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I am asking for Mathematica package that given an input of: symmetric matrix $G_{\mu\nu}$, antisymmetric matrix $B_{\mu\nu}$ and a scalar function $\Phi$ will check whether it is a solution to the one-loop string theory constraints of Weyl invariance: $\beta_{\mu\nu}(G)= \beta_{\mu\nu}(B) = \beta(\Phi)=0$ which is given in David Tong lectures section 7.2.3 or more explicitely for non-critical string theory with cosmological constant $\Lambda$ (which described in section 7.4.4) are given by:

$$R_{\mu\nu}+2\nabla_{\mu}\nabla_{\nu}\Phi-\frac{1}{4}H_{\mu\lambda k}H_{\nu}^{\lambda k}=0$$

$$-\frac{1}{2}\nabla^{\lambda}H_{\lambda \mu \nu}+\nabla^{\lambda}\Phi H_{\lambda \mu \nu}=0$$

$$\frac{\Lambda}{2}-\frac{1}{2}\nabla^2\Phi+\nabla_{\mu}\Phi\nabla^{\mu}\Phi-\frac{1}{24}H_{\mu\nu\lambda}H^{\mu\nu\lambda}=0$$

Where $H$ is a 3-form given in terms of antisymettric $B_{\mu\nu}$: $$H_{\mu\nu\rho}=\partial_{\mu}B_{\nu\rho}+\partial_{\nu}B_{\rho\mu}+\partial_{\rho}B_{\mu\nu}$$

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  • $\begingroup$ Is the function $\Phi$ given as a formula? $\endgroup$
    – Wolfgang Bangerth
    Commented Jun 10 at 2:54
  • $\begingroup$ @WolfgangBangerth yes it is a function of a given coordinate system, for example in the system $(t,r,\theta,\phi)$ the linear dilaton is given by $\Phi(r)=Q*r,\quad Q\in \mathbb{R}$ $\endgroup$
    – Daniel Vainshtein
    Commented Jun 10 at 5:33
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    $\begingroup$ But then you can compute all derivatives. Don't you just have to make sure the left and right hand sides of these equations all match? $\endgroup$
    – Wolfgang Bangerth
    Commented Jun 10 at 17:21
  • $\begingroup$ @WolfgangBangerth Yes, Of course $\endgroup$
    – Daniel Vainshtein
    Commented Jun 12 at 12:45
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    $\begingroup$ In that case, you can do all you need to do in symbolic math software such as Maple or Mathematica. $\endgroup$
    – Wolfgang Bangerth
    Commented Jun 13 at 0:01

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