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An exact beta function exists for Super-Yang-Mills theories in 4D without matter - the so-called NSVZ beta function.

Does a similar exact beta-function exist in gravity or supergravity theories? In string theory? Please provide references too.

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I will present the simplest example of beta functions arising in string theory, specifically within bosonic string theory. The states transform in the $24 \otimes 24$ representation of $SO(24)$, equivalent to three irreducible representations; schematically,

$$(\mathrm{traceless \;symmetric} )\otimes (\mathrm{antisymmetric}) \otimes (\mathrm{singlet})$$

To each we associate a massless field, the scalar dilaton $\Phi(X)$, a field $G_{\mu\nu}(X)$ and another commonly denoted the Kalb-Ramond field, $B_{\mu\nu}$ which can be interpreted as a generalization of the 4-potential in electromagnetism. Note these fields 'live' on the worldsheet of the string. The action of a string in the background of these fields is given by,

$$S = \frac{1}{4\pi \alpha'}\int \! \mathrm{d}^2 \sigma \, \sqrt{|g|} \, \left[ G_{\mu\nu} \partial_\alpha X^\mu \partial_\beta X^\nu g^{\alpha \beta} +i B_{\mu \nu}\partial_\alpha X^\mu \partial_\beta X^\nu \epsilon^{\alpha \beta} + \alpha' \Phi R^{(2)}\right]$$

We can compute the beta functions of the string theory in the standard manner, which are given by$^{\dagger}$,

$$\beta_{\mu\nu}(G) = \alpha'R_{\mu\nu} + 2\alpha'\nabla_\mu\nabla_\nu \Phi - \frac{\alpha'}{4}H_{\mu\lambda \kappa}H^{\lambda \kappa}_\nu$$

$$\beta_{\mu\nu}(B) = -\frac{\alpha'}{2}\nabla^\lambda H_{\lambda\mu\nu} + \alpha' \nabla^\lambda \Phi H_{\lambda \mu \nu}$$

$$\beta(\Phi) = -\frac{\alpha'}{2}\nabla^2 \Phi + \alpha' \nabla_\mu \Phi \nabla^\mu \Phi -\frac{\alpha'}{24}H_{\mu\nu\lambda}H^{\mu\nu\lambda}$$

To preserve scale invariance, we must demand these are all vanishing. We can therefore construct an action, known as the low energy effective action of bosonic string theory, whose equations of motion are equivalent to the beta functions, i.e.

$$S=\frac{1}{2\kappa^2_0} \int \! \mathrm{d}^{26}X \, \sqrt{|G|} \, e^{-2\Phi} \, \left( R - \frac{1}{12}H_{\mu\nu\lambda}H^{\mu\nu\lambda} + 4 (\partial_\mu \Phi)^2 \right)$$

Therefore, the beta functions can be viewed as equations of motion. Notice the action takes the convenient form of the Einstein-Hilbert action, with a 2-form and scalar field coupled to gravity. (A transformation to Einstein frame makes this evident.)


$\dagger$ Beta functions only to one loop order. Higher order calculations give rise to further corrections to the Einstein field equations; at one loop order they are in agreement, $\alpha' R_{\mu\nu}=0$. The field $H$ is a field-strength; in forms, $H = \mathrm{d}B$.


Resources:

  1. For a complete derivation of the beta functions, see Sigma Models and String Theory TASI lecture notes by Callan and Thorlacius.
  2. Becker, Becker and Schwarz's M-Theory and String Theory provide a discussion of solutions to the equations of motion of low energy effective actions, include higher-dimensional black holes.
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  • $\begingroup$ Thank you very much for the answer. What I originally was searching for examples of exact/non-perturbative beta functions in string theory, if any exist. If there are, I will be glad to know. $\endgroup$ – Ten Jun 13 '14 at 7:17
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@Ten The NSVZ beta function exists for theories with matter as well. Just read the scholarpedia article carefully. What happens is that the NSVZ beta function for the gauge coupling constants depends on the anomalous dimensions of the matter fields.

A very nice example is to consider $\mathcal{N}=4$ SYM theory and write it out as a $\mathcal{N}=1$ theory -- the spectrum then consists of one $\mathcal{N}=1$ vector multiplet and three chiral multiplets. Deform the superpotential into the most general cubic superpotential. Leigh and Strassler use the vanishing of the NSVZ beta function for the gauge coupling imposes an additional constraint that anomalous dimensions of the chiral scalars should vanish leading to a theory which is conformal with $\mathcal{N}=1$ supersymmetry. This theory generalises the beta-deformation of $\mathcal{N}=4$ SYM. There are many more examples in their paper. Two more papers by Arkani-Hamed and Murayama would provide further examples. Paper 1 and Paper 2.

(@Ten I realise that you want examples from string theory/supergravity but it is important to note the generality of the NSVZ beta function. So I will post my answer. I hope you don't mind. If so, let me know and I will delete my answer. Of course, one knows that any two-dimensional CFT has a vanishing beta function.)

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  • $\begingroup$ Thank you very much for the answer. Please see my comment above. $\endgroup$ – Ten Jun 13 '14 at 7:18

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