# Physics in torus, cylinder, Klein bottle and mobius strip

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?

• 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 – doetoe Oct 11 '14 at 6:45