**6. Discussion and conclusions**

22 Will-be-set-by-IN-TECH

• The process of changing the exotic smoothness on **R**<sup>4</sup> is capable of encoding a) the change in the configurations of specific D- and NS branes (Sec. 5.1), b) the change of the 4D part of the string background from flat to curved one in closed string theory (see Sec. 3). • All exotic **R**4's appearing in this setup are *small exotic* **R**4*'s*, i.e. those which embed

Given the fact that every small exotic **R**<sup>4</sup> from the radial family (see Sec. 2) determines the codimension-1 foliation of *S*3, we have natural *C*�-algebra assigned to this 4-exotic. Namely this is the noncommutative convolution *C*�-algebra of the foliation. Let us, following Brodzki et al. (2008b), represent every D-brane by suitable separable *C*�-algebra replacing, in the same time, spacetime by the correspponding separable *C*�-algebra as well. The usual semiclassical embedding of D-branes in spacetime is now reformulated in the language of morphisms between *C*�-algebras. In fact, taking into account the isue of stability of D-branes, we define

2. A � homomorphism *<sup>φ</sup>* : A → **<sup>B</sup>**(H) (a homomorphisms of the algebras preserving their � structure), generates the embedding of the D-brane world-volume *M* and its

3. D-branes embedded in a spacetime A are represented by the spectral triple (H, A*M*, *T*); 4. Equivalently, a D-brane in *A* is given by an unbounded Fredholm module (H, *φ*, *T*).

Thus, the classification of stable D-branes in A is given by the classification of Fredholm modules (H, *φ*, *T*) where **B**(H) are bounded operators on the separable Hilbert space H and *T* the operators corresponding to tachyons. In general, to every foliation (*V*, *F*) one can associate its noncommutative *C*� convolution algebra *C*�(*V*, *F*). The interesting connection with exotic

**Theorem 4.** *The class of generalized stable D-branes on the C*� *algebra C*�(*S*3, *F*1) *(of the codimension*

**Theorem 5.** *Let e be an exotic* **R**<sup>4</sup> *corresponding to the codimension-1 foliation of S*<sup>3</sup> *which gives rise to the C*�*algebra* <sup>A</sup>*e. The exotic smooth* **<sup>R</sup>**<sup>4</sup> *embedded in e determines a generalized quantum D-brane*

It is interesting to note that the tame subspace interpretation of D-branes can be recovered for the special class of the topological quantum D-branes. However, the embedding is replaced now by the wild embedding into spacetime, which historically appeared in the description of

*foliation of S*3*. Each wild embedding i* : *<sup>K</sup>*<sup>3</sup> <sup>→</sup> *<sup>S</sup><sup>p</sup> for p* <sup>&</sup>gt; <sup>6</sup> *of a 3-dimensional polyhedron determines a class in Hn*(*Sn*, **<sup>R</sup>**) *which represents a wild embedding i* : *<sup>K</sup><sup>p</sup>* <sup>→</sup> *<sup>S</sup><sup>n</sup> of a p -polyhedron into Sn.*

Now, a class of *topological quantum Dp-branes* are these branes which are determined by the wild embeddings *<sup>i</sup>* : *<sup>K</sup><sup>p</sup>* <sup>→</sup> *<sup>S</sup><sup>n</sup>* as above and in the classical and flat limit correspond to tame embeddings. In fact, *B*-field on *S*<sup>3</sup> can be translated into wild embeddings of higher

*<sup>H</sup> be some exotic* **<sup>R</sup>**<sup>4</sup> *determined by element in H*3(*S*3, **<sup>R</sup>**)*, i.e. by a codimension-1*

smoothly in the standard smooth **R**<sup>4</sup> as open subsets.

the setup:

and

*in* A*e.*

**Theorem 6.** *Let* **R**<sup>4</sup>

1. Fix the (spacetime) *<sup>C</sup>*� algebra <sup>A</sup>;

4-smoothness then emerges:

noncommutative algebra A*<sup>M</sup>* as A*<sup>M</sup>* := *φ*(A);

*1 foliation of S*3*) determines an invariant of exotic smooth* **R**4*,*

the horned Alexander's spheres, known from topology.

dimensional objects and generates quantum character of these branes.

Superstring theory (ST) appears in fact as very rich mathematics. The mathematics which is designed especially for the reconciling classical gravity, as in GR, with QFT. The richness of mathematics involved is, however, the limitation of the theory. Namely, to yield 4D physics from such huge structure is very non-unique and thus problematic. We followed the idea, proposed at the recent International Congress of Mathematician ICM 2010 [Asselmeyer-Maluga & Król (2010)], that the mathematics of ST refers to and advance understanding of the mathematics of exotic smooth **R**4. Conversely, exotic **R**4's provide important information about the mathematics of superstrings. Exotic **R**4's are non-flat geometries, hence contain gravity from the point of view of physics. ST is the theory of QG and gravity of exotic geometries is quantized by methods of ST. The 4-geometries also refer to effective correlated states of condensed matter as in Kondo effect. Thus, the approach presented in this Chapter indicates new fundamental link between *gravity, geometry and matter* at the quantum limit and exclusively in dimension 4. The exotic smoothness of **R**4, when underlies the 4-Minkowski spacetime, is a natural way to quantum gravity (given by superstring techniques) from the standard model of particles. On the other hand, exotic **R**4's serve as factor reducing the ambiguity of 10D superstring theory in yielding 4D physical results. The work on these issues should be further pursued.
