# Topological sigma model, Euler Lagrange equations

1. May 29, 2012

### physicus

1. The problem statement, all variables and given/known data
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: $H^i_\alpha=\partial_\alpha u^i \epsilon_{\alpha\beta}J^i{}_j\partial^\beta u^j$

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

3. 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.

Best regards, physicus

Last edited: May 29, 2012
2. May 29, 2012

### 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.

3. May 29, 2012

### physicus

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