# How to get the equations of motion with the Kalb-Ramond 3-form field strength by varying the bosonic string action?

+ 1 like - 0 dislike
144 views

I have the bosonic string action:

$S= - {1 \over 4 \pi \alpha'} \int_\Sigma d^2\sigma \sqrt{-g} g^{ab} \partial_a X^\mu \partial_b X^\nu G_{\mu\nu}(X)\, +\epsilon^{ab} B_{\mu\nu}(X) \partial_a X^\mu \partial_b X^\nu$

where $G_{\mu\nu}$is the metric for the background spacetime, $g_{ab}$is the worldsheet metric, $B_{\mu\nu}$is the Kalb-Ramond 2-form field and $\epsilon^{ab}$is the totally antisymmetric tensor density.

I'm supposed to get the equations of motion:

$\partial_a \partial^a X^\mu + \Gamma^\mu_{\nu\rho}\partial_a X^\nu \partial^a X^\rho - \frac{1}{2} H^\mu_{\nu\rho} \partial_a X^\nu \partial_b X^\rho \epsilon^{ab}= 0$

where $\Gamma^\mu_{\nu\rho}$are the connection components for the background spacetime and $H^{\mu}_{\mu\rho} = G^{\mu\xi}H_{\xi\nu\rho}$are the components of the Kalb-Ramond field strength $H=dB$. When I vary the second term of the action I get a term of the form  $\nabla_a B_{\mu\nu} \partial_b X^\nu \epsilon^{ab} = \nabla_\rho B_{\mu\nu} \partial_a X^\rho \partial_bX^\nu \epsilon^{ab}$, but I don't know how to get the other terms of $H_{\mu\nu\rho} = \frac{1}{2} \nabla_{[\mu} B_{\nu\rho]}$.

What I do instead is assume the worldsheet is the boundary of some 3-dimensional region. Then the second term of the action turns into an integral of $H$ over this region and the variation gives the desired equations of motions.

This seems a bit hacky, as I'm not sure I can assume the worldsheed is a boundary even for a closed string, and obviously not for the open string. I'd also like to analyze the two terms of the action in parallel. Is there a way to get the equations of motion in this form by direct variation of the action?

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysic$\varnothing$OverflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.