**5.1 NS and D branes in type II**

Let us consider again the bosonic, i.e. nonsupersymmetric, *SU*(2)*<sup>k</sup>* WZW model and follow Asselmeyer-Maluga & Król (2011c) closely. The semi-classical limit of it corresponds to taking *k* → ∞ as in Sec. 3.3. In that limit D-branes in group manifold *SU*(2) are determined by wrapping the conjugacy classes of *SU*(2), i.e. are described by 2-spheres *S*2's and two poles (degenarate branes) each localized at a point. Owing to the quantization conditions, there are *k* + 1 D-branes on the level *k SU*(2) WZW model Alekseev & Schomerus (1999b); Fredenhagen & Schomerus (2001); Schomerus (2002). To grasp the dynamics of the branes one should deal with the gauge theory on the stack of *N* D-branes on *S*3, quite similar to the flat space case where noncommutative gauge theory emerges Alekseev & Schomerus (1999a). Let *J* be the representation of *SU*(2)*<sup>k</sup>* i.e. *J* = 0, <sup>1</sup> <sup>2</sup> , 1, ... , *<sup>k</sup>* <sup>2</sup> . The non-commutative action for the dynamics of *N* branes of type *J* (on top of each other), in the string regim (*k* is finite), is then given by:

$$\mathcal{S}\_{N,l} = \mathcal{S}\_{YM} + \mathcal{S}\_{\text{CS}} = \frac{\pi^2}{k^2 (2l+1)N} \left( \frac{1}{4} \text{tr} (F\_{\mu\nu} F^{\mu\nu}) - \frac{i}{2} \text{tr} (f^{\mu\nu\rho} \text{CS}\_{\mu\nu\rho}) \right). \tag{29}$$

Here the curvature form *<sup>F</sup>μν*(*A*) = *iLμA<sup>ν</sup>* <sup>−</sup> *iLνA<sup>μ</sup>* <sup>+</sup> *<sup>i</sup>*[*Aμ*, *<sup>A</sup>ν*] + *<sup>f</sup>μνρA<sup>ρ</sup>* and the noncommutative Chern-Simons action reads CS*μνρ*(*A*) = *LμAνA<sup>ρ</sup>* + <sup>1</sup> <sup>3</sup> *Aμ*[*Aν*, *Aρ*]. The fields *Aμ*, *μ* = 1, 2, 3 are defined on a fuzzy 2-sphere *S*<sup>2</sup> *<sup>J</sup>* and should be considered as *N* × *N* matrix-valued, i.e. *A<sup>μ</sup>* = ∑*j*,*<sup>a</sup>* a *μ j*,*aY<sup>j</sup> <sup>a</sup>* where *<sup>Y</sup><sup>j</sup> <sup>a</sup>* are fuzzy spherical harmonics and a*<sup>μ</sup> <sup>j</sup>*,*<sup>a</sup>* are Chan-Paton matrix-valued coefficients. *L<sup>μ</sup>* are generators of the rotations on fuzzy 2-spheres and they act only on fuzzy spherical harmonics Schomerus (2002). The noncommutative action *SYM* was derived from Connes spectral triples from the noncommutative geometry, and they will be crucial in grasping quantum nature of D-branes in the next subsection. Originally the action (29) was designed to describe Maxwell theory on fuzzy spheres Carow-Watamura & Watamura (2000). The equations of motion derived from (29) read:

$$L\_{\mu}F^{\mu\nu} + [A\_{\mu\nu}F^{\mu\nu}] = 0 \,. \tag{30}$$

The solutions of (30) describe the dynamics of the branes, i.e. the condensation processes on the brane configuration (*N*, *J*) which results in another configuration (*N*� , *J*� ). A special class of solutions, in the semi-classical *k* → ∞ limit, can be obtained from the *N*(2*J* +1) dimensional representations of the algebra su(2). For *J* = 0 one has *N* branes of type *J* = 0, i.e. *N* point-like branes in *S*<sup>3</sup> at the identity of the group. Given another solution corresponding to *JN* = *<sup>N</sup>*−<sup>1</sup> <sup>2</sup> , one shows that this solution is the condensed state of *N* point-like branes at the identity of *SU*(2) Schomerus (2002):

$$(N, l) = (N, 0) \to (1, \frac{N - 1}{2}) = (N', l') \tag{31}$$

Turning to the finite *k* string regime of the *SU*(2) WZW model one makes use of the techniques of the boundary CFT, the same as was applied to the analysis of Kondo effect in Sec. 4. It follows that there exists a continuous shift between the partition functions governed by the Verlinde fusion rules coefficients *N <sup>l</sup> JN j* : *Nχj*(*q*) and the sum of characters ∑*<sup>j</sup> N <sup>l</sup> JN j χl*(*q*) where 20 Will-be-set-by-IN-TECH

governed by K-theory of the background, or in the presence of *H*-field, by, twisted by *H*, K-theory classes. This is briefly summerized in the next subsection where D and NS branes

Let us consider again the bosonic, i.e. nonsupersymmetric, *SU*(2)*<sup>k</sup>* WZW model and follow Asselmeyer-Maluga & Król (2011c) closely. The semi-classical limit of it corresponds to taking *k* → ∞ as in Sec. 3.3. In that limit D-branes in group manifold *SU*(2) are determined by wrapping the conjugacy classes of *SU*(2), i.e. are described by 2-spheres *S*2's and two poles (degenarate branes) each localized at a point. Owing to the quantization conditions, there are *k* + 1 D-branes on the level *k SU*(2) WZW model Alekseev & Schomerus (1999b); Fredenhagen & Schomerus (2001); Schomerus (2002). To grasp the dynamics of the branes one should deal with the gauge theory on the stack of *N* D-branes on *S*3, quite similar to the flat space case where noncommutative gauge theory emerges Alekseev & Schomerus (1999a). Let *J* be the

<sup>2</sup> . The non-commutative action for the dynamics

tr(*f μνρ*CS*μνρ*)

*<sup>J</sup>* and should be considered as *N* × *N*

, *J*�

) (31)

*JN j*

*χl*(*q*) where

). A special class

<sup>2</sup> ,

*<sup>a</sup>* are fuzzy spherical harmonics and a*<sup>μ</sup>*

<sup>3</sup> *Aμ*[*Aν*, *Aρ*]. The fields

. (29)

*<sup>j</sup>*,*<sup>a</sup>* are

2

*LμFμν* + [*Aμ*, *Fμν*] = 0 . (30)

, *J* �

will be understood also classically as subsets in specific CFT backgrounds.

<sup>2</sup> , 1, ... , *<sup>k</sup>*

*k*2(2*J* + 1)*N*

*<sup>a</sup>* where *<sup>Y</sup><sup>j</sup>*

noncommutative Chern-Simons action reads CS*μνρ*(*A*) = *LμAνA<sup>ρ</sup>* + <sup>1</sup>

*μ j*,*aY<sup>j</sup>*

& Watamura (2000). The equations of motion derived from (29) read:

the brane configuration (*N*, *J*) which results in another configuration (*N*�

*JN j*

of *N* branes of type *J* (on top of each other), in the string regim (*k* is finite), is then given by:

1 4

Here the curvature form *<sup>F</sup>μν*(*A*) = *iLμA<sup>ν</sup>* <sup>−</sup> *iLνA<sup>μ</sup>* <sup>+</sup> *<sup>i</sup>*[*Aμ*, *<sup>A</sup>ν*] + *<sup>f</sup>μνρA<sup>ρ</sup>* and the

Chan-Paton matrix-valued coefficients. *L<sup>μ</sup>* are generators of the rotations on fuzzy 2-spheres and they act only on fuzzy spherical harmonics Schomerus (2002). The noncommutative action *SYM* was derived from Connes spectral triples from the noncommutative geometry, and they will be crucial in grasping quantum nature of D-branes in the next subsection. Originally the action (29) was designed to describe Maxwell theory on fuzzy spheres Carow-Watamura

The solutions of (30) describe the dynamics of the branes, i.e. the condensation processes on

of solutions, in the semi-classical *k* → ∞ limit, can be obtained from the *N*(2*J* +1) dimensional representations of the algebra su(2). For *J* = 0 one has *N* branes of type *J* = 0, i.e. *N* point-like branes in *S*<sup>3</sup> at the identity of the group. Given another solution corresponding to *JN* = *<sup>N</sup>*−<sup>1</sup>

one shows that this solution is the condensed state of *N* point-like branes at the identity of

Turning to the finite *k* string regime of the *SU*(2) WZW model one makes use of the techniques of the boundary CFT, the same as was applied to the analysis of Kondo effect in Sec. 4. It follows that there exists a continuous shift between the partition functions governed by the

<sup>2</sup> )=(*N*�

: *Nχj*(*q*) and the sum of characters ∑*<sup>j</sup> N <sup>l</sup>*

(*N*, *<sup>J</sup>*)=(*N*, 0) <sup>→</sup> (1, *<sup>N</sup>* <sup>−</sup> <sup>1</sup>

tr(*FμνFμν*) <sup>−</sup> *<sup>i</sup>*

**5.1 NS and D branes in type II**

representation of *SU*(2)*<sup>k</sup>* i.e. *J* = 0, <sup>1</sup>

matrix-valued, i.e. *A<sup>μ</sup>* = ∑*j*,*<sup>a</sup>* a

*SU*(2) Schomerus (2002):

Verlinde fusion rules coefficients *N <sup>l</sup>*

*SN*,*<sup>J</sup>* <sup>=</sup> *SYM* <sup>+</sup> *SCS* <sup>=</sup> *<sup>π</sup>*<sup>2</sup>

*Aμ*, *μ* = 1, 2, 3 are defined on a fuzzy 2-sphere *S*<sup>2</sup>

*N* = 2*JN* + 1. In the case of *N* 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

$$Z\_{(N,0)}(q) = N^2 \chi\_0(q)$$

which is continuously shifted to *<sup>N</sup>χJN* (*q*) and next to <sup>∑</sup>*<sup>j</sup> <sup>N</sup> <sup>j</sup> JN JN χj*(*q*). As the result, we have the decay process:

$$\begin{aligned} Z\_{(N,0)}(\eta) &\to Z\_{(1,l\_N)} \\ (N,0) &\to (1,l\_N) \end{aligned} \tag{32}$$

which extends the similar process derived at the semi-classical *k* → ∞ limit (31), and the representations 2*JN* are bounded now, from the above, by *k*.

Thus, there are *k* + 1 stable branes wrapping the conjugacy classes numbered by *J* = 0, <sup>1</sup> <sup>2</sup> , ..., *<sup>k</sup>* 2 . The decaying process (32) says that placing *N* point-like branes (each charged by the unit 1) at the pole *e*, they can decay to the spherical brane *JN* wrapping the conjugacy class. Taking more point-like branes to the stack at *e*, gives the more distant *S*<sup>2</sup> branes until reaching the opposite pole −*e*, where we have single point-like brane with the opposite charge −1. Having identified *k* + 1 units of the charge with −1, we obtain the correct shape of the group of charges, as: **Z***k*<sup>+</sup>2. More generally, the charges of branes on the background *X* with non-vanishing *<sup>H</sup>* <sup>∈</sup> *<sup>H</sup>*3(*X*, **<sup>Z</sup>**) are described by the twisted *<sup>K</sup>* group, *<sup>K</sup> <sup>H</sup>*(*X*). In the case of *SU*(2), we get the group of RR charges as (for *K* = *k* + 2):

$$K\_H^\star(\mathbb{S}^3) = \mathbb{Z}\_K \tag{33}$$

Now, based on the earlier discussion from Secs. 2,3, let us place the *<sup>S</sup>*<sup>3</sup> � *SU*(2) above, at the boundary of the Akbulut cork for some exotic smooth **R**<sup>4</sup> *<sup>k</sup>* . Then, we have: *Certain small exotic* **<sup>R</sup>**4*'s generate the group of RR charges of D-branes in the curved background of S*<sup>3</sup> <sup>⊂</sup> **<sup>R</sup>**4*.*

We have yet another important correspondence:

**Theorem 2** (Asselmeyer-Maluga & Król (2011c))**.** *The classification of RR charges of the branes on the background given by the group manifold SU*(2) *at the level k (hence the dynamics of D-branes in S*<sup>3</sup> *in stringy regime) is correlated with the exotic smoothness on* **R**4*, containing this S*<sup>3</sup> = *SU*(2) *as the part of the boundary of the Akbulut cork.*

Turning to the linear dilaton geometry, as emerging, in the near horizon geometry, from the stack of *<sup>N</sup>* NS5-branes in supersymmetric model, i.e. **<sup>R</sup>**5,1 <sup>×</sup> **<sup>R</sup>***<sup>φ</sup>* <sup>×</sup> *SU*(2)*k*, we obtain next important relation:

**Theorem 3** (Asselmeyer-Maluga & Król (2011c))**.** *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 the smoothness structure on the transversal* **R**4*.*

### **5.2 Quantum and topological D-branes**

The recognition of the role of exotic **R**<sup>4</sup> in string theory, in the previous and in Sec. 3, relied on the following items:

• Standard smooth **R**<sup>4</sup> appears as a part of an exact string background;

**6. Discussion and conclusions**

**7. References**

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results. The work on these issues should be further pursued.

*Topology* 2: 40–82. arXiv:0806.3010.

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Singapore.

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

Quantum Gravity Insights from Smooth 4-Geometries on Trivial <sup>4</sup> 75

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Kondo effect: Asymptotic three-dimensional space- and time-dependent multi-point


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 the setup:


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 4-smoothness then emerges:

**Theorem 4.** *The class of generalized stable D-branes on the C*� *algebra C*�(*S*3, *F*1) *(of the codimension 1 foliation of S*3*) determines an invariant of exotic smooth* **R**4*,*
