Based on my recent study on non-Abelian gauge theory in physics, I encounter an identity that should be correct physically but I don't know how to prove it mathematically.
Consider a $n\times n$ unitary matrix $U(x,y)\in U(n)$ smoothly depending on two real parameters $x,y$, let $u_{ij}$ be its entities, then does the following relation $$\sum_{i,j=1}^n (\partial_xu_{ij}^*)(\partial_yu_{ij})=\sum_{i,j=1}^n (\partial_xu_{ij})(\partial_yu_{ij}^*)$$ hold ? Which means the sum is a real number (where $u^*$ means the complex conjugate). And how to prove this ?
Thank you very much.
The quantity you're considering on the left is $ {\rm tr}(\partial_x U^* \partial_y U)$.
Since $U U^* = I$ for all $x,y$, the product rule tells us that
$$ \partial_x U \ U^* + U\ \partial_x U^* = 0$$
and thus $$\partial_x U^* = -U^*\ \partial_x U\ U^*.$$
Very similarly we can obtain $$\partial_y U = -U\ \partial_y U^*\ U.$$ Substituting these in, we have (using $U^* U = U U^* = I$ and the cyclic invariance of the trace)
$$ \begin{align} {\rm tr}(\partial_x U^* \partial_y U) &= {\rm tr}(U^*\ \partial_x U\ U^*U\ \partial_y U^*\ U) \\&= {\rm tr} (U^*\ \partial_x U\ \ \partial_y U^*\ U) \\&= {\rm tr} (U U^*\ \partial_x U\ \ \partial_y U^*) \\&= {\rm tr} (\partial_x U\ \ \partial_y U^*)\end{align}$$ which is your RHS.