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Consider $bc$-system which is 2-dimensional CFT of fermions:

$S = \int_\Sigma d^2 z \ b \bar{\partial} c + h.c. $

where $\Sigma$ - 2-dimensional manifold of genus $p$, fields $b, c$ have dimensions $(\lambda, 0)$ and $(1-\lambda, 0)$ respectively.

The question is to find number of globally defined solutions of equations of motion $\bar{\partial} b = 0, \bar{\partial} c = 0$ and for Sphere and Torus find these solutions explicitly.

The problem is that I don't even understand, what does "globally defined" mean, and why solutions may be not globally defined, what is the problem here.

I've been told that it's somehow connected to Riemann-Roch theorem, but I don't understand how.

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Comments to the question (v2):

  1. Given a manifold $M$, the word locally is associated with an open neighborhood $U\subseteq M$, while the word globally refers to the whole manifold $M$.

  2. If $E\to M$ is a fiber bundle, let $E_{|U}\to U$ denote the restriction of the bundle $E\to M$ to the neighborhood $U\subseteq M$. For instance, if $s\in\Gamma(E_{|U}\to U)$ is a locally-defined section, it may or may not be possible to extend $s$ to a globally defined section $\tilde{s}\in\Gamma(E\to U)$ such that the restriction $\tilde{s}_{|U}=s$.

  3. Now the ghost fields $b$ and $c$ are sections in appropriate vector bundles over a Riemann surface $M=\Sigma$.

  4. The Riemann-Roch theorem (and more generally, the Atiyah–Singer index theorem) yields information about the dimension of the space of globally defined sections over certain vector bundles.

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