## February 16, 2004

We had Matt Headrick visting last week. He gave a really nice talk about his work on on the Adams, Polchinski & Silverstein conjecture for the fate of the $ℂ/{ℤ}_{n}$ orbifold.

APS noted that this nonsupersymmetric orbifold has a closed string tachyon in the twisted sector, and conjectured that it would decay, via condensation of this tachyon, into a supersymmetric ground state (usually, the unorbifolded $ℂ$). They were able to find evidence for this conjecture by studying a D-brane probe in this background, and noting that the closed-string perturbation would tend to drive the nonsupersymmetric quiver gauge theory of the D-brane in a $ℂ/{ℤ}_{n}$ background to flow, via the renormalization group, towards a supersymmetric one.

Matt (and also Gregory and Harvey) studied the late-time behaviour, where the probe analysis is untrustworthy, but where supergravity should be a good approximation. The 2+1 dimensional dilaton gravity problem turns out to be soluble, leading to a beautiful picture of the final state of the decay of this orbifold.

Is there any hope to connect the early- and late time behaviours? It’s not clear that there’s a sensible stringy formalism for doing so. But there’s an old paper by Vafa, which proposes a worldsheet renormalization group flow which interpolates between the early and late-time behaviours: consider a gauged linear $\sigma$-model, with two charged fields, with charges $\left(n,-1\right)$. The D-term constraint, $n\mid {\varphi }_{1}{\mid }^{2}-\mid {\varphi }_{2}{\mid }^{2}=r\left(\mu \right)$ means that, for $r\gg 0$, the theory corresponds to the $ℂ/{ℤ}_{n}$ orbifold, whereas for $r\ll 0$, it corresponds to $ℂ$. And $r\left(\mu \right)$ decreases under renormalization group flow…

Posted by distler at February 16, 2004 12:05 AM

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