This is probably a dumb question, but I'm a mathematician who's been trying to understand the equations of motion for Type IIB supergravity, and I'm not quite sure I understand what's going on with the Einstein equations. Specifically, I'm following the appendix A of this paper, and the equation I'm having trouble with is A.4. Here it is for clarity:

$$R_{\mu \nu} + 2 D_{\mu}D_{\nu}\Phi - \frac{1}{4}H_{\mu \nu}^2 = e^{2\Phi}\left[ \frac{1}{2}(F_1^2)_{\mu \nu} + \frac{1}{4}(F_3^2)_{\mu \nu} + \frac{1}{96}(F_5^2)_{\mu \nu} - \frac{1}{4}g_{\mu \nu}\left( F_1^2 + \frac{1}{6}F_3^2 \right )\right]$$

My understanding is that $F_1$ is a differential 1-form, which we would write in components as say $F_1 = f_i dx^i$, and that $F_1^2$ is defined then to be:
$$F_1^2 = f_i f_j g^{ij} $$
My problem is therefore that I expect $F_1^2$ to be a scalar. Indeed, I expect all the terms on the RHS of the equation to be scalars, and so I don't know what $(F_1^2)_{\mu \nu}$ means.

Small addendum: I am fairly sure that $D_{\mu }$ is the covariant derivative with respect to the metric connection, something I would normally call $\nabla_{\mu}$, but I just wanted to confirm this.

This post imported from StackExchange Physics at 2017-02-15 08:35 (UTC), posted by SE-user Mark B