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January 8, 2005


Well, seems we’re on a bit of a Cosmic Strings kick. Koji Hashimoto, who commented at some length on my last post on the subject, has a lovely new paper out with Ami Hanany.

In it, they take up the question of why the reconnection probability for colliding D-strings is so much smaller than for vortex strings. (For fundamental strings, the reconnection probability, at weak string coupling, is even more suppressed because it’s essentially a perturbative string effect.)

Their approach is to study reconnection from the point of view of the worldvolume effective field theory on a pair of long straight strings, which are nearly parallel, and moving towards each other with relative transverse velocity v r1v_r\ll 1.

A pair of D-strings is described by a 1+1 dimensional SU(2)SU(2) supersymmetric gauge theory. The bosonic part of the action is

(1)S=2πl s 2g sdtdxTr[14F μνF μν12D μΦ iD μΦ i14[Φ i,Φ j] 2] S = \frac{2\pi l_s^2}{g_s} \int dt dx Tr\left[-\frac{1}{4} F_{\mu\nu}F^{\mu\nu} - \frac{1}{2} D_\mu \Phi_i D^\mu\Phi^i - \frac{1}{4} [\Phi_i,\Phi_j]^2\right]

where the Φ i\Phi_i, i=2,,9i=2,\dots,9 are the adjoint scalars whose eigenvalues encode the transverse separations of the branes.

The remarkable fact about this system is that there exists a classical solution representing the two strings passing through one another without reconnecting,

(2)A μ=0,2πl sΦ 2=(tan(θ/2)x 0 0 tan(θ/2)x),2πl sΦ 3=(v rt/2 0 0 v rt/2) A_\mu=0,\quad 2\pi l_s \Phi_2 = \left(\array{ \tan(\theta/2)x & 0\\ 0 & -\tan(\theta/2)x }\right),\quad 2\pi l_s \Phi_3 = \left(\array{ v_r t/2 & 0\\ 0 & -v_r t/2 }\right)

where θ\theta is the relative angle between them, and v rv_r is the relative velocity.

Reconnection, in this system, is a quantum phenomenon. Consider two static strings, separated by a distance yy (i.e., replace Φ 3\Phi_3 above by 2πl sΦ 3=diag(y/2,y/2)2\pi l_s \Phi_3=\text{diag}(y/2, -y/2)). When you analyze the spectrum of small fluctuations about this configuration, you find the lowest mode has a frequency-squared,

(3)ω 2=θ2πl s 2+y 2(2πl s 2) 2 \omega^2 = -\frac{\theta}{2\pi l_s^2} +\frac{y^2}{(2\pi l_s^2)^2}

which is tachyonic for sufficiently small yy. The spatial dependence of this mode is localized near x=0x=0 and, at y=0y=0 takes a simple Gaussian form (in A 0=0A_0=0 gauge),

(4)Φ 2=T(t)2(0 1 1 0)exp(tan(θ/2)2πl sx 2),A 1=T(t)2(0 i i 0)exp(tan(θ/2)2πl sx 2) \Phi_2 = \frac{T(t)}{2} \left(\array{0 & 1\\ 1 & 0}\right)\exp\left(-\frac{\tan(\theta/2)}{2\pi l_s}x^2\right),\quad A_1 = \frac{T(t)}{2} \left(\array{0 & -i\\ i & 0}\right)\exp\left(-\frac{\tan(\theta/2)}{2\pi l_s}x^2\right)

This mode is the well-known worldsheet tachyon for branes at angles. The condensation of this mode represents reconnection.

Since the strings are moving, there’s only a finite amount of time during which the wave functions Ψ(T)\Psi(T) can spread out, before the mode again ceases to be tachyonic. The reconnection probability is given by the probability of finding

(5)T>θ2πl s 2 T\gt \sqrt{ \frac{\theta}{2\pi l_s^2} }

which they proceed to try to estimate.

Vortex strings have a somewhat more complicated realization in string theory. The worldvolume effective action, according to Hanany and Tong, has a bosonic part

(6)S v=dtdxTr[14g 2F μνF μνD μZ D μZD μϕ D μϕg 22(ϕ ϕ[Z,Z ]r) 2] S_v = \int dt dx Tr \left[-\frac{1}{4g^2}F_{\mu\nu}F^{\mu\nu} - D_\mu Z^\dagger D^\mu Z - D_\mu \phi^\dagger D^\mu \phi - \frac{g^2}{2}\left(\phi^\dagger\phi -[Z,Z^\dagger] - r\right)^2\right]

Here ZΦ 2+iΦ 3Z\propto \Phi_2+i\Phi_3 is a complex scalar in the adjoint, ϕ\phi is a scalar in the fundamental representation of SU(2)SU(2), and rr is a Fayet-Iliopoulos parameter. The gauge coupling and Fayet-Iliopoulos parameter of the (N=2 supersymmetric) 4D abelian Higgs model are

(7)e AH 2=2πr,r AH=1(2π) 3l s 4g 2 e_{\text{AH}}^2 = \frac{2\pi}{r},\quad r_{\text{AH}} =\frac{1}{(2\pi)^3l_s^4 g^2}

Unlike the previous case, there is no classical solution to this system which corresponds to the two strings passing through each other without reconnecting. The nonzero Fayet-Iliopoulos parameter prevents you from finding solutions with [Φ 2,Φ 3]=0[\Phi_2,\Phi_3]=0.

At low energies, the dynamics of these vortex strings reduces to a 1+1 dimensional nonlinear σ\sigma-model whose target space is the space of solutions to

(8)ϕ ϕ[Z,Z ]r=0 \phi^\dagger\phi -[Z,Z^\dagger] - r = 0

This space is a smooth one-dimensional complex manifold (a smoothed cone of deficit angle π\pi). Classical trajectories of strings in this background correspond to reconnection of the original vortex strings. This happens with probability 1, as long as the moduli space approximation holds (i.e. for sufficiently low velocities and for sufficiently small θ\theta).

There’s actually believed to be a cutoff on the relative velocity, beyond which the reconnection probability for vortex strings drops. Hashimoto and Hanany use their model to provide an estimate of this cutoff, which they say looks like

(9)v r<sin(θ/2)1+sin 2(θ/2) v_r \lt \frac{\sin(\theta/2)}{\sqrt{1+\sin^2(\theta/2)}}
Posted by distler at January 8, 2005 1:21 AM

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