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Natural Science Forum / Physics / Research / February 2006Modular Invariance
I am confused about the significance of modular invariance in Conformal Field Theory. What do statements like "a good conformal theory, has to be modular invariant", really mean? I understand that any CFT on the torus has to be modular invariant, to respect the discrete symmetries of the torus. If the conformal field theory is defined on the plane, does it still need to be modular invariant to be a "good" theory? Why? Thanks for any help, Marvin > I am confused about the significance of modular invariance in Conformal > Field Theory. What do statements like "a good conformal theory, has to [quoted text clipped - 4 lines] > If the conformal field theory is defined on the plane, does it still > need to be modular invariant to be a "good" theory? Why? Modular invariance on the sphere and on the torus imply that the theory can be extended to any Riemann surface (cf the original papers of Moore and Seiberg). There is a similar invariance under the mapping class group of each higher genus Riemann surface.  Signaturerusty Hi Rusty, Thanks for the reply. Modular invariance on the torus means invariance under the group SL(2,Z). I suppose there are other groups corresponding to higher genus surfaces. The mapping class group of the Riemann sphere is just 1. So if I want to do conformal field theory only on the plane/Riemann sphere, then I do not have to impose any discrete/modular symmetry on my theory. Is that correct? If I stay confined to the sphere, will such a theory be inconsistent/unphysical in any way? Thanks, Marvin > Hi Rusty, > [quoted text clipped - 12 lines] > If I stay confined to the sphere, will such a theory be > inconsistent/unphysical in any way? Apart from the mapping class groups, there are also the symmetries of N-point functions, namely the fundamental groups of the Riemann surfaces with the N points removed. Thus for example for the sphere one gets the braid groups. Invariance under the braid group for 4 and 5 point functions is another of the requirements of Moore and Seiberg. In examples the braid group appears through the monodromy of a system of complex partial differential equations with regular singular points. In the simplest cases, this reduces to the monodromy of the hypergeometric equation. Signaturerusty > If I stay confined to the sphere, will such a theory be > inconsistent/unphysical in any way? >From the point of view of CFT applied to statphys, theories defined on different Riemann surfaces are not different theories, but the same theory living on different manifolds. E.g., an Ising model is always an Ising model, whether you choose periodic boundary conditions (a torus), or you demand that all boundary spins be equal (a sphere). Good theories can be defined on every Riemann surface. |
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