10-dimensional String Theory and the Einstein Frame

We will look at how to obtain the Einstein frame \(S_{E}\) in \(10\)-dimensional AdS spacetime. Let's look at applying the effective low-energy string action for type II (A or B) strings for the string frame \(S_{s}\):

\(S_{s}= -s \frac{1}{16 \pi G_{D}} {\displaystyle \int } d^{D} x \sqrt{|g|} \) \( \left( e^{-2 \phi}\left(R+4 g^{\mu \nu} \partial_\mu \phi \partial_\nu \phi\right) \right. \) \( -\frac{1}{2} \sum_n \frac{1}{n !} F_n^2 +\ldots\big)\)

where the Newton constant is in \(D\)-dimensions. The \(\ldots\) represent fermionic terms and NS-NS 3-form field strength term, \(\phi\) is the dialation, and \(n\)-form field strengths \(F_{n}\) that are apart of the sector \(RR\). For the Newton constant in \(D\)-dimensions \(16 \pi G_D=2 \kappa_D^2\). When the Minkowski Signature \(s=-1(+1)\) flips to the "mostly minus" signature, an additional \((-)^{n}\) is added in front of \(F^{2}_{n}\). For IIA strings (IIB strings) only odd frames exist for \(n\) and for IIB string \(n=5\) in Minkowski space the field strength tensor is self-dual. Where a self-dual tensor satisfies \(*F=F\Rightarrow F=**F\).

Now, we will show that by adopting the low energy string action action above, we are able to derive the equations of motion while imposing self-duality and ensuring that the normalization of \(F^{2}_{5}\) is unchanged. It is convient to represent the actions for fields in the Einstein frame by first rescaling the strings, which can be accomplished using a specific type of Wely rescaling.

To validate it is sufficient to adopt the effective low energy action for type II strings in order to derive the equations of motions while imposing self-duality for \(D\)-spacetime dimensions, we consider the following implication:

\( g_{\mu \nu} \rightarrow e^{2\ \sigma \phi}\Rightarrow \) \( \sqrt{|g|} e^{-2 \phi} R \rightarrow\) \( \sqrt{|g|} e^{-\phi(\sigma(D-2)+2)} \left\{ R+2 \sigma(D-1)\right. \) \( \frac{1}{\sqrt{|g|}} \partial_\mu\big(\sqrt{|g|} \partial^{\mu} \phi\big) -\sigma^{2} (D-1) \) \( \left. (D-2)(\partial \phi)^{2} \right\} \)

Using this implication, we can see it is sufficient to adopt the string action to derive the equations of motions and imposing self-duality. Thus, we can now continue in good conscience.

We will let \(\sigma=-\frac{2}{D-2}\). By choosing this value for \(\sigma\), we can remove a total derivative when applied in \(10\)-dimensions, which gives:

\(g_{\mu \nu}(\text { Einstein })=e^{-\frac{1}{2} \phi} g_{\mu \nu}(\text { string })\)

From the equation above, we can see that the result for a string action is equivalent to result of the Einstein-Hilbert action in AdS spacetime. Continuing in \(10\)-dimensional spacetime, we obtain our Einstein frame as:

\( S_{E}= -s \frac{1}{16 \pi G_{10}} {\displaystyle \int } d^{10} x \sqrt{|g|} \) \( \left( R-\frac{1}{2} g^{\mu \nu} \partial_\mu \phi \partial_\nu \phi\right. \) \( \left.-\frac{1}{2} \sum_n \frac{1}{n !} e^{a_{n}\phi} F_n^2+\ldots\right) \)

where \(a_n=-\frac{1}{2}(n-5)\). From this equation, we can recognize the Minkowksi metric \(g_{\mu \nu}\) and the Minkowski Euclidean signature \(s=-1(+1)\) paired with the Newton constant in \(D\)-dimensions \(16 \pi G_D\). We also see the familiarity of this equation to the Einstein-Hilbert action, where, if you recall, the Einstein-Hilbert action is:

\(S=-s \frac{1}{16 \pi G_D} {\displaystyle \int } d^D x \sqrt{|g|}(R+\Lambda)\)

All-in-all, in this section we validated that adopting the low-energy string action for the Einstein-Hilbert action is sufficent for calcuating Einsteins equations of motions and imposing self-duality, specifically, in \(10\)-dimensional AdS spacetime. Therefore, yay! We have now derived an Einstein frame \(S_{E}\) in \(10\)-dimensional AdS spacetime implementing string theory!

M-theory as 11-dimensional Super Gravity and the Einstein Frame

We will look at low-energy M-theory in the form of \(11\)-dimensional super gravity, where the bosonic part of the action in the Einstein frame is:

\(S_{\text {bosonic }}(11 \text {-dim SUGRA })=\) \(-s \frac{1}{2 \kappa_{11}^2} \left({\displaystyle \int } d^{11} x \sqrt{|g|} \right.\) \(\left\{R-\frac{1}{48} K^2\right\}-\frac{1}{6}\) \(\left.{\displaystyle \int } C \wedge K \wedge K \right)\)

such that there is no dilation \(\phi\) like we saw with \(10\)-dimensional spacetime. The bosonic field is the metric with a \(3\)-form guage potential \(C\) with a \(4\)-frame field strength tensor. We can denote this bosonic field as:

\(K=dC\)

Meaning that when moving from \(10\)-dimensional spacetime to \(11\)-dimensional spacetime, we switch from the fermionic field to the bosonic field.

Note, that we can still provide classical based solutions of the above theories. An example of a classic based solution, is by considering static solutions to flat translationally invariant \(p\)-branes, isotropic in transverse directions. For these static solutions and to cover all cases for classical based solutions in \(11\)-dimensional supergravity, we can use the following generic action:

\(S= -s \frac{1}{2 \kappa_D^2} {\displaystyle \int } d^D x \sqrt{g} \) \( \left\{R-\frac{1}{2} g^{\mu \nu} \partial_\mu \phi \partial_\nu \phi \right. \) \( \left. -\frac{1}{2} \sum_n \frac{1}{n !} e^{a_n \phi} F_n^2+. .\right\} \)

where \(a=0\) and \(\phi \equiv 0\).

To continue.

Boundary on (n+1)-dimensional Anti-de Sitter Space

AdS space has a projective boundary to allow for the embedding space \((y^{0},y^{\mu})\) in \(AdS_{n+1}\) for some very large \(y\). These new variables are defined as \(y^a=R \tilde{y}^a\), \(u=R \tilde{u}\), \(v=R \tilde{v}\), where \(R \rightarrow \infty\). Next, given the implication \(y^2=b^2 \Rightarrow \tilde{u} \tilde{v}-\vec{\tilde{y}}^2=\)\(b^2 / R^2 \rightarrow 0\), we can infer that the boundary in \(AdS_{d+1}\) space is a manifold, defined as \(\tilde{u} \tilde{v}-\vec{\tilde{y}}^2=0\). Since \(t\in\mathbb{R}\) is sufficent for \(R\), we then continue by considering the boundary to be the projective equivalence classes:

\(\begin{aligned}u v-\vec{y}^2 & =0 \\ (u, v, \vec{y}) & \sim t(u, v, \vec{y})\end{aligned}\)

Meaning, that, as defined above, since our initial state of the manifold boundary is asympotically equivalent to that manifold state on any \(t\) the boundary is \(n\)-dimensional. Next, we apply equivalence scaling on the boundary, such that the boundary can be represented with the Minkowski signature as \(\left(y^0\right)^2+\left(y^{n+1}\right)^2=1\)\(=\vec{y}^2\), where topologically the boundary is \(S^{1}\times S^{n+1}\).

Sometimes when scaling, the points with \(v\neq 0\) scale to \(v=1\), so we define \(u=\vec{y}^2\) and can then use \(\vec{y}\) as input coordinates for the boundary. In the case, however, that when scaling the points with \(u\neq 0\), we can instead scale \(u=1\) and use \(\vec{\tilde{y}}\) as input coordinatesfor the boundary. This gives us \(v=\vec{\tilde{y}}^2\), from which we can draw the following connection between the two sets:

\(\vec{\tilde{y}}=\frac{\vec{y}}{y^2}\)

Note, that the above conditions allow only one of the two set may be used, dependent on if \(v=0\) or \(u=0\). Such that when \(v=0\) then \(\vec{\tilde{y}}=\vec{0}\), where as when \(u=0\) then \(\vec{y}=\vec{0}\). In these equations regarding the manifold boundary of \(AdS_{n+1}\) space, the one point \(v=0\) is the point at infinity for the coordinates \(\vec{y}\) and similiarily for the point \(u=0\). Thus, the boundary for the space is automatically compactified with the application of the given walk through.

All-in-all, the important aspect to take away from this section is that he isometry group \(SO(2,n)\), or \(SO(2,n+1)\) for the Eucledian Signature, acts on the boundary as the conformal group acting on Minkowski Euclidean space!

Eucledian Representation of Boundary on (n+1)-dimensional Anti-de Sitter Space

Given \(\mathbb{R}^{d+1}\) with coordinates \(y_{0},...,y_{d}\), let \(B_{d+1}\) represent the open unit ball \(\sum_{i=0}^d y_i^2 < 1\). \(AdS_{d+1}\) can be represented as \(B_{d+1}\) with the metric:

\({\displaystyle d s^2=\frac{4 \sum_{i=0}^d d y_i^2}{\left(1-|y|^2\right)^2} }\)

The compactification of \(B_{d+1}\) yields the closed unit ball \(\bar{B}_{d+1}\), defined as \(\sum_{i=0}^d y_i^2\leq1\), with boundary \(\mathbf{S}^d\) given as \(\sum_{i=0}^d y_i^2=1\). Where \(\mathbf{S}^d\) is the Eucledian conformal compitification of Minkowski space, meaning the boundy of \(AdS_{d+1}\) is Minkowski space.

To continue.

Massless Field Equations

Consider a scalar field \(\phi\) where the massless field equation is a naive Laplace equation \(D_i D^i \phi=0\). For any function \(\phi(\Omega)\) on the boundary \(\mathbf{S}^d\) in \(AdS_{d+1}\) space, there exists an extension of \(\phi\) to \(\bar{B}_{d+1}\) that perserves the boundary values and obeys the field equation.

\(\begin{aligned}0 & =-\int_{B_{d+1}} d^{d+1} y \sqrt{g} \phi D_i D^i \phi \\ & =\int_{B_{d+1}} d^5 y \sqrt{g}|d \phi|^2\end{aligned}\)

Using the square-integrable solution above, we can validate the given property of \(AdS_{d+1}\) is preserved and that \(d\phi=0\) and therefore \(\phi=0\). The reason this validates the property of a unique extension, is that we can see if we integrate the naive Laplace equation by parts respective to the square-intergrable solution, there does not exist a nonzero solution. This validation is important, because if a nonzero solution did exist, it would imply that any given solution works. Which tells us there is not an unique extenion and, therefore, in the space we can not properly calculate field equations. Since we see that for \(AdS_{d+1}\) space, there does exist an unique extension from \(\phi\) to \(\bar{B}_{d+1}\) for any function \(\phi(\Omega)\) on the boundary \(\mathbf{S}^d\), we can continue with calculating the massless field equation in good conscious.

Next, let's implement the Laplace equation with respect to \(B_{d+1}\) which can be written as:

\(\left(-\frac{1}{(\sinh y)^d} \frac{d}{d y}(\sinh y)^d \frac{d}{d y}+\frac{L^2}{\sinh ^2 y}\right)\)\(\phi=0\)

where the angular momentum squared \(L^{2}\) represents the angular of the Laplacian. If we then set the value for the scalar field \(\phi=\sum_\alpha \phi_\alpha(y)\), where \(f_\alpha\) are spherical harmonics, the equation for any \(\phi_{\alpha}\) on any large \(y\) is denoted as:

\(\frac{d}{d y} e^{d y} \frac{d}{d y} \phi_\alpha=0\)

We can see that two solutions for the Laplace equations are \(\phi_\alpha \sim 1\) and \(\phi_\alpha \sim e^{-d y}\). Therefore, yay! We captured two solutions. These two solutions tell us that for every partial wave, we get a unique solution for the Lapalace equation with a given infinite value for the constant.

To continue

AdS/CFT Correspondence

Anti-de Sitter/conformal theory correspondence AdS/CFT in theoretical physics is a conjecture that describes the relationship between two kinds of physical theories. AdS used in quantum gravtiy and is formulated in terms of string theory of M-theory, while, CFT are quantum field theories that include theories such as Yang-Mills theories describing elementatry particles.

Lorentzian Space

Lorentzian \(n\)-space is the inner product space of \(\mathbb{R}^{n}\) vector space with \(n\)-dimensional Lorentzian inner product. Where the vector space is a set that is closed under finite vector addition and scalar multiplication and inner product is defined as a vector space with an inner product on it.

Minkowski Space

The Minkowski Space is a particular type of Lorentzian space, specifically \(4\)-dimensional Lorentzian space, with a Minkowski metric or Minkowski tensor. Where the Minkowski metric is a type of metric tensor denoted as \(d \tau^2\) with the form \(-\left(d x^0\right)^2+\left(d x^1\right)^2+\left(d x^2\right)^2+\left(d x^3\right)^2\). Minkowski space forms the basis of the study of spacetime within special relativity. Another relative feature of this space is that it unifies \(\mathbb{R}^{3}\) plus time (the "fourth dimension") in Einstein's theory of special relativity.

Minkowski Metric

The Minkowski metric is defined as:

\(g_{\mu \nu} \approx \eta_{\mu \nu}=\left[\begin{array}{cccc}-1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1\end{array}\right]\)

Such that it is a metric of generally curved spacetime. One application of the Minkowski metric is modeling the cosmological constant term in Einstein's field equation with stress-energy from a vaccum or not.

Metric Signature

In theoretical physics, the metric signature counts the number of time-like or space-like characters are in the spacetime. For example, in the case Minkowski metric signature is \((1,3,0)^{+}\) or \((+,-,-,-)\) if the time direction is defined in the time direction. If the eigenvalue is defined in three spatial directions \(x,y,z\), then the metric signature is \((1,3,0)^{-}\) or \((-,+,+,+)\).