Musings

Eva Silvertein talked about her recent paper with McGreevy about tachyon condensation in big bang/big crunch backgrounds. She and collaborators have written a series of papers attempting to lay out the case for localized tachyon condensation. Imagine you have a compactification on a manifold $M$ with a nontrivial first homology, $H^1(M,\mathbb{R})$. If

1. There’s some region where the length, $l$, of the minimal geodesic around this nontrivial cycle is substringy, $l\lt l_s$.
2. The spin structure of the spacetime fermions, restricted to this $S^1$ have the bounding spin structure.

then there is a Scherk-Schwarz tachyon localized near this minimal geodesic. From the behaviour under worldsheet RG flow, and by analogy with Liouville theory, they argued that this leads the manifold to pinch off this cycle.

In the more recent paper with McGreevy, this setup is applied to cosmological scenarios, where one has an FRW cosmology, in which the radius of the circle is time-dependent $$ds^2 = -dt^2 + a(t)^2 d\vec{x}^2 + ds_\perp^2$$ (one of the $\vec{x}$ directions is compactified on a circle). They, again argue that at early (late) times, when $l(t)\lt l_s$, tachyon condensation — localized in time — cuts off the classical big bang (crunch) singularity, replacing it with an intrinsically stringy regime.

A very attractive story, but my inability to make sense of the details kept me from writing about this before. I can’t say that my understanding has improved enough to say anything really insightful now.

Eric Verlinde talked about his paper with Craps and Sethi. They consider a linear dilaton background, where the dilaton increases linearly along a null direction (producing no net shift in the worldsheet CFT’s central charge). The worldsheet stress tensor looks like $$T= -\partial X^+ \partial X^- + \frac{1}{2} \partial X_i \partial X_i + Q\partial^2 X^+$$ In the Einstein frame, you clearly see the null singularity $$d s_E^2 = -2d u d v + u d x_i^2,\quad \phi(u) = -2\log u$$ at $u=0$, where $u=e^{Q x^+ /2}$, $v=2 x^- /Q$. In the string frame, the metric is flat, but strings become infinitely strongly coupled at $u=0$. Lifting to M-theory, one gets $$\array{ \arrayopts{\colalign{right left}} d s^2_{11} &= e^{-\frac{4}{3}Q x^+} d y^2 + e^{\frac{2}{3}Q x^+}(-2 d x^+ d x^- + d x_i^2)\\ &= u^{-2} d y^2 -2d u d v +u d x_i^2 }$$ which also has a null singularity at $u=0$, where $u=e^{\frac{2}{3} Q x^+}$, $v= \frac{3}{2Q}x^+$.

Verlinde and collaborators write down a DLCQ matrix string theory for this background, $$S = Tr \int d\sigma d\tau (D X_i)^2 + \theta^T D \theta + e^{2Q\tau/R} F + e^{-2Q\tau/R} \sum [X_i,X_j]^2 - e^{-Q\tau/R} \theta^T \Gamma^i [X_i,\theta]$$ where $p^+ = N/R$, $x^-=x^-+2\pi R$. This is a $U(N)$ supersymmetric YM theory on a nontrivial 1+1 dimensional manifold, a piece of Milne space. At early times, the effective Yang Mills theory is weakly coupled. At late times, the weakly-coupled string theory is valid. This is an interesting nonperturbative model for the big bang (albeit, with a null, rather than a spacelike singularity).

On the other hand, the Milne space is non-Hausdorff, so the physics of the SYM theory in this background is going to be a bit … interesting.