What are the constraint equations for covariant quantization of supermembrane theory?

Question

I'm self-taught when it comes to string theory, so forgive me if I misspeak a bit here.

As I understand it, in order to quantize string theory covariantly, once we have defined the state space, we introduce the constraints that the only physical states are the ones annihilated by the generators $L_n, G_r$ of the Super-Virasoro algebra for $N,r>0$ as well as $L_0 - 1$ (plus some additional constraints for closed strings).

The question I had is, what is the equivalent procedure for the supermembrane theory? Since the Super-Virasoro algebra is the algebra of a 2D superconformal field theory, does that mean for supermembranes we should require the states to be annihilated by the generators of the 3D superconformal algebra? I.e., if we introduce states of the form $$|ψ> \ = |X_0,X_1,...,X_{10}>,$$ would the relevant constraint equations be something like:

$$M_{μν} |ψ> \ = 0$$ $$P_{μ} |ψ> \ = 0$$ $$D |ψ> \ = 0$$ $$K_μ |ψ> \ = 0$$ $$Q_{αβ} |ψ> \ = 0$$ $$S_{αβ} |ψ> \ = 0$$

for $μ,ν$ ranging over the dimensions of spacetime and $α,β$ ranging over (both dotted and undotted) spinor indices?

I tried to look this up in the literature, but I had trouble finding many papers with general information on supermembranes at all, let alone information about covariant quantization.

This post imported from StackExchange Physics at 2024-09-08 20:28 (UTC), posted by SE-user Ian MathWiz

Comments

See second paragraph arxiv.org/abs/hep-th/0201151

This post imported from StackExchange Physics at 2024-09-08 20:28 (UTC), posted by SE-user Mitchell Porter

---

to answer your question, the covariant quantization of the string requires us to keep all unphysical degrees of freedom. As explained by Polchinski, the mass-shell condition in string theory is equivalent to the condition that unphysical degrees of freedom are kept. better to keep in mind that we are working in 10 flat dimensions. you have to enlarge the world-sheet constraint algebra. The enlarged algebra with T and Tbar annihilate physical and unphysical states. verify that it satisfies the N = 1 algebra with c = 3/2 D

XXbar ~ -1/z^2

as you said, the second question it is possible to carry it out using the super virasoro algebra. the conformal and superconformal transformations close to form the superconformal algebra. see Polchinski, only one copy of the superconformal algebra.