Toroidal Compactification, $T$-duality and D-branes

D-branes are, despite of the dual fundamental degrees of freedom in string theory, extremely interesting and useful tools to study nonperturbative properties of string and field theories (for some reviews see [10,11]). Non-perturbative properties of supersymmetric gauge theories can be better understanding as the world-volume effective theory of some configurations of intersecting D-branes (for a review see [14]). D-branes also are very important to connect gauge theories with gravity. This is the starting point of the AdS/CFT correspondence or Maldacena's conjecture. We don't review this interesting subject in this paper, however the reader can consult the excellent review [15]. Roughly speaking D-branes are static solutions of string equations which satisfy Dirichlet boundary conditions. That means that open strings can end on them. To explain these objects we follow the traditional way, by using T-duality on open strings we will see that Neumann conditions are turned out into the Dirichlet ones. To motivate the subject we first consider T-duality in closed bosonic string theory. truecm T-duality in Closed Strings The general solution of Eq. (4) in the conformal gauge can be written as $X^I (\sigma , \tau ) = X^I_R(\sigma^-)+X^I_L (\sigma^+)$, where $\sigma^{\pm}=\sigma \pm \tau$. Now, take one coordinate, say $X^{25}$ and compactify it on a circle of radius $R$. Thus we have that $X^{25}$ can be identified with $X^{25} + 2\pi R m$ where $m$ is called the winding number. The general solution for $X^{25}$ with the above compactification condition is

$$X^{25}_R(\sigma^-)=X^{25}_{0R} +\sqrt{\frac{\alpha '}{2}}P^... ...frac{1}{l}\alpha^{25}_{R,l}exp \bigg(-il(\tau - \sigma )\bigg)$$

$$X^{25}_L(\sigma^+)=X^{25}_{0L}+\sqrt{\frac{\alpha '}{2}}P^{... ...frac{1}{l}\alpha^{25}_{L,l}exp\bigg(-il(\tau +\sigma )\bigg),$$ (38)

where

$$P^{25}_{R, L}=\frac{1}{\sqrt{2}} \bigg(\frac{\sqrt{\alpha '}}{R}n \mp \frac{R}{\sqrt{\alpha '}}m\bigg).$$ (39)

Here $n$ and $m$ are integers representing the discrete momentum and the winding number, respectively. The latter has not analogous in field theory. While the canonical momentum is given by $P^{25}=\frac{1}{\sqrt{2\alpha '}}(P^{25}_L+P^{25}_R)$. Now, by the mass shell condition, the mass of the perturbative states is given by $M^2 = M_L^2 + M_R^2$, with

$$M^2_{L,R}=-\frac{1}{2}P^I P_I =\frac{1}{2}(P^{25}_{L,R})^2+\frac{2}{\alpha '}(N_{L,R}-1).$$ (40)

We can see that for all states with $m \neq 0$, as $R \to \infty$ the mass become infinity, while $m = 0$ implies that the states take all values for $n$ and form a continuum. At the case when $R \to 0$, for states with $n \neq 0$, mass become infinity. However in the limit $R \to 0$ for $n =0$ states with all $m$ values produce a continuum in the spectrum. So, in this limit the compactified dimension disappears. For this reason, we can say that the mass spectrum of the theories at radius $R$ and $\frac{\alpha '}{R}$ are identical when we interchange $n \Leftrightarrow m$. This symmetry is known as T-duality. The importance of T-duality lies in the fact that the T-duality transformation is a parity transformation acting on the left and right moving degrees of freedom. It leaves invariant the left movers and changes the sign of the right movers (see Eq. (39))

$$P^{25}_L \to P^{25}_L, \ \ \ \ \ \ \ \ P^{25}_R \to -P^{25}_R.$$ (41)

The action of T-duality transformation must leave invariant the whole theory (at all order in perturbation theory). Thus, all kind of interacting states in certain theory should correspond to those states belonging to the dual theory. In this context, also the vertex operators are invariant. For instance the tachyonic vertex operators are

$${\cal V}_L= exp(iP_L^{25}X^{25}_L), \ \ \ \ \ \ \ \ \ {\cal V}_R= exp(iP^{25}_RX^{25}_R).$$ (42)

Under T-duality, $X^{25}_L \to X^{25}_L$ and $X^{25}_R \to -X^{25}_R$; and from the general solution Eq. (38), ${\alpha}^{25}_{R,i} \to -{\alpha}^{25}_{R,i}$, $X^{25}_{0R} \to -X^{25}_{0R}$. Thus, T-duality interchanges $n \Leftrightarrow m$ (Kaluza-Klein modes $\Leftrightarrow$ winding number) and $R \Leftrightarrow {\alpha ' \over R}$ in closed string theory. truecm T-duality in Open Strings Now, consider open strings with Neumann boundary conditions. Take again the $25^{th}$ coordinate and compactify it on a circle of radius $R$, but keeping Neumann conditions. As in the case of closed string, center of mass momentum takes only discrete values $P^{25}=\frac{n}{R}$. While there is not analogous for the winding number. So, when $R \to 0$ all states with nonzero momentum go to infinity mass, and do not form a continuum. This behavior is similar as in field theory, but now there is something new. The general solutions are

$$X^{25}_R =\frac{X_0^{25}}{2} -\frac{a}{2} + \alpha 'P^{25}(... ...frac{1}{l} \alpha^{25}_l exp \bigg(-i2l(\tau -\sigma ) \bigg),$$

$$X^{25}_L =\frac{X_0^{25}}{2} +\frac{a}{2} +\alpha 'P^{25}(\... ...frac{1}{l}\alpha^{25}_l exp \bigg(-i2l(\tau +\sigma ) \bigg)$$ (43)

where $a$ is a constant. Thus, $X^{25}(\sigma ,\tau ) = X^{25}_R(\sigma^-) +X^{25}_L(\sigma^+)=X_0^{25} +\frac{2\alpha 'n}{R}\tau + oscillator \ terms.$ Taking the limit $R \to 0$, only the $n =0$ mode survives. Because of this, the string seems to move in 25 spacetime dimensions. In other words, the strings vibrate in 24 transversal directions. T-duality provides a new T-dual coordinate defined by $\widetilde{X}^{25}(\sigma ,\tau )=X^{25}_L(\sigma^+)-X^{25}_R(\sigma^-)$. Now, taking $\widetilde{R}=\frac{\alpha '}{R}$ we have $\widetilde{X}^{25}(\sigma ,\tau )=a +2 \widetilde{R} \sigma n + oscillator \ terms.$ Using the boundary conditions at $\sigma =0,\pi$ one has $\widetilde{X}^{25}(\sigma ,\tau ) \mid_{\sigma =0} = a$ and $\widetilde{X}^{25}(\sigma ,\tau )\mid_{\sigma =\pi}=a +2\pi \widetilde{R}n.$ Thus, we started with an open bosonic string theory with Neumann boundary conditions, and T-duality and a compactification on a circle in the $25^{th}$ dimension, give us Dirichlet boundary conditions in such a coordinate. We can visualize this saying that an open string has its endpoints fixed at a hyperplane with 24 dimensions. Strings with $n =0$ lie on a 24 dimensional plane space (D24-brane). Strings with $n=1$ has one endpoint at a hyperplane and the other at a different hyperplane which is separated from the first one by a factor equal to $2\pi \widetilde{R}$, and so on. But if we compactify $p$ of the $X^i$ directions over a $T^{p}$ torus ($i=1,...,p$). Thus, after T-dualizing them we have strings with endpoints fixed at hyperplane with $25-p$ dimensions, the D$(25-p)$-brane. Summarizing: the system of open strings moving freely in spacetime with $p$ compactified dimensions on $T^p$ is equivalent, under T-duality, to strings whose enpoints are fixed at a D$(25-p)$-brane i.e. obeying Neumann boundary conditions in the $X^i$ longitudinal directions ($i=1,\dots ,p$) and Dirichlet ones in the transverse coordinates $X^m$ ($m=p+1,...,25$). The effect of T-dualizing a coordinate is to change the nature of the boundary conditions, from Neumann to Dirichlet and vice versa. If one dualize a longitudinal coordinate this coordinate will satisfies the Dirichlet condition and a D$p$-brane becomes a D$(p-1)$-brane. But if the dualized coordinate is one of the transverse coordinates the D$p$-brane becomes a D$(p+1)$-brane. T-duality also acts conversely. We can think to begin with a closed string theory, and compactify it on to a circle in the $25^{th}$ coordinate, and then by imposing Dirichlet conditions, obtain a D-brane. This is precisely what occurs in Type II theory, a theory of closed strings. truecm Spectrum and Wilson Lines Now, we will see how does emerges a gauge field on the D$p$-brane world-volume. Again, for the mass shell condition for open bosonic strings and because T-duality $M^2=(\frac{n}{\alpha '}\widetilde{R})^2+\frac{1}{\alpha '}(N-1)$. The massless state ($N=1$, $n =0$) implies that the gauge boson ${\alpha}^{I}_{-1}\mid 0 \rangle$ ($U(1)$ gauge boson) lies on the D24-brane world-volume. On the other hand, ${\alpha}^{25}_{-1}\mid 0 \rangle$ has a vev (vacuum expectation value) which describes the position $\widetilde{X}^{25}$ of the D-brane after T-dualizing. Thus, we can say in general, there is a gauge theory $U(1)$ over the world volume of the D$p$-brane. Consider now an orientable open string. The endpoints of the string carry charge under a non-Abelian gauge group. For Type II theories the gauge group is $U(N)$. One endpoint transforms under the fundamental representation ${\bf N}$ of $U(N)$ and the other one, under its complex conjugate representation (the anti-fundamental one) ${\bf N}^*$. The ground state wave function is specified by the center of mass momentum and by the charges of the endpoints. Thus implies the existence of a basis $\mid k; ij \rangle$ called Chan-Paton basis. States $\mid k; ij \rangle$ of the Chan-Paton basis are those states which carry charge 1 under the $i^{th}$ $U(1)$ generator and $-1$ under the $j^{th}$ $U(1)$ generator. So, we can decompose the wave function for ground state as $\mid k;a \rangle=\sum_{i,j=1}^N \mid k;ij \rangle \lambda^a_{ij}$ where $\lambda^a_{ij}$ are called Chan-Paton factors. From this, we see that it is possible to add degrees of freedom to endpoints of the string, that are precisely the Chan-Paton factors. This is consistent with the theory, because the Chan-Paton factors have a Hamiltonian which do not posses dynamical structure. So, if one endpoint to the string is prepared in a certain state, it always will remains the same. It can be deduced from this, that $\lambda^a \longrightarrow U\lambda^aU^{-1}$ with $U\in$ $U(N)$. Thus, the worldsheet theory is symmetric under $U(N)$, and this global symmetry is a gauge symmetry in spacetime. So the vector state at massless level ${\alpha}^{I}_{-1}\mid k,a \rangle$ is a $U(N)$ gauge boson. When we have a gauge configuration with non trivial line integral around a compactified dimension (i.e a circle), we said there is a Wilson line. In case of open strings with gauge group $U(N)$, a toroidal compactification of the $25^{th}$ dimension on a circle of radius $R$. If we choice a background field $A^{25}$ given by $A^{25} =\frac{1}{2\pi R} diag(\theta_1,...,\theta_N)$ a Wilson line appears. Moreover, if $\theta_i=0$, $i=1,...,l$ and $\theta_j \neq 0$, $j=l+1,...,N$ then gauge group is broken: $U(N) \longrightarrow U(l) \times U(1)^{N-l}$. It is possible to deduce that $\theta_i$ plays the role of a Higgs field. Because string states with Chan-Paton quantum numbers $\mid ij \rangle$ have charges $1$ under $i^{th}$ $U(1)$ factor (and $-1$ under $j^{th}$ $U(1)$ factor) and neutral with all others; canonical momentum is given now by $P^{25}_{(ij)} \Longrightarrow \frac{n}{R} + \frac{(\theta_j -\theta_i )}{2\pi R}.$ Returning to the mass shell condition it results,

$$M^2_{ij}=\bigg(\frac{n}{R} +{\theta_j - \theta_i \over 2\pi R}\bigg)^2 +\frac{1}{\alpha '}(N-1).$$ (44)

Massless states ($N=1, n=0$) are those in where $i=j$ (diagonal terms) or for which $\theta_j=\theta_i$ $(i \neq j)$. Now, T-dualizing we have $\widetilde{X}^{25}_{ij}(\sigma ,\tau )=a +(2n +\frac{\theta_j- \theta_i}{\pi})\widetilde{R}\sigma + oscillator \ terms.$ Taking $a=\theta_i \widetilde{R}$, $\widetilde{X}^{25}_{ij}(0,\tau )= \theta_i \widetilde{R}$ and $\widetilde{X}^{25}_{ij}(\pi ,\tau )= 2\pi n \widetilde{R} + \theta_j \widetilde{R}.$ This give us a set of $N$ D-branes whose positions are given by $\theta_j\widetilde{R}$, and each set is separated from its initial positions ($\theta_j=0$) by a factor equal to $2\pi \widetilde{R}$. Open strings with both endpoints on the same D-brane gives massless gauge bosons. The set of $N$ D-branes give us $U(1)^N$ gauge group. An open string with one endpoint in one D-brane, and the other endpoint in a different D-brane, yields a massive state with $M \sim (\theta_j -\theta_i)\widetilde{R}$. Mass decreases when two different D-branes approximate to each other, and are null when become the same. When all D-branes take up the same position, the gauge group is enhanced from $U(1)^N$ to $U(N).$ On the D-brane world-volume there are also scalar fields in the adjoint representation of the gauge group $U(N)$. The scalars parametrize the transverse positions of the D-brane in the target space $X$. truecm D-Brane Action With the massless spectrum on the D-brane world-volume it is possible to construct a low energy effective action. Open strings massless fields are interacting with the closed strings massless spectrum from the NS-NS sector. Let $\xi^a$ (with $a=0, \dots , p$) be the world-volume coordinates on $W$. The effective action is the gauge invariant action well known as the Dirac-Born-Infeld (DBI)-action

$$S_D = - T_p \int_W d^{p+1} \xi e^{-\Phi} \sqrt{ det\big( G_{ab} + B_{ab} + 2 \pi \alpha ' F_{ab} \big)},$$ (45)

where $T_p$ is the tension of the D-brane, $G_{ab}$ is the world-volume induced metric, $B_{ab}$ is the induced antisymmetric field, $F_{ab}$ is the Abelian field strength on $W$ and $\Phi$ is the dilaton field. For $N$ D-branes the massless fields turns out to be $N \times N$ matrices and the action turns out to be non-Abelian DBI-action (for a nice review about the Born-Infeld action in string theory see [16])

$$S_D = - T_p \int_W d^{p+1} \xi e^{- \Phi} Tr\bigg(\sqrt{ de... ...2 \pi \alpha ' F_{ab}\big)} + O\big( [X^m,X^n]^2 \big) \bigg)$$ (46)

where $m,n = p+1, \dots , 9$. The scalar fields $X^m$ representing the transverse positions become $N \times N$ matrices and so, the spacetime become a noncommutative spacetime. We will come back later to this interesting point. truecm Ramond-Ramond Charges D-branes are coupled to Ramond-Ramond (RR) fields $G_p$. The complete effective action on the D-brane world-volume $W$ which taking into account this coupling is

$$S_D = - T_p \int_W d^{p+1} \xi \bigg\{e^{-\Phi} \sqrt{ det\big( G_{ab} + B_{ab} + 2 \pi \alpha ' F_{ab} \big)}$$

$$+ i \mu_p \int_W \sum_p C_{(p+1)} Tr\bigg( e^{2 \pi \alpha ' (F+B)}\bigg) \bigg\}$$ (47)

where $\mu_p$ is the RR charge. RR charges can be computed by considering the anomalous behavior of the action at intersections of D-branes. Thus RR charge is given by

$$Q_{RR} = ch(j!E) \sqrt{ \widehat{A}(TX)},$$ (48)

where $j : W \hookrightarrow X$. Here $E$ is the Chan-Paton bundle over $X$, $\widehat{A}(TX)$ is the genus of the spacetime manifold $X$. This gives an ample evidence that the RR charges take values not in a cohomology theory, but in fact, in a K-Theory. This result was developed by Witten in Ref. [17] in the context of non-BPS brane configurations worked out by Sen [18]. Finally, RR charges and RR fields do admit a classification in terms of topological K-theory. The inclusion of a $B$-field turns out the effective theory non-commutative and a suitable generalization of the topological K-theory is needed. The right generalization seems to be the K-Homology and the K-theory of $C^*$ algebras [19]. This subject is right now under intensive investigation. truecm