Journal of Applied Mathematics and Physics, 2014, 2, 150162 Published Online April 2014 in SciRes. http://www.scirp.org/journal/jamp http://dx.doi.org/10.4236/jamp.2014.25019 How to cite this paper: Bulnes, F. (2014) Framework of Penrose Transforms on DpModules to the Electromagnetic Carpet of the SpaceTime from the Moduli Stacks Perspective. Journal of Applied Mathematics and Physics, 2, 150162. http://dx.doi.org/10.4236/jamp.2014.25019 Framework of Penrose Transforms on Modules to the Electromagnetic Carpet of the SpaceTime from the Moduli Stacks Perspective Francisco Bulnes Department of Research in Mathematics and Engineering, TESCHA, Federal Highway MexicoCuautla, Tlapala La Candelaria, Chalco, State of Mexico, Mexico Email: francisco.bulnes@tesch.edu.mx Received January 2014 Abstract Considering the different versions of the Penrose transform on Dmodules and their applications to different levels of DMmodules in coherent sheaves, we obtain a geometrical reconstruction of the electrodynamical carpet of the spacetime, which is a direct consequence of the equivalence between the moduli spaces ˆ Gequivariant Gequivariant G HCC (/, )(CY, ),≅ MM that have been demon strated in a before work. In this case, the equivalence is given by the Penrose transform on the quasicoherent Dλmodules given by the generalized Verma modules diagram establish ed in the Recillas conjecture to the group and consigned in the Dpmodules on which have been obtained solutions in field theory of electromagnetic type. Keywords Penrose Transform, Moduli Stack, DpModules, Verma Modules, Field Theory, E l ec troDynamical Stacks 1. Introduction We consider the Penrose transformation, [1] through the correspondence (1) Let be the Penrose transform [2] associated to the double fibration (1), used to represent the holomorphic solutions of the generalized wave equation [2], with parameter of helicity [3,4]: (2)
F. Bulnes on some open subsets in terms of cohomological classes of bundles of lines [2], on (where is the superprojective space). In the modules language the direct and inverse images of the double fibration comes given to mod ules that are modules through of the right functor equivalence: F R(;())R hom((),)[1], Yy Y yk kΓ≅Φ− −OO ppDD (3) where module transform of is the module associated to the wave equation given in the equa tion (2), whose helicity is Now in the context of the generalized modules (inside of the derived categories) we can use one version generalized of the SchmidRadon transform, to cycles in the derived category and finally obtain for the Radon Penrose Transform, the functor [5] , S additional geometrical hypothesisΦ+ (4) establishing an equivalence between the category X M(modules/ flat connection),D and the category M(DY modules/Singularities to along of the involutive manifold V/flat connection), [2,5,6]. Then our moduli space that we constructed is the categorization of equivalences: d[ /Y]Y {M(modules/Singularities to along of the involutive manifold / flat connection) (U, )}, SD V H• = O M (5) considering the moduli space as base [2,6], d[/]( ) {M(modules/ flat connection) ker(U,)}, SX Xhk D= M (6) The additional geometrical hypothesis in the functor (4), comes established by the geometrical duality of Langlands [5], which says that the derived category of coherent sheaves on a moduli space where , is the complex given by [6] 11 11 1 : jj ji1i3 ddddd jjj j C , −+ ++ + → →→→ → EEE E (7) which is equivalent to the derived category of modules on the moduli space of holomorphic vector bundles given by [7], this shaped by the geometrical extension of a derived functor corresponding to one Langlands ramification with corresponding stack moduli given by [8]. Inside of this study, the transform on modules (see Kashiwara [9]), that arises is the RadonPenrose transform for generalized flag manifolds in the framework of this quasi equivariant modules (that are a class of the coherent modules useful in the manager of cycles to reconstruct and model the geometrical im ages [10,11] of the electrodynamics of the spacetime through of the different photon fields that interact as strings, heterotic strings and branes [12]), is the generalized version that we want. Likewise, in particular their Radon integral operator defines and proves the necessary invariance of the geometrical images obtained by the modules transform, predicted by the Kashiwara theorem [12], and that guarantee the invariance in the solutions of the wave equation given in (2) in the frame of conformally [13] in the integral geometry context. By the Recillas conjecture [14], and their application in the obtaining of Cěch complexes (complexes obtained in the development by derived sheaves (ver Kapustin [7])) in the tacking of strings through superconformal spaces given in the corollary by [5 ], we have that the PenroseWard transform is done evident when the inverse images of the modules that are quasicoherent modules established by the diagram [14] (Verma modules diagram in the conjecture Recillas to the group [4,15]) result naturally inside of vector bundles language as image of the degeneration of cycles given inside the manifold signed in the equiva lences of moduli spaces in the theorem 4.1, given in [16]. Likewise the duality between string theories (string and branes) stay established through of the intert wining operators of the Penrose transform and their isomorphism in all different string dualities, field/particle and the conformally and holonomy levels required in invariance of the spacetime field theory. 2. Framework Penrose Transforms on Modules that are Modules As we want the framework of the spacetime through the different versions of the Penrose transforms and their
F. Bulnes relation between them considering the corresponding intertwining operators (integral transforms that are iso morphism between cohomological spaces of orbital spaces [17,18] of the spacetime), as for example, the twis tor transform to the Gorbits of the complex holomorphic bundle of the complex Riemannian manifold. Since solutions of differential equations on manifolds and cohomological classes are part of the bases of this generalized Penrose transform, the theory of sheaf cohomology and modules was perfectly considered for their modern study. In [13], we used this theory to generalize and study the Penrose correspondence in (1). More recently in [15] has been studied the generalized Penrose transform between generalized flag manifolds over a complex algebraic group , using . The Kashiwara’s correspondence (see [14]) between quasi equiva riant modules and some kind representation spaces which are modules (loosely, they are com plex vector spaces endowed together with an action of the Lie algebra , associated to , and an action of , which are compatible in some way) establishes the solid criteria to obtain equivalences between cohomo logical classes when , is a closed algebraic subgroup of . Conjecture. 1. [14] We want a generalization of Penrose transform on the homogeneous space or a Radon transform on equivariant modules on generalized flag manifolds [13,15]. Let , be a reductive algebraic group over , , and , two parabolic subgroups containing the same Borel subgroup of . Let and let , be the unique closed Orbit in for the diagonal action. Then we can identify , with The natural correspondence (8) where , and π, are the restriction to , of the projections of on and , induces an integral transform from to , which generalizes the classical RadonPenrose transform. This subject has been in vestigated intensively both in the complex and real domains (to see BastonEastwood [1], D’AgnoloSchapira [6], Marastoni [11], Gindikin [13]). Our goal is to obtain one version of this transform in the framework of quasi equivariant modules (see Kashiwara [7]), i.e. the functor 1 bb GG * :M()M(),()gf, XY − →=RRMMM M (9) called the Radon transform, which is predicted by the conjecture 1, given by [15] and completed by [11], where denotes the derived category of quasi equivariant modules with bounded cohomologies, and and are the operations of direct image (integration) and inverse image (pullback) for modules. More precisely, we consider a module of type , where , is an irreducible equivariant locally free module. In this case is had that The Grothendieck group of the category of quasi equivariant modules of finite length is spanned by elements corresponding to the objects of the form . This means that we can generate solutions in complex holomorphic manifolds as , through conformal solutions defined on bundle of lines, at least to each a of the derived category components of quas i equivariant modules. From BGGresolution and using some fact of the Dolbeault cohomology on , discussed by Cousin complexes and their sheaves we obtain a resolution of the quasi equivariant module of the form: 0 1 0 11 0f ()0, n rr nk k kk DD = = →→→→ → ⊕⊕ LL DL (10) To obtain a generalization of the Kashiwara theorem in the context of the equivariant modules is necessary the geometric Zuckerman functor. This is the localization of the equivariant Zuckerman functor to the derived equivariant module categories on generalized flag varieties (this to apply in our study an appropri ate module transform of the Penrose transform type). To this case, we define our categories of interest and construct the geometric Zuckerman functor from the basic module functors. The category of (left) equivari ant module is denoted by The following theorem is well known : Theorem 2.1. If acts on , freely, there is an equivalence of categories For homogeneous spaces, we can make a stronger statement. Let , be a complex linear group, let be the category of algebraic representations of .
F. Bulnes Proof. [15]. Theorem 2.2. If there is an equivalence of categories Proof. [15]. Let a quasi equi va r ia nt  module belonging to the subcategory of modules given by (category of the equiva r iant  modules). The idea is to obtain equivalences between the different classes of modules establishing the isomor phism between the categories ( ) RS( )/ M() Mod, G GH υ =MD and where must be the restricted Zuckerman functor to the subcategory of left equiva r ia nt  modules to obtain the de rived category that is product of a trianguled subcategory (as the given in [16]) for a factor category that deter mine us the HarishChandra functor between the categories. These equivalences in analogy to the established in (9) take the shape to the specific modules // / {(){derived category of modulesequivariants}M(, )M()}, G GHGGHGHGG MDG H==−=DMMg (11) considering the moduli space as base, / {M(ker( ,)), equivG G / HGHBRST Dmodules G equivariantsUQ= −M (12) If we consider then , from the group . Bequivariant module on the flag manifold provide integral kernels and thus integral transforms, to know 0BRST (,)ker( ,),H XUQ λ ≅ L (13) where such that where and , are Higgs fields on either side of the open string [7], which are implicitly these kernels? The condition establis hed in the kernel of equivalences inside the moduli space , is analogous to the given by the isomorphism of the Penrose transform discussed in foot of page 2, in the section 2, in [5]. The idea is extend the harmonic condition to the functions to differential operators of derived sheaves that are in BRST cohomology, such that the equivalences in (12) are defined by certain functors due to the duality between the BGGresolution and meromorphic version of the Cousin complex1 associated to (bundle of lines asso ciated to the flag manifold , then , is a Borel subgroup). These is the way of the classification of the differential operators with their moduli stacks. Now we will dedicate to obtain these moduli stacks to elec tromagnetism in field theory as representation of the spacetime. 3. Electromagnetic Moduli Stacks to SpaceTime A concrete application of the moduli space , on bundles and their complexification establishes that the moduli space of the relations between hyperbolic waves (horocycles) [18], and the Haar measure of the group action in on 2 is the moduli space of the functors (2) 1 0 (Hom(( ), (2)) ( )((( ))),,, () SU jX Y H XU Dk HDkD DM Dmodules • Γπ =−=Φ− ∀ ∈− P FP P M where the cohomological space is the space of the equivariant functors related with the functors given in the relation (3) where these last come from the use of Penrose transform on  modules [6]. But these functors are induced on orbits of conserving their equivariance through of one integral transform that relates HarishChandra modules from the string le vel like branes ( modules) with one intertwining operator of cohomological description as the given in geometrical Langlands correspondence established in [8]. Then by the Langlands ramifications the moduli stack correspond to the holomorphic bundles on where is the module defined in the complexes succession (7). Here the group , is de 1Cousin Dolbeault theory refers to the complex Dolbeault infinite dimension. 2In measurements of observables like curvature and torsion the corresponding Radon transform on spaces classic Penrose transform, and their image points are used to measure the curvature through of light waves (macroscopic case) [19].
F. Bulnes fined by the condition that the lattice of homomorphisms from to a maximal torus of , be isomorphic to the weight lattice of . Studies on representations to noncompactness Minkowski spaces such as that appear in electromagnetism to characterize the gauge fields in and establish that when the gauge group is and is a noncompact maximal subgroup of the unique nonzero invariant solutions are Abelian Maxwell fields [20]. Then this fact produces (zero branes stability) that is to say, the module of branes are the torus [7] which is isomorphic to .3 What is the mosaic of the field theories of the spacetime in this case? The mosaic is very simple and only includes two theories of the electromagnetic nature shaping the frame of the study Figure 1. In this case the two different electromagnetic theories, one in the commutative case (that includes the elec tromagnetic macroscopic fields) and other that includes the weakly electromagnetic type that conforms, joined with others considerations the gauge theory, can be the same if their semigroup operation is not invertible. To the dimensional scheme of the conformation of the one only theory from two different theories (for example to the gauge theories) the quotients of the homogeneous space can be visualized under the branes square only (blue square in the Figure 1). We consider the description of the grading (or classification) that appears physically in terms of the (the renormalizing of the group ). Considering the TQFT, and CFT, for branes and antibranes wrapped on the entire space, the analysis from the gauge theories is straightforward. The tachyon , is a degree zero operator. The term that we have that add to the boundary to describe a tachyon is the field descendant where , is the topological ly twisted boundary supercharge (that is to say a term in the TFT OR TQFT). The operator , has , charge , so has charge . But a necessary condition to preserve supersymmetry is that boundary terms must be neutral under (otherwise the , is broken which breaks the boundary su per s y mmet ry4). For other side, to lower dimensional sheaves, the relationship between the charge and grading is more subtle considering the dimension of the cohomological spaces to different vector bundles considered. For example, to the state corresponding to an element of for two sheaves , and , we need not have charge , equal to , which could be the degree of the , group will not match the charge of the state. Then the use of the twistor transform results very useful until certain point when there exist correspondence between sheaves that are corresponded to mutually supersymmetric branes. For example, the inverse twistor transform applied to the sheaf corresponding to the power of the tautological line bundle of superprojec tive space [21], for example the bundle of lines when is considered a twistorstring case with Nevertheless one important aspect that born of these relation that are explained by the twistor transform is the field consideration that gives origin to the shape of the geometry (at least in lower dimension) from the quantum states and their orbifolds visualized as the lines of their corresponding vector bundles shaping the physical stacks of the first topological electromagnetic theories to some classes of the space . But, fur thermore using the “ramifications” to certain Langlands classes [8] we can to obtain the visualizing of the sin gularities of the field as sources of the proper fields, through of their extensions as rational curves. Remember Figure 1. Renormalizing of the Group in semi groups of electrodynamical quotients from when [7] . The coweight lattice of is defined as the lattice of homomorphisms from to a maximal torus , of ,is the lattice of homomorphisms from , to 4Twisting SuperYangMills theory.
F. Bulnes that we want discomposes singular regions of the space in regions of the conformal nature that can give the val ue of the singularity evaluated as zeros of polynomials in algebraic geometry and reinterpreted by the Penrose transform as varieties (varieties that are rational curves or algebraic curves). This is as to take a microscope and observe the flatness of the space in these singular regions (as the study frame given in the first square of the Figure 1) to these scales (for example to the QFT). To this last is very useful establish dualities between of set of equivalences from the moduli space and the moduli space of the corresponding classes in (12) [5] (see Figures 2 (a) and (b)). In the case of the complexes (7) the idea is discompose a complex with nonconformal properties in “factora ble” pieces in complexes where flatness due to the holomorphicity exists and also conformability of , in these scales. A promissory result in that sense is the due to KashiwaraPoicaréMebkhout on local cohomology where in the resolution proposed in the Bulnes corollary [14], we can investigate the cohomology5 of , through the moduli space To the electrodynamical context, the moduli space is the corres ponding to the superprojective (superminitwistor space). What happen in the case when the spacetime group is measured from electromagnetic fields in the TQFT, or TFT contexts as ? 4. Results As has been mentioned before, to calibrate the spacetime through the electromagnetic fields is necessary re place the Abelian group , in the invariant structure of the principal bundle of , given by for the nonAbelian group since we want to realize an action through electromagnetic fields on a representation of the spacetime given as , where , is the torus . Then , where , is a compact group of For the isomorphism with , a representation through of connections of the spacetime will comes given for representations of the group . An example of the application of these representa tions is the dimensional hyperbolic space whose curvature can represents through the value of certain invariant integrals on lines of a bundle of lines on the principal bundle of , and whose that come from of the relation between cocycles of the cohomology of with an open orbit and fields of (for example, the fields on horocycles as solutions of the wave equation on the hyperbolic plane when and ). The case that interest us is consider a as flag manifold and with and Here is a dimensional hyperbolic space. The way to define the corresponding moduli stacks of electromagnetic nature is consider the following impli cations that are consequences of cohomology of derived sheaves of modules using the Cousin cohomology on bundles [8] and after extend them to the complex case: (a) (b) Figure 2. Mirror equivalences to sources and holes of the quantum electro magnetic fields. The first image corresponds to the hole or singularity of the connections. The two images correspond to the source of the gauge field. Both geometrical objects are equivalent inside of the dualities of the moduli spaces and Hitchin moduli space 5Remember that the twistor transform is an intertwining operator of cohomological description of isomorphisms from the different images of Penrose transforms of the same physical object (for example a hole can be a source and vice versa inside the bundle of lines context [5]).
F. Bulnes ˆ Cohomology of CechCohomology Complexification of the Twistor model to Space Time ˆ hyperfunctional of Cech, →∂− → → (14) But this only can be realized directly through of different classes of the bundles of the space that can be given only on those states inside the Lorentz norm. Then by ingredients from the field theory we have that the Penrose transform gives the following identifica tion: 2(,22){hyperfunctions fields on , of helicity },Hh h−=PM (15) But this space is too large to our purposes of field theory since it contains states of infinite norm like many radiative interactions that come from the background. Then we consider the cohomological space 1 (,22){real analytic fields on , of helicity },Hh h−=PM (16) which, using the p ositive/n ega tive frequency decomposition due the standard polarization takes the form 11 ( ,22)( ,22).HhHh +− −⊕ −PP Then by the extension of functors given in the category [8], we have equivariant (1), (2)(1), (2) (R(( )))R( ), jP U SUU SU HV VΓ=Γ D (17) and , a HarishChandra module, where is an equivariant image of the functor modulo , given in the moduli space , and these are the functors , of the derived cate gory .Observe the modules as modules inside of the sheaf , that is to say, we want do correspond modules that come from a derived sheaf to certain class branes. These branes must be that come from of the electromagnetic fields to our research. Theorem (F. Bulnes) 4.1. [2]. The following isomorphism is valid quasi equivariantquasi equivariant (/,)(Y, ), GG G HCCC ∧• ≅ MM being a Cousin Dolbeault complex 1 , , [/ X] (M)[ ]((( ))). kk I np np XX HRkH R •• ⊗≅⊗ λOOEEDD Proof. [5,24]. From the before theorem given in [5,8] we can give the application to a geometric electrodynamical represen tation of the spacetime considering conformally invariant twodimensional theories6: Theorem (F. Bulnes). 4.2. Using the isomorphism between the moduli space and the mod uli space to the sta ck , we can obtain the moduli stacks 7 We want , [23] and obtain a result in real form spaces because we want use gauge theory, that is to say, “Abelianize” the theory to find a decorated of the moduli stack where we can use the branes to tack the field carpet as a image of the corresponding CFTspace (crystal field theory space) obtained of the corresponding gerbe8 [7]: via a homomorphism , to know; ( )(1) 22 ( ,())( ,(1)), ZGU HXZG HXU − → (18) Proof. Considering the twistor transform that appears in the composition [22] we have the Penrose 6Usually that stack is presented as where to this presentation one associates a nonconformally invariant two of the gauged sigma model. As stacks they are the same: 7For example considering a decomposition given by 8A grebe is a construct in homological algebra and topology. A gerbe on a topological space is a stac k of groupoids over X is locally none mpty (each point in has an open neighbourhood over which the section category of the gerbe is not empty) and transitive (for any two objects and of for any open set there is an open covering of such that the restri tions of and to each are connected by at least one morphism).
F. Bulnes transform as a map congruent with , modulo . The congruence of these maps defines the gerbe de fined in (18) and shapes the composition of bundles [7,21,22]9 (19) where , is the sheaf of the smooth invertible functions on X. Of this way the cocycles obtained by the Pe nrose transform are Čech cocycles that come from the smooth principal bundle. The space , is a fine sheaf such that Thus a principal bundle is a geometrical realization of one element of the cohomological space But we need establish the isomorphism between the moduli spaces to modules and of branes such and as is necessary to apply the geometrical Langlands correspondence between of the (bundles characterized on ) and the structure shea ves like the given in (with , a Hitchin base given by the cone )10 to the bo sonic moduli space that we want (remember is the electromagnetic group of weak interactions such as ). Remember that we want obtain a photonic carpet as a carpet of the spacetime. Then to the Čech cocycles we need that come from holomorphic bundles. Then the correspond ing sheaf of inverfible holomorphic functions on X, will be Then a geometrical realization of a gerbe is the cohomological group 11 But the gerbe corresponds to a principal bundle (in this case a gerbe equal to bundle with connection) where their topological realization is a cohomological space Then considering the twisted image of this cohomological space we obtain a characteristic class of gerbe in the space which is the preim age of the Penrose transform { } 2 21 (, )(,)/,(,)HXUd U θθ ≅Φ∈Γ Ω Φ=∈Γ ΩO (20) in the stack . But we want extend the electromagnetic field solution to the stacks such that (21) But our moduli space of the corresponding stack moduli must be the orbifolds in , since we want field extensions given by the Higgs field Different bundles implies fields that have different zero modes, which implies different anomalies which implies different physics measured from the electromagnetic gauges. All these different field physics obtained are ramifications of the same field following different patrons given by . For example on a noncompact worldsheet (see the Fig ure 3), the instantons that have charge corresponding to electrons with charge , and massive particles can appear with periodicity es tablished by the closed forms whose cohomological classes lands in lattice (22) Then the Čech cocyles are who represents the kernel , of the map (18) which is the space of the flat bundles, that is to say, the moduli space who is image of the functor of finite range12 In this point we have in the flat complex space that a in in 4 [1][1] , a ab O TTO ∇ →→ →≅ M C C which is conformally invariant (since the vector bundles of , in , need of the twistor space deduced to the derived wave operators of the affine connections let be conformally invariants). Then considering the corresponding to the forms to curved case ( is the curvature of (essentially the Weyl curvature of ) , a 0form in ) we find an analogous connection to 2forms, proving the obtaining of elements of the space In this composition , is a twstor image of bundle using the connection . Then by the Penrose transform there is isomorphism between respective vector bundles. But from the congruence between (18) their Penrose transform we have a stack modulo 10 is the Weyl group. 11This cohomological space to all spacetime , must be accord with the map (18). 12 , is the homomorphism given by . Also is explicitly given to this case (Higgs fields) by the map 01 (, ) (, ) H Higgs GC X GC ππ Φ → ↓ M M
F. Bulnes Figure 3. branes Moduli Stack calculated with the action of on , lifting to an action where usually to the Riemannian surface we can consider the causal structure of . The phy sics are different and are corresponded to different “tim es” by a phase . The different physics can be viewed as sources (charges), holes (scattering or mas sive particles), warm holes (perturbative spaces), flat sp aces (de Sitter regions, for example neighborhoods in Lorentzian manifolds). 22 1 (Hom( (),(1)))( ,()), L HZGUHCG π Γ≅ (23) But the MOduality implies that the moduli spaces and (with symplectic struc ture ) are a mirror pair [25]. Then the flat fields from take values in the space . But is well known that is naturally isomorphic to . Newly by the argu ments of the MOduality we can map the class defining the field on to the cor responding element in , labeling the connected component of (and vice versa). Then to our case when then the moduli space is connected and has two possible flat fields labeled by Then, for other side and therefore haves two components labeled by . Finally the sheaf of the openstring states (states that come from the branes) is the sheaf of holomorphic functions on in the complex structure of the symplectic complex structure 13. Then can be identified with , which the bosonic moduli space that we want. Example 4.1. We consider the Abelian Hodge theory approach, and let , with , the curve of their corresponding ramification with the same geometry to construct the pair or Langlands data where and (24) This connection is a meromorphic relative flat connection acting along of fibers of the Hitchin map: (25) Furthermore by construction the bundle with connection is a Hecke eigen module with eigen value (as the found in [8]) but with respect to an Abelianized version 13Since has been used the Hitchin base (to involve a the torus , and relate the symplectic form with Then the isomorphism given by the Penrose transform must be (because we want include the Hitchin base to singularities of the field seen these as source 2 (,)ker( ,), n HMU QΩ ⊗≅∂+L where is the new connection considering the Langlands parameter to regularize the connection given by the global Langlands category of sheaves [8] to elements in the Penrose transform problem when we want obtain the functor (4).
F. Bulnes Higgs Higgs (1),(2),(2).:D (Bun,)D (Bun,), Li bb Abelian U USOCΦ →×OO (26) of Hecke functors. These are defined again for as integral transforms with respect to the tri vial local system on the abelianized Hecke correspondences with and , in the classic scheme of the double fibration of the integral geometry (Penrose transforms!) on microlocal objects of these bundles. Their stack moduli is . Corolla ry (F. Bulnes) 4.1. An dimensional brane model of the electromagnetic carpet is given by the moduli space with dimension with and charge such that 2 1( ). IC Tr AA e ωδδδφ δφ =∧ −∧ ∫ (27) Proof. Applying [26], and considering the example 4.1, with the moduli space with the hypo thesis given for our corollary, the image under the Penrose transform on double fibration in , that is to say, for example, the cohomological space , (this formulation is through the integral geome try) can establish a 2dimensional image given as fibration (not vector bundle (minitwistor surface)) in [5]14. Then the 2dimensional image of the electromagnetic carpet is the generated by the rotations and sources such as in the F ig ure 4. Figur e 4. Electromagnetic carpet of spacetime ob tained by quantum electromagnetic waves using the coefficients from the energy of the waves to quantum level [27]. The model is obtained simulating the waves in the corresponding model, using a twopoints function on the torus. Of this way, is obtained the framework using this orbital space pre dicted by the gauge theory. In the carpet (blue surface) the nodes are states of energy and lines (or routes) are the paths framework. The nodes are the singulari ties or sources given in the Figure 2. 14 , is the Riemann sphere .
F. Bulnes For other way, the gauge field can be naturally coupled to sources which are transformed to unitary re presentations from of These are electrical sources, since they create a charge like a Cou lumb charge of the form (28) where , and are generators of in the representation from Then if we con sider to the spacetime with the distance from the origin in , and we assume that the wor ldline of the source is given for [7], then we can establish that magnetic sources are labeled by ir reducible representations of different groups as We can establish using a path integral as the given by (29) the traces inside the symplectic structure given in (27), and draw the electrophysical carpet of the spacetime with all constitutive sources “charges”. Here , is the principal bundle on where is the com pact component of . For example, to vector bundle given by which is basically the functor image , (that appeared in the demonstration of the theorem 4.2) necessary to represent the spacetime in their string model. Note that in QFT one sums over all isomorphism classes of . It’s necessary the BRSTcohomology and the equivalence between QFT, with a reinterpretation in graph quantum theory using the string theory with a photon of the class of the boson field as calculated by . Joining the paths of this boson floor we obtain the corresponding fields to every instanton in path integral as given by [7] 3 exp{ } j Q j ZAAA i ν = ℑ ∑∑ ∫ D DD (30) having the tra ns formatio n exp{ /2},exp{/2}/,exp{}/, aa aa QiQQiQi ττ → −Φ→ −ΦΦ= (31) Then the three actions contemplated in (30) are consigned in [7]. From the corollary 4.1, and the before discussion with the frame (31) we have the following electromagnetic carpet and the corresponding their framework. 5. Conclusions Develop a electromagnetic theory of spacetime is equivalent to extend the electromagnetic forces to the gauge context using the concepts as supersymmetry from the conformality and holonomicity of the electromagnetic fields always present from the bosonic symmetry generators given from the dualities of the moduli spaces and using their corresponding physical stacks that comes from their connections with ramifications [8]. Using the hypothesis of that all singularities of the spacetime , are sources in different moduli stacks (mirror theory) of a CalabiYau manifold of the spacetime , corresponding to the superprojective flat spaces [5,14], we have that in their supermi nitwistor spaces the flatness is traduced in superconformality to these spaces, and their line integrals give the values of the singularities. The singularities coshape varieties of different degrees of genus. The functor given in (17) say us that the equivalences necessaries to comply the dualities established in the theor em 4.2, must be considering the HarishChandra modules that are quasi equiva r ia nt s modules such and as must do it on moduli space , in the level of the functors obtained by the Penrose transform to the problem of ramification signed in the theo rem 4.2, on geometrical Langlands correspondences and their corresponding derived category (the Hecke category) given in [8]. Inside of the set of dualities established on moduli stacks, the dualities (MontonenOlive duality) are of special attention due to their application in twisted theory of the different classes that appear when is used the twistor transform inside the Penrose trans form, this in frame of the field problem (for example the correspondences between branes and branes [7] or the behavior of the field in different orbits (see Figure 5) to after extend it to all spacetime, and of this
F. Bulnes Figure 5. Hyperbolic flat model of the spacetime with one singularity of space (could be one diagram in Fig ure 4 or 6) This model is product of two orbital spaces [17,28], and conform the surface of photon backreac tion in presence of singularity, this visualized as a su permassive particle. This simulation is shaped for com plex hyperbolic surface ( ,)(1/1.9)tanh(2/)zxyx y= [14, 27] . The massive particle corresponds to the sources of photons and these are their nodes in the Figure 4, cor responds to the energy states average given by the path integrals that connect them. Every node of energy that appears in the electromagnetic framework is the singu larity of the hyperbolic space used by the twistor trans form [21], and their neared space to the linkwave in Feynman diagrams in Figure 6, is the surface that is showed. Figure 6. From the way established in the Figure 4, is obtained the framework using this orbital space pre dicted by the gauge theory. In the carpet (blue surface) the nodes are states of energy and lines (or ruts) are the paths framework. manner to obtain the electromagnetic field carpet that we want. The Theorem 4.2, is the first intent through the elements and methods of the integral geometry to obtain different geometrical models, considering different electromagnetic interactions or adding they until to arrive to the curved case, which will can be included consi dering the bundle Acknowledgements Thanks to the TESCHA and the State of Mexico Government for the financial support to this research work. References [1] Basto n, R.J. and Eastwood, M.G. (1989) The Penrose Transform. The Clarendon Press, Oxford University Press, New
F. Bulnes York. http://dx.doi.org/10.1007/BF01942327 [2] Eastwood, M.G. , Penrose, R. a nd Wells, R.O. (1981 ) Cohomology and Massless Fields. Communications in Mathe matical Physics, 78, 305351. [3] P enr o se, R. (1969) Solutions of the ZeroRestMass Equations. Journal of Mathematical Physics, 10, 3839. http://dx.doi.org/10.1063/1.1664756 [4] Bailey, T. N. and Eastwood, M.G. (1991) Complex Par aConformal Manifolds Their Differential Geometry and Twi s tor Theory. Forum Mathematicum, 3, 61103. http://dx.doi.org/10.1515/form.1991.3.61 [5] Bulnes, F. (2013) Penrose Transform on Induced DG/HModules and Their Moduli Stacks in the Field Theory. Advan ces in Pure Mathematics, 3, 246 253. [6] D’Agnol o, A. and Shapira, P. (1996) R ado n Penrose Transform for DModules. Journal of Functional Analysis, 139, 349382. http://dx.doi.org/10.1006/jfan.1996.0089 [7] Kapust in, A., Kreu ser, M. and S chlesin ger, K.G. (2009) Homological Mirror Symmetry: New Developments and Pers pectives. Springer. Berlin, Heidelberg. [8] Bulnes, F. (2013) Geometrical Langlands Ramifications and Differential Operators Classification by Coherent DMo dules in Field Theory. Journal of Mathematics and System Sciences, 3, 491507. [9] Kashi wara, M. (1989) Representation Theory and DModules on Flag Varieties. Astérisq ue, 173 17 4 , 55109. [10] Helgaso n, S. (1999) The Radon Transform. Birkhauser Boston, Mass. http://dx.doi.org/10.1007/9781475714 63 0 [11] Marast on i, C. and Tan isaki, T. (2003) Radon Transforms for QuasiEquivariant DModules on Generalized Flag Manifolds. Differential Geometry and its Applications, 18, 147176. http://dx.doi.org/10.1016/S09262245(02)001456 [12] Maso n, L. and S kinn er , D. (2007) Heterotic TwistorString Theory. Oxford University. [13] Gindikin, S. Penrose Transform at Flag Domains . The Erwin Schrödinger International Institute for Mathematical Physics. Boltzmanngasse 9, A1090, Wien Austria. [14] Bulnes, F. (2011 ) Cohomology of Moduli Spaces in Differential Operators Classification to the Field Theory (II). Pro ceedings of FSDONA11 (Function Spaces, Differential Operators and Nonlinear Analysis), Tabarz Thur, Germany, 1, 001022. [15] Kashi wara, M. and Schmid, W. (1994) Quas iEquivariant DModules, Equivariant Derived Category, and Rep resen ta tions of Reductive Lie Groups, in Lie Theory and Geometry. Progr. Math., Birkhäuser, Boston, 123, 45 7488. [16] Bulnes, F. (20 12 ) Penrose Transform on DModules, Moduli Spaces and Field Theory. Advances in Pure Mathematics, 2, 379390. http://dx.doi.org/10.4236/apm.2012.26057 [17] Bulnes, F. (2013) Orbital Integrals on Reductive Lie Groups and Their Algebras. Intech, Rijeka, Croatia. http://www.intechopen.com/books/orbitalintegralsonr edu cti veliegroupsandthe i r algeb ras/o rb italintegralsonred uctiveliegroupsandtheiralgebrasB [18] Ratcliffe, J.G. (2006) Foundations of Hyperbolic Manifolds, Graduate Texts in Mathematics. 2nd Edition, 149, Sprin  gerVer lag, Berlin, New York. [19] Bulnes, F. (2009) Design of Measurement and Detection Devices of Curvature through of the Synergic Integral Opera tors of the Mechanics on Light Waves. ASME, Internal. Proc. Of IMECE, Florida, 91102. [20] Antoine, J.P. and Jacques , M. ( 198 4) Class. Quantum Grav., 1, 431. http://dx.doi.org/10.1088/02649381/1/5/002 [21] Dunne, E.G. and Eastwood, M. (1990) The Twistor Transform. In: B ailey, T.N. and Baston , R.J., Ed s . , Twistor in Mathematics and Physics, Cambridge University Press, 110127. [22] Bulnes, F. (2006) Doctoral Course of Mathematical Electrodynamics. Internal. Proc. Appliedmath, 2, 398447. [23] Benz vi, D. and Nadler, D. (2011) The Character Theory of Complex Group. [24] Mebkhout, Z. (19 77) Local Cohomology of Analytic Spaces. Rubl. RIMS, Kioto, Univ., 12, 247256. [25] Hausel , T. and Thaddeus, M. (2003) Mirror Symmetry, Langlands Duality and the Hitchin System. Inventiones Mathe maticae., 153, 197229. http://dx.doi.org/10.1007/s002220030286 7 [26] Bershadsky, M., Johansen, A., Sadov, V. and V a fa , C. (1995) Topological Reduction of 4d SYM to 2d Sigma Models. Nuclear Physics B, 448, 166. ht tp:/ /d x. do i.o rg/ 1 0.10 16 / 05 50 3213(95)00242K [27] Weylman , H. (2012 ) Path Integrals for Photons: The Framework for the Electrodynamics Carpet of the SpaceT ime. Journal on Photonics and Spintronics, 1, 2127. [28] Bulnes, F. (2012 ) Electromagnetic Gauges and Maxwell Lagrangians Applied to the Determination of Curvature in the SpaceTime and their Applications. Journal of Electromagnetic Analysis and Applications, 4, 252266. http://dx.doi.org/10.4236/jemaa.2012.46035
