Exotic smooth , geometry of string backgrounds and quantum D-branes
In this paper we make a first step toward determining 4-dimensional data from higher dimensional superstring theory and considering these as underlying structures for the theory. First, we explore connections of exotic smoothings of and certain configurations of NS and D-branes, both classical and (generalized) quantum using algebras. Effects of some small exotic ’s, when localized on , correspond to stringy geometries of -fields on . Exotic smoothness of acts as a non-vanishing B-field on in . The dynamics of D-branes in WZW model at finite indicates the exoticness of ambient .
Next, based on the relation of exotic smooth with integral levels of WZW model we show the correspondence between exotic smoothness on 4-space, transversal to the world volume of NS5 branes, and the number of these NS5 branes. Relation with the calculations in holographically dual 6-dimensional little string theory is discussed.
Generalized quantum D-branes in the noncommutative algebras corresponding to the codimension-1 foliations of are considered and these determine the KK invariants of exotic smooth for the case of non-integral . Moreover, exotic smooth ’s embedded in some exotic as open submanifolds, are shown to correspond to generalized quantum D-branes in the noncommutative algebra of the foliation. Finally, we show how exotic smoothness of is correlated with D6 brane charges in IIA string theory.
In the last section we construct wild embeddings of spheres and relate them to D-brane charges as well to KK theory. Wild embeddings as constructed by using gropes are basic to understand exotic smoothness as well Casson handles. Finally we conjecture that a quantum D-brane is wild embedding. Then we construct an action for a quantum D-brane and show that the classical limit (the usual embedding) agrees with the Born-Infeld action.
- 1 Introduction
- 2 Geometry of string backgrounds and exotic
- 3 Quantum D-branes and 4-exotica
- 4 From wild embeddings to quantum D-branes
- 5 Conclusion
- A Remarks on K-matrix string theory and BSFT
- B Tachyons and D brane charges
Despite the substantial effort toward quantizing gravity in 4 dimensions, this issue is still open. One of the best candidates till now is the superstring theory formulated in 10 dimensions. A way from superstring theory to 4-dimensional quantum gravity or standard model of particle physics (minimal supersymmetric extension thereof) is, at best, highly non-unique. Many techniques of compactifications and flux stabilization along with specific model-building branes configurations and dualities, were worked out toward this end within the years. Possibly some important data of a fundamental character are still missing.
The point of view advocated in this paper is that indeed we have not respected till now 4-dimensional phenomena of different smoothings of Euclidean which presumably are very important for the program of QG. There are strong evidences that exotic 4-smoothness on compact manifolds should be taken into account by any QG theory Asselmeyer-Maluga2010 (). Here we refer to open 4-manifolds and try to consider exotic ’s as serving a link between higher dimensional superstring theory and 4-dimensional ,,physical´´ theories and 4-dimensional QG. String theory would describe directly 4-dimensional structures at the fundamental level. This paper serves as a step toward seeing exotic smooth as fundamental objects underlying higher dimensional (super)string theory. Further results regarding compactification and realistic 4-dimensional models of various brane configurations in string theory and their relation to exotic 4-smoothings, will be presented separately.
The problem with successful inclusion of the effects of 4-open-exotics into any physical theory, is the notorious lack of an explicit coordinate-like description of these smooth manifolds. In the series of our recent papers we addressed this issue and worked out some relative techniques allowing for analytical treatment of small exotic ’s AsselmeyerKrol2009 (); AsselmeyerKrol2009a (); AsselmeyerKrol2010 (); Krol2010 (). In this paper we show that description of D-branes in some exact string backgrounds are related with 4-smoothness of . Moreover, the deep quantum regime of the D-branes is also 4-exotic sensitive.
However the connection of abstract, generalized quantum D-branes to the actual superstring theory D-branes (in the manifold limit) is not directly given. The Witten limit of superstring theory where D-branes yield their noncommutative world-volumes is only the midway and in fact motivates the full algebra approach Szabo2008a (). This last serves as a possible partial solution to the problem of describing quantum D-branes in superstring theory. The connection with exotic at this, quantum level is unexpected and shows that 4-dimensionality may get into the game in string theory through back-door of nonperturbative quantum regime. In the last section of the paper we use the algebra approach to quantum D-branes to construct a manifold model of a quantum D-brane as wild embedding. Then we show that the algebra of the wild embedding is isomorphic to the algebra of the quantum D-brane. Furthermore we construct a quantum version of an action using cyclic cohomology and get the right limit to the classical D-brane described by the Born-Infeld action.
The basic technical ingredient of the analysis of small exotic ’s enabling uncovering many applications also in string theory is the relation between exotic (small) ’s and non-cobordant codimension-1 foliations of the as well gropes and wild embeddings as shown in AsselmeyerKrol2009 (). The foliation are classified by Godbillon-Vey class as element of the cohomology group . By using the -gerbes it was possible to interpret the integral elements as characteristic classes of a -gerbe over AsselmeyerKrol2009a ().
The main line of the topological argumentation can be briefly described as follows:
In Bizacas exotic one starts with the neighborhood of the Akbulut cork in the K3 surface . The exotic is the interior of .
This neighborhood decomposes into and a Casson handle representing the non-trivial involution of the cork.
From the Casson handle we construct a grope containing Alexanders horned sphere.
Akbuluts construction gives a non-trivial involution, i.e. the double of that construction is the identity map.
From the grope we get a polygon in the hyperbolic space .
This polygon defines a codimension-1 foliation of the 3-sphere inside of the exotic with an wildly embedded 2-sphere, Alexanders horned sphere.
Finally we get a relation between codimension-1 foliations of the 3-sphere and exotic .
This relation is very strict, i.e. if we change the Casson handle
then we must change the polygon. But that changes the foliation and
vice verse. Finally we obtained the result:
The exotic (of Bizaca) is determined by the codimension-1 foliations with non-vanishing Godbillon-Vey class in of a 3-sphere seen as submanifold .
2 Geometry of string backgrounds and exotic
In this section we take the point of view that exotic smoothness of some small exotic ’s when localized on , correspond to some stringy geometry given by so-called -fields on . The localization is understood as the representation of the exotics by 3-rd integral or real cohomologies of . This correspondence takes place in fact for a classical limit of the geometry of string backgrounds, i.e. curved Riemannian manifold with B-field. One can say that localized exotic smooth on is described by stringy geometry of -fields on this . The correspondence can be extended over string regime of finite volume of WZW model.
2.1 WZW model, D-branes and exotic
We want to focus on changing the smoothness of and considering the changes as localized on . As follows from AsselmeyerKrol2009 (); AsselmeyerKrol2009a () this gives rise to stringy effects, since the changes can be described by computations in some 2D CFT, namely WZW models on at finite level.
First we are going to discuss bosonic WZW model and dynamics of branes in it. We deal here with hence the nonzero metric of string background. In general, non-vanishing curvature , where is a non-constant metric, of the background manifold on which bosonic string theory is formulated, enforces that -field on cannot vanish. This is since the string field equations gives rise to (see e.g. Schomerus2002 ())
where is the NSNS 3-form, is the B-field, and dilaton is fixed to be constant. Also in the case of superstring theory this equation still holds true provided all RR background fields vanish Schomerus2002 ().
D-branes in group manifold (at the semi-classical limit) are determined as wrapping the conjugacy classes of , which are 2-spheres plus 2 points-poles, seen as degenerated 2-spheres. Due to the quantization conditions there are D-branes on the level WZW model Schomerus2000 (); Schomerus2002 (); Alekseev1999 (). To grasp the dynamics of the branes one should deal with the gauge theory on the stack of D-branes on which is quite similar to the flat space case where noncommutative gauge theory emerges Alekseev1999b ().
For branes of type on top of each other, where is the representation of i.e. , the dynamics of the branes is described by the noncommutative action:
Here the curvature form and the noncommutative Chern-Simons action reads . The fields are defined on fuzzy 2-sphere and should be considered as matrix-valued, i.e. where are fuzzy spherical harmonics and are Chan-Paton matrix-valued coefficients. are generators of the rotations on fuzzy 2-spheres and they act only on fuzzy spherical harmonics Schomerus2002 (). The noncommutative action was derived from Connes spectral triples of the noncommutative geometry and was aimed to describe Maxwell theory on fuzzy spheres Watamura2000 ().
One can solve the equations of motion derived from the stationery points of (2) and the solutions describing the dynamics of the branes, i.e. the condensation processes on the brane configuration which results in another configuration . Namely the equation of motion derived from (2) read:
A class of solutions for (3), in the semi-classical limit, can be obtained from the dimensional representations of the algebra . For one has branes of type , i.e. point-like branes in at the identity of the group. Given another solution corresponding to the one can show that this corresponds to the brane wrapping the sphere and is obtained as the condensed state of point-like branes at the identity of Schomerus2002 ():
Turning to the finite stringy regime of the WZW model one can make use of the techniques of the boundary CFT when applied to the analysis of Kondo effect Schomerus2002 (). It follows that there exists a continuous shift at the level of partition function, between and the interfered sum of characters where (in the vanishing value of the coupling constant) and are Verlinde fusion rule coefficients. In the case of point-like branes one can determine the decay product of these by considering open strings ending on the branes. The result on the partition function is
which is continuously shifted to and next to . As the result we have the decay process Schomerus2002 ()
which extends the similar process derived at the semi-classical limit in the effective gauge theory (4), however the representations are bounded now, from the above, by .
Given the above dynamics of branes in the WZW model at stringy regime, one can address the question of brane charges in a direct way. This is based on the decay rule (5) in the supersymmetric WZW model. In this case we have a shift of the level namely which measures the units of the NSNS flux through . One can see the supersymmetric model as strings moving on with units of NSNS flux. From the CFT point of view there exist currents which satisfy level of the Kac-Moody algebra and free fermionic fields in the adjoint representation of . However it is possible to redefine the bosonic currents as
which fulfill the current algebra commutation relation at the level . Here are the structure constants of . The fields commute with such currents, thus we have the splitting of the supersymmetric WZW model at level as WZW model at level times the theory of free fermionic fields.
Thus there are stable branes wrapping the conjugacy classes numbered by . The decaying process (5) says that placing point-like branes (each charged by the unit ) at the pole they can decay to the spherical brane wrapping the conjugacy class. Taking more point-like branes to the stack at gives the more distant branes until reaching the opposite pole where we have single point-like brane with the opposite charge . Having identify units of the charge with we arrive at the conclusion that the group of charges is . More generally the charges of branes on the background with non-vanishing are described by the twisted group (see e.g. MathaiMurray2001 ()). In the case of we get the group of RR charges as above for
Based on AsselmeyerKrol2009 (), the following important observation is in order: certain small exotic ’s generate the group of RR charges of D-branes in the curved background of . This observation is based on the integral classes from which one can construct the exotic as corresponding to the codimension-1 foliation of (determined by the class ). In AsselmeyerKrol2009 () we showed that twisted K-theory of by the class can be seen as the effect of the exotic smoothness on the ambient 4-space, when is understood as the part of the boundary of the Akbulut cork of .
Thus we arrive at the correspondence:
The classification of RR charges of the branes on group manifold background at the level , hence the dynamics of D-branes in in stringy regime, is correlated with exotic smoothness on containing this as the part of the boundary of the Akbulut cork.
We can give yet another interpretation of the 4-exoticness which appears on flat in this context. Exotic smoothness of , , determines the collection of stable D-branes on at the level of the WZW model, where . Thus, the stringy, finite , level of WZW model characterizes exotic 4-smoothness. Recall that in the case of (e.g. constant in a flat space, i.e. in limit) the smooth structure on is the standard one AsselmeyerKrol2009 (). Thus the exotic smoothness on translates the 4-curvature to the non-zero H-field on of finite volume in string units. This is similar to the effect of string field equations relating and as in (1), though it holds now between different spaces ( and ).
2.2 WZW model in the geometry of the stack of NS5-branes
The group manifold is the only manifold which became relevant so far for the description of small exotic . From the other side it is the only one which appears directly as part of a string background (namely one generated by NS5-branes). The reason is given by the connection of 4-exotics and string theory as it can be naturally formulated in the geometry of the stack of NS5-branes. Let us briefly describe this string background Schomerus2000 (); Schomerus2002 (); Bachas2000 ().
We consider a configuration of coincident supersymmetric NS5-branes in type II theory. The full fivebrane background is (in string frame)
where are the longitudinal coordinates along NS5-branes referred to by indices , , etc., being 4 transverse coordinates referred to by indices , , … and , string tension. The fields of this background reads as
where are the positions of the NS5-branes. When the branes coincide at 0, , the near horizon solutions , are
In the near-horizon limit , the background factorizes into a radial component and a and flat 6-dimensional Minkowski spacetime. Strings propagating at this limiting background are described by the exact world-sheet CFT with the target . Here is the real line with the parameter which is a scalar corresponding to the ,,linear dilaton”
The flat Minkowski space is longitudinal to the directions of NS5-branes, is and is a level WZW supersymmetric CFT (SCFT) on as discussed in the previous subsection. This corresponds to the angular coordinates of the transversal . We see that infinite geometrical ,,throat” , emerges. The metric of the background (in the string frame) thus reads
This background is obtained in the near horizon, (), geometry of the stack of NS5-branes in type II string theory and is in fact a SCFT on the throat. The NS5-branes are placed at and string theory is strongly coupled there, . In the opposite limit , or , gives asymptotically flat 10-space and string theory is weakly coupled in that limit. This is essentially the CHS (Callan, Harvey, Strominger CHS1991 ()) exact string theory background where WZW model appears at suitable level .
Given the CHS limiting geometry of NS5-branes we have the 4-dimensional tube . The volume of in string units is finite and correlated with the number of NS5-branes by Bachas2000 (). We take an exotic for . This can be achieved more directly by considering the Akbulut cork with the boundary, , the homology 3-sphere. As was shown in AsselmeyerKrol2009 () contains such that the codimension-1 foliations of it generates the foliations of . The foliations in turn are generated by Casson handles attached to . Thus the attached Akbulut cork and Casson handle(s) determine the small exotic smoothness of GomSti:1999 (); AsselmeyerKrol2009 (). Moreover, the cobordism classes of codimension-1 foliations of are classified by the Godbillon-Vey invariants which are elements of . In our case we deal with integral 3-rd cohomologies . Thus, a way of embedding the Akbulut cork, for some class of exotic ’s, in the ambient is determined by the integral classes . Taking the above from the boundary of the Akbulut cork, as in the string background of NS5-branes we arrive at the following result:
In the geometry of the stack of NS5-branes in type II superstring theories, adding or subtracting a NS5-brane is correlated with the change of smoothing on transversal .
Now the tube of the limiting geometry can be embedded in the ambient standard . Taking this as lying in the boundary of the Akbulut cork for some exotic smooth , the embedding of the tube in this exotic 4-space is determined by the embedding of the Akbulut cork. But this embedding is determined by Casson handles attached to the cork and corresponds to the integral class . Thus the background is geometrically realized as . We propose here a general heuristic rule:
R1. D-branes probing exotic 4-dimensional Euclidean space, , times 6-dim. Minkowski spacetime ,, are described equivalently by the D-branes of type II string theory probing the transversal 4-space, , to NS5-branes in the background of these 5-branes. Here . Since appears in both sides of the correspondence we say that D-branes explore exotic Euclidean .
Rule R1 is based on the assumption that various nonstandard smoothings of can be grasped by the effects of . This follows from the correlation of the classes and 4-exotics as proved in AsselmeyerKrol2009 (). Following this rule we can consider many examples of D-branes in the above background (see e.g. GiveonAntoniadis2000 (); GiveonKutasov2000 (); YunKwon2009 (); Ribault2003 (); ChenSun2005 ()), as referring to 4-exoticness.
Furthermore type II string theory on is given by the SCFT on the infinite ,,throat” of the background, i.e. . Then this theory was proposed to be approachable via holography by using duality. The holographically dual theory appears to be so called 6-dimensional little string theory (LST) GiveonKutasov2000 (); Aharony2002 (). This is a very interesting situation for us since LST was analyzed as having possible experimental signatures at the TeV scales after the compactification on torus GiveonAntoniadis2000 (). By the rule above this refers to 4-exotics as well. We do not deal here with the details and refer the interesting reader to a separate paper devoted to (flux) compactification in string theory and exotic 4-smoothness. But we will present some general remarks here.
LST’s are non-local theories without gravity and can be described in the limit in the theory on NS5-branes. In that limit the bulk degrees of freedom decouple, hence gravity does. This 6-dim. LST without gravity is holographically dual to the type II string theory formulated on the background GiveonAntoniadis2000 (). From the rule R1 it follows that LST is referred to exotic and calculations in LST should lead to invariants of the 4-exotics. The perturbative calculations, however, are hardly performed in LST since the string coupling diverges in the dual string background along the tube, and LST is sensitive on that. One usually regulates the geometry via chopping the tube. But the decomposition of the SCFT on can be performed. Here is a minimal model at the level and is the Cartan subalgebra of with the parameter . The dependence on is crucial at this reformulation since this refers to 4-exotics by theorem 2 and the rule R1. Thus we have the SCFT instead of the tube . The chopping of the strong coupling region is now performed by taking the SCFT instead of which means replacing the background by . This means, on the level of NS5-branes, the separation of these 5-branes along the transverse circle of radius . Now the double-scale limit of LST is the one when taking both and to zero while remains constant.
Following GiveonKutasov2000 () we can take systems of D4, D6-branes between separated NS5-branes. The various expressions like correlation functions can be now calculated perturbatively in the holographically dual 6-dimensional LST theory. Besides, suitable compactifications may refer to the spectra with the TeV scale of the standard model of particles. The dependence on of some of these expressions can be seen as the signature of the existence of exotic structure in the 4-space transversal to the branes.
Exoticness of the 4-space transversal to the worldvolume of NS5-branes, is reflected in specific perturbative spectra of D-branes when calculated in dual 6-dimensional LST. When compactifying this LST on 2 directions longitudinal to the 5-brane one gets spectra which could be sensitive on transversal exoticness of . From the point of view of physics, the calculations refer to the TeV scale Aharony2002 ().
The important observation can be made: Some LST calculations refer not only to holographically dual string theory but also to exotic smoothness on . This is the indication that one can try, at least in some cases, to replace higher dimensional string theory effects by 4-dimensional phenomena.
This is in fact the reformulation of the rule R1. The NS5-branes backgrounds show that string theory computations ,,feel” the 4-exoticness.
3 Quantum D-branes and 4-exotica
In this section we want to show that D-branes of string theory, as in the previous sections, are related with exotic smooth ’s also beyond the semi-classical limit, i.e. in the quantum regime of the theory where one should deal rather with quantum branes. What quantum branes mean in general is still an open and hard problem. One appealing proposition, relevant for this paper, is to consider branes in noncommutative spacetimes rather than on commutative manifolds or orbifolds. This leads to abstract D-branes in general noncommutative separable algebras as counterparts for quantum D-branes. In the next section we will present a definition using wild embeddings.
3.1 D-branes on spaces: K-homology and KK theory
The description of systems of stable Dp-branes of IIA,B string theories via K-theory of topological spaces can be extended toward the branes in noncommutative algebras. A direct string representation of the algebraic and K-theoretic ideas is best seen in K-matrix string theory where, in particular, tachyons are elements of the spectral triples representing the noncommutative geometry of the world-volumes of the configurations of branes AsakawaSugimotoTerasima2002 (). The elements of the formulation of type II strings as K matrix theory is presented in the Appendix A.
First let us consider the case of vanishing -field on . The charges of D-branes are classified by suitable theory groups, i.e. in IIB and in IIA string theories, where is the background manifold. On the other hand, world-volumes of Dp-branes correspond to the cycles of K homology groups, , , which are dual to the theory groups. Let us see how -cycles correspond to the configurations of D-branes.
A - cycle on is a triple where is a compact manifold without boundary, is a complex vector bundle on and is a continuous map. The topological -homology is the set of equivalence classes of the triples respecting the following conditions:
when there exists a triple (bordism of the triples) such that is isomorphic to the disjoint union where is the reversed structure of and is a compact manifold with boundary.
Vector bundle modification . is even dimensional sphere bundle on , projection, is a vector bundle on which gives the generator of on every over each Szabo2002a ().
The topological K-homology as above has an abelian group structure with disjoint union of cycles as sum. The triples with being even dimensional determines . Similarly, corresponds to odd dimensions. Thus decomposes into a direct sum of abelian groups:
Now the interpretation of cycles as D-branes HarveyMoore2000 () is the following: is the world-volume of brane, the Chan-Paton bundle on it and gives the embedding of the brane into spacetime . Moreover, has to wrap manifold FreedWitten1999 () and classifies stable D-branes configurations in IIB, and in IIA, string theories. The equivalences of K-cycles as formulated in the conditions (i)-(iii) correspond to natural relations for D-branes AsakawaSugimotoTerasima2002 (); Szabo2008b ().
The topological K-homology theory above can be obtained analytically (analytic K-homology theory) as a special commutative case of the following construction on general algebras AsakawaSugimotoTerasima2002 ().
A Fredholm module over a algebra is a triple such that
is a separable Hilbert space,
is a homomorphism between algebras and of bounded linear operators on ,
is self-adjoint operator in satisfying
where are compact operators on . Now let us see how a Fredholm module describes certain configuration of IIA K matrix string theory directly related to D branes. To this end we consider the operators of the K-matrix theory (infinite matrices) acting on the Hilbert space as generating the algebra (see the Appendix A and AsakawaSugimotoTerasima2002 ()). In the case of commuting , hence commutative , we have the following correspondence (explaining the index in ):
Every commutative algebra is isomorphic to the algebra of continuous complex functions vanishing at infinity on some locally compact Hausdorff space (Gelfand-Najmark theorem). A point is determined by a character of which is a homomorphism .
serves as a common spectrum for and the choice of a point from as the eigenvalue of fixes the position of the non BPS instanton along .
In this way is covered by the positions of infinite many non BPS instantons and serves as the world-volume of some higher dimensional D brane AsakawaSugimotoTerasima2002 ().
Now let us explain the role of the tachyon . is a self-adjoint unbounded operator acting on the Chan-Paton Hilbert space . is a unital algebra generated by which can be now noncommutative. The corresponding geometry of the world-volume would be noncommutative and given by some spectral triple. The spectral triple is in fact which means that the following conditions are satisfied AsakawaSugimotoTerasima2002 ():
These conditions indeed hold true in our case of K matrix string theory for a tachyon field , Chan-Paton Hilbert space and algebra generated by (see Appendix B). The extension of spacetime manifold toward noncommutative algebra and noncommutative world-volumes of branes, represented by spectral triples, is thus given by AsakawaSugimotoTerasima2002 ():
Fixing the spacetime algebra ;
A homomorphism generates embedding of the D-brane world-volume and its noncommutative algebra as ;
D-branes embedded in a spacetime are represented by the spectral triple ;
Equivalently, D-brane in is given by unbounded Fredholm module .
In particular the classification of stable D-branes in is the classification of Fredholm modules given by analytical K-homology. Given the isomorphisms of the topological and analytical K homology groups, we have the classification of stable D-branes in terms of K-cycles, as we discussed at the beginning of this section. In terms of K matrix string theory we can say that stable configurations of D-instantons determine the stable higher dimensional D-branes which are K-homologically classified as above.
Now let us turn to a more general situation than K-string theory of D-instantons, i.e. backgrounds given by non-BPS Dp-branes or non-BPS Dp--branes in type II string theory. The stable configurations of Dq-branes are then classified by generalized K-theory namely Kasparov KK-theory. As in the above case of D-branes in a algebra corresponding to Fredholm modules, one defines an odd Kasparov module , where is an countable Hilbert module over algebra , as
a -homomorphism from to the algebra of bounded linear operators on , ;
a self-adjoint operator from satisfying:
where is . is in fact a family of Fredholm modules on the algebra . When is we have an ordinary Fredholm module as before. The homotopy equivalence classes of odd Kasparov modules determine elements of . Also one defines an even Kasparov classes as homotopy equivalence classes of the triples . A natural grading appears due to the involution .
Now the classification pattern for branes in spaces emerges. There are non-BPS unstable Dp-branes wrapping the -dimensional world-volume . Then stable Dq-branes configurations embedded in a space transverse to correspond to (are classified by) the classes of . Similarly, given non-BPS unstable Dp--branes system, then stable Dq-branes embedded in transverse to (-dimensional world-volumes) are classified by elements of . The case of even contains the grading as corresponding to the Chan-Paton indices of Dp and -branes.
3.2 D-branes on separable algebras and KK theory
The classification of D-branes in a spacetime manifold given by KK theory as sketched in the previous subsection, can be extended over noncommutative spacetimes and noncommutative D-branes both represented by separable algebras. Let us first recapitulate the ,,classic” case of spaces allowing the extension over algebras Szabo2008c ().
In the case of type II superstring theory, let be a compact part of spacetime manifold, i.e. is a compact manifold again with no background -flux. As we saw, a flat D-brane in is a Baum-Douglas K-cycle . Here is the embedding of the closed submanifold of and is a complex vector bundle with connection (Chan-Paton gauge bundle). As follows from Baum-Douglas construction, determines the stable class in the K-theory group and all K-cycles form an additive category under disjoint union. Now, the set of all K-cycles classes up to a kind of gauge equivalence as in Baum-Douglass construction, gives the K-homology of . This K-homology is also the set of stable homotopy classes of Fredholm modules which are taken over the commutative algebra of continuous functions on . This defines the correspondence (isomorphism) where a K-cycle corresponds to unbounded Fredholm module . Here is the separable Hilbert space of square integrable spinors on taking values in the bundle , i.e. , is the representation of the algebra in such that where is the operator of point-wise multiplication of functions in by the function on , , and . is the Dirac operator twisted by corresponding to the structure on . Given the K-theory class of the Chan-Paton bundle , i.e. , then the dual K-homology class of a D-brane, uniquely determines . In that way D-branes determine K-homology classes on which are dual to K-theory classes from where is the transversal dimension for the brane world-volume . This K-theory class is derived from the image of by the Gysin K-theoretic map . As we discussed already, the odd and even classes of K-homology correspond to the parity of the dimension of . The K-cycle corresponds to a Dp-brane and its gauge equivalence is given by Baum-Douglas construction using the conditions (i)-(iii) in Sec. 3.1. Thus we have Szabo2008b ():
Fact 1: There is a one-to-one correspondence between flat D-branes in , modulo Baum-Douglas equivalence, and stable homotopy classes of Fredholm modules over the algebra .
In the presence of a non-zero -field on , which is a -gerbe with connection represented by the characteristic class in Szabo2008b (); AsselmeyerKrol2009 (), one can define twisted D-brane on as Szabo2008b ():
A twisted D-brane in a B-field is a triple , where is a closed, embedded oriented submanifold with , and is the Chan-Paton bundle on , i.e. , and is the 3-rd integer Stiefel-Whitney class of the normal bundle of , .
The condition in the definition is in fact required by the cancellation of the Freed-Witten anomaly, where is the NS-NS -flux. Since is the obstruction to the structure on , in the case of one has flat D-branes in . Thus equivalence classes of twisted D-branes on are represented by twisted topological K-homology which is dual to the twisted K-theory . As was argued in AsselmeyerKrol2010 (), in case of , one has some exotic ’s which can be twisted by leading to the K-theory . We can represent the gerbes with connection on , by the bundles of algebras over , such that the sections of the bundle define the noncommutative, twisted algebra and the Dixmier-Douady class of , , is AsselmeyerKrol2009a (); AtiyahSegal2004 (); Szabo2002a (). The important relation is the following (Szabo2008b (), Proposition 1.15):
Fact 2: There is a one-to-one correspondence between twisted D-branes in and stable homotopy classes of Fredholm modules over the algebra .
Since the algebra certainly determines its stable homotopy classes of the Fredholm modules on it, then in the case one has the following observation:
A. Let the exotic smooth ’s are determined by the integral third classes . Then, these exotic smooth ’s correspond one-to-one to the set of twisted D-branes in .
In principle, given the complete collection of twisted D-branes in , which take values in , one can determine the corresponding exotic . This is simply the exotic corresponding to the class and makes the twist in the K-homology as dual to the twisted K-theory AsselmeyerKrol2009a (); AsselmeyerKrol2010 (); Szabo2002a (). In this paper we collect further evidences that this is also the case more generally, and the relation D-branes - 4-exotics is closer.
Remembering that as part of the Akbulut cork of the exotic structure, our previous observation can be restated as:
B. The change of the exotic smoothness of , , , , , corresponds to the change of the curved backgrounds hence the sets of stable D-branes.
This motivates the formulation:
C. Some small exotic smoothness on , , can be destabilize (or stabilize) D-branes in , where lies at the boundary of the Akbulut cork of . We say that D-branes in are 4-exotic-sensitive.
Turning to the generalization of spaces to noncommutative algebras, there were developed recently impressive counterparts of many topological, geometrical and analytical results, like Poincaré duality, characteristic classes and the Riemann-Roch theorem. Also the generalized formula for charges of quantum D-branes in a noncommutative separable algebras was worked out Szabo2008a (); Szabo2008b (). Thus the suitable framework for considering the quantum regime of D-branes emerged. In next subsection we will try to find a relation to 4-exotics also in this quantum regime of D-branes.
Following AsakawaSugimotoTerasima2002 (); Szabo2008a (); Szabo2008b (); Szabo2008c () one can take as an initial substitute for the category of quantum D-branes, the category of separable algebras and morphisms being elements of KK theory groups. This means that for a pair of separable algebras the morphisms is lifted to the element of the group . Thus we can consider a generalized D-branes in a separable algebra as corresponding to the lift where represents a quantum D-brane.
More precisely following Szabo2008a (), let us consider a subcategory of the category of separable algebras and their morphisms, which consists of strongly K-oriented morphisms. This means that there exists a contravariant functor such that is mapped to , here is the category of separable algebras with KK classes as morphisms. Strongly K-oriented morphisms and the functor are subjects to the following conditions:
Identity morphism is strongly K-oriented (SKKO) for every separable algebra and . Also, the 0-morphism is SKKO and .
If is SKKO then is either, and . is the opposite algebra to , i.e. one which has the same underlying vector space but reversed product.
Any morphism is SKKO, provided and are strong Poincaré dual (PD) algebras. Then is determined as:
here is the class of in . is the fundamental class in , its dual class in which exist by strong PD Szabo2008a ().
K-orientability was introduced, in its original form, by A. Connes in order to define the analogue of structure for noncommutative algebras (see also Connes1984 () and next subsections). Presented here formulation of K-orientability and strong PD algebras are crucial ingredients of noncommutative versions of Riemann-Roch theorem, Poincaré-like dualities, Gysin K-theory map and allows to formulate a very general formula for noncommutative D-brane charges Szabo2008b (); Szabo2008a (); Szabo2008c (). Let us notice that if both and are PD algebras then any morphism is K-oriented and the K-orientation for is given in (11).
In the particular case of the proper smooth embedding of codimension , where , are smooth compact manifolds, let the normal bundle over , of with respect to , be . When also is then the condition on when -flux is absent in type II string theory formulated on , is the Freed-Witten anomaly cancellation condition Szabo2008a (). In this case any D-brane in , given by the triple , determines the KK-theory element . The construction of K-orientation , between smooth compact manifolds, can be extended to smooth proper maps which are not necessary embeddings. Thus the general condition for K-orientability gives the correct analogue for stable D-branes in algebras.
A generalized stable quantum D-brane on a separable algebra , represented by a separable algebra , is given by the strongly K-oriented homomorphism of algebras, . The K-orientation means that there is the lift where fulfills the functoriality condition as in (11).
This kind of an approach to quantum D-branes is in fact a conjectural framework which exceeds both the dynamical Seiberg-Witten limit of superstring theory (where noncommutative brane world-volumes emerges) and geometrical understanding of branes, and places itself rather in a deep quantum regime of the theory Szabo2008c ().
3.3 Exotic and stable D-branes configurations on foliated manifolds
Now we want to approach the problem of description of stable states of D-branes in a more general geometry than spaces, namely the geometry of foliated manifolds. The case of our interest is a codimension-1 foliation of . This is a noncommutative geometry. In general, to every foliation one can associate its noncommutative algebra , on the other hand a foliation determines its holonomy groupoid and the topological classifying space . Both cases, topological K-homology of and algebraic K-theory, are in fact dual. Analogously to our previous discussion of branes as K-cycles on , let us start with K-homology of and define D-branes as K-cycles in :
A - cycle on a foliated geometry is a triple where is a compact manifold without boundary, is a complex vector bundle on and is a smooth K-oriented map. Due to the K-orientability in the presence of canonical -bundle on , the condition of structure on is lifted to the structure on Connes1984 ().
The topological -homology of the foliation is the set of equivalence classes of the above triples, where the equivalence respects the following conditions:
when there exists a triple (bordism of the triples) such that is isomorphic to the disjoint union where is the reversed structure of and is a compact manifold with boundary.
Vector bundle modification similarly as in the case of manifolds.
As in the case of spaces (manifolds) and the corresponding K-homology groups representing stable D-branes of type II superstring theory (see Sec. 3.1), also here, in the case of the geometry of foliated manifolds we generalize stable D-branes as being represented by the above triples.
The class of generalized stable D-branes on the algebra (of the codimension 1 foliation of ) which correspond to the K-homology classes , determines an invariant of exotic smooth . Such an exotic contains this foliated as a generalized (noncommutative) smooth subset AsselmeyerKrol2009a ().
Let us note that this approach allows for considering a kind of string theory and branes also beyond the integral levels of WZW model given by . The relation with exotic smooth ’s extends over this as well.
3.4 Net of exotic ’s and quantum D-branes in
The extension of string theory and D-branes over general noncommutative separable algebras where also D-branes are represented by noncommutative separable algebras, can be considered as an approach to quantum D-branes. A category of D-branes in a quantum regime, is the category of separable algebras and morphisms which are elements of KK theory groups. For a pair of separable algebras the morphisms belong to . Abstract quantum D-branes in a separable algebra correspond to where is the algebra representing a quantum D-brane and is a strongly K-oriented map. For such branes a general formula for RR charges in noncommutative setting was worked out Szabo2008a (); Szabo2008b ().
D-branes considered in the previous subsection, correspond to the lifted KK-theory classes, i.e. where D-brane corresponds to the triple and