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In string theory, or supersymmetric gauge theory, they often calculate the partition function on specific Riemann surfaces, such as torus, cylinder, Klein bottle, Mobius strip.

Refer to the Polchinski chapter 7, these surfaces are Euler number zero type surfaces.

In organizing them, they can be parametrized by the complex plane with different period and different boundary conditions.

Why do we distinguish them and what is the physical interpretation of each surface?

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  • $\begingroup$ One remark: not all of these surfaces are Riemann surfaces. They all locally look like the complex plane, but for the Klein bottle and the Möbius strip this cannot be done in such a way that the change of coordinate maps are complex differentiable $\endgroup$ – doetoe Oct 11 '14 at 6:45
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Two-dimensional surfaces are the Feynman diagrams of string theory.

In quantum field theory one sums over one dimensional objects in order to calculate quantities like scattering cross sections or decay rates. This is due to the fact that in this framework, particles are represented as zero dimensional objects, i.e. their world lines are points.

In string theory, the fundamental objects are (as the name says) strings, i.e. one dimensional objects with two-dimensional "world lines". Specifying in- and out-states results in diagrams that are two-dimensional surfaces, and in order to calculate scattering amplitudes, one has to sum over surfaces of various topologies, which is analogous to the one-dimensional QFT case.

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