing Operators, acting on the space of superforms and on the space of integral forms), and to their relation with the cohomology. In light of these achievements, integral forms are crucial in a consistent geometric (superspace) approach to supergravity actions. It is here worth remarking that in [58] the use of integral forms, in the framework of the group manifold geometrical approach [77, 78] (intermediate between the super eld and the component approaches) to supergravity, led to the proof of the aforementioned EIT, showing that the origin of that formula can be understood by interpreting the super eld action itself as an integral form. Subsequent further developments dealt with the construction of the super Hodge dual, the integral representation of Picture Changing Operators of string theories, as well as the construction of the super-Liouville form of a symplectic supermanifold [79]. A third context in which super geometry may be relevant is mirror symmetry. In [80], Sethi proposed that the extension of the concept of mirror symmetry to super Calabi-Yau manifolds (SCY's) could improve the de nition of the mirror map itself, since supermanifolds may provide the correct mirrors of rigid manifolds. Such a conjecture has been strengthened by the works of Schwarz [84, 85] in the early days, but it seems to have been almost ignored afterwards, at least until the paper of Aganagic and Vafa [86] in 2004, in which a general super mirror map has been introduced and, in particular, it has been shown that the mirror of the super Calabi-Yau space P3j4 is, in a suitable limit, a quadric P3j3. This is a quite interesting case, since these SCY's are related to amplitude computations in (super) quantum eld theories, see e.g. [87]. Since then, a number of studies on mirror symmetry for SCY's has been carried on, see for example [88]{[91]. However, a precise de nition of SCY is currently missing, and, consequently, the de nition of mirror symmetry and its consequences is merely based on physical intuition. The aim of the present paper is to provide a starting point for a systematic study of SCY's, by addressing the lowest dimensional case: SCY's whose bosonic reduction has complex dimension one. In section 2 we collect some de nitions in supergeometry and introduce the projective super spaces, which will play a major role in what follows. We will not dwell into a detailed exposition, and we address the interested reader e.g. to [11] and [12] for a mathematically thorough treatment of supergeometry. We also recall that an operative exposition of supergeometry, aimed at stressing its main connections with physics, is given in [49]. In section 3 we will be concerned with the geometry of the projective super space P1j2 and of the weighted projective super space WP(12j1). Cech and de Rham cohomology of super di erential forms are computed for these super varieties: here some interesting phenomena occur. Indeed we will nd that on the one hand one there might be some in nite-dimensional Cech cohomology groups as soon as one deals with more than one odd coordinate (as in the case of P1j2); on the other hand this pathology gets cured at the level of de Rham cohomology, where no in nite dimensional groups occur. Our interest in these two particular supermanifolds originates from the fact that, together with the class of the what follows, P1j2 and WP(12j1) are indeed the unique (non-singular) SCY's2 having reduced manifold given by P1. These are therefore the simplest candidates to be considered, as one is interested into extending the mirror symmetry construction in dimension 1 to a super geometric context, pursuing a task initially suggested in [80]. Moreover, despite we keep super spaces having generic dimension. In section 4 we will then construct the mirrors of the projective super spaces P1j2 and WP(2j), following a recipe introduced in [86]. Moreover, we will show that, surprisingly, by Finally, the main results and perspectives for further developments are discussed in section 5, whereas an appendix is devoted to illustrating the coherence of the adopted rule Supermanifolds and projective super spaces De nitions and notions in supergeometry In general, the mathematical basic notion that lies on the very basis of any physical supersymmetric theory is the one of Z2-grading: algebraic constructions such as rings, vector spaces, algebras and so on, have their Z2-graded analogues, usually called in physics super rings, super vector spaces and super algebras, respectively. A ring (A; +; ), for example, is called a super ring if (A; +) has two subgroups A0 and A1, such that A = A0 2In the sense speci ed further below. By additivity, this extend to a map [ ; ] : A By de nition, a super ring is super commutative if and only if all the super commutators among elements vanish (or in other words, if and only if the center of the super ring is the for all a 2 Ai; b 2 Ai with i 2 Z2. Supergeometry only deals with this class of super rings, allowing for anti-commutativity of odd elements. This has the following obvious fundamental consequence: all odd elements are nilpotent. A basic but fundamental example of super commutative ring (actually algebra) is provided by the polynomial superalgebra over a certain eld k, denoted as k[x1; : : : ; xp; 1; : : : ; q], where x1; : : : ; xp are even generators, and 1; : : : ; q are odd generators. The presence of the odd anti commuting part implies the following customary picture for this super algebra: k[x1; : : : ; xp; 1; : : : ; q] = k[x1; : : : ; xp] k which makes apparent that the theta's are generators of a Grassmann algebra. Even and odd superpolynomials might be expanded into the odd (and therefore nilpotent) generators Peven(x; ) = f0(x) + The generalisation of vector spaces to super vector spaces, as well as of algebras to super algebras, follows the same lines. Given an homogeneous element with respect to the Z2-grading of a super ring, we can de ne an application, called parity of the element, as follows: a 7 ! jaj ..= jaj = 1 (a 2 A1) are called odd or fermionic. It should be pointed out that so far no supersymmetric structure has been related to any super commutativity of elements, provided by the super commutator, i.e. a bilinear map acting on two generic homogeneous elements a; b in a super ring A, as follows: Podd(x; ) = where the f 's are usual polynomials in k[x1; : : : ; xp] and we have written i j instead of i ^ j for the sake of notation. As one wishes to jump from pure algebra to geometry, in physics it is customary to consider a supermanifold M of dimension pjq (that is, of even dimension p and odd dimension q) as described locally by p even coordinates and q odd coordinates, as a generalisation of the standard description of manifolds in di erential geometry. Even if this is feasible [49], due to the presence of nilpotent elements, in supergeometry it is preferable to adopt an algebraic geometric oriented point of view, in which a supermanifold is conceived as a locally ringed space [11, 12, 14, 54]. Within this global point of view, we de ne a super space M to be a Z2-graded locally ringed space, that is a pair (jM j; OM ), consisting of a topological space jM j and a sheaf of super algebras OM over jM j, such that the stalks OM ;x at every point x 2 jM j are local rings. Notice that this makes sense as a requirement, for the odd elements are nilpotent and this reduces to ask that the even component of the stalk is a usual local commutative ring. Morphisms between super spaces become morphisms of locally ringed spaces, i.e. they are given by a pair where i] : OM ! idjM j OM /JM = OM /JM . : jM j ! jN j is a continuous function and ] : ON ! OM is morphism of sheaves (of super rings or super algebras), which needs to preserve the Z2-grading. Clearly, OM contains the subsheaf JM of ideals of all nilpotents, which is generated by all odd elements of the sheaf: this allows us to recover a purely even super space, (jM j; OM /JM ), which is called reduced space Mred underlying M . There always exists a closed embedding Mred ,! M , given by the morphism (idjM j; i]) : (jM j; OM/JM ) ! (jM j; OM ), A special super space can be constructed as follows: given a topological space jM j and a locally free sheaf of OjM j-modules E , one can take OM to be the sheaf V E _: this makes OM out of a super commutative sheaf whose stalks are local rings. Similarly to [54], super spaces constructed in this way are denoted as S(jM j; E ). Ap is the ordinary p-dimensional a ne space over A and OAp is the trivial bundle over it. Super spaces like these are common in supersymmetric eld theories, where one usually works with Rpjq or Cpjq. A supermanifold is de ned as a super space which is locally isomorphic3 to S(jM j; E ) for some topological space jM j and some locally free sheaf of OjM j-module E . Along this line of reasoning, then, one recovers (out of a globally de ned object!) the original di erential geometric induced view that physics employs, in which a real supermanifold of dimension pjq locally resembles Rpjq and, analogously, a complex supermanifold of dimension pjq locally resembles Cpjq, de ned above: the gluing data are encoded in the cocycle condition satis ed by the structure sheaf. Given a supermanifold M , we will call Mred the pair (jM j; OM /JM ), which is an ordinary manifold conceived as a locally ringed space of a certain type: as above, a closed embedding Mred ,! M will always exist. It is here worth pointing out that, on the contrary, the de nition of supermanifold does of (idjM j; ]) : (jM j; OM ) ! (jM j; OM /JM ), where not imply the existence of a projection M ! Mred: this would correspond to a morphism ] is a sheaf morphism that embeds 3In the Z2-graded sense: here indeed isomorphisms are isomorphisms of super algebras. OM /JM into OM ; moreover, one should also endow the sheaf OM with the structure of sheaf of OMred -modules. When such a projection exists, the supermanifold is said to be projected. Conceiving a supermanifold in terms of the gluing data between open sets covering the underlying topological space, the projectedness of such a supermanifold implies the even transition functions to be written as functions of the ordinary local coordinates on the reduced manifold only: there are no nilpotents (e.g. bosonisation of odd elements) at all. Obstruction to the existence of such a projection for the case of the supermoduli space of super Riemann surfaces has been studied in [54] and it is an issue that has striking consequences in superstring perturbation theory, as mentioned in the introduction. A stronger condition is realised when the supermanifold is globally isomorphic to some supermanifold suggests by itself. local model S(jM j; E ). Such supermanifolds are said to be split. If this is the case, not only the even transition functions have no nilpotents, but the odd transition functions can be chosen in such a way that they are linear in the odd coordinates. This bears a nice geometric view of split supermanifolds: they can be globally regarded as a vector bundle Mred on the reduced manifold having purely odd bers, as the above de nition of Projective super spaces and weighted projective super spaces The supermanifolds known as (complex) projective super spaces, denoted by Pnjm, have been discussed extensively in the literature and introduced from several di erent points of view, both in mathematics and in physics, being of fundamental importance in twistor In most cases, complex projective spaces are regarded as a quotient of the super spaces Cnjm by the even multiplicative group C , so realising a super analogue of the set of homogeneous coordinates [X1 : : : : : Xn : m] obeying [X1 : : : : : Xn : 1 : : : : : m] = m], where (see for example [49, 54]). In contrast with this global construction by a quotient, a popular local construction realises Pnjm mimicking the analogous constructions of Pn as a complex manifold, namely by specifying it as n + 1 copies of Cnjm glued together by the usual relations. This construction relies on the possibility to pair the usual bosonic local coordinates with q fermionic anticommuting local coordinates: such an intuitive approach can be made rigorous using the functor of points formalism [15]. A more rigorous treatment, connecting the requested invariance under the action on C with the structure of the sheaf of super commutative algebra characterising the projective super space, can be found in [56]. An elegant construction of the projective super space consistent with the notions introduced above is given in [12]. Pnjm can actually be presented as a (split) complex superV1 of rank n + 1jm. As one can imagine, the topological space underlying the super projective space coincides with the usual one and it is given by the projectivization of the even part of V , called simply Ui ..= f[X0 : : : : : Xn] 2 Pn : Xi 6= 0g ; in such a way that one can form a system of local a ne coordinates on Ui given by z(i) ..= Xj /Xi for j 6= i. Intuitively, as above, we would like to have something similar for j the odd part of the geometry: this is achieved by realising a sheaf of super algebras on Pn, This is isomorphically mapped to the structure sheaf OPnjm of the projective super space, by a map induced by (i) ..= 2 OPnjm (Ui); where it should be stressed that over Ui and the (i), where the projective super space Pnjm. section, Pnjm = S(Pn; V1 This construction makes apparent that, in the notation introduced in the previous One can also read out the transition rules on Ui \ Uj , even and odd, that are usually is a generator for V1_, X 1 is a section of OPn ( 1) i z(i) = (i) = In the language of morphisms of ringed spaces, we would have an isomorphism ( Ui\Uj ; ]Ui\Uj ) : (Ui \ Uj ; OPnjm (Ui)bUj ) ! (Ui \ Uj ; OPnjm (Uj )bUi ); Ui\Uj : Ui \ Uj ! Ui \ Uj being the usual change of coordinate on projective space Ui\Uj : OPnjm (Uj bUi ) ! ( Ui\Uj ) OPnjm (UibUj ), so that (zk(i)) = ( (i)) = We note, incidentally, that the cocycle relation is indeed satis ed. Before proceeding further with our treatment, we here generalise a little the construction above, allowing us to deal in a somehow uni ed way with the weighted projective super spaces, as well. We will actually be interested in the case in which the odd part of the geometry carries di erent weights compared to the even part, which is made by an ordinary projective space. Since above we considered V = V0 V1 to be a super C-vector space, then it yields a well de ned notion of dimension, namely n + 1jm, and one can actually take a basis for it. Focusing on the odd part, we take f g =1;:::;m as a system of generators for V1. Then, we might realise a more general sheaf of super algebras by In other words, each odd variable has been assigned a weight w , which a ects the transition functions: the ordinary case of Pnjm is recovered assigning w = 1 for each = 1; : : : ; m. We will call this space weighted projective super space and we will denote it by WP(nwjm1;:::;wm), where the string (w1; : : : ; wm) gives the fermionic weights. In this paper, we will be particularly concerned with low dimensional examples of projective and weighted 1 1 projective super space, namely P1j2 and WP(2j), whose geometry will be studied in some details in the following section. Vector bundles over P1, Grothendieck's theorem and cohomology of OPn (k)-bundles In this section we recall and comment a classi cation result due to Grothendieck, which will then be exploited to study projective super spaces whose reduced space is P1. Moreover, for future use, the cohomology of OPn (k)-bundles is given. The main result about vector bundles on P1 is that any holomorphic vector bundle E of rank n is isomorphic to the direct sum of n line bundles and the decomposition is unique up to permutations of such line bundles, that is : E = where the ordered sequence k1 kn is uniquely determined (see [92] for a complete proof). We will refer at it as Grothendieck's Theorem. Basically, it states that the only interesting vector bundles on P1 are the line bundles on it, which in turn are all Concretely, since every (algebraic) vector bundle over C is trivial, the restriction of a and U1 ..= f[X0 : X1] : X1 6= 0 g = C is trivial. Choosing the coordinates z ..= X1 on U0 X0 of OP1 (U0) = C[z]-modules, as follows: 0 : E (U0) = ! OP1 (U0) n = C[z] n: 1 : E (U1) = ! OP1 (U1) n = C z 1 Clearly, two such isomorphisms i and 0i yields to an automorphism 0i determines an invertible n matrix, having coe cients in C[t]. The composition 01 ..= 1 gives the glueing relation between the two trivial bundles over U0 and U1: it is again given by an invertible n n matrix having coe cient in C[z; z 1], with determinant zk for some k 2 Z, up to a non-zero constant. classifying rank n vector bundles over P M 2 GL(n; C[z; z 1]) up to the following equivalence: 1 corresponds to classifying invertible matrices A(z)M (z; z 1)B(z 1) A(z) 2 GL(n; C[z]); B(z 1) 2 GL(n; C[z 1]): By a theorem due to Birkho , M (z; z 1) belongs to the same class of a diagonal matrix direct sum of line bundles OP1 (k1) By looking at vector bundles over P1 as sheaves of locally free OP1 -modules, the theorem reduces the problem of computing sheaf cohomology over P1 to computing the sheaf cohomology of OP1 (k), which is well-known. In general, for k C[x0; : : : ; xn](k), the degree-k linear subspace of the polynomial ring, therefore 0 one has H0(Pn; OPn (k)) = h0(Pn; OPn (k)) = exercise in combinatorics to see that and if k < 0 one has Hn(Pn; OPn (k)) = xi0 : : : xinn : ij < 0; 0 Pin=0 ij = k C. It is an hn(Pn; OPn (k)) = These results will be used to compute the cohomology in the following section. Super Calabi-Yau varieties The physical approach to Calabi-Yau geometries in a supersymmetric context is based on the di erential geometric point of view, de ning a super Calabi-Yau manifold (SCY) as a Ricci- at supermanifold. Indeed, a generalisation of tensor calculus to a supersymmetric context exists, making use of the notion of super Riemannian manifold, super curvature tensor and super Ricci tensor. The crucial observation concerning projective super spaces is that there exists a generalisation of the Fubini-Study metric to the supersymmetric context. Considering Pnjm, one can de ne the super Kahler potential s = log @ X XiXi + X Ks = log @ everywhere on Pnjm, which reduces to the ordinary Fubini-Study potential as one restrict it to the underlying reduced manifolds. Locally, on a patch of the projective super space, it takes the form Notice that, in this complex di erential geometric context, the variables are paired with their anti-holomorphic counterparts, as customary in theoretical physics. The super Kahler form is de ned in the local patch to be s ..= @@Ks or analogously s ..= @A@BKsdXAdXB where @ and @ are the holomorphic and anti-holomorphic super derivatives; we refer to the appendix A for details. Then, the super metric tensor is simply given by and the Ricci tensor reads HAsB ..= @A@BKs; RicAB ..= @A@B log (BerHs) : Notice that, in comparison to the ordinary complex geometric case, the unique modi cation is that the determinant of the metric has been substituted by the Berezinian [11, 49] of the Remarkably, as one chooses the projective super spaces of the form Pnjn+1 for n then one gets a vanishing super Ricci tensor! We stress that, as it is common in the context of super geometry, even an easy calculation might present some di culties, due to the anti-commutativity of some variables. One needs to establish and keep coherent conventions throughout the calculations. As an example, a detailed computation of the vanishing of the super Ricci tensor in the case of P1j2 is reported in appendix A. The same calculation can be easily generalised to any Actually, de ning a SCY manifold by requiring that its super Ricci tensor vanishes turns out to be a very strong request. Moreover, it is not that useful, for it is often hard to write down super metrics for interesting classes of supermanifolds: for example, there is not straightforward generalisation of the super Fubini-Study metric to the case of weighted projective super spaces. Moreover, by the result in [88] a Ricci- at Kahler supermetric on WP(12j1) does not exist. Still, it is possible to give a weaker, but certainly more useful, De nition 1 We say that an orientable super projective variety X is super Calabi-Yau if its Berezinian sheaf is trivial, that is Ber X = OX . It is here worth remarking that this de nition is again the super analogue of the usual algebraic geometric de nition of an ordinary CY variety, that calls for a trivial canonical sheaf. Indeed, the Berezinian sheaf is, in some sense (see [11, 12, 49]), the super analogue of the canonical sheaf, since the sections of the Berezinian transform as densities and they are the right objects to de ne a meaningful notion of super integration, i.e. the Berezin integral. In other words, then, a SCY variety is one whose Berezinian sheaf has an everywhere nonzero global section. We now start using of Grothendieck's Theorem in order to prove the triviality of the Berezinian sheaf of the supermanifolds P1j2 and WP(12j1), and hence to con rm that they are both SCY varieties. To compute the cohomologies, we will consider the varieties as split supermanifolds; in this case, this amounts to consider the total space of vector bundles over P1 (the reduced manifold, having odd line bundles over P1 and we will compute their cohomology. bers). Then, we will achieve the splitting into We start considering P1j2. Following the above line of reasoning, we have two patches, namely Uz and Uw: switching from one patch to the other yields the following 8> w 7 ! z = w1 The structure sheaf, OP1j2 , is therefore locally generated by OP1j2 (Uz) = 1 where in the last equality we are conceiving it as a locally free OP1 -module. Considering the transformation rules in the intersection, we nd the following factorisation as OP1 -modules: OP1j2 = OP1 Using a notation due to Witten [49], the Berezinian sheaf over P1j2 is locally generated Ber P1j2 (Uz) = [dzjd 0; d 2] OP1j2 (Uz): Under a coordinate transformation, call it , taking local coordinates wj 0; 1 to zj 0; 1 as above, the Berezinian transforms as h1(P1j2; OP1j2 ) = 1. by an element of the form Therefore one gets: ! z = w1 = w2 1=w2 0=w2 1=w 1=w2 0 1=w 0 C7 = 1=w2 1=w 1=w This trivial transformation implies the triviality. More precisely, viewing Ber P1j2 as a OP1 module, one nds the following factorisation: Ber P1j2 = OP1 OP1 ( 2) = OP1j2 ; and under the correspondence 1 7! [dzjd 0; d 1], one has Ber P1j2 = OP1j2 ; as expected. The cohomology is obviously the same as the one of the structure sheaf. One can reach the same conclusions for the structure sheaf and the Berezinian sheaf of WP(12j1), remembering that in such a case one has transformations of the following form Again, one nds that the Berezinian has a trivial transformation on the intersection and we have a correspondence 1 7! [dzjd ] and an isomorphism WP(12j1) = OWP(12j1) : This con rm that also the weighted projective space WP(12j1) is a SCY variety, in the The sheaf cohomology of di erential and integral forms To a large extent, generalisation of the ordinary commuting geometry to the richer context of supergeometry is pretty straightforward and it boils down to an application of the \rule of sign" [11, 12]. One issue stands out for its peculiarity: the theory of di erential forms and integration. The issues related to this topic have been recently investigated by Catenacci et al. in a series of papers (here we will particularly refer to [56]) and reviewed by Witten in [49]. Below, we brie y sketch the main points, addressing the reader to the literature for details of the constructions. As one tries to generalise the complex of forms ( ; d ) to supergeometry using the 1-superforms f products such as d 1 d igi2I constructed out of the i, then it comes natural to de ne wedge ^ : : : ^ d n to be commutative in the d 's, since the 's are odd elements. This bears a very interesting consequence: the complex of superforms ( s; ds) d i ^ : : : ^ d i do make sense and they are not zero, such as their bosonic counterparts. The troublesome point is that there is no notion of a top-form, therefore a coherent notion of \super integration" is obtained only at the cost of enlarging the complex of superforms and supplementing it with the so-called integral forms. Using the notation of [56], the basic integral form are given by f (d i)gi2I and its higher derivatives f (the degree of the integral form). Here the use of the symbol should remind the Dirac delta distribution | and indeed an integral form satis es similar properties [56] |: it sets to zero terms in d i and therefore, in some sense, it lowers the degree of a superform. For this reason, an integral form is assigned a non-positive degree: in fact, in the context of supergeometry one can also have forms with a negative degree. This fact can be better understood considering an example. Let us take the super space C2j2, and consider the following superform !s = dz1dz2(d 2)4 (2)(d 1); where the wedge products are understood. Then dz1dz2(d 2)4 carries a degree of 6, while the integral form (2)(d 1) lower the degree by 2, so as a whole, we say that !s has degree 4 and we signal the presence of an integral form (of any degree) by saying that it has picture number equal to 1. Therefore, this enlarged complex of superforms is characterised by two numbers, the degree of the form n and their picture number s. The degree of the superform is the homogeneity degree in the di erentials lowered by the total degree of the integral forms appearing in the monomial, whereas the picture number is the total number of the integral forms appearing in the monomial, regardless their degree. For example, we have dimension of the supermanifold and operators linking complexes having di erent picture numbers | called picture changing operators | can be de ned. In [56], the sheaf cohomology of superforms and integral forms of P1j1 has been studied, proving that just by adding an anti-commuting dimension, the cohomology becomes far richer. There is, though, a substantial hole in the literature: no sheaf cohomology of superforms and integral forms has ever been computed for supermanifolds having extended supersymmetry, that is more than one odd dimensions. In this scenario the computation of the cohomology for the case of P1j2 acquires value, besides being an example of cohomology of a SCY variety. We will see indeed that as soon as one has more than a single odd dimension, when the picture number is middle-dimensional (that is, it is non-zero and not equal to the odd dimension of the manifold), then one nds that the space of superforms is in nitely generated and its cohomology may be in nite-dimensional! This result calls, from a mathematical perspective, for a better understanding of the (algebraic) geometry of the complex of superforms and integral forms. Moreover, on the physical side, the possible usage and purposes of forms having middle-dimensional picture number should be investigated and clari ed. We now proceed to compute the sheaf cohomology of superforms of P1j2. In order to elucidate our method, we will carry out the computation in some details for the rst case, namely for the space of superforms having null picture number, reader all the other cases, that follow the same pattern. As a OP1j2-module, P1j2; we then leave to the gj=0;:::;n 1 OP1j2(Uz): By looking at it as a (locally free) OP1-module, we can nd the transformations of its generators. The rst block of generators transform as (up to unimportant constants and signs) 0 1d 0id 1n i = The second block, instead, has only diagonal terms: d 0id 1n i = 0d 0id 1n i = 1d 0id 1n i = dzd 0jd 1n 1 j = 0dzd 0jd 1n 1 j = 1dzd 0jd 1n 1 j = 0 1dzd 0jd 1n 1 j = terms in the factorisation; indeed, this is the dimension as a vector bundle/locally free sheaf of OP1-modules. We will pursue the strategy to group together pieces having similar form, evaluating their transformations and afterwards factorising them into a direct sum of line bundles 1 by means of Grothendieck's Theorem, by treating the o -diagonal terms in the transition functions matrix: we recall that, in the notation above, we will be free to perform To this end, we now focus on the diagonal terms that do not need any further indzd 0jd 1n 1 j, so these contribute to the factorisation with terms of the form vestigation: we will get n + 1 terms and n standing-alone terms out of 0 1d 0id 1n i and 1. Before proceeding further, we observe that, for n xed, looking at Pn1;0j2 as a OP1-module, there are dimOP1 Pn1;0j2 = 4(n + 1) + 4n = 8n + 4 So we are left with 8n + 4 n = 6n + 3 terms to give account to. The other terms need some careful treatment. We start dealing with the terms coming from the transformation of d 0id 1n i: these couples with the ones coming from d 1n 1 j. Since this holds true in the 0 1=wn 1=wn+1 1=wn+1 1 1=wn+2 B 1=wn+1 1=wn+2 1=wn+1 1=wn+1 1 0 1=wn+1 B 1=wn+2 0 1=wn+1 0 1=wn+1 1=wn+1 1=wn+1 1=wn+1 1=wn+2 A 1=wn+2 A 1=wn+1 1 1=wn+2 A 1=wn+1 1 1=wn+2 C 1=wn+2 A 1=wn+2 C 1=wn+2 A C1 wC2 B 1=wn+1 1=wn+2 0 1=wn+1 B 1=wn+2 0 1=wn+1 0 1=wn+1 0 1=wn+1 1=wn+1 1=wn+1 1 1=wn+1 1=wn+1 1=wn+1 1=wn+1 1=wn+2 A 1=wn+1 1 1=wn+2 A 1=wn+1 1 1=wn+2 C ; 1=wn+2 A 1=wn+2 C ; 1=wn+2 A 1=wn+2 A C R2 1=wR1 B So this bit contributes with terms of the following form: to be added to the previous ones. This boils the number of the remaining pieces down to 3) = 3n + 6. It is here worth pointing out that we have not given account for some terms in the that is 1dzd 1n 1. This gives a 2 2 matrix of the form: counting above yet: we need indeed to consider separately 4 terms that group into two 1=wn 1=wn+1 ! 1=wn+2 1=wn+1 ! 1=wn+1 1=wn+2 1=wn+1 1=wn+2 1=wn+1 ! R2 1=wR1 1=wn+1 ! 1=wn+1 1=wn+2 1=wn+1 1=wn+2 1=wn+1 1=wn+1 1=wn+1 sums up to the ones already accounted: . So, we have a pair of identical contributions that All in all, this adds up 4 terms to the counting above, leaving us with 3n + 2 terms to be still accounted for. the cases i = 0; : : : n The terms 0d 0id 1n i and 1d 0id 1n i couple with the last term, 0 1dzd 0jd 1n 1 j, in j. Therefore we have 3n identical 3 3 matrices of the form: 1=wn+1 1=wn+2 C 0 1=wn+1 0 1=wn+1 B 1=wn+1 1=wn+1 0 1=wn+1 1=wn+2 0 1=wn+2 1=wn+3 1=wn+1 1=wn+1 1=wn+1 1=wn+2 1 1=wn+3 A 1=wn+2 1 1=wn+3 A 1=wn+2 1 1=wn+3 A 1=wn+2 1 1=wn+3 A 1=wn+2 A 1=wn+2 0 1=wn+2 1=wn+3 0 1=wn+2 C R1 1=wR3 B 1=wn+1 1=wn+1 1=wn+1 1=wn+2 1 1=wn+3 A 1=wn+2 1 1=wn+3 A 1=wn+2 1 1=wn+3 A 1=wn+2 A 1=wn+2 A Consequently, we have the following contribution to the factorisation: These last 2n + 2 terms complete the enumeration. Summing it all up, we are therefore ready to write down the whole factorisation for n > 0: Pn1;0j2 = OP1 ( n factorisation as OP1 -modules. Finally, we can count the dimensions of the cohomology groups: h0( Pn1;0j2 ) = 0; h1( Pn1;0j2 ) = 8n2 + 8n: This terminates the discussion of the di erential forms with null picture number. All the other cases, having non-null picture number are treated in an analogous way, by remembering the transformation of the integral forms of type (n)(d i) [56]. Below, we list their The space of superforms having maximal picture number and degree is locally generIts transformation among the two charts yields the factorisation 1P;12j2 (Uz) = dz (d 0) (d 1) 1P;12j2 = OP1 OP1 ( 2) = OP1j2 : Thus, we can easily compute the dimensions of the cohomology groups, which are exactly the same as the ones of the structural sheaf: h0( 1P;12j2 ) = 1; h1( 1P;12j2 ) = 1: It is not surprising that this is the same as the Berezinian line bundle over P1j2: indeed, elements of this sheaf are in some sense the supersymmetric analogues of the ordinary topform for a manifold, and we (Berezin-)integrate them, as one can integrate sections of the Berezinian sheaf. These two peculiar supersymmetric sheaves are fundamental in theory of integration on supermanifolds. We are left with the group keeping such as Pn1;0j2 ). It is locally generated by 0 (which deserve some attention and bookP1nj2;2(Uz) = f (i)(d 0) (n i)(d 1)gi=0;:::;n; dzf which give the following factorisation P1nj2;2 = OP1 (n + 1) 4n+5 The dimensions of the cohomology groups then read h0( P1nj2;2) = 8(n + 2)(n + 1); h1( P1nj2;2) = 0: Summarising, we have the following results for all cohomologies: h0( Pn1;mj2 ) = <> 0 0; m = 2 n = 0; m = 0 n = 1; m = 2 n = 0; m = 0 n > 0; m = 0 0; m = 2 n = 1; m = 2 Notice that so far we have not carried out the computation of superforms having picture number equal to 1: as anticipated, these are in nitely generated as a locally free sheaf, and they give in nite dimensional cohomology. The generators read Similarly, one nds f (jnj+i)(d 1)d 0igi2N; dzf (jnj+i+1)(d 1)d 0igi2N OP1j2(Uz) This is factorised as having factorisation which again gives in nite dimensional cohomology. The computation of the cohomology of WP(12j1) is much easier, and it can be performed following the same lines as above. Also, having no middle picture number, there are no in nitely generated modules, and no in nite cohomologies. By means of Grothendieck's Theorem, the complete sheaf cohomology can be computed 0; m = 1 n = 0; m = 0 n = 1; m = 1 n = 0; m = 0 n > 0; m = 0 0; m = 1 n = 1; m = 1 of the \top-superform" WP(12j1) ) = 0. By the way, we signal a pathology, which looks like it may apply to any weighted projective space. While the Berezinian sheaf is isomorphic to the structural sheaf of WP(12j1) | and indeed it has analogous factorisation and cohomology | one nds instead that the sheaf factorisation and therefore the di erent cohomologies: one nds indeed that h1( 0;0 WP(12j1) ) = 1 De Rham cohomology of WP(12j1) and P1j2 After calculating the sheaf cohomology of superforms on the super varieties WP(12j1) and P1j2, we now aim at computing their holomorphic de Rham cohomology. Before we start, a remark on the adopted notation is due: given a supermanifold M , we will denote its de Rham cohomology groups as HdnR;m(M ), where n refers to the usual degree of the forms and m refers to their picture number. We also stress that the boundary operator of the complex acts as d : AM n;m is the freely generated module of the n-forms having xed picture number m that are de ned everywhere, that is AM n;m = H0( nM;m). In other words, the boundary operator d does not change the picture number of the form, and it just raises the degree of the form, so | as in ordinary, purely bosonic geometry | we are just moving horizontally on the complex, and we cannot jump from one complex to the other, by picture changing procedure. For clarity's sake, we start from the end of the treatment of the previous subsection, and we compute the de Rham cohomology of the weighted projective super space WP(12j1), whose sheaf cohomology of superforms is always nite. We will adopt a cumbersome but e ective method, that has the advantages to display explicitly a basis of generators for the various de Rham groups. This is remarkable, for it possibly sets a more concrete ground for the observations in [82] and especially in [83], where it is observed that the BRST cohomology of a (super) A-model is isomorphic to the cohomology of the superforms on the target space, that is on a supermanifold M . The most interesting group is the zeroth one: we nd that deg F0 = 2; deg F1 = Grouping together the terms having the same coe cients, we obtain the following basis: F0(z) = az2 + bz + c; F1(z) = d; G0(z) = ez + f; G1(z) = dz (1)(d ) = d z (1)(d ) ; zdz (1)(d ) = d Thus, one can verify that the module of the closed forms is generated by Zd0R;1(WP(12j1)) = (0)(d ); dz (1)(d ); zdz (1)(d ) Actually, the forms dz (1)(d ); zdz (1)(d ) are easily seen to be exact; indeed: and both the forms on the right-hand sides are everywhere de ned, that is they belong to the closed form Writing explicitly the forms, we can see that all the other groups HdnR;1(WP(12j1)) for n > 0 are trivial: one nds that ZdnR;1(WP(12j1)) is actually non-zero | there are closed forms 1 1 1 1 |, but ZdnR;1(WP(2j)) = BdnR;1(WP(2j)) |, namely that all closed forms are exact and do not contribute to the de Rham cohomology. Summing it all up, we have: hdnR;m(WP(12j1)) = <> 0 n > 0; m = 0 8 1 n = 0; m = 0 > 1 n = 0; m = 1 >: 0 n 6= 0; m = 1: We now proceed to consider the holomorphic de Rham cohomology of P1j2: again, the starting point will be to look at the forms de ned everywhere. By recalling the results on de ned forms). implies that Hd1R;2(P1j2) = 0. top-form |: the relative group is locally generated by the superform dz (0)(d 0) (0)(d 1), which extends globally: this is certainly closed and moreover, one can easily see, it is exact, Next, we consider the groups HdnR;2(P1j2) for n 0. The most interesting case is given by Hd0R;2(P1j2): the relative Cech cohomology group has dimension 16 and we will study it carefully. We should be considering forms of the kind (G0(z) + G1(z) 0 + G2(z) 1 + G3(z) 0 1)dz (0)(d 1) (1)(d 1)+ (H0(z) + H1(z) 0 + H2(z) 1 + H3(z) 0 1)dz (1)(d 1) (0)(d 1); where the F 's, G's and H's are all polynomials, whose degree is identi ed as above, by studying whenever the form remains de ned everywhere under a change of local chart, say from Uz to Uw. There are 10 closed forms: Zd0R;2(P1j2) = (0)(d 0) (0)(d 1); 0dz (0)(d 0) (1)(d 1); 1dz (1)(d 0) (0)(d 2); (0)(d 0) (0)(d 1); zdz (0)(d 0) (1)(d 1); dz (0)(d 0) (1)(d 1); zdz (1)(d 0) (0)(d 1); dz (1)(d 0) (0)(d 1); The unique closed form that it is not exact is 0 1 (0)(d 0) (0)(d 1), which is therefore a generator for the group Hd0R:2(P1j2) = C. Indeed, considering for example the closed form dz (1)(d 0) (0)(d 1), one has: dz (1)(d 0) (0)(d 1) = d( where d 1 (1)(d 1) = (0)(d 1) has been used. As in the case of the weighted projective super space, proceeding in the negative degree This is ultimately to be connected to the dimension of the space H0( Pn1;2j2 ) for n and in turn to the transformation properties of the integral forms, giving rise to a huge space of globally de ned forms. We now consider the space of everywhere de ned forms having picture number equal to 1, which is somehow the most sensitive one, because, as we have seen above, it yields to an in nite dimensional sheaf cohomology. Before proceeding further, we recall that is in nitely generated as a locally free sheaf, and its generators read The factorisation is Firstly, we observe that there are no globally de ned forms for n > 0. Thus, it follows that the de Rham cohomology is HdR n>0;1(P1j2) = 0. Secondly, all the other modules, for n 0, gives an in nite dimensional zeroth (Cech) cohomology group. we can just deal with the rst two blocks, and the other ones are symmetric up to the In this case, i = 0, one has: (G0(z) + 0G1(z) + 1G2(z) + 0 1G3(z))dz (1)(d 0): 0P;11j2 (Uz) = f From Cech cohomology computations, we expect 4 free parameters that yield: H0( 0P;11j2 )bi=0= z (0)(d 0) The last three forms are closed, but only 0 (0)(d 0) is not exact, indeed (0)(d 0) = d( dz (1)(d 0) = d(z (1)(d 0)) 0 (1)(d 0); z (1)(d 0) are globally de ned. Analogously, we have that 1 (0)(d 1) is closed and not exact, therefore it is non-zero in the quotient. (G0(z) + G1(z) 0 + G2(z) 1 + G3(z) 0 1) dz (i+1)(d 0)d 1i: (0)(d 0)d 1i + 1 (i 1)(d 0)d 1i 1; dz (i+1)(d 0)d 1i : It can then be seen that (i)(d 0)d 1i and dz (i+1)(d 0)d 1i are closed forms for every i but they are also exact, because (i)(d 0)d 1i = d( dz (i+1)(d 0)d 1i = d(z (i+1)(d 0)d 1i); so there is no contribution to the cohomology. This applies to each n < 0, so there are no closed and not exact forms, and the complete holomorphic de Rham cohomology of P1j2 reads hdnR;m(P1j2) = <> 2 n = 0; m = 1; 8 1 n = 0; m = 0; >>> 0 n > 0; m = 0; > 0 n 6= 0; m = 1; >>> 1 n = 0; m = 2; >: 0 n 6= 0; m = 2: Hd0R;0(P1j2) = 1 Hd0R;1(P1j2) = Hd0R;2(P1j2) = The generators in the holomorphic case are given by a straightforward generalisation of the case P1j2 displayed above. In the real case, they are f0; 1; : : : ; mg has cardinality j, and !FS is the ordinary Fubini-Study form. As anticipated above, this is an interesting result, showing that the in nite dimensionality of Cech cohomology is cured at the level of the de Rham cohomology, which is the relevant one for physical applications, since it is connected to the physical observables and it enters the evaluation of correlation functions [83]. We would expect this kind of behaviour to be a feature of supermanifolds with more than one fermionic dimension. The complete de Rham cohomology of Pnjm For completeness' sake as well as for future reference, we write down the whole holomorphic and real de Rham cohomology for general projective superspaces Pnjm. This can be computed by using the same tedious direct method as above (see also [56]). Hdi;Rj (Pnjm) = (m) i = 0; j = 0; : : : ; m; i 6= 0; j = 0; : : : ; m: In the real case, one obtains instead Hdi;Rj (Pnjm) = (m) i = 2k; k = 0; : : : ; n; j = 0; : : : ; m; i = 2k + 1; k = 0; : : : ; n 1; j = 0; : : : ; m: The generators of the non-trivial groups are !k;Ij ..= ^k!FS The method developed for the computation of the cohomology of projective super spaces over P1 easily allows us to evaluate the cohomology of the super tangent space, as well. Calculating the super Jacobian of the change of coordinates, we get @z = @ i = w@ i TP1jn Uz = @z; f J @zgJ=(j1;:::;jm); f@ i g i=1;:::;m i=1;:::;m; f J @ i gJ=(j1;:::;jm) OP1 (Uz) of generators is (m + 1) 2m. These have the following transformation rules: @z = J @z = J @ i = @ i = w@ i bUz : (z; 1; 2) 7 ! (z + 1 2; 1; 2); where we stress that, depending on J , many terms might be zero in the transformation of J @z (namely, all the terms in the sum over i such that i 2 J ). Using Grothendieck's Theorem as above, one can compute the zeroth cohomology group of the tangent sheaf, whose dimension is: h0(TP1jm ) = (m + 2)2 Notice that (m + 2)2 1 is just the number of generators of the Lie algebra associated Mobius group PGL(2; C), the automorphisms group of the projective line P1. to the super group PGL(2jm), which is the supersymmetric generalisation of the ordinary It is worth noticing the presence of the \correction" n;2, which, incidentally, makes its very appearance in the case of the super CY variety P1j2. This correspond to the presence of a further global vector eld, (locally) given by 1 2@z 2 H0(TP1j2 ), which clearly does not belong to sl(2j2), the Lie algebra of PGL(2j2), as already noticed in [13] and more recently in [72]. Integrating this global vector eld, we get the \ nite" version of the automorphism : P1j2 ! P1j2, called a \bosonisation" in physics; locally, it is given by: Before we go on, it is important to stress that among all the projective super spaces Pnjm | not only among P1jm! |, the case of P1j2 represents, remarkably, a unique exception: indeed, it is the only case in which the automorphism group is larger than PGL(n+1jm; C),4 unlike to what stated in [72]. For reduced dimension 1 this exception has been rst observed in [13], page 41. This and other issues will be the subject of a forthcoming paper, where di erent methods to compute the cohomology of projective super spaces in a more general setting will be introduced and discussed. As for the deformations, given by h1(TP1jm ), one nds h1(TP1jm ) = (m + 2) (m + 2) + (m We can see therefore that P1j1, together with P1j3 and the super CY variety P1j2 are rigid as they have no deformations, while in the case m For instance, for m = 4 we nd h1(TP1j4 ) = 19. investigation of the structure of these deformations. 4, we start nding a non-zero h1(TP1jm ). We leave to future works a careful A super mirror map for SCY in reduced dimension 1 In [80] the conjecture has been put forward that the puzzle of mirror of rigid (ordinary) CY manifolds could be solved by enlarging the relevant category for mirror symmetry, including also super manifolds, in particular SCY manifolds. Later on, triggered by previous studies in [93] and [87], Aganagic and Vafa proposed a path integral argument to obtain the mirror of Calabi-Yau supermanifolds as super Landau-Ginzburg (LG) theories [86]: the construction is exploited to compute the mirror of SCY manifolds in toric varieties and in particular to compute the mirror of the \twistorial" (actually super) Calabi-Yau P3j4 [87]. Remarkably, after a suitable limit of the Kahler parameter t, the mirror has a geometric interpretation: indeed, it is a quadric in the product space P3j3 P3j3, and it is again a SCY manifold. Since we are interested into enlarging the mirror symmetry map for elliptic curves to a supersymmetric context, here we will apply the construction of [86] to the case of bosonic dimension equal to 1 and reduced manifold given by P1, i.e. to the two SCY's P1j2 and WP(12j1). In doing that, in contrast with [86], we will not need to take any limit of the Kahler parameter: in fact, a further geometric investigation, carried out by some suitable change of coordinates, shows that P1j2 is actually self-mirror and it is mapped to itself. The mirror of the weighted projective super space WP(12j1) instead is not a geometry. Before proceeding to the actual computation, it should be here remarked that a further, mathematically oriented, analysis needs to be carried out. Despite the e ort in [86], many issues are still unsettled, as for example the role of the Kahler parameter t. It is indeed a matter of question how to de ne, mathematically and in full generality, a super analogue of the ordinary Kahler condition, and therefore how to identify a super Kahler variety. 4The bosonic reduction of PGL(n + 1jm). Following [86], we construct the dual of the LG model associated to P1j2: it turns out this is given by a -model on a super Calabi-Yau variety in P1j1 P1j1, which is again a SCY variety given by P1j2. In other words, P1j2 gets mapped to itself! lowest component of their expansion is a bosonic eld and a fermionic eld, respectively), while t is the Kahler parameter, mentioned above. This is given by WP1j2 (X; Y; ; ) = By a eld rede nition, the path-integral above can be recast as follows: X1 = X^1 + Y0; Y1 = Y^1 + Y0; D I D I (Y0 Integrating in X0, the delta imposes the following constraint on the bosonic elds: X0 = Y0 + (Y1 Plugging this inside the previous path integral one gets D I D I exp ne Y0 + e Y0 (Y1 X1)+t + e Y^1 Y0 + e X^1 Y0 o The fermionic D 0D 0 integration reads o = and therefore one obtains that DY0DY^1DX^1D 1D 1e Y0 (Y1 X1)+t e Y0 might be interpreted as a multiplier, and we perform the coordinate charge e Y0 = ; DY0 = such that the integral reads D DY^1DX^1D 1D 1 e (Y1 X1)+t 1 + e (Y1 X1)+t + e Y^1 + e X^1 + 1 1e X^1 o Finally, by performing another eld rede nition, namely e X^1 = x1; e Y^1 = x1y1; 1 = DX^1 = DY^1 = = x1D ~; we notice that the Berezinian enters the transformation of the measure! In fact, the pathintegral acquires the following form: WP1j2 = 1 + ety1 + x1 + x1y1 + ~1 1 D Dy1Dx1D ~1D 1et exp 1 + ety1 + x1 + x1y1 + ~1 1 : By noticing that the factor et is not integrated over, and performing the integration over the Lagrange multiplier , one obtains that the theory is constrained on the hypersurface 1 + x1 + x1y1 + ~ + ety1 = 0: 1 + x1y~1 + ~ + et(y~1 1) = 0: Casting the equation in homogeneous form, we have X0Y~0 + X1Y~1 + ~ + et(X0Y~1 X0Y~0) = 0: This is a quadric, call it Q, in P1j1 P1j1, with homogeneous coordinates [X0 : X1 : ~] and [Y~0 : Y~1 : ] respectively, and it is a super Calabi-Yau manifold. In the following treatment, we will drop the tildes and we will just call the homogenous coordinates of the super projective spaces [X0 : X1 : ] [X0 : X1 : ~] and [Y0 : Y1 : ] now re-write the equation for Q in the following form: et)Y0 + etY1) + X1Y1 + = 0: `(Y0; Y1) ..= (1 it is not hard to see that the reduced part Qred in P1 odd coordinates to zero, as P1 is obtained just by setting the X0 `(Y0; Y1) + X1Y1 = 0; and one can realize that Qred = P1. We are interested into fully identifying Q as a known variety; to this end, we observe that, as embedded into P1j1 P1j1, it is covered by the Cartesian product of the usual four Moreover, one needs all the above four open sets to cover Q, because V0 = f[X0 : X1 : ] : X0 6= 0g V1 = f[X0 : X1 : ] : X0 6= 0g V0 = f[X0 : X1 : ] : X1 6= 0g V1 = f[X0 : X1 : ] : X1 6= 0g f[Y0 : Y1 : ] : Y0 6= 0g; f[Y0 : Y1 : ] : Y1 6= 0g; f[Y0 : Y1 : ] : Y0 6= 0g; f[Y0 : Y1 : ] : Y1 6= 0g: Qred \ fX0 = 0g = [0 : 1] [1 : 0] 2 U1 Qred \ fX1 = 0g = [1 : 0] [1 : 1 Qred \ fY0 = 0g = [1 : et] [0 : 1] 2 U0 Qred \ fX0 = X1 = 1g = [1 : 1] [et + 1 : et Therefore, we would like to nd a suitable change of coordinates allowing us to use fewer open sets. It turns out that one can reduce to use only two open sets. Indeed, by switching coordinates to the equation for Q becomes following equation for Q: Then, by exchanging Y00 with Y10 and dropping the primes for convenience, one obtains the Y00 ..= `(Y0; Y1); X00 ..= X0; 0 ..= ; Y10 ..= Y1; X10 ..= X1; 0 ..= ; X00Y00 + X10Y10 + 0 0 = 0: X0Y1 + X1Y0 + = 0: this change of coordinates allows us to cover Q by just two open sets, namely by: Therefore, by choosing the following (a ne) coordinates: UQ ..= Q \ (U0 VQ ..= Q \ (U1 UQ : z ..= VQ : w ..= ; u ..= ; v ..= 0 ..= 0 ..= 1 ..= 1 ..= the following two a ne equations for Q of U Q are respectively obtained: describing lines in C2j2. We notice that these two equations are glued together using the z + u + 0 1 = 0; 0 1 = 0; v = 1 = v 1: w = 0 = z = w = Finally, we would like to characterise the variety Q by its transition functions, in order to identify it with a known one. By the previous equation, we may take as proper bosonic coordinates u and v, as 0 = 0 = u2 = implying that the variety Q P1 is actually nothing but P1j2. This shows that the super mirror map proposed by Vafa and Aganagic makes the supermanifold P1j2 self-mirror, actually it is mapped to itself. This goes along well with what holds for elliptic curves: indeed, an elliptic curve is the mirror of another elliptic curve. P1j2 as a N = 2 super Riemann surface manifold M such that the super tangent sheaf TM has two 0j1 subbundles D1 and D2, function, and D1 D2; D1; D2 generate TM at any point. We address the interested reader to [81] and [50] for details, as well as to the more recent articles [82] and [83] for further developments and some physical interpretations. nd the needed 0j1 line bundles D1 and D2, we adopt the method proposed in [50] at page 107, namely we will nd two maps p1 : P1j2 ! X1 and p2 : P1j2 ! X2, with X1; X2 two suitable 1j1 supermanifolds, and we will de ne Di as the sheaf kernel of the di erential dpi : TP1j2 ! pi TXi. These two maps can immediately be determined from the model of P1j2 contained in P1j1 P1j1 found in the previous section, in which we computed the mirror i-th projection i : P1j1 ! P1j1 to P1j2. In order to give explicit local calculations of the vector elds D1; D2 that generate the line bundles D1; D2 and to show that they have all the required properties, we can exploit eqs. (4.34) and (4.35) of the open sets U Q as sub-supermanifolds of the open a ne A2j2 For example, from the equation P1j1 with coordinates z; u; 0; 1. in A2j2, we see that z + u + 0 1 = 0 p1(z; u; 0; 1) = (z; 0) p2(z; u; 0; 1) = (u; 1): D1 = @ 1 + 0@u: D2 = @ 0 Then, D1 has sections given by those vector elds @z + @u + @ 0 + @ 1 that vanish on the elements z; 0; z + u + 0 1. This implies = 0 and 0, and therefore they are multiples of Similarly, one nds that the vector eld D12 = D22 = 0, and that generates all the vector elds on U Q that vanish on u; 1; z + u + 0 1. Since D1 and D2 vanish on z + u + 0 1, they are tangent vector elds on U Q that, by construction, generate the kernels D1 and D2 of the di erentials dp1 and dp2. The reader can easily check that fD1; D2g = D1D2 + D2D1 = @u moreover, this latter is equal to @u when evaluated on an element of OUQ. Since u is a bosonic coordinate for UQ, one ralizes that fD1; D2g; D1; D2 generate TP1j2 at any point of UQ. Similar formulas can be obtained for the open VQ. Mirror construction for WP(12j1) In the case of weighted projective super space, we need to evaluate the following (super)potential in order to nd the dual theory: (DY1DY2)DXD D Performing the integration in the fermionic variables, one obtains Next, we can integrate the eld X. Up to factors to be removed by the normalisation, the delta yields to the following result: The measure changes as We can then de ne the new variables 2 yi 1Dyi = DYi, and therefore, up to factors in the normalisation, 1 (Dy1Dy2) exp ny12 + y22 + et=2y1y2o : One can then state that in the case of WP(12j1) one does not get directly a geometry. However, we can further introduce the new variables and x, de ned by y1 = y2x; y22 = ; in such a way that, omitting an inessential constant factor, the nal result can be achieved: x2 + 1 + et=2x is a multiplier and the geometric phase reduces to two points parametrized by t. This is a zero dimensional bosonic model in accordance with the results of Schwarz [84]. In the present paper we have investigated some basic questions about super Calabi-Yau varieties (SCY's). We have introduced a very general de nition of a SCY, which encompasses a large class of varieties, including the usual Calabi-Yau manifolds and several projective super spaces. We then restricted our analysis to the SCY with complex bosonic dimension | into the Cartesian product of two copies of P1j1. A comment is in order here. In the proof of triviality of the Berezinian bundle is given in [82]. Nevertheless, there exists a completely obvious that the two de nitions do actually coincide: indeed the de nition of this topic deserve some more study. Next, we have computed the super cohomology groups, which include integral forms, showing that for extended supersymmetric varieties a puzzle arises: when the picture number is not maximal nor vanishing, then the corresponding Cech cohomology groups are in nitely generated. Surely, this result will deserve a much deeper investigation; for instance, it would be interesting to understand if it enjoys a geometrical interpretation. Anyway, remarkably, we have shown that this kind of pathology is cured whenever one considers the de Rham cohomology of superforms, which is always nite, even when the corresponding group in Cech cohomology is in nite-dimensional. The same phenomenon occurs in arbitrary dimension njm as we have seen by explicitly computing the de Rham cohomology of Pnjm. The computation of the sheaf cohomology also allowed us to determine the automorphisms of P1j2 and WP(12j1), which, on the other hand, are rigid manifolds. It is interesting to note that for SCY with fermionic dimension larger than 1, the automorphism supergroup is never larger than the superprojective group. As announced, a more systematic analysis of the automorphism group will be presented in a separate paper. Finally, we have applied the mirror map de ned by Aganagic and Vafa in [86], showing that P1j2 is self-mirror (and, indeed, mapped to itself), whereas WP(12j1) is mapped to a zero dimensional bosonic model. Even though we have chosen to investigate an apparently elementary framework, we realize that highly non-trivial aspects appear and some questions remains unanswered. For example, we have not been able to provide a suitable de nition of Kahler structure (or having P1 as reduced space are simple enough in order to allow a complete analysis, as well as to shed some light on new interesting properties of supermanifolds; on the other hand, they are too simple for providing a rich list of examples hinting to suitable solutions to the unanswered questions. The natural prosecution would then be to include properly dimension 1, and, more interestingly, to analyse SCY's with bosonic dimension 2, i.e. super Despite the results discussed above, we still cannot take our de nition of SCY manifold as a de nitive one. At the moment, indeed, the triviality of the Berezinian bundle alone appears as a provisional condition, maybe allowing for too many varieties to belong to the class. From this point of view, our de nition might be considered as a pre-SCY condition. In this context, one might wonder whether the existence of a Ricci- at metric is a natural condition to add, but in some meaningful example, such as WP(12j1), it does not even exist. This seemingly suggests that Ricci- atness is not the natural condition to add to the triviality of the Berezinian bundle. These and other topics are currently under SN would like to thank Ron Donagi for having suggested this stimulating research topic. SN and SLC would like to thank Gilberto Bini and Bert van Geemen for valuable discussions. AM and RR would like to thank the Department of Science and High Technology, Universita dell'Insubria at Como, and the Departments of Mathematics and Physics, Universita di Milano, for kind hospitality and inspiring environment. Super Fubini-Study metric and Ricci atness of P1j2 We take on the computation of the super Ricci tensor for P1j2 starting from the local form, say in Uz, of the Kahler potential, given by Ks = log(1 + zz + 1 1 + 2 2): This can of course be expanded in power of the anticommuting variables as in [80], but it is not strictly necessary to our end. In the following we will adopt this convention: we use latin letters i; j; : : : for bosonic indices, Greek letters ; ; : : : for fermionic indices and capital Latin letters A; B; : : : will gather both of them. The convention on the unbarred and barred indices goes as usual. The holomorphic and anti-holomorphic super derivatives are de ned as follows (in the @ ..= @zdz + @ d ; @ ..= @zdz + @ d ; holomorphic derivative @ acts as usual from the left to the right, the anti-holomorphic derivative @ acts from the right to the left instead (even if it is written on left of the function acted on). We also stress that @ and @ behave as a standard exterior derivative d on forms. As such the derivatives \do not talk" at all with the forms and only acts on functions, while the forms in @ or @ are moved to the right and in turn do not talk to the functions acted by the derivatives. This means that, for example, considering the local expression for a (holomorphic) 1-form acted on by @, we will nd: @(f (zj 1; 1)d 1) = (@Bf (zj 1; 1))dXBd 1: Coherently, we will never consider expression of the kind dXBf (zj )d 1, so that we will never have to commute or anti-commute a form with a function to get forms and functions just don't talk to each other and the form in @ and @ are moved We now de ne the super Kahler form as s ..= @@Ks or analogously s = @A@BKsdXAdXB: The super metric tensor HAsB can then be red out of it, similarly to the ordinary complex geometric case: HAsB = @A@BKs: We now deal with the derivative of the super Kahler potential Ks. Remembering that @B acts from the right, it is straightforward to check that: @BKs = @B log(1 + zz + 1 1 + 2 2) = We now have a product of functions: since we are dealing with anti-commuting objects we need to make a careful use of the generalized Leibniz rule @(f g) = (@f ) g + ( 1)j@jjfjf (@g): While the rst bit of the @ derivative is pretty straightforward and simply gives (@f ) g = dzdz + d 1d 1 + d 2d 2 ; the second contribution need some extra care: to avoid errors, we may split the derivatives in @ by linearity, bearing in mind the non-trivial commutation relation in the generalised Leibniz rule above. We have the following contribution from ( 1)j@jjfjf (@g): not come from the commutation relation, but just from the derivative: the commutation relation gives contribution when @ i is involved @@Ks = so the supermetric reads z 2d 2dz + 2 1d 1d 2 + 1 2d 2d 1 ; HAsB = BBB Using the metric one can generalise the expression for the Ricci tensor one has in ordinary complex geometry, by substituting the determinant with the Berezinian: RicAB = @A@B log (Ber Hs) : So the rst thing we need to evaluate to prove the (super) Ricci atness of P1j2 is the Berezinian of the super metric above. We recall that in general, considering a generic square matrix X valued in a super commutative ring, we have Ber(X) = det(A) det(D where A; B; C; D are the blocks as enlightened above. Notice that A and D are even while B and C are odd. We underline that in our case, to make sense out of the expression above we have to look at CA 1B as Kronecker product, as follows: A 1 consisting of a single even element. We start from the computation of A 1 C B. We have: A 1 = C = B = = (1 + 2)(1 + jzj2 + 2)2 @B where the overall minus sign comes from the commutation relation of the theta's. It is actually convenient to multiply (1 + 2 ) 1 out: rst of all we observe Therefore one has the following expression: D CA 1B = (1 + jzj2 + 2)2 1 + jzj2 + (2 + jzj2) 1 1 + 2 2 jzj2 4 1 + jzj2 + 1 1 + (2 + jzj2) 2 2 jzj2 4 : We now need to evaluate the determinant of the square matrix above: h det(D CA 1B) = (1 + jzj2 + 2)4 (1 + jzj2)2 + (1 + jzj2) 1 1 + (1 + jzj2)(2 + jzj2) 2 2 + (1 + jzj2)(2 + jzj2) 1 1 + (1 + jzj2) 2 2+ where we have isolated on di erent lines the zeroth, quadratic and quartic contribution in the theta's. We can simplify a little the expression above to get: To evaluate the full Berezinian we need to invert the determinant we just got. 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One-dimensional super Calabi-Yau manifolds and their mirrors, Journal of High Energy Physics, 2017, DOI: 10.1007/JHEP04(2017)094