September 15, 2008

Tidbit

One concept emphasized in Harry Collins’s talk at the Science in the 21st Century Conference was that of “tacit knowledge,” something brought home to me, later in the week, in a conversation with Rob Myers. He and collaborators noticed that, in AdS/CFT backgrounds which admit (at most) $\mathcal{N}=2$ supersymmetry in the UV, curvature-squared terms are allowed in the effective bulk supergravity, and that this modifies the lower bound on the ratio of shear viscosity to entropy density, $\eta/s$, in the boundary theory. They found $\frac{\eta}{s}\geq \frac{1}{4\pi} (1-4\lambda_{\text{GB}})$ where $\lambda_{\text{GB}}$ is the coefficient of the Gauss-Bonnet density in the bulk effective action. They also argued that

(1)$\lambda_{\text{GB}}\leq \frac{9}{100}$

because, otherwise, one finds acausal behaviour in the boundary SCFT.

What I didn’t know was that this bound was related to familiar bounds on the central charges $a,c$ of the SCFT. In a curved background, these are given by $\tensor{T}{_{\mu}_^{\mu}} = \frac{c}{16\pi^2} {(\text{Weyl})}^2 -\frac{a}{16\pi^2} (Euler)$ where \begin{aligned} {(\text{Weyl})}^2 &= R_{\mu\nu\lambda\rho}^2 -2 R_{\mu\nu}^2 +\tfrac{1}{3} R^2\\ (\text{Euler}) &= R_{\mu\nu\lambda\rho}^2 -4 R_{\mu\nu}^2 + R^2 \end{aligned} In a supersymmetric gauge theory, the coefficients, $a$, $c$ are one-loop exact, being related by supersymmetric Ward identities to certain R-current anomalies. For an $\mathcal{N}=2$ gauge theory, with gauge group, $\mathcal{G}$, and hypermultiplets in representation $R_{\text{hyper}}$,

(1)\begin{aligned} a &= \tfrac{1}{24}\left(5 dim(adj) + dim(R_{\text{hyper}})\right)\\ c &= \tfrac{1}{12}\left(2 dim(adj) + dim(R_{\text{hyper}})\right) \end{aligned}

and, for an $\mathcal{N}=1$ gauge theory, with gauge group, $\mathcal{G}$, and chiral multiplets in representation $R_{\text{chiral}}$,

(2)\begin{aligned} a &= \tfrac{1}{48}\left(9 dim(adj) + dim(R_{\text{chiral}})\right)\\ c &= \tfrac{1}{24}\left(3 dim(adj) + dim(R_{\text{chiral}})\right) \end{aligned}

In each case, $dim(R)$ and $dim(adj)$ are non-negative numbers. Hence we have inequalities

(3)\begin{aligned} \frac{1}{2} \leq \frac{a}{c} \leq \frac{5}{4} \end{aligned}

for $\mathcal{N}=2$ and

(4)\begin{aligned} \frac{1}{2} \leq \frac{a}{c} \leq \frac{3}{2} \end{aligned}

for $\mathcal{N}=1$.

While I stated the results for supersymmetric gauge theories, the bounds (3) and (4) follow from superconformal Ward identities, as shown in Appendix C of Hofman and Maldacena (see also Shapere and Tachikawa). For $\mathcal{N}=1,2$, these bound are saturated by free field theories (the lower bound, by free abelian vector multiplets, the upper bound by free massless chiral/hyper multiplets). If this continues to hold for $\mathcal{N}=0$, we can write

(5)$\frac{1}{3} \leq \frac{a}{c} \leq \frac{31}{18}$

where the lower limit comes from free scalar field theory and the upper limit from free Maxwell theory.

The connection that I didn’t make was that $a,c$ also depend on the coefficient of the curvature-squared terms in the bulk theory and, for Gauss-Bonnet gravity, one has

(6)\begin{aligned} a&\propto \left(3(1-4\lambda_{\text{GB}})^{1/2}-2\right)\\ c&\propto (1-4\lambda_{\text{GB}})^{1/2} \end{aligned}

so that the lower limit $a/c\geq 1/2$ corresponds to $\lambda_{\text{GB}}\leq 9/100$, as above.

Posted by distler at September 15, 2008 10:47 AM

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