# Generalization of Chern-Simons Wilson Line

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 $$B=B(X)$$?

• FWIW, the 5-form $B\wedge H$ can only be integrated over a 5-dim manifold. – Qmechanic Jan 18 '19 at 13:18
• @Qmechanic Yes I want the target space to be five dimensional. – Libertarian Monarchist Bot Jan 18 '19 at 13:48

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.