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The Wilson line in Abelian Chern-Simons theory is

$$\langle\exp\left(i\int dt\dot{x}^{\mu}A_{\mu}\right)\rangle=\int\mathcal{D}A\exp\left(i\int dt\dot{x}^{\mu}A_{\mu}+\frac{i}{4\pi}\int A\wedge F\right).$$

The above path-integral is Gaussian and can be solved. I want to generalize this on a worldsheet, but I don't know much about string theory.

Let $B_{\mu\nu}$ be a $2$-form field, and $H_{\mu\nu\rho}$ is a $3$-form field such that $H=dB$. $B$ and $H$ are defined in the target space.

Does it make sense to write the following action,

$$\int dzd\bar{z} B_{\mu\nu}\epsilon^{\alpha\beta}\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}+\int B\wedge H,$$

where $B=B(X)$?

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  • $\begingroup$ FWIW, the 5-form $B\wedge H$ can only be integrated over a 5-dim manifold. $\endgroup$
    – Qmechanic ♦
    Commented Jan 18, 2019 at 13:18
  • $\begingroup$ @Qmechanic Yes I want the target space to be five dimensional. $\endgroup$
    – Xenomorph
    Commented Jan 18, 2019 at 13:48

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Two-form fields like your $B_{\mu\nu}$ are know as Kalb-Ramond fields. There is a Wikipwedia article on them, and a large literature.

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  • $\begingroup$ Thanks. Could you please say something about the action? $\endgroup$
    – Xenomorph
    Commented Jan 18, 2019 at 13:50
  • $\begingroup$ I don't use these things myself, so I have little useful to say. But knowing their name should give you access to the literature where your question may be addressed. $\endgroup$
    – mike stone
    Commented Jan 18, 2019 at 13:53

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