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in  @xcite , witten studied @xmath2 heterotic string theory on @xmath3 with an @xmath4 , @xmath5 gauge instanton supported on @xmath6 with instanton number @xmath7 . the effective @xmath8 theory manifests @xmath9 supersymmetry and has gauge symmetry @xmath10 . the boundary of instanton moduli space contains singular points , at each of which one unit of instanton number is concentrated . these are called `` small '' instantons and the local moduli space @xmath11 was shown to be @xmath12 dimensional . it was demonstrated in  @xcite that @xmath11 can be described as the flat directions of a potential energy function . this is constructed from @xmath13 hypermultiplets , each transforming as a fundamental @xmath13 representation of @xmath4 and as a doublet under an additional @xmath14 gauge symmetry . at the origin of field space , all of these fields become massless and the @xmath14 symmetry becomes exact . away from this point , however , the @xmath14 symmetry is spontaneously broken and only the @xmath12 moduli of @xmath15 remain massless . one concludes that as one approaches a small instanton point in moduli space , the theory develops extra ( massless ) species and an enhanced symmetry . such points are often referred to as being @xmath16 .    small instantons also occur in @xmath1-theory on @xmath17 , where @xmath18 is a calabi - yau threefold  @xcite . as first discussed in  @xcite , to be consistent these vacua must have , supported on the calabi - yau manifold at each end of the @xmath0 interval , a gauge instanton indexed by a subgroup of @xmath19 . one can , for simplicity , assume a trivial instanton at one end of the interval . the non - trivial instanton on the other end is a connection on a holomorphic vector bundle @xmath20 over @xmath18 with structure group @xmath21 . quantum consistency requires that the instanton number , here generalized to the 2nd chern class @xmath22 , must satisfy the constraint @xmath23 , where @xmath24 is the tangent bundle of the calabi - yau threefold and @xmath25 is an effective class corresponding to m - five - branes in the @xmath26 `` bulk '' space . the effective @xmath27 theory manifests @xmath28 supersymmetry . its gauge symmetry depends on the choice of structure group @xmath29 . in a series of publications  @xcite , holomorphic vector bundles with @xmath30 were constructed . we will refer to these when specificity is required . for example , when @xmath31 , @xmath32 and @xmath33 , the effective @xmath27 gauge symmetry is @xmath34 , @xmath35 and @xmath33 respectively . the boundary of instanton moduli space contains singular points where a portion of the vector bundle becomes concentrated as a torsion free sheaf over an isolated holomorphic curve with effective class @xmath36 . these are the heterotic @xmath1-theory analogs of small instantons . here , however , the local moduli spaces @xmath11 are more complicated , changing with the structure of the calabi - yau threefold , the holomorphic vector bundle and the choice of @xmath37 . the moduli spaces of small instantons on elliptic calabi - yau threefolds over hirzebruch bases @xmath38 with vector bundles constructed from spectral covers were analyzed in several publications  @xcite . for example , consider @xmath39 , @xmath31 and @xmath40 , with @xmath41 a section of @xmath42 . then , it was shown that @xmath43 , where @xmath44 are integers satisfying the constraints @xmath45 and @xmath46 . it is expected that any @xmath11 can again be described as the flat directions of a potential energy function . this would be constructed from chiral supermultiplets transforming in some representation @xmath47 of @xmath29 and , possibly , as a representation @xmath48 of an additional gauge symmetry @xmath49 . at the origin of field space , all of these fields become massless and the @xmath49 symmetry becomes exact . away from this point , the @xmath49 symmetry is spontaneously broken and only the moduli of @xmath15 remain massless . therefore , as one approaches a small instanton in moduli space , the theory develops extra ( massless ) species and , possibly , an enhanced symmetry . that is , small instantons in heterotic @xmath1-theory are @xmath16 . unfortunately , such potential functions have not been explicitly constructed for most moduli spaces . limited simple examples indicate that such potentials exist , but a general proof is still lacking . be that as it may , in this paper we will assume that small instanton moduli spaces @xmath15 in heterotic @xmath1-theory can be so described .    in heterotic @xmath1-theory , strings wrapping a holomorphic curve in the calabi - yau threefold will generate a non - perturbative superpotential for both geometric and vector bundle moduli  @xcite . as first discussed in  @xcite , the superpotential created by a such a `` worldsheet instanton '' has a specific form ; namely , an exponential of geometric and curve moduli multiplied by a `` pfaffian '' . the pfaffian is a function of the determinant of the dirac operator twisted by the vector bundle restricted to the curve . it is generically true  @xcite that the pfaffian evaluated at any holomorphic curve is a homogeneous polynomial of the moduli of the restricted bundle . this has important implications for small instantons . it was shown in  @xcite that the moduli of the vector bundle restricted to the small instanton curve @xmath36 are precisely those which parameterize @xmath11 . hence , strings wrapped on @xmath36 generate a superpotential on @xmath11 which `` lifts''some , or all , of the flat directions . furthermore , new local minima may now occur substantially away from the @xmath16 at the zero of field space . within the context of @xmath38 , @xmath31 and @xmath50 , pfaffians were explicitly computed in  @xcite . for example , for @xmath51 , @xmath52 and @xmath53 , @xmath54 and the pfaffian was shown to be an explicit homogeneous polynomial of degree @xmath55 in @xmath56 of the @xmath57 moduli of @xmath11 .    at a small instanton in heterotic @xmath1-theory , the vector bundle @xmath20 degenerates into the `` union '' of a new holomorphic vector bundle @xmath58 over @xmath18 and a sheaf over the curve @xmath36 . because of the existence of the @xmath26 interval , the sheaf can `` detach '' from @xmath18 , leaving only @xmath58 behind , and become a component of a new m - five - brane curve in the bulk space . that is , @xmath59 . the physical interpretation of this is that the small instanton `` emits '' a five - brane into the bulk space . of course , this can occur in reverse , a component @xmath59 of a bulk space five - brane being `` absorbed '' into @xmath58 via a small instanton to become @xmath20 . such a process , either absorbing or emitting a bulk five - brane , is called a small instanton phase transition and was defined and studied in  @xcite . it was shown that there are two types of small instanton transitions , the first in which the structure group of the vector bundle is unchanged while the number of chiral zero modes is altered and a second where the reverse is true , the structure group changes while the number of chiral zero modes remains fixed . for an elliptically fibered calabi - yau threefold @xmath18 , the first type of transition occurs when @xmath36 is the pull - back of a curve in the base @xmath60 . this is analogous to the heterotic @xmath2 case . the second type of transition takes place when @xmath36 is proportional to the elliptic fiber @xmath61 . in this paper , we will consider only those small instanton transition for which the structure group is unchanged . an important aspect of any small instanton transition is that the @xmath1-five - brane wrapped on @xmath36 in the bulk space itself has moduli . the most conspicuous of these is the translation modulus , the real part of which describes the separation of the five - brane from the end of the @xmath62 interval and whose imaginary part is an axion . for a calabi - yau threefold @xmath18 elliptically fibered over any base @xmath60 , the complete moduli space @xmath63 of an @xmath1-five - brane wrapped on @xmath36 , both its dimension and geometry , was computed in  @xcite . this space depends strongly on the curve @xmath36 and typically contains many moduli in addition to the translation modulus . combining this with the discussion above , we see that in heterotic @xmath1-theory the full moduli space of a small instanton phase transition is @xmath64 , where @xmath11 is the instanton phase and @xmath63 the five - brane phase . generalizing the above discussion , we assume that the complete phase transition can be described by a potential energy function constructed from chiral supermultiplets in some representation @xmath47 of @xmath29 and , possibly , in a representation @xmath48 of an additional gauge symmetry @xmath49 . the origin of field space is an @xmath16 . away from this point , however , the @xmath49 symmetry is spontaneously broken and only the moduli parameterizing @xmath65 remain massless . the existence of small instanton phase transitions , from the point of view of holomorphic vector bundles and topology , was demonstrated in  @xcite . in the present paper , by studying the motion of heterotic @xmath1-theory vacua in moduli space coupled to an expanding friedmann - robinson - walker ( frw ) spacetime , we will show that these phase transition can also take place dynamically . specifically , we will show that an @xmath1-five - brane in the bulk can dynamically evolve through the esp , the point in moduli space where the five - brane collides with the end of the @xmath66 interval , continuing until the vacuum comes to rest at a local minimum of the non - perturbative superpotential . at this point , which is substantially distant from the esp , the sheaf over @xmath36 has combined with @xmath58 to form a smooth vector bundle @xmath20 , completing the small instanton phase transition . the dynamical system is trapped at this local minimum by the hubble friction of the frw space . we ignore the effect of moduli trapping due to radiation into massless fields at the esp  @xcite , but will argue that this is insufficient to alter our results . our results are consistent with those found , in different contexts , in  @xcite . these papers ignore any non - perturbative potentials that might arise and discuss flat moduli space . they find that the moduli are trapped , both by hubble friction and quantum radiation , exactly at the esp , which , they go on to argue , gives it a special physical meaning . we note that were we to turn off the non - perturbative superpotential in heterotic @xmath1-theory , one would arrive at precisely the same conclusion . however , turning on this potential completely alters this result and allows the completion of the small instanton phase transition .    throughout this paper , for notational and computational simplicity , we will take all fields to be dimensionless and choose units in which @xmath67 . in this section , we consider the dynamics of small instanton transitions ignoring the non - perturbative superpotential generated by worldsheet instantons . in this case , the moduli space @xmath65 is the flat locus of the potential . as discussed above , the dimension and geometry of this moduli space is completely understood , at least for a wide range of base spaces @xmath60 and small instanton curves @xmath36 . however , @xmath11 and @xmath63 are both , generically , multi - dimensional . for example , the small instanton discussed in the introduction had @xmath11 parameterized by @xmath57 complex fields . unfortunately , a detailed calculation of dynamics in a multi - dimensional moduli space is not only unenlightening in its complexity , but is far beyond our computational power . for this reason , we will restrict our discussion to a simplified moduli space , one in which both @xmath11 and @xmath63 are taken to be one - dimensional . we denote the complex modulus of @xmath11 , the instanton phase , by @xmath68 . physically , this modulus describes the `` smoothing out '' of the small instanton sheaf into the smooth vector bundle @xmath20 . the complex modulus of @xmath63 , the five - brane phase , is written as @xmath69 and represents the translation modulus of the five - brane . we want to emphasize that the literature describing transitions in different contexts  @xcite all employ very similar simplifications . we begin by considering the dynamics of the moduli @xmath69 and @xmath68 in static flat spacetime ; that is , we turn off their interaction with gravity . the simplest @xmath28 supersymmetric self - interactions that have precisely a one - dimension instanton phase and a one - dimension five - brane phase are described by the superpotential @xmath70 recall that both @xmath69 and @xmath68 are taken to be dimensionless and , hence , @xmath71 has mass dimensions @xmath72 in planck units . furthermore , we choose the planck mass @xmath73 to be unity . the potential function describing the small instanton phase transition is then given by @xmath74 the instanton phase is the one - dimensional complex submanifold defined by @xmath75 , whereas the five - brane phase is the one - dimensional space given by @xmath76 . the dynamical calculation is carried out in the four - dimensional real space parameterized by @xmath77 , @xmath78 , @xmath79 and @xmath80 . without loss of generality , one can take @xmath81 . other values of @xmath71 lead to the same conclusions . in this paper , we will assume that the calabi - yau radius and the interval length are chosen , in units where @xmath67 , to be in the phenomenologically acceptable range @xmath82 this implies that in order to achieve the correct phenomenological values for the newton and gauge coupling parameters , the dimensionless volume and interval moduli should be stabilized at ( or be slowly rolling near ) the values of order one . in this paper , we will assume that this is the case . this limits @xmath69 to be less than or of the order of unity . for reasons to become clear in the next subsection , we will choose both @xmath69 and @xmath68 moduli to have values much smaller than one . let us compute the dynamics with the following initial conditions . take the initial moduli to be @xmath83 and their initial velocities as @xmath84 these values represent the following physical situation . first , there is a five - brane in the bulk space substantially separated from the end of the @xmath0 interval with a much smaller displacement in the axionic direction . a similar very small displacement occurs in the vector bundle modulus . the five - brane has a small negative velocity in the @xmath85 direction which will cause it to collide with the end of the interval and a similar velocity in the axion direction . substantially larger velocities are given to both @xmath86 and @xmath87 which enable a possible transition from the five - brane branch to the instanton branch . the equations of motion are solved numerically on mathematica , with the results for @xmath85 and @xmath87 presented in figures 1 and 2 .    curve ( a ) in figure 1 shows that the five - brane collides with the end of the interval but then continues to oscillate around that point . similarly , curve ( b ) shows that the collision initiates a transition in the vector bundle modulus direction which , however , quickly terminates . the resulting motion is an oscillation around the small instanton in the vector bundle direction as well . figure 2 shows the overall motion in moduli space near the small instanton at the origin of field space . note that although the amplitude of any individual real modulus increases and decreases quasi - periodically , the overall amplitude remains constant in time , as is clearly indicated in figure 2 . the dynamics of the remaining two real moduli , @xmath88 and @xmath86 , is similar , so we will not display them . these results are easy to explain . first of all , in both the five - brane and instanton branches , the potential energy function has walls which become progressively steeper as one moves away from the origin of field space . hence , the moving moduli see a `` concave '' potential that always reflects them back toward the origin . this accounts for the oscillation around the small instanton at the origin of field space . secondly , the lack of any damping , be it hubble friction or any other mechanism , means that the overall oscillation amplitude persists . the final conclusion is that the phase transition is never completed . instead the system oscillates endlessly around the small instanton . note that these results are consistent with those found in  @xcite when analyzing undamped motion in different contexts . let us now consider the moduli @xmath69 and @xmath68 interacting with gravity . in this case , the dynamics is described by @xmath28 chiral multiplets coupled to supergravity and one must specify both a superpotential @xmath89 and a kahler potential @xmath90 . the superpotential remains the same , namely @xmath91 in general , the kahler potential of moduli in heterotic @xmath1-theory is unknown . however , near the origin of field space , that is , when the magnitudes of the moduli are much smaller than unity , this can be taken to be `` flat '' . therefore , as long as we restrict @xmath69 and @xmath68 describing our small instanton phase transition to be small , the kahler potential is given by @xmath92 this is one reason why we chose the initial conditions for @xmath69 and @xmath68 in the previous subsection to be much less than one . the potential function describing the small instanton phase transition is now @xmath93 where @xmath89 and @xmath90 are given in  ( [ 5 ] ) and  ( [ 6 ] ) respectively and @xmath94 is the kahler covariant derivative . this is a rather complicated function of the moduli . however , for @xmath69 and @xmath68 much smaller than unity , one need only consider the lowest order interactions . all terms that are higher order in @xmath95 can safely be ignored . the result is that one can analyze the dynamics with the same potential as in the non - gravitational case , that is , @xmath96 as previously , we will choose @xmath81 and only consider the region of moduli space where @xmath69 and @xmath68 are much smaller than one . additionally , one must give an ansatz for the spacetime geometry . we will assume a metric of the form @xmath97 corresponding to a spatially flat manifold of the frw type .    furthermore , we will only consider metrics that are monotonically expanding . there are , in total , five equations of motion that must be solved simultaneously . to begin with , there is a single einstein equation for the scale parameter @xmath98 . additionally , there are the four equations of motion for the real and imaginary parts of the moduli @xmath69 and @xmath68 . despite the fact that the potential in  ( [ 8 ] ) is independent of gravitationally induced interactions , non - trivial gravity effects do enter each equation as a hubble damping term . let us compute the dynamics with the following initial conditions . take the initial value of the scale parameter to be @xmath99 it is unnecessary to fix an initial radial velocity . we will continue to choose the initial conditions for the @xmath69 and @xmath68 moduli and their velocities to be those given in  ( [ 3 ] ) and  ( [ 4 ] ) respectively . these values represent exactly the same physical situation as in the previous subsection . here , however , the moduli evolve with respect to an expanding frw background spacetime . the equations of motion are solved numerically on mathematica . the result for the scale parameter @xmath98 is shown in figure 3 , while those for @xmath85 and @xmath87 are presented in figures 4 and 5 . figure 3 shows that @xmath98 increases monotonically from its initial value , indicating an expanding frw universe .    from curve ( a ) in figure 4 , one sees that the five - brane collides with the end of the interval and then continues to oscillate around that point . however , the average amplitude and frequency of the oscillation are now strongly damped in time , with @xmath85 converging toward zero . similarly , curve ( b ) in figure 4 shows that the collision initiates a transition into the vector bundle modulus direction which then terminates , reverses and begins to oscillates around @xmath100 . once again , the average amplitude and frequency of the oscillation in the vector bundle modulus direction are damped in time and @xmath87 converges to zero . figure 5 shows the overall motion in moduli space near the small instanton at the origin of field space . note that the overall amplitude decreases in time . the dynamics of the remaining real moduli , @xmath88 and @xmath86 , is similar , so we do not display them . the physical explanation of these results is identical to that given in the previous subsection , with one important exception . the expansion of the universe produces `` friction '' in the motion of the moduli . this causes their oscillation around the origin of field space to have ever diminishing amplitude and frequency . the physical conclusion is that after collision and a period of damped oscillation , the five - brane becomes fixed at the end of the interval . the associated bundle data is the `` union '' of @xmath58 and a torsion free sheaf over the curve @xmath36 . that is , the system approaches the small instanton @xmath16 and the phase transition to a smooth bundle @xmath20 is never completed . note that these results are completely consistent with those found in  @xcite when analyzing damped motion in different contexts . having established our preliminary results , we now turn to the main discussion in this paper ; namely , the dynamics of small instanton transitions in the presence of a non - perturbative superpotential generated by worldsheet instantons . in this case , the moduli space @xmath63 of the curve remains flat . however , the potential for the vector bundle moduli is now substantially altered , with the result that , generically , the flat moduli space @xmath11 is replaced by one or more isolated local minima . to proceed , we must first be more explicit about the form of the non - perturbative interactions . it was shown in  @xcite that the superpotential generated by a string wrapped on an isolated curve @xmath101 has the form @xmath102 where @xmath103 are a subset of vector bundle moduli and we have assumed , for simplicity , that the moduli space of the five - brane wrapped on @xmath36 is one - dimensional and parameterized by the dimensionless translation modulus @xmath69 . the coefficient @xmath104 is then dimensionless and given by @xmath105 where @xmath106 is the size of the @xmath0 interval , @xmath107 is the area of @xmath36 and @xmath108 is the eleven - dimensional planck scale . when the calabi - yau radius and the interval length are chosen , in units where @xmath67 , to be in the phenomenologically acceptable range eq . , @xmath104 is found  @xcite to be of order @xmath109 . let us recall that in our notation the position of the five - brane is dimensionless and bounded by the length of the interval which is of order one .    as mentioned in the introduction , @xmath110 is the pfaffian of the dirac operator twisted by the vector bundle @xmath20 restricted to the curve @xmath36 . it is found to be a homogeneous polynomial of the moduli @xmath103 of the restricted bundle . in general , the pfaffian is difficult to compute . however , it has been explicitly calculated in  @xcite for vector bundles with structure group @xmath111 restricted to an isolated curve @xmath36 in an elliptic calabi - yau threefold with base @xmath38 . for example , choosing @xmath51 , a specific @xmath112 vector bundle and @xmath50 , where @xmath41 is an isolated curve in @xmath42 , it was shown that the pfaffian is a homogeneous polynomial of degree @xmath55 in @xmath56 out of the @xmath57 moduli of @xmath11 . an important pattern that emerges from the results in  @xcite is that the smaller the dimension of @xmath11 , the smaller the degree of the homogeneous polynomial . a final comment is that the coefficient of proportionality has not been explicitly computed . now consider the simplified moduli space in which both @xmath11 and @xmath63 are one - dimensional and parameterized by the moduli @xmath68 and @xmath69 respectively . in this case , the non - perturbative superpotential @xmath113 in  ( [ 11 ] ) takes the form @xmath114 where @xmath115 is a small integer and @xmath116 is a coefficient of mass dimension @xmath117 in planck units . we now proceed to discuss the moduli dynamics in the presence of superpotential ( [ 14 ] ) . again consider the dynamics of the moduli @xmath69 and @xmath68 in static flat spacetime . however , the complete superpotential is now given by @xmath118 the potential function describing the small instanton phase transition is @xmath119 where @xmath120 the minimum of this potential has two distinct types of separated loci . the first , specified by @xmath121 is the moduli space @xmath122 of @xmath36 . however , the moduli space @xmath11 is now replaced by @xmath123 isolated points located at @xmath124 where @xmath125 .    to carry out an explicit calculation of the moduli dynamics , one must specify the values of the parameters in superpotential ( [ 15 ] ) . without loss of generality , choose @xmath81 . for specificity , we will take @xmath126 the choice for @xmath104 is consistent with the phenomenologically acceptable values of @xmath127 and @xmath128 given in  ( [ 13 ] ) . similarly , in view of the discussion above , @xmath129 is a reasonable value for the degree of the pfaffian polynomial . the choice of @xmath116 in  ( [ 20 ] ) is less well - motivated . it is taken because it enhances the distance between the local minimum and the esp in moduli space , as can be seen from  ( [ 19 ] ) . for the parameters in  ( [ 20 ] ) , the local minima in  ( [ 19 ] ) become @xmath130 similar physical conclusions will be reached for a wide range of all three parameters .    with these choices , we compute the moduli dynamics using initial conditions similar to those given in  ( [ 3 ] ) and  ( [ 4 ] ) . the physical situation is identical to that described in the previous section . the moduli equations of motion are solved numerically on mathematica , with the results for @xmath77 and @xmath80 shown in figure @xmath131 . figure @xmath132 indicates that the five - brane collides with the end of the interval but then , eventually , continues to oscillate without damping around that point . now , however , there is a new phenomenon . note that immediately after the collision the @xmath77 modulus gets `` hung up '' at the negative value corresponding to the local minima in  ( [ 21 ] ) . it lingers there for a considerable time , before rolling back through the origin and continuing its oscillation around the small instanton point . similar , but more dramatic , behaviour for @xmath80 is shown in figure @xmath133 . after a few small oscillations around the origin of field space , the @xmath80 modulus rolls all the way out to the value @xmath134 corresponding to one of the local minima in  ( [ 21 ] ) . it stays in the vicinity of this minimum for some time , before rolling back through the origin and beginning an undamped oscillation around the small instanton . the choice of @xmath134 rather than the other minimum at @xmath135 is due to the initial conditions . these can be changed to select the other local minimum . the behaviour of both moduli is graphically presented in figure @xmath136 , where the excursion of the moduli to the local minimum and back is clearly visible . the other two real moduli , @xmath78 and @xmath79 , simply oscillate with undamped motion around the origin of field space , so we will not display them .    comparing these results with the static flat case in section @xmath137 , we conclude that the non - perturbative superpotential has a dramatic effect on the dynamics of the moduli . specifically , the moduli now evolve far from the origin of field space , seeking , and reaching , an isolated local minimum of the potential . however , since there is no mechanism for damping , the moduli roll through the isolated minimum , returning to oscillate around the small instanton . were one to wait for a sufficiently long time , the moduli might once again briefly return to one of the two local minima , only to roll out again and oscillate around the origin of field space . the lack of any damping means that the overall oscillation amplitude persists . we conclude that , despite the existence of isolated local minima , the phase transition is never completed . now let the moduli @xmath69 and @xmath68 interact with gravity . the superpotential remains the same , namely @xmath138 since the values of all moduli we consider , including the isolated local minima , are much smaller than unity , one can continue to choose the kahler potential to be @xmath139 furthermore , since @xmath69 and @xmath68 are much smaller than one , the complicated potential function in  ( [ 7 ] ) simplifies to @xmath140 that is , all terms that are higher order in @xmath95 can be ignored . the potential function is then identical to the non - gravity case with identical minima ; namely , the five - brane branch  ( [ 18 ] ) and the @xmath123 isolated minima given in  ( [ 19 ] ) . as in the previous subsection , we choose @xmath81 and take @xmath141 for these parameters , the two isolated minima in  ( [ 19 ] ) are again @xmath142 additionally , one must give an ansatz for the spacetime geometry . we will continue to assume an expanding frw metric of the form given in  ( [ 9 ] ) . let us now compute the dynamics of the moduli @xmath69 and @xmath68 , as well as the spacetime expansion parameter @xmath98 , using the same initial conditions as in section @xmath137 , namely , ( [ 3 ] ) ,  ( [ 4 ] ) and  ( [ 10 ] ) . the physical situation remains the same as previously . we use mathematica to numerically solve the equations of motion . the result for the scale parameter @xmath98 is similar to that shown in figure @xmath72 and , hence , we wo nt present it here . the dynamics of @xmath85 and @xmath87 is given in figure @xmath56 . figure @xmath143 indicates that the five - brane collides with the end of the interval . immediately after the collision , the @xmath85 modulus gets `` hung up '' at the negative value corresponding to the local minima in  ( [ 26 ] ) . now , however , the modulus is `` trapped '' near this value and does not return to the origin of field space . rather , it oscillates with ever decreasing amplitude and frequency around @xmath144 , eventually coming to rest at this point . the behaviour of @xmath87 is similar . figure @xmath145 shows that after a few small oscillations around the origin of field space , the @xmath87 modulus rolls all the way out to the value @xmath134 corresponding to one of the local minima of  ( [ 26 ] ) . again , this local minimum , rather than @xmath135 , is selected by the initial conditions . however , unlike in the previous subsection , the modulus is now `` trapped '' near this minimum , oscillating around it with ever decreasing amplitude and frequency . it eventually comes to rest at @xmath146 . the behaviour of both moduli is graphically presented in figure @xmath147 . here , the excursion of the moduli to the local minimum , and the phenomenon of their being trapped at that point , is clearly visible . the other two real moduli , @xmath88 and @xmath86 , simply oscillate with decreasing amplitude and frequency around the origin of field space . hence , we will not display them .    comparing these results with those in the previous subsection , we see that the effect of the expanding frw geometry on the dynamics of the @xmath69 and @xmath68 moduli is dramatic . previously , without gravitation , these moduli evolved out to the local minimum of the non - perturbative superpotential . however , because of the absence of any mechanism for damping , the moduli eventually rolled out of the local minimum and back to the origin of field space . they oscillated forever with no loss of amplitude or frequency around the small instanton , occasionally , perhaps , temporarily revisiting one of the two local minima . when coupled to the expanding frw spacetime , however , this behaviour is completely modified . the results in figure@xmath56 graphically confirm that the moduli now evolve out to the local minimum and are trapped there by gravitational damping . the physical reason is clear . immediately after collision , the moduli have sufficient `` energy '' to overcome a potential barrier and roll into a valley surrounding the local minimum . however , the energy is rapidly `` red - shifted '' by the expanding geometry . the result is that the moduli can no longer cross the potential barrier to return to the origin of field space . the continued loss of energy to gravitational expansion causes the moduli to oscillate with smaller and smaller amplitude and frequency around the minimum , eventually coming to rest there . that is , they are gravitationally trapped at the isolated minimum .    before drawing our final conclusions , we would like to comment on the effect of radiative damping on the moduli dynamics . as discussed in  @xcite , radiative damping occurs at an esp , where the number of massless states and , hence , the quantum mechanical amplitude to produce light particles is greatly enhanced . this leads to radiative `` friction '' on the moduli dynamics , damping their motion . thus far in this paper we have ignored this effect , considering only gravitational damping . could the inclusion of radiative damping change our conclusions , perhaps trapping the moduli at the esp ? the answer is no . the results reported in  @xcite show that although radiative damping is a strong effect , it requires the moduli to oscillate several times around the esp before their amplitude is substantially reduced . however , as is clear from figures @xmath143 and @xmath145 , the transition of @xmath69 and @xmath68 from the vicinity of the esp to that of the isolated minimum occurs immediately upon collision , within less than one oscillation ( note that the small amplitude oscillations of @xmath87 in figure @xmath145 all occur prior to the collision and not near the esp ) . furthermore , they never return to the esp . it is safe to conclude , therefore , that radiative damping at the esp simply has no opportunity to effect the moduli dynamics and can be safely ignored for the process under discussion in this paper . in fact , radiative damping helps enable our transition to the isolated minimum . the reason is that there is also an enhanced number of light states at the local minimum . once trapped in its vicinity , radiative damping will work with gravitational damping to quickly bring the moduli to rest at the isolated minimum . to conclude : in an expanding frw spacetime , the collision of a five - brane with an end of the @xmath0 interval passes through the small instanton point , described by the union of a vector bundle @xmath58 and a torsion free sheaf over the five - brane curve , and then continues to an isolated minimum of the non - perturbative superpotential where the vector bundle is smooth . gravitational ( and radiative ) damping traps the moduli at the isolated minimum , thus completing the small instanton phase transition to the smooth vector bundle @xmath20 . the moduli are _ not _ attracted to the esp in this process . it has been demonstrated in this paper that not only is a small instanton phase transition mathematically consistent , as shown in  @xcite , but it can also be realized dynamically . the results were presented for a specific choice of parameters and initial conditions . however , we have explored the dynamics of phase transitions for a range of parameters and initial conditions and found that the transition is completed for many different values . however , this is not always the case . some choices do lead to trapping at the esp , presumably because the energy is either too small to roll into the valley of an isolated minimum or so large that it rolls out again . be this as it may , initial conditions leading to a completed phase transition are ubiquitous and demonstrate that a nonperturbative addition to the potential can dramatically alter the results of  @xcite . we are grateful to volker braun for discussions of the intricacies of mathematica . the work of evgeny buchbinder is supported by nsf grant phy-0070928 . the research of alexander borisov and burt ovrut is supported in part by the department of physics and the math / physics research group at the university of pennsylvania under cooperative research agreement de - fg02 - 95er40893 with the u.  s.  department of energy and an nsf focused research grant dms0139799 for `` the geometry of superstrings . 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the small instanton transition of a five - brane colliding with one end of the @xmath0 interval in heterotic @xmath1-theory is discussed , with emphasis on the transition moduli , their potential function and the associated non - perturbative superpotential . using numerical methods , 
 the equations of motion of these moduli coupled to an expanding friedmann - robertson - walker spacetime are solved including non - perturbative interactions . 
 it is shown that the five - brane collides with the end of the interval at a small instanton . 
 however , the moduli then continue to evolve to an isolated minimum of the potential , where they are trapped by gravitational damping . 
 the torsion free sheaf at the small instanton is `` smoothed out '' into a vector bundle at the isolated minimum , thus dynamically completing the small instanton phase transition . 
 radiative damping at the origin of moduli space is discussed and shown to be insufficient to trap the moduli at the small instanton point .