Onedimensional super CalabiYau manifolds and their mirrors
Received: September
Onedimensional super CalabiYau 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, I20133 Milano , Italy
1 Via Valleggio 11, I22100 Como , Italy
2 Centro Studi e Ricerche `Enrico Fermi'
3 Via Saldini 50, I20133 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, I35131 Padova , Italy
9 and INFN , Sezione di Padova
10 Via Panisperna 89A, I00184 Roma , Italy
We apply a de nition of generalised super CalabiYau 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 selfmirror, 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 CalabiYau 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 FubiniStudy metric and Ricci atness of P1j2
Introduction
\Supermathematics" 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 socalled \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 dcomplex. In order to solve this
problem, distributionlike 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 1forms, admitting distributionlike
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 FourierBerezin 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 socalled 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 superLiouville 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 CalabiYau
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 CalabiYau 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 nitedimensional 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 (nonsingular) 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 Z2grading: algebraic constructions such as rings, vector
spaces, algebras and so on, have their Z2graded 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 anticommutativity 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 Z2grading 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 Z2graded 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 Z2grading.
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 jmodules 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 pdimensional 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 jmodule 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 Z2graded 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 Cvector 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 nonzero 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 wellknown. In general, for k
C[x0; : : : ; xn](k), the degreek 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 CalabiYau varieties
The physical approach to CalabiYau geometries in a supersymmetric context is based on
the di erential geometric point of view, de ning a super CalabiYau 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 FubiniStudy 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 FubiniStudy 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 antiholomorphic 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 antiholomorphic 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 anticommutativity 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 FubiniStudy 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 CalabiYau 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
1superforms 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 topform, therefore a coherent notion of
\super integration" is obtained only at the cost of enlarging the complex of superforms and
supplementing it with the socalled 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 nonpositive 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 anticommuting 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 middledimensional (that is, it is nonzero 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 nitedimensional!
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 middledimensional 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 OP1j2module,
P1j2; we then leave to the
gj=0;:::;n 1 OP1j2(Uz):
By looking at it as a (locally free) OP1module, 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 OP1modules.
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 standingalone 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 OP1module, 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 nonnull 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 \topsuperform"
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 nforms 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) Amodel 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 righthand 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 nonzero  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.
topform : 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 nonzero 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 FubiniStudy 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 nontrivial 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 nonzero 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 CalabiYau supermanifolds as super LandauGinzburg (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) CalabiYau 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 selfmirror 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 CalabiYau 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 pathintegral 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 CalabiYau 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 rewrite 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 selfmirror, 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
ith 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 subsupermanifolds 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 CalabiYau
varieties (SCY's). We have introduced a very general de nition of a SCY, which encompasses
a large class of varieties, including the usual CalabiYau 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 nitedimensional. 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 selfmirror (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 nontrivial 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 preSCY
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 FubiniStudy 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 antiholomorphic 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 antiholomorphic
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) 1form 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 anticommute 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 anticommuting 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 nontrivial 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. This yields:
Putting together the pieces, we can evaluate the full Berezinian:
this leads us the the conclusion:
RicAB = 0:
P1j2 is Ricci at and therefore it is a super CalabiYau manifold in the strong sense.
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