Worldsheet SCFT on a lattice My question is clear from the title. I'm curious whether it is possible to put the string world sheet SCFT on a lattice. I expect when the world sheet theory is chiral, then it's not possible.
But I remember I heard somewhere that it is impossible in general, and GLSMs are actually the way out of this problem. I mean the (worldsheet) UV theory is indeed the GLSM.
I don't remember the reason though. I'm not sure even it's correct. So I would appreciate any comments.
 A: It's pretty difficult and not very promising . Marginally all of the important features of the superstrings are lost if one tries to put them on a lattice.
Examples:

*

*Conformal symmetry is broken by the introduction of the mesh parameter of the lattice.


*The very idea of holomorphic function (the algebra generated by the Virasoro Algebra) is absent. Although discrete complex analysis already exists, nothing truly useful (as far as my ignorance can tell) has emerged from those structures (except in the thermodynamic limit where actually conformal symmetry and smoothness is recovered).


*The CFT gadgets (operators, vertex algebras, OPEs, the operator-sate correspondence etc.) have no useful analogues.


*What is supersymmetry on a graph?


*Perturbative expansion lost. What would be a genus zero or genus one superstring? Graphs with no 1-homology (a tree) and one dimensional 1-homology (a tree with a loop) respectively? You cannot attach to much structure to something so simple .
Nevertheless, some efforts have been done, although not exactly as "discretizing" the world sheet. They incorporate some interesting discrete structures (to the spacetime of the GS superstring).
References:
Valentina Forini - Green Schwarz Superstring on a Lattice.
Valentina Forini - Worldsheet superstring on the lattice and AdS/CFT
