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Register a new userOn a coordinate independent description of string worldsheet theory
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Partha Mukhopadhyay
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We study worldsheet conformal invariance for bosonic string propagating in a curved background using the hamiltonian formalism. In order to formulate the problem in a background independent manner we first rewrite the worldsheet theory in a language where it describes a single particle moving in an infinite-dimensional curved spacetime. This language is developed at a formal level without regularizing the infinite-dimensional traces. Then we adopt DeWitts (Phys.Rev.85:653-661,1952) coordinate independent formulation of quantum mechanics in the present context. Given the expressions for the classical Virasoro generators, this procedure enables us to define the coordinate invariant quantum analogues which we call DeWitt-Virasoro generators. This framework also enables us to calculate the invariant matrix elements of an arbitrary operator constructed out of the DeWitt-Virasoro generators between two arbitrary scalar states. Using these tools we further calculate the DeWitt-Virasoro algebra in spin-zero representation. The result is given by the Witt algebra with additional anomalous terms that vanish for Ricci-flat backgrounds. Further analysis need to be performed in order to precisely relate this with the beta function computation of Friedan and others. Finally, we explain how this analysis improves the understanding of showing conformal invariance for certain pp-wave that has been recently discussed using hamiltonian framework.
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Although the energy spectrum of the Heisenberg spin chain on a circle defined by $H=frac{1}{4}sum_{k=1}^M(sigma_k^xsigma_{k+1}^x+sigma_k^ysigma_{k+1}^y +Deltasigma_k^zsigma_{k+1}^z)$ is well known for any fixed $M$, the boundary conditions vary according to whether $Min 4mathbb{N}+r$, where $r=-1,0,1,2$, and also according to the parity of the number of overturned spins in the state, In string theory all these cases must be allowed because interactions involve a string with $M$ spins breaking into strings with $M_1<M$ and $M-M_1$ spins (or vice versa). We organize the energy spectrum and degeneracies of $H$ in the case $Delta=0$ where the system is equivalent to a system of free fermions. In spite of the multiplicity of special cases, in the limit $Mtoinfty$ the spectrum is that of a free compactified worldsheet field. Such a field can be interpreted as a compact transverse string coordinate $x(sigma)equiv x(sigma)+R_0$. We construct the bosonization formulas explicitly in all separate cases, and for each sector give the Virasoro conformal generators in both fermionic and bosonic formulations. Furthermore from calculations in the literature for selected classes of excited states, there is strong evidence that the only change for $Delta eq0$ is a change in the compactification radius $R_0to R_Delta$. As $Deltato-1$ this radius goes to infinity, giving a concrete example of noncompact space emerging from a discrete dynamical system. Finally we apply our work to construct the three string vertex implied by a string whose bosonic coordinates emerge from this mechanism.
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