Derivatives of parallel transport operator

Question

Let $M$ be a Riemannian manifold, $\sigma$ the world function, $E$ a vector bundle and $\nabla$ a covariant derivative. The parallel transport operator $P$ satisfies$$\sigma^\mu\nabla_\mu P=0.$$ Now let $F$ be the curvature of $\nabla$. For the calculation of the heat kernel coefficients of a generalized Laplacian it is relevant that $$[P_{;\mu}{}^\mu{}_\nu{}^\nu]=\frac12 F_{\mu\nu}F^{\mu\nu}$$ (see e.g. $(17.83)$ in DeWitt's Dynamical Theory of Groups and Fields). I assume that one way to get this equation is to calculate the LHS of $$[(\sigma^\alpha\nabla_\alpha P)_{;\mu}{}^\mu{}_\nu{}^\nu]=0$$ but this yields twelve terms and is a huge a mess. That being said, I would like to know 1) whether the calculation can be found in the literature and 2) whether there are more convenient ways to calculate $[P_{;\mu}{}^\mu{}_\nu{}^\nu]$ (and possibly higher derivatives).

Answer 1

In general, published papers don’t show all the steps of a tedious but straightforward calculation; they just give the final result.

You might as well calculate the more general result $[P_{;abcd}]$, which you can find in this review by Barvinsky and Vilkovisky if it’s not in DeWitt’s book.

Expand $[(g^{mn}\sigma_{;m} P_{;n})_{;abcd}]=0$. There will be $2^4$ or $16$ terms after taking the four covariant derivatives of the product.

One vanishes because $[\sigma_{;a}]=0$, six vanish because $[\sigma_{;abc}]=0$, and one vanishes because $[P_{;a}]=0$.

In four of the remaining eight terms, use known results for $[\sigma_{;abcd}]$ (in terms of $R_{abcd}$) and $[P_{;ab}]$ (in terms of $F_{ab}$). (These terms will vanish when you do your two contractions, either because you’ll contract a Riemann tensor on antisymmetric indices or you’ll get $R^{ab}F_{ab}$, which is a symmetric tensor times an antisymmetric one.)

Using the known result for $[\sigma_{;ab}]$ (in terms of $g_{ab}$), the other four terms reduce to a sum of permutations of $[P_{;abcd}]$. You’ll have to use the commutation rules for covariant derivatives to express $[P_{;bacd}]$, $[P_{;cabd}]$, and $[P_{;dabc}]$ in terms of $[P_{;abcd}]$ plus additional terms involving $R_{abcd}$ and $F_{ab}$. (This is the hardest part of the calculation.) Many of the extra terms will vanish in the coincidence limit and others will vanish when you do your two contractions.

This hand-calculation should take less than six pages of paper, and you shouldn’t need to use a computer algebra system.

If you are mainly interested in the coincidence limits $[a_n]$ of the heat kernel coefficients themselves, rather than their covariant derivatives, then there is an alternate approach used by Gilkey. You can construct the most general possible form as some linear combination of a basis of invariants, and then determine the coefficients by specializing to a particular geometry. He uses a torus to calculate the 46 terms of $[a_3]$.