For a reasearch project I am studying the paper by L. Alvarez-Gaumé, D. Freedman and S. Mukhi called "The Background Field Method and the UV Structure of the Supersymmetric Nonlinear $\sigma$ model" (doi: https://doi.org/10.1016/0003-4916(81)90006-3). I don't know how to obtain the correct covariant expansion for the fermionic part of the supersymmetric action and I hope someone can help me with obtaining it.

My question regards the $\frac{1}{2}R_{ijkl}\partial_{\mu}\phi^l\xi^k(\bar{\psi^i}\gamma^{\mu}\psi^j)$ term in the second line of equation (2.20). It is obtained by expanding the mixed term in equation (1.4) using the background field expansion. If we omit the numerical factor in front ($i/2$) and the integral $\int d^2\xi$ then the background field expansion of this mixed term is equation (2.20): $$ g_{ij}(\phi)\bar{\psi^i} D\!\!\!\!/ \psi^j = [g_{ij}(\phi)-\frac{1}{3}R_{ik_1jk_2}\xi^{k_1}\xi^{k_2}]\bar{\psi^i}D\!\!\!\!/\psi^j + \frac{1}{2}R_{ijkl}\partial_{\mu}\phi^l\xi^k(\bar{\psi^i}\gamma^{\mu}\psi^j). $$

It is easy to obtain the first term on the right side, namely with the covariant expansion of $g_{ij}$ in equation (2.16) one easily sees how this term arises. My problem is with the $\frac{1}{2}R_{ijkl}\partial_{\mu}\phi^l\xi^k(\bar{\psi^i}\gamma^{\mu}\psi^j)$. In the parantheses below this equation it is said that contributions from the Jacobian occur in the expansion of $\partial_{\mu}\psi^i$. There is also a transformation given for $\psi^i$ to normal coordinates, namely: $$ \psi^i = \frac{\partial\phi^i}{\partial\xi^i}\psi^{-j} $$ but this equation given in the paper does not make sense. There are two $i$ indices on the right and one on the left, furthermore what is a negative index supposed to mean... I feel like the Jacobian that is refered to is the $\frac{\partial\phi^i}{\partial\xi^i}$ term in the expression above but I don't know what to do with it. Could someone point me in the right direction on how to obtain this $\frac{1}{2}R_{ijkl}\partial_{\mu}\phi^l\xi^k(\bar{\psi^i}\gamma^{\mu}\psi^j)$ term? I would appreciate it a lot since I've been stuck on it for over a month. Since I'm working on this alone I would appreciate some peer feedback and maybe some explicit formulas.

An overview of all quantities:

  • $R_{ijkl}$ is the Riemann curvature tensor of the target space manifold.
  • $g_{ij}$ is the target space metric.
  • $\phi^i$ is the background bosonic field that satisfies the field equations that follow from the Euler-Lagrange equation. In the paper the same quantity $\phi$ is used for an arbitrary bosonic field so take care there.
  • $\xi^i$ are the normal coordinates developed in section 2 of the paper. They are used to obtain a covariant, perturbative expression for all terms in the nonlinear sigma model (see for example equation (2.15) for the covariant expansion of any second rank tensor).
  • $\psi^i$ is the fermionic field and there no background fermion field.
  • $\gamma^{\mu}$ are the two-dimensional gamma matrices (conventions are in section 6).
  • a slashed quantity means contraction with gamma matrices $\gamma^{\mu}$.
  • a barred quantity means $\bar{\psi^i} = \psi^*\gamma^0$ where a star is the complex conjugate.
  • $i,j,\dots$ indices refer to the ten-dimensional target space.
  • $\mu,\nu,\dots$ indices refer to the two-dimensional flat worldsheet.
  • $D_{\mu}$ is the covariant derivative defined as $D_{\mu}\psi^j=\partial_{\mu}\psi^j + \Gamma^j_{kl}\partial_{\mu}\phi^k\psi^l$ (where $\Gamma$ is the Christoffel connection)


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