In string theory, the Ramond–Neveu–Schwarz (RNS) formalism is an approach to formulating superstrings in which the worldsheet has explicit superconformal invariance but spacetime supersymmetry is hidden, in contrast to the Green–Schwarz formalism where the latter is explicit.
It was originally developed by Pierre Ramond, André Neveu and John Schwarz in the RNS model in 1971,[1][2] which gives rise to type II string theories and can also give type I string theory.
Heterotic string theories can also be acquired through this formalism by using a different worldsheet action.
[4][5] While these are S-matrix theories rather than quantum field theories, Yoichiro Nambu, Holger Bech Nielsen, and Leonard Susskind gave them a string interpretation, whereby mesons behave as strings of finite length.
In 1970 Pierre Ramond was working at Yale trying to extend the dual resonance models to include fermionic degrees of freedom through a generalization of the Dirac equation.
At the same time, Andre Neveu and John Schwarz were working at Princeton to extend existing dual resonance models by adding to them anticommutating creation and annihilation operators.
[7] At the time, the main issue with the RNS model was that it contained a tachyon as the lowest energy state.
It was only in 1976 with the introduction of GSO projection by Ferdinando Gliozzi, Joël Scherk, and David Olive that the first consistent tachyon-free string theories were constructed.
[9][10] The last approach starts from the Euclidean partition function where
that represents an overcounting of the physically distinct configurations of the fields that the action depends on.
This overcounting is eliminated by dividing by the volume of the gauge group
BRST quantization proceeds by gauge fixing the path integral via the Fadeev–Popov procedure, which gives rise to a ghost action in addition to the now gauge fixed action.
The physical states of this theory split up into a number of sectors depending on the periodicity condition of the fermionic fields.
supergravity action gives rise to heterotic string theories.
One way to classify all possible string theories that can be constructed using this formalism is by looking at the possible residual symmetry algebras that can arise.
To give rise to a physical theory, this algebra must be imposed on the Hilbert space by projecting out unwanted states.
Topological string theory is not found in this classification because for it the spin-statistics theorem does not hold in the conformal gauge which was required in the full argument.
superconformal field theory on the string worldsheet with an action of the form where
[12] These bosonic fields have a physical interpretation as the coordinates of the string worldsheet embedded in spacetime, with
defined according to where now the R and NS sectors correspond to a periodicity or antiperiodicity condition on this extended field.
The operator product expansion (OPE) of the fermionic theory translate to anticommutation relations for the modes given by The states in the Hilbert space can then be built up by acting with these modes on the vacuum state.
The Lorentz covariant, diffeomorphism invariant action for the fermionic superstring is found by coupling the bosonic and fermionic fields to two-dimensional supergravity, giving the action where
There are holomorphic and antiholomorphic ghosts in the gauge fixed superstring action.
This action gives rise to additional ghost contributions to the overall stress energy tensor
is the corresponding charge associated with this current The physical spectrum is the set of BRST cohomology classes.
This condition truncates the ghost spectrum for kinematic reasons.
is the level, counting the creation operators used to create the state.
[12] First, the vertex operators of the theory have to be mutually local, meaning that their OPEs have no branch cuts.
Type I string theory can be constructed from type IIB theory that has gauged its worldsheet parity symmetry and has been combined with the GSO projected open RNS string.
This last condition arises from a requirement to make the theory non-anomalous.