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abstract: 'We study the time evolution of freely rolling moduli in the context of M-theory on a $G_2$ manifold. This free evolution approximates the correct dynamics of the system at sufficiently large values of the moduli when effects from non-perturbative potentials and flux are negligible. Moduli fall into two classes, namely bulk moduli and blow-up moduli. We obtain a number of non-trivial solutions for the time-evolution of these moduli. As a generic feature, we find the blow-up moduli always expand asymptotically at early and late time.'
author:
- |
    [André Lukas[^1] and Stephen Morris[^2]]{}\
    [*Department of Physics and Astronomy, University of Sussex*]{}\
    [Falmer, Brighton BN1 9QH, UK]{}\
title: '\'
---

Introduction
============

Seven-dimensional manifolds with holonomy $G_2$ provide the general framework for relating M-theory to four-dimensional models with $N=1$ supersymmetry. Moreover, it has been shown that for certain singular limits of such manifolds four-dimensional models with phenomenologically interesting properties can be obtained [@Papadopoulos:1995da]–[@Acharya:2002kv].

Given these promising features it seems worthwhile to have a closer look at the cosmology of such $G_2$ compactifications of M-theory. More specifically, in this paper, we will be interested in the time-evolution of $G_2$ moduli. This problem has not previously been analysed since the four-dimensional effective theories, specifically the kinetic terms, for these fields were unknown. However, in a recent paper [@Lukas:2003dn], the four-dimensional moduli Kähler potential for M-theory on a $G_2$ manifold has been computed explicitly and our analysis will be based on these results.

The $G_2$ manifold considered in Ref. [@Lukas:2003dn] is based on the compact examples due to Joyce [@joyce1]–[@joyceb], constructed by blowing-up a seven-dimensional $G_2$ orbifold. The moduli for such a $G_2$ manifold split into two groups, namely the bulk moduli (related to the radii of the underlying seven-torus) and the blow-up moduli (related to the volume of the blow-ups). The Kähler potential of Ref. [@Lukas:2003dn] was obtained in the usual large-radius supergravity approximation and for blow-up volumes small compared to the total volume of the space. In the limit of a vanishing blow-up modulus the space develops a singularity (which corresponds to a singularity of the underlying orbifold) leading to new fields in the low-energy theory. It is singular limits such as this which are essential for a phenomenologically interesting low-energy theory.

In this paper, we will focus on the simple case of free moduli evolution based on the Kähler potential of Ref. [@Lukas:2003dn]. That is we will neglect the effect of possible non-perturbative potentials and flux. Hence our results constitute a good approximation at large moduli values where effects of a non-trivial moduli (super)-potential can be neglected. Particular consideration is given to the evolution of the blow-up moduli; particularly as to whether they may contract, thereby approaching one of the phenomenologically interesting singular limits.

Let us summarise our main results. Starting from the given Kähler potential and consistently truncating off the axions in the chiral moduli multiplets, we show that, after a suitable reparametrisation of fields, the sigma-model metric becomes independent of the blow-up moduli. As a result, these fields can be integrated out and the problem reduces to one of studying the evolution of the bulk moduli in an effective potential. Based on this approach, we show that a number of explicit analytic solutions can be obtained using methods from Toda theory. For all these solutions, we find the relative volume of the blow-up moduli increases asymptotically at early and late times. As a consequence, the small blow-up approximation underlying the Kähler potential breaks down after a finite proper time. This feature can also be intuitively understood from the properties of the effective bulk-moduli potential and can, therefore, be viewed as generic. Hence, once moduli are large it appears to be difficult to evolve towards a singular state of the internal space. This suggests that a successful cosmological evolution leading to phenomenologically interesting singular $G_2$ spaces should stabilise moduli at small values from the outset.

Kähler potential for $G_2$ moduli
=================================

In Ref. [@Lukas:2003dn] the moduli Kähler potential has been calculated for a specific manifold with $G_2$ holonomy constructed from a $G_2$ orbifold by blowing up the orbifold singularities. The chiral moduli multiplets for this manifold split into two categories, namely the bulk moduli $T^A$, where $A,B,C,\dots =
1,\dots ,7$, which encode the radii of the seven-torus $T^7$ underlying the orbifold, and the blow-up moduli $U^i$, which measure the volume of the blow-ups. The general structure of this Kähler potential is $$K = -\sum_{A=1}^7\ln\left(T^A+\bar{T}^A\right) + \sum_iF_i(T^A,\bar{T}^A)
     \left( U^i+\bar{U}^i\right)^2+c\; , \label{K}$$ where the functions $F_i$ are given by $$F_i = \frac{8}{(T^{A_i}+\bar{T}^{A_i})(T^{B_i}+\bar{T}^{B_i})}$$ and $c$ is a constant. Here, $A_i$ and $B_i$ specify the two particular bulk moduli by which the blow-up modulus $i$ is divided in the above Kähler potential. The values for these indices reflect the underlying structure of the orbifold and they can be conveniently encoded in constant seven-dimensional vectors ${\rm p}_i$ by writing $$F_i = 2e^{-{\bf p}\cdot\bbe}$$ where the new fields $\bbe = (\b_A)$ are defined by [^3] $${\rm Re}(T^A)=e^{\b_A}\; . \label{beta}$$ We expect this form of the Kähler potential to apply to a wide class of $G_2$ manifolds constructed by blowing up orbifolds, with the specifics of each example encoded in the vectors ${\bf p}_i$.

In the following we will, for concreteness, focus on the particular example of Ref. [@Lukas:2003dn], which is based on the orbifold $T^7/\ZM_2^3$. The blow-ups for this particular manifold can be labeled by a triple $(i)=(\t ,n,a)$ of indices where $\t =\a ,\b ,\g$ indicates the type, that is, under which of the three $\ZM_2$ orbifolding symmetries the associated fixed point remains invariant, $n=1,2,3,4$ labels the fixed points of equal type and $a=1,2,3$ is an index associated to each fixed point that describes the orientation of the blow-up relative to the bulk. Hence, there are $12$ fixed points, labeled by $(\t
,n)$, each with three associated blow-up moduli and, consequently, $36$ blow-up moduli in total. For simplicity of notation, we will use the single index $i$ to label the blow-ups whenever possible and only split into the triple $(\t ,n,a)$ when required. It turns out that the vectors ${\bf p}_i$ for this case only depend on the type $\t$ and the orientation $a$. The resulting nine vectors are given in Table 1.

  ${\bf p}_{(\t ,a)}$   $a=1$               $a=2$               $a=3$
  --------------------- ------------------- ------------------- -------------------
  $\t = \a$             $(1,0,0,0,0,1,0)$   $(0,1,0,0,1,0,0)$   $(0,0,1,1,0,0,0)$
  $\t = \b$             $(1,0,0,0,0,0,1)$   $(0,0,1,0,1,0,0)$   $(0,1,0,1,0,0,0)$
  $\t = \g$             $(1,0,0,1,0,0,0)$   $(0,0,1,0,0,1,0)$   $(0,1,0,0,0,0,1)$

  : Values of vectors ${\bf p}_{(\t ,a)}$ which define the moduli Kähler potential for the $G_2$ manifold based on $T^7/\ZM_2^3$.

It will be useful for the subsequent discussion to have an interpretation for the moduli and the various parts of the Kähler potential  in terms of the geometry of the underlying $G_2$ manifold. To this end, one notes that the moduli Kähler potential for M-theory on a $G_2$ manifold is generally related to the volume $\cV$ of this manifold by [@Beasley:2002db] $$K = -3\ln\left(\frac{\cV}{2\p^2}\right)\; . \label{KV}$$ The first, $T^A$–dependent, part of the Kähler potential  then corresponds to the volume $\cV_0$ of the orbifold, while the second, $U^i$–dependent, part measures the reduction of this volume due to the blow-ups. For the orbifold volume $\cV_0$ one easily finds from Eqs.  and that $$\cV_0 = \frac{1}{8}\exp\left(\frac{1}{3}\sum_A\beta_A\right)\; ,
 \label{V0}$$ where we have used the value $c=6\ln (8\p )+\ln (2)$ found in Ref. [@Lukas:2003dn]. From a similar calculation one finds that the fraction $\e_{(\t ,n)}$ by which the blow-up $(\t ,n)$ reduces the orbifold volume $V_0$ is given by $$\e_{(\t ,n)} = \frac{8}{3}\sum_{a=1}^3e^{-{\bf p}_{(\t ,a)}\cdot\bbe}\,
                u_{(\t ,n,a)}^2\; ,$$ where we have introduced the real parts $${\rm Re}(U^{i})=u_i$$ of the blow-up moduli.

The Kähler potential  relies on two approximations so we should discuss its range of validity in moduli space. First, we require all moduli to be larger than one, that is $$T^A\gg 1\; ,\qquad  U^i\gg 1\; ,\label{app1}$$ for the supergravity approximation underlying the calculation of $K$ to be valid. Second, the Kähler potential has been calculated to leading (quadratic) order in $U^i/T^A$ and terms of order four in these ratios have been neglected. Consequently, application of the Kähler potential  should be confined to the region of moduli space where the ratios $U^i/T^A$ are smaller than one, or more precisely, where $$\e_{(\t ,n)} \ll 1\; . \label{app2}$$ From the above interpretation of $\e_{(\t ,n)}$ this means that the volume taken away by the blow-ups should be small compared to the volume of the orbifold. In subsequent calculations, we will consistently apply this approximation in that we neglect higher order terms in $\e_{(\t ,n)}$.

A simple form of the Lagrangian
===============================

The Kähler potential described in the previous section depends on $43$ chiral superfields. We, therefore, have $43$ real moduli fields from the real parts of theses superfields, which are associated to the geometry of the $G_2$ manifold, plus $43$ axions from the imaginary parts. It is clear that the axions can be consistently set to constants and for simplicity this is what we will do in the following. We are, hence, left with the real scalar fields $\b_A$ and $u_i$.

However, their kinetic terms, as computed from Eq. , are still fairly complicated. A substantial simplification can be achieved by introducing a new set of fields $\f_A$ and $z_i$ defined by the Ansatz $$\begin{aligned}
 \b_A &=& \f_A+\sum_i c_{iA}e^{{\bf q}_i\cdot\bph}z_i^2 \\
 u_i &=& e^{{\bf s}_i\cdot\bph}\; .\end{aligned}$$ Here, ${\bf c}_i=(c_{iA})$, ${\bf q}_i=(q_{Ai})$ and ${\bf
s}_i=(s_{Ai})$ are constants to be determined shortly. The kinetic terms for $\b_A$ and $u_i$ can now be rewritten using this field reparametrisation. For the particular choice of constants $$\begin{aligned}
 {\bf c}_i &=& -4\m_i{\bf p}_i\\
 {\bf s}_i &=& \frac{1}{2}(\m_i+1){\bf p}_i\\
 {\bf q}_i &=& \m_i{\bf p}_i\; ,\end{aligned}$$ where $\m_i\in\{ -1,+1\}$ are arbitrary signs, we find that the Lagrangian simplifies to $$L = \sqrt{-g}\left( R+\frac{1}{2}\sum_A{\partial}_\m\f_A{\partial}^\m\f_A+8
     \sum_ie^{-{\bf P}_i\cdot\bph}{\partial}_\m z_i{\partial}^\m z_i\right) \label{L}$$ where $${\bf P}_i = -\m_i{\bf p}_i\; .$$ The obvious advantage of this form is that the rescaled blow-up moduli $z_i$ (unlike their counterparts $u_i$) only appear through their derivatives. This implies that their equations of motion can be immediately integrated once, leading to a set of first integrals. We will use this property in the following section when we discuss the cosmological evolution based on the Lagrangian . Since we have neglected higher order terms in $\e_{(\t ,n)}$ in its derivation, it is equivalent to the one directly obtained from the Kähler potential  only provided fields are in the region of moduli space defined by Eq. . Also note that this equivalence holds for an arbitrary choice of the signs $\m_i$.

Cosmological evolution equations
================================

To study the cosmological evolution based on the Lagrangian  we start with a metric $$ds^2=-e^{2\n (\t )}d\t^2+e^{2\a (\t )}d{\bf x}^2$$ of Friedmann-Robertson-Walker type, with the spatial sections taken to be flat for simplicity, and time-dependent moduli fields $$\f_A=\f_A (\t )\; ,\qquad z_i=z_i(\t )\; .$$ Here, $\a (\t )$ is the scale factor of the universe and $\n (\t
)$ is the lapse function, which we will determine later. From Eq.  the equations of motion for the blow-up moduli $z_i$ are given by $$\frac{d}{d\t}\left(e^{-{\bf P}_i\cdot\bph +3\a -\n}\dot{z}_i\right)=0\; ,$$ where the dot denotes the derivative with respect to $\t$. They can be easily integrated to $$\dot{z}_i = \z_i\, e^{{\bf P}_i\cdot\bph -3\a +\n}\; , \label{zsol}$$ with arbitrary integration constants $\z_i$. Let us now choose the particularly convenient gauge $\n = 3\a$. In this gauge, the sum of the $(00)$ component and the spatial components of the Einstein equations reads $\ddot{\a}=0$ and, we have $$\a = v\t +\a_0 \label{a0}$$ where $v$ and $\a_0$ are arbitrary integration constants. Proper time $t$ is obtained by integrating $$dt = e^{3\a}d\t$$ which leads to $$v\t +\a_0 = \frac{1}{3}\ln\left(\sqrt{3E}|t-t_0|\right)\; .
 \label{pt}$$ Here $t_0$ is another integration constant and $$E =3v^2\; .$$ The solution  for the scale factor, written in proper time, then takes the form $$\a = \frac{1}{3}\ln\left(\sqrt{3E}|t-t_0|\right)$$ and shows the expected power-law behaviour, with power $1/3$, characteristic of evolution driven by kinetic energy. As usual, we have two branches, the negative-time branch, $t-t_0<0$, which ends in a future curvature singularity, and the positive-time branch, $t-t_0>0$, which starts out in a past curvature singularity.

We still have to consider the equations of motion for the bulk moduli $\f_A$. Replacing the blow-up moduli with Eq.  and working in the gauge $\n =3\a$ they take the form $$\frac{1}{2}\frac{d^2\f_A}{d\t^2}+\frac{{\partial}V}{{\partial}\f_A} = 0\; . \label{feom}$$ The “effective" potential $$V = 4\sum_i\z_i^2\, e^{{\bf P}_i\cdot\bph} \label{V}$$ in these equations of course originates from integrating out the blow-up moduli. The above equations of motion can be obtained from the Lagrangian $$\cL = \frac{1}{4}\dot{\bph}\cdot\dot{\bph}-V\; . \label{Lf}$$ They have to be supplemented by the Hamiltonian constraint $$\cH = \frac{1}{4}\dot{\bph}\cdot\dot{\bph}+V=E\; , \label{H}$$ which is simply the Friedmann equation rewritten in our language.

For the subsequent discussion it is useful to express the quantities measuring the geometry of the $G_2$ manifold in terms of the redefined fields. The volume $\cV_0$ of the underlying orbifold, defined in Eq. , can then be written as $$\cV_0 = \frac{1}{8}\exp\left(\frac{1}{3}\sum_A\f_A\right)\; ,
 \label{V1}$$ where we have neglected higher order terms. Likewise, the fraction $\e_{(\t ,n)}$ of the total volume taken up by the blow-up $(\t ,n)$ now takes the form $$\e_{(\t ,n)} = \frac{8}{3}\sum_ae^{-{\bf P}_i\cdot\bph}\,
                z_{(\t ,n,a)}^2\; .
 \label{e1}$$

Let us summarise what we have achieved so far. We have integrated out the blow-up moduli and decoupled the scale factor by choosing a particular gauge. This has reduced the problem of analysing the evolution of $43$ freely rolling $G_2$ moduli to one that only involves the seven bulk moduli subject to the potential . The remaining problem is, therefore, to solve the seven equations of motion  for the bulk moduli and the constraint . Each solution for the bulk moduli can then be inserted into Eq. , which determines the corresponding evolution of the blow-up moduli.

There exists a well-developed solution theory [@Kostant:qu], [@Lu:1996hh]–[@Lukas:1996zq] for the Lagrangian  and the associated equations of motion  for the bulk moduli if the system is of Toda type. To discuss this more specifically we introduce the matrix $$A_{ij}={\bf P}_i\cdot{\bf P}_j\; , \label{A}$$ which consists of one or more irreducible blocks along the diagonal. The system is called Toda if each of these blocks is proportional to the Cartan matrix of a simple Lie-group. One can now inspect the vectors ${\bf P}_i=-\m_i{\bf p}_i$ from Table 1, where we recall that the $\m_i$ are arbitrary signs. It is clear that the complete nine-dimensional matrix obtained from  is not proportional to a Cartan matrix for any choices of the signs $\m_i$ and, hence, the system is not of Toda type in this case. However, note that the potential  does not necessarily contain all possible terms. In fact, each term is multiplied by an arbitrary integration constant $\z_i$ that can be set to zero. Such a vanishing $\z_i$ implies, from Eq. , that the corresponding blow-up modulus $z_i$ is constant. Hence, we learn that we can consistently freeze an arbitrary subset of blow-up moduli $z_i$ and thereby select an arbitrary subset of the vectors ${\bf P}_i$ that appear in the potential . Then, for suitable subsets and choices of signs $\m_i$, the associated matrix $A$ can well be proportional to a Cartan matrix, as we will see for several explicit examples studied further below. The system is then Toda and can be integrated analytically. The resulting solutions are, of course, special in that they rely on a number of blow-up moduli being frozen. However, in the present context, this seems to be the best one can do by analytic methods.

Before we go on to analyse explicit examples, it is worth making a general observation about the structure of the solutions. It is well-known [@Lukas:1996ee; @Lukas:1996iq], for Lagrangians of the type , that the fields $z_i$ approach constant values asymptotically at early and late times (both in the negative and the positive time branch). This behaviour will also be confirmed in the explicit examples. Further note that the potential  consists of a sum of positive terms. Therefore, one expects the modes ${\bf P}_i\cdot\bph$ to roll up the exponential slope at early time, then turn around and roll down at late time. In other words, the exponentials $\exp ({\bf
P}_i\cdot\bph )$ decrease both in the past and in the future, at least if they explicitly appear in $V$, due to their associated integration constants $\z_i$ being non-zero. Hence, for the moduli that are not completely frozen, the associated relative volumes of the blow-ups, $\e_{(\t ,n)}$, always increases asymptotically at early and late times. We will confirm this general feature explicitly in our examples.

A universal solution
====================

As a warm-up, we would first like to consider a universal solution where all bulk moduli and all blow-up moduli evolve in the same way. It appears that a natural way to obtain such a solution is to start with an Ansatz where the seven bulk moduli are equal, that is where there is only a single breathing mode. However, it turns out that such an Ansatz is incompatible with the equations of motion . The reason behind this is that not all bulk moduli are on the same footing because of their differing couplings to the blow-up moduli, as is evident from the coupling vectors in Table 1. Instead, we start with the slightly more general Ansatz $$\f_A = c_A\f \label{A1}$$ where $\f$ is the breathing mode and $c_A$ are constants to be determined. For this Ansatz to be successful we need the seven equations of motion  for $\f_A$ to reduce to a single equation for $\f$. This leads to the conditions $$\begin{aligned}
 {\bf p}_{(\t ,a)}\cdot{\bf c} &=& {\rm const} \label{c1} \\
 \sum_{\t ,a}\z_{(\t ,a)}\, p_{(\t ,a)A} &=& \frac{1}{2}c_A\z^2 \label{c2}\end{aligned}$$ where we have defined $$\z_{(\t ,a)}^2 = \sum_n \z_{(\t ,n,a)}^2\; ,$$ and $\z$ is a constant. These conditions lead to a unique solution $${\bf c} = \frac{2}{7}(4,4,4,3,3,3,3)$$ for the vector ${\bf c}$ in our Ansatz . It can further be shown that all conditions , can be satisfied for specific choices of coefficients $\z_{(\t ,a)}$ within a three-parameter family of solutions. The details of this are inessential for our subsequent discussion. For such a solution, the seven equations of motion  reduce to the single equation $$\frac{1}{2}\ddot{\f}-2\m\z^2 e^{-2\m\f} = 0\; ,$$ where $\m$ is an arbitrary sign. This equation can be easily integrated once, and the integration constant can be fixed from the constraint . This leads to the first integral $$E=|{\bf c}|^2\left(\frac{1}{4}\dot{\f}^2+\z^2e^{-2\m\f}\right)$$ and the solution $$\f = \m\ln\cosh (y)
      +\frac{\m}{2}\ln\left(\frac{|{\bf c}|^2\z^2}{E}\right)$$ for the breathing mode $\f$, where $$y=\frac{2\sqrt{E}}{|{\bf c}|^2}(\t -\t_1)$$ is a rescaled time variable and $\t_1$ is a constant. The seven bulk moduli are proportional to $\f$ and can be obtained by inserting this result into Eq. . From Eq.  one can then, in turn, obtain the solutions for the blow-up moduli. After another integration one finds $$z_i = \frac{d_i}{2}\left(\tanh (y)+1\right)+z_{0i}\;$$ where $$d_i = \frac{\sqrt{E}\z_i}{\z^2}$$ and $z_{0i}$ are independent integration constants. As advertised earlier, the blow-up moduli $z_i$ indeed approach constants asymptotically. More precisely, at early time, $y\rightarrow -\infty$, we have $z_i\rightarrow z_{0i}$ and at late time, $y\rightarrow\infty$, we have $z_i\rightarrow z_{1i}\equiv z_{0i}+d_i$, that is, $d_i$ is the distance by which the blow-up modulus $z_i$ moves. We also note that the non-trivial evolution of $z_i$ happens around the time $y\simeq 0$.

Let us now interpret this solution. From Eq.  the orbifold volume is directly measured by the breathing mode $\f$ via $$\cV_0 = \frac{1}{8}\exp\left(\frac{16}{7}\f\right)\; .$$ Further, from Eq. , the relative volume of a blow-up is given by $$\e_i = \frac{8}{3}e^{2\m\f}z_i^2=\frac{2}{3}|{\bf c}|^2\frac{\z_i^2}{\z^2}
        \left(\frac{z_{0i}}{d_i}e^{-y}+\frac{z_{1i}}{d_i}e^y\right)^2\; .$$ We recall that our solution can only be trusted as long as $\e_i\ll 1$. Since, from Eq. , the pre-factor on the RHS of the above equation is of order one (for at least one $i$) we have to require the term in bracket be smaller than one. This leads to two cases, namely

-   $\e_i\ll 1$ if $|z_{0i}/d_i|\ll 1$ and $\ln |z_{0i}/d_i|\ll y\ll -1$. In this time range the breathing mode evolves as $\f\simeq -\m y +{\rm const}$.

-   $\e_i\ll 1$ if $|z_{1i}/d_i|\ll 1$ and $1\ll y\ll\ln |d_i/z_{1i}|$. In this time range the breathing mode evolves as $\f\simeq \m y +{\rm const}$.

In both cases the breathing mode and, hence, the orbifold volume $\cV_0$, can increase or decrease depending on the choice of the sign $\m$. However, independent of this choice, the relative blow-up volume $\e_i$ will always leave the allowed range $\e_i\ll
1$ when one of the limits of the given time ranges is approached. This means that after a finite proper time, both in the past and in the future, at least one of the blow-ups will take up a significant portion of the space and the approximation on which our Kähler potential  is based brakes down. In particular this means an evolution towards a state with small blow-ups at late time, $y\rightarrow\infty$ is not possible.

Potential with a single exponential
===================================

Let us now analyse the solutions more systematically, starting from simple patterns of evolution of the blow-up moduli and moving to more complicated ones.

Certainly, the simplest possibility is to freeze all blow-up moduli by setting all $\z_i=0$ in Eq. . In this case, the effective potential  for the bulk moduli vanishes identically and the equations of motion  for $\f_A$ can be easily integrated. In proper time $t$, related to $\t$ by Eq. , one easily finds as the general solution in this case $$\f_A = q_A\ln\left(\frac{|t-t_0|}{T}\right)+k_A\label{frees}$$ where $t_0$, $T$ and $k_A$ are constants. The expansion powers $q_A$ satisfy $$|{\bf q}|^2=\frac{4}{3}\; ,$$ which follows from the constraint , and are otherwise arbitrary. These are simply solutions describing power-law evolution of the bulk moduli, which, in fact, are identical to the ones that can be obtained for M-theory on a seven-dimensional torus.

We now move to the next more complicated case where blow-up moduli of only one particular type $(\t ,a)$ evolve non-trivially and all the others have been set to constants. We write ${\bf P}={\bf
P}_{(\t ,a)}$ and $\z^2=4\sum_n \z_{(\t ,n,a)}$ for simplicity of notation. The effective potential consists of only one term and takes the form $$V=\z^2 e^{{\bf P}\cdot\bph}\; .$$ This situation corresponds to an $SU(2)$ Toda model so the general solution to Eq.  can be found. It is given by $$\bph = {\bf p}^{(i)}\ln (x) +({\bf p}^{(f)}-{\bf p}^{(i)})
        \ln (1+x^\d )^{1/\d } + {\bf k} \label{singles}$$ subject to the constraints $$\begin{aligned}
 |{\bf p}^{(i)}|^2 &=& \frac{4}{3} \label{C1}\\
 \d &=& {\bf P}\cdot{\bf p}^{(i)} \label{C2}\\
 {\bf p}^{(f)} &=& {\bf p}^{(i)}-\frac{2{\bf P}\cdot{\bf p}^{(i)}}
                   {|{\bf P}|^2}{\bf P} \label{C3}\\
 \exp ({\bf P}\cdot{\bf k}) &=& \frac{3E\d^2}{|{\bf P}|^2\z^2}\; .
 \label{C4}\end{aligned}$$ Further, $$x=\frac{|t-t_0|}{T}$$ is the rescaled proper time and $t_0$ and $T$ are constants. Note that Eqs.  and imply that $$|{\bf p}^{(f)}|^2= \frac{4}{3}\; .$$ This solution is invariant under the exchange of ${\bf p}^{(i)}$ and ${\bf p}^{(f)}$ and we remove this ambiguity by requiring $\d\geq 0$ in the positive-time branch and $\d\leq 0$ in the negative-time branch. The interpretation of these solutions is well-known [@Lukas:1996ee; @Lukas:1996iq]. They interpolate between two, generally different, “free" solutions , one with ${\bf q}={\bf p}^{(i)}$ at early time and one with ${\bf q}={\bf p}^{(f)}$ at late time. In these asymptotic regions the blow-up moduli $z_{(\t ,n,a)}$ are constant while they move around the time $x\simeq 1$ to facilitate the transition between the two free solutions.

One can now compute the relative volume $\e_{(\t ,a)}$ of the blow-ups by inserting the solution  into Eq. . As before, we find that the required condition $\e_{(\t ,a)}\ll 1$ is satisfied only in two finite time windows at $x\ll 1$ and $x\gg 1$. At the endpoints of these windows the volume of the blow-ups becomes sizeable and control over the approximation is lost.

More complicated cases
======================

An obvious generalisation of the previous example is to consider a situation that corresponds to an $SU(2)^n$ Toda model, for some integer $n$. This amounts to having a subset of blow-up moduli evolve non-trivially for which the associated characteristic vectors ${\bf P}_i$ are orthogonal, that is, $${\bf P}_i\cdot {\bf P}_j = 2\d_{ij}\; . \label{ortho}$$ Inspection of Table 1 shows such cases can indeed be realised. For example, allowing the moduli of a certain type $\t$ to evolve non-trivially, while moduli of the other two types are being kept constant by setting $\z_{\t ',n,a}=0$ in Eq.  for $\t
'\neq\t$, leads to an $SU(2)^3$ Toda model with three exponentials appearing in the potential . From Table 1 there are a number of other options and in the following we will simply assume the existence of $n$ vectors ${\bf P}_i$ satisfying  to cover all possibilities.

The standard procedure to deal with such a system is to introduce a new constant basis ${\bf e}_A=({\bf e}_i,{\bf e}_a)$ in field space, where $i =1,\cdots ,n$ and $a=n+1,\cdots ,7$ such that $${\bf e}_i = \frac{1}{\sqrt{2}}{\bf P}_i$$ and the remaining vectors ${\bf e}_a$ are chosen to complete the set to an orthonormal basis satisfying $${\bf e}_A\cdot{\bf e}_B = \d_{AB}\; .$$ The bulk fields $\bph$ can then be expanded as $$\bph = \sum_A\r_A{\bf e}_A\; ,$$ where $\r_A$ is a new set of fields. The Lagrangian  and the Hamiltonian , written in terms of these new fields, take the form $$\cL = \frac{1}{4}\sum_A\dot{\r}_A^2-V\; ,\qquad
 \cH = \frac{1}{4}\sum_A\dot{\r}_A^2+V=E\; , \label{LH}$$ with the potential $$V = 4\sum_i\z_i^2e^{2\r_i}\; .$$ Note that the modes $\r_a$ have decoupled from this potential, which is, of course, one of the motivations behind introducing the basis ${\bf e}_A$.

The general solution to the Lagrangian in  can be easily obtained as $$\bph = \sum_A\r_A{\bf e}_A$$ with $$\begin{aligned}
 \r_a &=& k_a\t +\t_a \\
 \r_i &=& \sqrt{2}\ln\cosh (y_i)
          -\frac{1}{\sqrt{2}}\ln\left(\frac{16\z_i^2}{k_i^2}\right)\; ,\end{aligned}$$ where $\t_A$ and $k_A$ are constants and $$y_i = \frac{k_i}{\sqrt{2}}(\t -\t_i)$$ are rescaled time coordinates. The Hamiltonian constraint in  amounts to the condition $$\sum_Ak_A^2 = 4E\; .$$ As expected the modes $\r_a$ evolve freely. Each of the other modes $\r_i$ evolves similarly to what has been found for the single $SU(2)$ Toda model in the previous section. This can be seen explicitly by converting the above solution to proper time using  and comparing with Eq. . This means that each mode interpolates between two regions of simple power-law evolution at early and late time. The blow-up moduli $z_i$ are constant in these asymptotic regions and their evolution at intermediate times facilitates the transition. This can be seen explicitly from their solution $$z_i=\frac{d_i}{2}\left(\tanh (y_i)+1\right)+z_{0i}\; ,$$ with $z_{0i}$ and $d_i$ constants representing the value of the modulus at early time and its total change respectively. This solution is obtained by inserting the above solution for $\bph$ into Eq. .

From Eq.  one finds the relative volume $$\e_i = \frac{\sqrt{2}}{3}\left(\frac{z_{1i}}{d_i}e^{y_i}+\frac{z_{0i}}{d_i}
        e^{-y_i}\right)^2$$ of the blow-ups. As in all previous cases, $\e_i$ can only be kept small, $\e_i\ll 1$, for a finite proper time at either $y_i\ll -1$ or $y_i\gg 1$. For a valid solution, all $\e_i$ need to be small, which amounts to arranging an overlap between those regions by choosing the time shifts $\t_i$ appropriately. Outside this overlap region control of the approximation is lost as one or more blow-ups become large and take up a significant part of the internal space.

There are also cases leading to a Toda model associated with a higher-rank simple group. Consider, for example, the three vectors $$\begin{aligned}
 {\bf P}_{(\a ,1)} &=& (-1,0,0,0,0,-1,0) \label{P1}\\
 {\bf P}_{(\b ,1)} &=& (1,0,0,0,0,0,1) \label{P2}\\
 {\bf P}_{(\g ,3)} &=& (0,-1,0,0,0,0,-1)\; ,\label{P3}\end{aligned}$$ obtained from Table 1 with the sign choice $\m_{(\a ,1)}=-1$, $\m_{(\b ,2)}=1$ and $\m_{(\g ,3)}=-1$ and set all blow-up moduli with $(\t ,a)$ different from the above to constants. The matrix $A$ in Eq.  is then given by $$(A_{\t\s}) = \left(\begin{array}{rrr}2&-1&0\\-1&2&-1\\0&-1&2\end{array}\right)\; ,$$ which is precisely the Cartan matrix of $SU(4)$. Hence, we are dealing with an $SU(4)$ Toda model.

As before, the first step in solving this model is to introduce a new basis $({\bf e}_A)=({\bf e}_\t ,{\bf e}_a)$, where $\t = 1,2,3$ and $a=4,\dots ,7$, in field space. Here the three vectors ${\bf e}_\t$ are identified with – and the remaining four vectors ${\bf e}_a$ are chosen to be orthonormal among themselves and orthogonal to all ${\bf e}_\t$. Consequently, we have a basis satisfying $${\bf e}_\t\cdot{\bf e}_{\s} = A_{\t\s}\; ,\qquad
 {\bf e}_\t\cdot{\bf e}_a = 0\; ,\qquad
 {\bf e}_a\cdot{\bf e}_b = \d_{ab}\; .$$ Expanding the fields $\bph$ $$\bph = \sum_A\r_A{\bf e}_A$$ as before one finds for the Lagrangian  $$\cL = \frac{1}{4}\sum_{\t ,\s}A_{\t ,\s}\dot{\r}_\t\dot{\r}_\s
       +\frac{1}{4}\sum_a\dot{\r}_a^2-V$$ with potential $$V = 4\sum_\t\z_\t^2\,\exp\left(\sum_\s A_{\t\s}\r_\s\right)\; .$$ The modes $\r_a$ are decoupled from this potential and their equations of motion immediately lead to the general solution $$\r_a = k_a(\t -\t_a )\; ,$$ where $k_a$ and $\t_a$ are constants. Following the methods of Ref. [@Kostant:qu], the solutions for the other modes are obtained as $$e^{-\r_\t} = \sum_{\bla\in\L_\t}b_\t (\bla )\exp\left(\bla\cdot ({\bf k}
              \t - \btau )\right)\; , \label{sol4}$$ where ${\bf k}$ and $\btau$ are constant vectors. The three sets $\L_\t$ contain the weights of the fundamental representation ${\bf 4}$, the vector representation ${\bf 6}$ and the anti-fundamental representation $\bar{\bf 4}$ of $SU(4)$, respectively, and are explicitly given by $$\begin{aligned}
 \L_1 &=& \{ (100),(-110),(0-11),(00-1)\} \\
 \L_2 &=& \{ (010),(1-11),(-101),(10-1),(-11-1),(0-10)\} \\
 \L_3 &=& \{ (001),(01-1),(1-10),(-100)\} \; .\end{aligned}$$ Finally, $b_\t (\bla )$ represents a set of constants which depends on $\z_\t$ as well as on ${\bf k}$. They can be calculated by inserting the solutions  into the equations of motion but their explicit form will not be of any relevance here.

The important feature of  is that asymptotically one of the exponentials in the sum will dominate leading to a power-law evolution. In these regions, the blow-up moduli are approximately constant and the relative blow-up volumes increase until they approach values of order one where the approximation underlying our analysis breaks down. Hence, this more complicated Toda solution also conforms with our general expectation of asymptotically increasing blow-up volumes.

[**Acknowledgments**]{}\
A. L. is supported by a PPARC Advanced Fellowship, and S. M. by a PPARC Postgraduate Studentship.\

[01]{}

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[^1]: email: a.lukas@sussex.ac.uk

[^2]: email: s.morris@sussex.ac.uk

[^3]: Here and in the following bold-face symbols denote seven-dimensional vectors.
