Topological sigma model, Euler Lagrange equations

[physicus]

Homework Statement

My question refers to the paper "Topological Sigma Models" by Edward Witten, which is available on the web after a quick google search. I am not allowed to include links in my posts, yet. I want to know how to get from equation (2.14) to (2.15).
We consider a theory of maps from a Riemann surface \Sigma with complex structure \varepsilon to a Riemannian manifold M with an almost complex structure J. h is the metric on \Sigma, g the metric on M.
The map \phi:\Sigma \to M is locally described by functions u^i(\sigma). H^{\alpha i} is a commuting field (\alpha = 1,2 is the tangent index to \Sigma and i=1,\ldots,n runs over a basis of \phi^*(T), which is the pullback of the tangent bundle of T to M). H^{\alpha i} obeys H^{\alpha i}=\varepsilon^\alpha{}_\beta {J^i}{}_j H^{\beta j}.
We consider the Lagrangian \mathcal{L}=\int d^2\sigma(-\frac{1}{4}H^{\alpha i}H_{\alpha i} + H^\alpha_i \partial_\alpha u^i + (\text{terms independent of H})).
H is a non-propagating field, since the Lagrangian does not depend on its derivative.
Using the Euler-Lagrange equations I want to show: H^i_\alpha=\partial_\alpha u^i \epsilon_{\alpha\beta}J^i{}_j\partial^\beta u^j

Homework Equations

\mathcal{L}=\int d^2\sigma(-\frac{1}{4}H^{\alpha i}H_{\alpha i} + H^\alpha_i \partial_\alpha u^i + (\text{terms independent of H}))
H^{\alpha i}=\varepsilon^\alpha{}_\beta {J^i}{}_j H^{\beta j}
\frac{\partial \mathcal{L}}{\partial H^\alpha_i(\sigma)}=0

The Attempt at a Solution

Since \mathcal{L} does not depend on the derivative of H, the Euler Lagrange equations simply state \frac{\partial \mathcal{L}}{\partial H^\alpha_i(\sigma)}=0. I tried to evaluate this:
\frac{\partial}{\partial H^\alpha_i(\sigma)}\left(\int d^2s(-\frac{1}{4}H^{\beta j}(s)H_{\beta j}(s) + H^\beta_j(s) \partial_\beta u^j(s))\right)
=\frac{\partial}{\partial H^\alpha_i(\sigma)}\left(\int d^2s(-\frac{1}{4}h_{\beta\gamma}g^{jk}H^{\beta}_k(s) H^{\gamma}_j(s)+ \epsilon^\beta{}_\gamma J_j{}^k H^\gamma_k(s) \partial_\beta u^j(s))\right) where the second equation of the realtions above is used
=-\frac{1}{4}h_{\beta\gamma}g^{jk}(h^{\beta\alpha} g_{ik} H^{\gamma}_{j}(\sigma) + H^{\beta}_{k}(\sigma) h^{\alpha\gamma} g_{jk}) + \varepsilon^\beta{}_\gamma J_j{}^k h^{\alpha\gamma} g_{ik} \partial_\beta u^{j}(\sigma)
=-\frac{1}{2}H^\alpha_i+\varepsilon^{\beta\alpha}J_{ji}\partial_\beta u^j(\sigma)

Unfortunately, that is not really close to the expression that I am looking for. Can someone find mistakes? I appreciate any help.

physicus

 

[fzero]

Use the constraint to write

$$-\frac{1}{4} H^{\alpha i}H_{\alpha i} + H_{\alpha i} \partial^\alpha u^i
=- \frac{1}{4} H^{\alpha i}H_{\alpha i} + \frac{1}{2} H_{\alpha i} \partial^\alpha u^i
- \frac{1}{2} \epsilon_{\beta\alpha} {J^i}_jH_{\alpha i} \partial^\beta u^j .$$

Note that the sign of the 3rd term changes when you transpose the indices on $\epsilon_{\beta\alpha}$ after computing the variation.

 

[physicus]

Thanks a lot, that was very helpful. I got it now.