Musings

Headrick

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 $\mathbb{C}/\mathbb{Z}_n$ 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 $\mathbb{C}$). 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 $\mathbb{C}/\mathbb{Z}_n$ 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 $(n,-1)$. The D-term constraint, $n|\phi_1|^2-|\phi_2|^2=r(\mu)$ means that, for $r\gg 0$, the theory corresponds to the $\mathbb{C}/\mathbb{Z}_n$ orbifold, whereas for $r\ll 0$, it corresponds to $\mathbb{C}$. And $r(\mu)$ decreases under renormalization group flow…