I've been taking a look at the paper arXiv:2202.07580 [math-ph] , which discusses the so-called Wetterich equation in Lorentzian manifolds. I believe the Physics details behind the Wetterich equation are irrelevant, so I'll skip them here. On p. 3, the authors write
[...] the Wetterich equation involves the $k$-derivative of $\Gamma_k(\phi) = \tilde{\Gamma}_k(\phi) - Q_k(\phi)$ and it takes the well-known form $$\partial_k \Gamma_k = \frac{i}{2} \left\langle (\Gamma^{(2)}_k + Q^{(2)}_k)^{-1}, \partial_k Q^{(2)}_k \right\rangle_2,$$ where $\langle \cdot, \cdot \rangle_2$ is the standard pairing on $\mathcal{M}^2 = \mathcal{M} \times\mathcal{M}$.
Question: What is meant by "the standard pairing on $\mathcal{M}^2 = \mathcal{M} \times\mathcal{M}$?
Extra information (if needed): In this context, $\mathcal{M}$ is the spacetime manifold, $\Gamma_k$ is known as the effective average action and it is a functional of the fields $\phi$, which are distributions on spacetime. $Q_k$ is also a functional of the fields. In Physics texts, it is common to see the "standard pairing" written as a functional trace, i.e., $\left\langle (\Gamma^{(2)}_k + Q^{(2)}_k)^{-1}, \partial_k Q^{(2)}_k \right\rangle_2 = \mathrm{Tr}\left[(\Gamma^{(2)}_k + Q^{(2)}_k)^{-1} \partial_k Q^{(2)}_k\right]$, where the objects inside the trace are understood as if they were matrices with entries labeled by spacetime points $x$ and $y$. Symbolically, given $A$ and $B$ functionals of the fields, $$\mathrm{Tr}[A^{(2)}B^{(2)}] \equiv \int_{\mathcal{M}^2} \frac{\delta^2 A}{\delta \phi(x) \delta \phi(y)} \frac{\delta^2 B}{\delta \phi(y) \delta \phi(x)} \sqrt{-g(y)}\mathrm{d}^d y \sqrt{-g(x)} \mathrm{d}^d x,$$ with $d$ being the dimension of spacetime and $\sqrt{-g(x)}\mathrm{d}^d x$ represents a volume element on the spacetime.