Penrose Transform on D-Modules , Moduli Spaces and Field Theory *

We consider a generalization of the Radon-Schmid transform on coherent D-modules of sheaves of holomorphic complex bundles inside a moduli space, with the purpose of establishing the equivalences among geometric objects (vector bundles) and algebraic objects as they are the coherent D-modules, these last with the goal of obtaining conformal classes of connections of the holomorphic complex bundles. The class of these equivalences conforms a moduli space on coherent sheaves that define solutions in field theory. Also by this way, and using one generalization of the Penrose transform in the context of coherent D-modules we find conformal classes of the space-time that include the heterotic strings and branes geometry.


Introduction
It is opportune to consider a generalization of the Radon-Schmid [1], on coherent D-modules [2], of the sheaf of holomorphic bundles into a cohomological context, [3] with the goal of establishing the equivalences between geometrical objects (vector bundles) and algebraic objects as they are the coherent D-modules, these last with the goal of establishing the conformal classes useful to define adequately the differential operators that define the field equations of all the microscopic and macroscopic phenomena of the space-time through the connections or shape operators [4], of the complex holomorphic bundles in gauge theory.The class of these equivalences is precisely in our moduli space, which has in consideration the differential operator that defines the connection of the corresponding vector bundle which establish the relation among dimensions of cohomological conformal classes [5], and the vector bundles corresponding via the differential operators of the equations of shape (connection) of the Riemannian manifold [5].
But studies in algebraic geometry have established that manifolds of Calabi-Yau have the integrals of Penrose type, or at least complex integrals on strings like equivalences of geometrical invariants under the philosophy of the mirror symmetry.From a point of view of coherent D-modules, the conformally comes established in automatic form through the use of Penrose transforms on Dmodules.Of this manner, we come to D-strings and Dbranes corresponding to specific classes of derived sheaves obtained from the appropriate generalized Penrose transform [6].
We define our moduli space as the space of the invariant differential operators of the  -cohomology modulo that are conformally invariants.We determine a cohomology between moduli spaces on the space of differenttial operators that accept a scheme of integral operators cohomology of Penrose type in the context of the coherent D-modules [7], since the scheme of the ireducible unitary representations to these operators are unitary representations of compact components of the group   4, SL C [5].

Conformally Invariant Operators, Penrose Transform and Derived Sheaves
There exists indictions that the class of differential operators that accept an re-interpretation of a integral operators cohomology (as the due for the Penrose transform, the Twistor transform, etc.) is accordingly the class of invariant differential operators, of the fact that the Penrose transform generates these conformal invariants operators [4], and thus we can identify the conformal classes to which they belong [8,9].Some examples of these dif- ferential operators are the existence of the massless field equations (to flat versions and curved of some of their similar ones [10]) and the conformal invariant wave operator given by the map [4]:   (1) or also the Einstein operator [4]: or the conformal invariant modification of the square of the wave operator     , that is to say; the wave operator who involves terms of the Ricci tensor [4], Then the integration of the partial differential equations corresponding to these invariant linear differential operators is realized as integrals transform of the Penrose type, because the context of the irreducible unitary representations for these operators are unitary representations of compact components of the group , [7], of the fact that in the flat case the classification of invariant differential operators like those described previously are a problem of representations theory of Lie groups applied to the group of   4, SL C SL  and its compact subgroups [7].Then one visualizes these operators through Lie group , as equi-variants operators among homogeneous vector bundles on M considering to   , like homogeneous space [5][6][7][8][9][10][11].The integral operators in this case are realizations of these representations and they are orbital integrals of one integral transform from the resolutions to these differential equations which in this concrete case is the Penrose transform.
The problem is solved using the machinery of representations theory and is given in [12], and with more generality in [13].In after studies, the local twistor connection [14] is used to investigate the questions as to whether these operators have curved analogues i.e. conformally invariant operators in the curved case.
For example, some of these operators that are mentioned in the Equation (2), and the Equation (3) are included in a conformal class that is obtained by the image of the Penrose transform on the corresponding sheaf whose germs are these differential operators in the holomorphic vector bundle given.The Penrose transform generates conformally differential operators.
We consider the Penrose transformation, [15] through the correspondence (4) where , is the manifold of flags of dimension one and two, associated to 4-dimensional complex vector space ), and let 2 , be such that (Grassmannian manifold of 2-dimensional complex subspaces), with , , , , , , is the 4-dimensional complex compactified Minkowski space [16].The projections of , are given for: and where 1

2
, are complex subspaces of dimension one and two, respectively, defined a element  

If
, is compactified Minkowski space [16] then that is to say, the spectral resolution of complex sheaves is needed [16], of certain class seated in the projective space , to give solution to the field equations modulo a flat conformally connection [16][17][18]       0 1 0, Let , be the Penrose transform [16] associated to the double fibration (4), used to represent the holomorphic solutions of the generalized wave equation [16], with parameter of helicity h [17,19]: , in terms of cohomological classes of bundles of lines [16], on ( P , is the super-projective space).It is necessary to mention that these cohomological classes are the conformal classes that we want to solve the phenomenology of the space-time to diverse interactions studied in gauge theory [20], and can construct a general solution of the general cohomological problem of the space-time.
With major precision, the tensors of the bundle of lines on , are given   Z , by the kth-tensor power

 
O k P , of the tautological bundle [21], (is the bundle that serves to explain in the context of the bundles of lines on , the general bundle of lines of ).
Then a result that establishes the equivalences on the cohomological classes of the bundle of lines on , and the family of solutions of the Equation ( 9), (equations of the massless fields family on the Minkowski space , with helicity h) is given by the following theorem.
Proof.[4]. Which are the classes that are extensions of the space of equivalences of the type Equation ( 12)?Why are these classes necessary to include more phenomena of the space-time?What version of the Penrose transform will be required?
Part of the object of our research is centered in the extension of the space of equivalences of the type (12), under a more general context given through the language of the D-modules, that is to say, we want to extend our classification of differential operators of the field equations to context of the G-invariant holomorphic bundles and obtain a complete classification of all the differential operators on the curved analogous of the Minkowski space of .Thus our moduli space will be one of the equivalences of the conformal classes given in the equivalence (12), in the language of the algebraic objects D-modules with coefficients in a coherent sheaf [13].
A way of answering the first and the second questions is analyzing the origin of the structure of the complexes that define the microscopic physical phenomena.Through this way and in natural form we can establish equivalences (isomorphisms), between derivative categories and categories of physical phenomena, considering a complex as the defined in the succession (10), for the micro-local context in which a regularity theorem subjacent for the Penrose transform on D-Modules is used, that is to say, given a D-Modules complex [13][14][15][16][17][18][19][20][21][22] 0 1  E E (13) we can map this complex to the system of branes/antibranes, in which every i , This bears to the application of a symplectic context, which also is a consequence of the foliation in Lagrangian submanifolds of , inside the problem of the uniqueness of the Radon-Penrose transform on the same coherent D-Modules [12].

Resolution on D-Modules
Consider the category of D-modules given by the space We consider a correspondence X where all the manifolds are analytic and complex,  , and  , odd defines branes and other sheaves define anti-branes [22].Such D-branes and anti-D-branes are defined of equal form in the space-time, although dynamically they annihilates for defining a pair of particle and anti-particle [22].
We define the transform of a sheaf F, on X, (more generally, of an object of the derived category of sheaves) like and we define the transform of a D X -module , like , where   , and    denotes the direct and inverse images of and  respectively, in the sense of the D-modules 1 [2], and we consider also to a sheaf , on Y. Then we have the formula of which is deduced the formula, to , the sheaf on Y, (coherent sheaf): 1 To define the images of direct functors to D-modules it is have that use derived categories.For it, is simpler defining them for right D-modules.Be

  
where , is the characteristic manifold and R, is the right derived functor following: Then , is a right D   -module to the right multiplication in the second functor.
This defines a categorical equivalence of the transformati t of the right derived D-modules, qc , because it is necessary to give an equivalence with a sub-category of the right D-modules that have support in Y, for this way guarantee the inverse image of P , and with it, to obtain an image of closed range of the operator P , conforming their uniqueness on the given context [15][16][17][18][19][20][21].
We formulate in the language of the D-modules and their sheaves, like the given in a resolution (10), the correspondence between the space of coherent D-modules and the space of equations of massless fields.It can be established if we can grant the uniqueness of the Penrose transform.But for it is necessary to include a result that it establishes the regularity in the analytical sense of the Riemannian manifold, which shapes the space-time, and that allows the application of the involutive distribution theorem on integral submanifolds as solutions of the corresponding equations of field without mass on sub-manifolds isomorphic in the Kaehlerian model inside of the flat model given on 2,4 Of the fact, an analogy in the obtaining of models of space-time (under the same reasoning) must be realized between special Lagrangian submanifolds and m-folds of Calabi-Yau.But to it, we need define the complex micro-local structure that defines all the phenomena of strings and branes in microscopic level, which happens in the 6-dimensional component of the universe (6-dimensional compact Riemannian manifold) with ratio of the order 10 33 cm (Max Plank longitude of a string).The Penrose transform is an integral geometric method that interprets elements of various analytic cohomology groups on open subsets of complex projective 3-space as solutions of linear differential equations on the Grassmannian of 2-planes in the 4-space.The motivation for such transform comes from the interpretation of this Grassmannian as the complexification of the conformal compactification of the Minkowski space and their differential equations being the massless field equations of various helicities.

A Micro-Local Analysis and Version of the Penrose Transform
Let , be a conic regular involutive submanifold.We say that a coherent D X -module , has regular singularities on V, if so we has ( M is regular if ).We denote by , whose objects have regular singularities on , and by X , the full triangulated subcategory of good , whose objects have cohomology groups belong to Associated to the correspondence given in ( 14), we give the microlocal correspondence: The manifold  , being Lagrangian, is well acquaintance that , in , then we have the following local model of correspondences of ( 14): , is a closed embedment and assumptions given by 1), and 2) above.
Then, for every   , there are open subsets X , X , with X  , and Y , a complex manifold Z, of dimension , and a contact transformation , is the graph of a contact trans-where , and , denotes the projection .

X X
Proof.[16]. The construction of characteristic cycles of Kashiwara associates to every object G , (coherent sheaf on Y) of this category, a Lagrangian cycle , is the structure sheaf of closed submanifold like the given by , this cycle is justly the conormal bundle and this tends to be singular.Then we can enounce the Kashiwara Theorem 3.2, also as: The construction of a smoothed characteristic cycle gives origin to an full embedment of derived categories (full subcategory of objects , have cohomology groups with regular singularities along V) into The X -modules shape a support ring of the D Xmodules.A X E -module is a ring of microdifferential operators of finite order on .It is clear that from the lemma 4.1, we can characterize the cycles propitious to determine an embedment full of derived categories that shapes the structure of our sheaves defined in bounded derived category of the full subcategory of sheaves of K-vector spaces comprising complexes with constructible cohomology.
Which are the integral transforms that can obtain the quantized contact transformation?Definition 4.1.1) Let V, be a closed conic regular involutive submanifold of , and let M , be a co- herent D X -module.We say that , is simple along V, if , can be endowed with a good filtration k , such that * T X * T X  , is locally isomorphic to , as an * T X  -module.We denote by coh X the full subcategory of whose objects are simple along V.
proper direct image and inverse image for Dmodules, and we denote by the  , th exterior tensor product.For sing the resolution (13) we associate their dual , where X , is the sheaf of holomorphic forms of maximal degree.We also set , be a simple X Y  -module along .In particular , is regular holonomic, and hence , and , we set where , and , is defined Assume that 1 q , q 1 , are proper on

 
supp K , and assume 1), and 2) from a a le ple D Y -mo (19).Let M , be simple D X -module along * T X , nd t N , be a sim- dule along V. Then 1) 0  K , and , and re isomorphisms modulo flat connections.In particular, the functor are quasi-inverse to each other, and thus esta ish the equivalences of categories.In D-modules theory the bl category given by V O , is of the simple D Xmodules to along V.This is due to the support of Radon transform of our R e transform that is required.
The following step is to give a result of equivalences between c adon-Penros ategories that suggest the extension to the befo the no valen re functors to the category of the vector bundle of lines of where it has the classification of differential operators belong to sheaves defined in the Section 2. Indeed.
Theorem 4.2.[12] With the same hypothesis as given by the theorem 4.1, assume also 3 d  .Then with tation of the theorem given by the Section 3, then the following correspondence is an equi ce of categories Precisely these equivalences shape a classification of the homogeneous vector bundles of lines [18,19].
As a corollary, using these Penrose transform, which is of Radon type on Lagrangian manifold  , we can o e complex Minkowski space M , simple D M -modules along the characteristic manifold V, of he wave equations that are classified by (half) tegers, so lled helicity   h k .The following section establishes these equivalences using geometrical additional hypothesis.
The last theorem concludes, category of holomorphic vector bundles with flat connections However, it is important in calculating image of a D X -module associated to a line bu w explicitly the ndle, one ay to do it consists in "quantizing" this equivalence.In the following results we will.
Let M , be a simple D X -module along * T X  , and let N , be a simple D Y -module along V.
, and a generator a f

Charact
Now in the context of the generalized D-modules to the use of the Schmid-Radon transform, and finally obtain Radon-Penrose Transform, the functor additional geometrical hypothesis, S   (29) establishes an equivalence between the category M(D X -modules/flat connection), and M(D Y -modules/Singularities along of the involutive manifold V/flat connection) [12].Then our moduli space that we construct is the categorization of equivalences: -modul es Singularities along of the considering the moduli space as base [12], then the cohomology on moduli spaces is the cohomology of the space-time with an equivalence like the given in the equivalence (12), to a more general cohomological group that the given by     Proof: [25]. Our space mo ant to characterize is the e that establish rm S  , between the category of coherent D-Modules on X, and the category of coherent D-Modules on Y, with regular singularities on V.
We define the category , by the thick sub-catego s X endowed with the ry of holomorphic bundle flat connection: In particular, the objects of pace [25]: The additional geometrical hypothesis in the functor (29), comes established by the geometrical duality o Langlands [26], which says that the derived categ co f ory of herent sheaves on a moduli space Flat , , where C, is the complex given by to eigen-sheaf of Hecke given by B C [22].

R
this way is the is the space of , o li space (35).Then these functors are G-invariants and their image under [23 Bun .The s must be ], which is an extension of the transformed cycles by the classic Penrose transform [15] equivalence under the G-invariance of holomorphic bundle demonstrated using a generalized Penrose transform for D-modules that are the composition of an inverse image functor and a direct image functor on the side D-module, which is foreseen by the geometrical duality of Langlands [26,28]. In electromagnetic interactions the functors are the objects that characterize the moduli space , are homotopies on the Riem n manifold X [9][10][11][12][13][14][15][16][17][18][19][20].In a more general co and using the flag domains that will be necessary to define th ravity ory (f annia ntext e quantum g phenomena [28], and equivalences inside of the string the or example heterotic strings, D-branes and others phenomena) to give solution a extensions of the wave equation on observables of curvature, boson and fermion equations, Schmid equation [10], and classify their differential operators on the same base of vector bundles, but now through holomorphic vector bundles that are G-invariants.
But this only covers some aspects of brane theory as those given outside of the homogeneous space G H , when H K  .For example, they are acknowledges the cases to heterotic strings that can develop on D-modules that are [11].In tegral o logy on (10) and generalise this resolution for co e g tra o at will conform a version of Penrose transfo this case the integral operators cohomology given on such complex submanifolds is also equivalent to the inperators cohomo submanifolds of a complex maximum torus.What happen when these flags are complex domains or their equivalents, Lagrangian submanifolds in F ?
To be able to establish a moduli space that could give a resolution for the equations of the mathematical physics given by the set herent D F -Modules that are D F -Modules induced by a bundle of lines F , with complex domains that are integral submanifolds of a Kählerian manifold is necessary to use on eneralized Penrose nsform with the conformal invariance n the D F -Modules, and that the differential operators who are not flat might write each one in accordance with similar operators.This last comes reflected when a piecewise linear manifold has a differential structure.This piecewise linear manifold, defines a non-symmetrical component curvature, which reflects their difficult expression by conformal operators of superior degree such as the Laplacian, and Dirac differential operators.
It is necessary consider some conjectures on integral geometry, that shows the construction of the geometrical hypothesis th rm more useful to generate quasi-conformal operators (quasi D equivariant operators) in an analytic cohomol-ogy as the given one by the  -cohomology but in a more generalized context, that it includes the Cĕch-cohomology.
Conjecture 1.A more generali ed Penrose transform is necessary to include differential operators in Hodge representative c z lass to obtain all complete holomorphic cohomological classes.
In other words, there is an isomorphism here Conjecture 2. The corresponding extended class of tia ormal operators are those of the Gelfand-Graev-Sh where S mani differen l operator that can be generated like generalized conf apiro type [29].
We want to compute the analytic cohomology Since solutions of differential equations on manifolds an generalize ry of sheaf coho d cohomological classes are part of the bases of this d Penrose transform, the theo mology and D-modules was perfectly considered for their modern study.In [5], we used this theory to generalize and study the Penrose correspondence given by the double fibration (4).More recently the Equation (17), in [1] has studied the generalized Penrose transform between generalized flag manifolds over a complex algebraic group G, using M. The Kashiwara's corresponddence (see [1][2][3][4][5][6]) between quasi G equivariant G H Dmodules and some kind of representation spaces which are   , -H g modules (loosely, they are complex vector spaces endowed together with an action of the Lie bra g , associated to G, and an action of H, which are com in some way) when H, is a closed algebraic subgroup of G. Proof.See [5,30]. Theorem 7.1.(F.Bulnes) Proof.We use the Radon transform of dimensions [30][31][32], given by , are the Lagrangian operators of instantons considered in flat space 4 8 P , of the corresponding supertwistor space PT [33].Then by Theorem 7.1, and using the large resolution re  coherent D-modules given by the images of Penrose transform in these pro-Some consequences and components of the space-time which is a special case of the solution (36), we obtain the equivalence between the moduli spaces.The long sequence induced by the jective spaces gives the string fabric by heterotic strings [33].
re included in the Calabi-Yau manifold, in the hole in the 3-dimensional space.lated under the isomorphism discussed in the previous theorem are those that establish the equivalences given by the Equation (12), with the microlocal structure of  , and w

Some Applications and New Research Developments
ith Floer cohomology group deduced from the coho bi mology group sketch by the theorem given in the search of the integral operators cohomology that encloses the Penrose transform class.
The corresponding category associated to a Cala -Yau threefold X, would be the Fukaya-Floer category of the moduli space of unitary flat bundles over holomorphic curves in X, denote   curve X `.We summarize these in the following Table 1.
We base in the scheme on Stein manifolds from Riemannian structure of the space-time [34], and using the generalization by Gindikin conjectures formulated in the Section 7, we obtained a result given in [30] , is a relative Abelia rface on 1 P , then their fibers are supersingular surfaces (D-branes of one . Proof: [18][19][20][21][22][23][24][25][26][27][28][29][30]. The affirmation 2), and 3), are generalized in the context of the moduli space . Using some cohomological classes of geometrical integral transforms that give the equivalences T , from the theorem 8.1, with their corresponding identifications in the D-modules context we can give the following Table 2.
Other applications and new results obtained recently [22][23][24][25][26][27], are the relatives to cohomology group classes whose objects have the same metrics in the Kählerian context: Proposition [27].The L 2 -cohomology groups of g `, de complete Kählerians metrics are all the same cohomology groups of D'Rham of . A concrete application of this result establishes that the moduli space of the relations between hyperbolic waves (horocycles) [35], and the Haar measure of the group action in SU (2.2), on , is the moduli space of the functors

o-cyles of the P ose transform on coherent D-modules. rose Transform on Derived Sheaves
The D-module transform of is the - Homogeneous Complex odule transform [17] vector Bundle Complexification of the twistor model 3 (QFT) [1,9,12 of the space-time M {hyperfunctions fields on M, of helicity s} Penrose-Ward Transform [32] Mini-Superambi-twistor space Yang-Mill-Higgs fields o braic group G, we enunciate the conjecture based on the result given by  which shows that the D-module inverse image functor ass ction  

8.
he skills used in this work are the developed maps of eful to realize con f the string theory and brane theory inside SUSY theory, for example that are germs of these sheaves and h k

Conclusions
T coherent sheaves of D-Modules, which establish conformal classes that are us texts o differential operators their corresponding geometric images in the complex holomorphic bundles, who receive sense in the QFT for the particles fields.These equivalences gives birth to the moduli spaces on the calculated conformal classes that can be useful in the securing of classes of solutions modulo a geometric characteristic of the different equations of field studied in theoretical physics and that an invariant turns out to be geometric of the application of the Penrose transform on classic elements via their classic version.
In this way, departing from some hypotheses obtained by Gindikin on the geometric invariance of cycles in a Kählerian manifold, we can obtain more general versions of the Radon transform, coming to the point of extrapolating these hypotheses to the geometric context of the Penrose transform on the coherent D-Modules in order to use for the generation of the isomorphisms via their double fibration and with it the moduli spaces securing that establish the equivalences of geometric objects that can be re-interpreted algebraically and vice versa.An example of it is the theorem 7.1, and their concrete applications in the description of equivalence of objects happened in the Table 1.
Since the generalized Penrose transform for D-mod- ules is the composition of an inverse image functor and a direct image functor on the side D-module, we first describe the algebraic analogs of these functors.More explicitely, if H K  , are two closed algebraic subgroups -modules (more precisely, objects of the derived categories).The next step is to obtain an analog of the Bott-Borel-Weil theorem for computing the Zuckerman functor image of "basic objects" of the category of   , H g -modules [10,26].In representation theory this is equivalent to computing explicitly the image by the derived Zuckerman functor KH R , of generalized Verma modules   H M  [36] ciated to weights , asso  , which are integral for g , and dominant for the Lie subalgebra of g , corresponding to H.Those generalized Verma modules are the objects which correspond M , Kashiwara correspondence to the D-modules which generate the Grothendieck group of the category of quasi-G-equivariant via G H D -modules of finite length [23].In this way, we can also obtain a classification of the differential operators considering extensions of the Verma modules considered [18], and showed to invariance of conformal operators by [4].
Nevertheless, from the point of view of the representation theory the demonstration of the theorem 7.1, leaves open questions as soon as to be able to generalize the Radon and Penrose transforms on G H , for not compact case of G, since conditions have not happened to guarantee an operator of this class that is of closed range on these spaces [36,37].
Through the corollary 7.1, we want to establish at least initially in physics, the form in how might there be established the equivalence of the isomorphism obtained by the different Penrose transforms between the dimensions of the singularities, these are seen and shaped by the st tion ry.
ring theory, and the projective spaces (in the context of the bundles of lines) [18,22,38] that give birth to the conformal classes via the derived sheaves and their stacks in physics, given through the applica of the Penrose transform.Likewise, the concrete applications that have been mentioned are studied for the mirror theory in the last advances on field theo omenclature and Abbreviations -Wave operator.This is a differential operator that omposes the wave equation on the space 4  R , also alled D'Alambert operator.

Theorem 3 . 1 .
(Classic Penrose transform) Let , be a open subset such that U  x , is connect and simply connect x U   .Then , the associated morphism to the twistor correspondence (1); which maps a 1-form on 0 k   U , to the integral along the fibers of  , of their inverse image for  , induce a isomorphism of cohomological classes: such that the category of quasi-coherent left D-modules on X, is isomorphic to the category of coherent D-modules.
 , the derived category bounded for right quasi-coherent D  -modules of the form are of the DL , type to some bu particular, if N , is a simp e along V, there is an unique ( o O X -linear isomorhisms) line bu dle L , on X, such t ndle of lines L .In le D Y -modul up t .In other words, the above theorem says that simple D -modules along V d, up t Y

Theorem 4 . 3 .
sections s, defines a D Y -linear morphism With the above notations, and with s,

OTheorem 5 . 1 .
, can be calculated by the intersection methods, to a comple[13], which is of the type x sheaf O   i O (see examples in Section 6).A version of the theorem 4.1, foreseen in the before section, that es b ishes the regularity through the transfo k P ta l rmations realized by the functor  , on the categories of derived D-modules adding the corresponding cohomological groups of zero dimension is: Let  , smooth and  , proper.Let induced by the transfo Y .
, if w, an er w, correspond to M .This is equivalent t : on X .In similar form we define M Y , and For the doble fibration(14), and the moduli spaces(30) and (31), these equivalences shape the moduli s the derived category of D-mod moduli space of holomorphic vector G-bundle a complex of certain special sheaf of holomorphic G-bundles (eigensheaf of H Proof.It is necessary to go from the o tained co-cycles for the Radon-Penrose transform on The equivalence of both spaces of cocycles demonstrates that the functors are y

Co jecture 3 .
We have a generalization of Penrose transform on the homogeneous space We need t ersion of Radon transform to compute dimensions (transform of dimensions).Their func e equivalences on the generalization mentioned in Conjecture 3.
2, the class given by   C G , is a complex whos lass is Proof.By the Kashiwara T e sub-c   C B , where B , is a D-brane, that is to say, a D-module  , is nsion of the brane space.M , is a D-module the dime P -m What happens with the orbifolds?It is necessary to conorbifolds like o u which is a odule induced by the bundle of lines P .sider the D-branes in the D-modules on bundles of lines in  P .Of the fact, by the theorem 6.1, we can dem nstrate that   , H G C `, under certain dality [22], is composed for objects of derived category of D-modules on   G B C .This vector bundle of lines under the same duality [22 is their wide version which is isomorphic to the moduli sp   , G C . Using a version of the Penrose transform predicted by Recillas's conjecture and etrical duality of Langlands on D-branes, is generated the isomorphism class given by Then by the propo-Y, is projective[31], that is to say, sition that says that the scheme whose m ds Y (that are orbifolds of CY-manifold), are P -modul e equivalence between the li spaces is completed with inverse Penrose transform. Corolla .(F. Bulnes).Moduli space = {homological relations of the black holes with the distinct cohomological dimensions obtained by lemma 5.1 and lemma 7.1}  moduli space = {given b lent to the forgetful functor from the category of   also shows that the derived direct image functor Dg , on the side D-module corresponds (up to a shift) to the algebraic Zuckerman functor KH R , which maps  

  4 ,D
C -Special linear group of the complex matrices p.This com--module.This D-module is the fundamental ingredient of the equivalences of the objects of moduli space.M -D X -module object of our research and their category is usef of linear differential sy M  X -module u □ c c SLul to realize the classification operators in field theory.Th of rank four.Have the structure of Lie grou pone is D-module represents a stem of PDE, in mathematical physics to which is n to nt is necessary to define the linear reductive groups of M .P -Penrose transform operator.

XD-
Shea of rings of holom ecessary give solution.G -Coherent Sheaf of the D X -module .Y D -Im f orphic linear differential operators (a D-module).

XO-
Sheaf of holomorphic functions on a complex manifold X. age of D nder Penrose transform.DL -D X -module of bundles of lines. of D-modules that are D Pmodules, and that can be induced to D F -modules.SUSY Cycles-Supersymmetry Cycles.QFT-Quantum field theory.qc M D -Category of D-modules.  co  D -Category of D-modul M h es that is subcategory of   qc M  .X E -Ring naturally endowed with a Z -filtration by the