**5. From smooth geometry of string backgrounds to quantum D-branes**

One could wonder what is, if any, suitable sense assigned to geometry of spacetime in various string constructions or backgrounds. As we know the geometry of GR, hence, classical gravity, is the one of (pseudo)-Riemannian differentiable manifolds. String theory has GR (10D Einstein equations) as its classical gravitational limit; however, string theory is the theory of QG and the spacetime geometry should be modified. What is the fate of this (pseudo) Riemannian geometry when gravity is quantized? To answer this question we should find correct classical limit for some quantum string constructions. The proper way is to consider the string backgrounds. These are semi-classical solutions in string theory or supergravity, around which one develops a purturbative theory. GR is not the only ingredient of classical geometry in string theory. There are other fields which are equally fundamental. In type II we have metric *Gμν*, antisymmetric *H*-field, i.e. three-form *Hμνρ*, and dilaton Φ. In heterotic strings we have additionally gauge field *F<sup>a</sup> μν* and the calculations of Sec. 3 made use of these. The presence of *B*-field such that *H* is represented by the non-zero cohomology class (see below), is a highly non-trivial fact and indicates that the correct, semi-classical, geometry for string theory is one based on *abelian gerbes* as supplementing Riemannian geometry Król (2010a;b); Segal (2001). Small exotic **R**4's show strong connections with abelian gerbes on *S*<sup>3</sup> Asselmeyer-Maluga & Król (2009a) which has many important consequences. Some of them are discussed in what follows.

Another crucial feature is the role assigned to D- and NS-branes. Closed string theory, as we made use of it in Sec. 3, is not complete in the sense that there are possible boundary conditions, Dirichlet (D) or Neveu-Schwarz (NS), for open strings, already appearing in closed string theories. These boundary conditions determine geometric subspaces on which open strings can end. In that sense open string theory complements the closed one and predicts the existence of D- or NS-branes. This tame picture of branes as subspaces has only very limited validity. In the quantum regime, or even in the non-zero string coupling *gs*, the picture of D-branes as above fails Aspinwall (2004). Nevertheless, interesting proposals were presented recently. They are based on the ideas from non-commutative geometry and aim toward replacing D-branes and spacetime by corresponding (sub) *C*-algebras Brodzki et al. (2008a;b); Szabo (2008). Surprisingly, such an *C*-algebraic setting again shows deep connections with exotic **R**4's.

The appearance of the codimension-one foliations of *S*<sup>3</sup> in the structure of small exotic **R**4, is the key for the whole spectrum of the connections of exotics, beginning with differential geometry and topology, up to non-commutative geometry. This opens very atractive possibilities for exploring both, 1) the classical limit of string geometry, as above and 2) quantum D-branes regim in string theory.

Let us comment on 1) above. The presence of non-zero *B*-field in a string background is crucial from the point of view of resulting geometry: in *σ*-model the *B*-field modifies metric as in (3). Moreover, supposing dilaton is constant and *F<sup>a</sup> μν* vanishes, the second equation of (8) (the *β*-function), enforces the background be non-flat, unless *H* = *dB* is zero. Given *S*<sup>3</sup> part of the linear dilaton background as in Sec. 3, we have non-trivial *H*-field on it. The topological classification of *H*-fields is given by 3-rd de Rham cohomology classes on background manifold *M*, *H*3(*M*, **R**). In order to avoid anomalies we restrict to the integral case *H*3(*S*3, **Z**) for *M* = *S*3. These classes however are equally generated by exotic **R**<sup>4</sup> *<sup>k</sup>* , *k* ∈ **Z** (see Sec. 2). On the other hand, the classification of D-branes in string backgrounds is

*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

Quantum Gravity Insights from Smooth 4-Geometries on Trivial <sup>4</sup> 73

*<sup>Z</sup>*(*N*,0)(*q*) = *<sup>N</sup>*2*χ*0(*q*)

*Z*(*N*,0)(*q*) → *Z*(1,*JN* )

which extends the similar process derived at the semi-classical *k* → ∞ limit (31), and the

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

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

**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*

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

**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*

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

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

Thus, there are *k* + 1 stable branes wrapping the conjugacy classes numbered by *J* = 0, <sup>1</sup>

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>*

*K*

**<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*.*

*JN JN χj*(*q*). As the result, we have

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

*<sup>H</sup>*(*X*). In the case of

(*N*, 0) <sup>→</sup> (1, *JN*) (32)

*<sup>H</sup>*(*S*3) = **<sup>Z</sup>***<sup>K</sup>* (33)

*<sup>k</sup>* . Then, we have: *Certain small exotic*

which is continuously shifted to *<sup>N</sup>χJN* (*q*) and next to <sup>∑</sup>*<sup>j</sup> <sup>N</sup> <sup>j</sup>*

representations 2*JN* are bounded now, from the above, by *k*.

*SU*(2), we get the group of RR charges as (for *K* = *k* + 2):

boundary of the Akbulut cork for some exotic smooth **R**<sup>4</sup>

We have yet another important correspondence:

*the part of the boundary of the Akbulut cork.*

*smoothness structure on the transversal* **R**4*.*

**5.2 Quantum and topological D-branes**

important relation:

on the following items:

the decay process:

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