Physics in torus, cylinder, Klein bottle and mobius strip

Question

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?

This post imported from StackExchange Physics at 2014-10-11 09:47 (UTC), posted by SE-user phy_math

Comments

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

This post imported from StackExchange Physics at 2014-10-11 09:47 (UTC), posted by SE-user doetoe

Answer 1

This is a short answer just trying to give some insight on what you are asking. Please correct any false statements and augment my response if needed.

About the Tori: Well, depends on the context. For example the torus is a very usual type of compactifications we make. Actually the first thing on learns on string compactifications is the torus compactification where you assume that the internal dimensions have the corresponding geometry. Furthermore tori appear whenever we have the complex plane modded out by some lattice, something that appears quite often in string theory and susy gauge theories. Understanding Seiberg-Witten theory is all about understanding the what the two cycles in the torus actually represent (check Tachikawa's book on N=2 susy (http://arxiv.org/abs/1312.2684) or if you want I can send you some personal "decently" and pedagogically written notes)

About the Moebius strip and the Klein bottle: Well I will just give you an example. There exists a class of string theories which have unoriented strings. These theories are defined by unorientable worldsheets in the genus expansion. Now, it is easier to picture and understand this when you consider some orientifold. Imagine you have a string theory whose left and right sectors (movers) are identical. An obvious example is bosonic string theory or Type IIB. Then it is possible to construct a new theory as the quotient of some symmetry called worldsheet parity \(\tilde{\Omega}\)whose action exchanges the left and right movers. So, in this theory we have the following states identified

\(|A\rangle_L \otimes |B\rangle_R \equiv |A\rangle_R \otimes |B\rangle_L\),

and this identification poses a modification of the genus expansion.  For example (and related to the torus) the usual contribution is the torus which can be understood as: string states evolve and then they glue back together to the original state up to the action of the worldsheet parity \(\tilde{\Omega}\). Depending on wether we are dealing with closed string or open string we get, as a worldsheet geometry, a Klein bottle or a Moebius strip respectively! Try to draw it! There is much more to say but I will stop here for now!