Questions about Neveu-Schwarz B-field

[synoe]

In string theory, the Neveu-Schwarz $B$-field appears in the action:
$$S_{NS}=\frac{1}{4\pi\alpha^\prime}\int d^2\xi\;\epsilon^{\mu\nu}B_{ij}\partial_\mu X^i\partial_\nu X^j.$$

In Polchinski's text, the antisymmetric tensor appears in the form of
$$\frac{1}{4\pi\alpha^\prime}\int d^2\xi\;\sqrt{\gamma}i\epsilon^{\mu\nu}B_{ij}\partial_\mu X^i\partial_\nu X^j,$$
where $\gamma$ is the world sheet metric.

(1)
These terms are same?
Terms apparently these terms seem to be called NS-NS 2-form, Wess-Zumino-Witten term, or just antisymmeteric tensor. These terminologies are the same meaning?

(2)
What's the definition of $\epsilon^{\mu\nu}$?

(3)
Is this term conformal invariant?
Could you show the invariance?

 

[john baez]

$\epsilon_{\mu \nu}$ is the Levi-Civita symbol. Click the link to learn about that.

You need the square root of the determinant of the metric in there to be doing the integral correctly, as far as I can tell.