Mathematica Package for validating effective string theory solution

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 $$B_{\mu\nu}$$: $$H_{\mu\nu\rho}=\partial_{\mu}B_{\nu\rho}+\partial_{\nu}B_{\rho\mu}+\partial_{\rho}B_{\mu\nu}$$

• Is the function $\Phi$ given as a formula? Commented Jun 10 at 2:54
• @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 $\Phi(r)=Q*r,\quad Q\in \mathbb{R}$ Commented Jun 10 at 5:33
• 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? Commented Jun 10 at 17:21
• @WolfgangBangerth Yes, Of course Commented Jun 12 at 12:45
• In that case, you can do all you need to do in symbolic math software such as Maple or Mathematica. Commented Jun 13 at 0:01