The GSO projection (named after Ferdinando Gliozzi,

Joël Scherk Joël Scherk (; 27 May 1946 – 16 May 1980) was a French theoretical physicist who studied string theory and supergravity. Work Scherk studied in Paris at the École Normale Supérieure (ENS). In 1969 he received his diploma (Thèse de troisièm ...

, and David I. Olive) F. Gliozzi, J. Scherk and D. I. Olive, "Supersymmetry, Supergravity Theories and the Dual Spinor Model", ''Nucl. Phys. B'' 122 (1977), 253. is an ingredient used in constructing a

consistent In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent ...

model in superstring theory. The projection is a selection of a subset of possible vertex operators in the

worldsheet In string theory, a worldsheet is a two-dimensional manifold which describes the embedding of a string in spacetime. The term was coined by Leonard Susskind as a direct generalization of the world line concept for a point particle in special a ...

conformal field theory A conformal field theory (CFT) is a quantum field theory that is invariant under conformal transformations. In two dimensions, there is an infinite-dimensional algebra of local conformal transformations, and conformal field theories can sometime ...

(CFT)—usually those with specific worldsheet fermion number and

periodicity Periodicity or periodic may refer to: Mathematics * Bott periodicity theorem, addresses Bott periodicity: a modulo-8 recurrence relation in the homotopy groups of classical groups * Periodic function, a function whose output contains values tha ...

conditions. Such a projection is necessary to obtain a consistent worldsheet CFT. For the projection to be consistent, the set ''A'' of operators retained by the projection must satisfy: * Closure — The

operator product expansion In quantum field theory, the operator product expansion (OPE) is used as an axiom to define the product of fields as a sum over the same fields. As an axiom, it offers a non-perturbative approach to quantum field theory. One example is the vert ...

(OPE) of any two operators in ''A'' contains only operators which are in ''A''. * Mutual locality — There are no branch cuts in the OPE of any two operators in the set ''A''. * Modular invariance — The partition function on the two-torus of the theory containing only the operators in ''A'' respects

modular invariance In theoretical physics, modular invariance is the invariance under the group such as SL(2,Z) of large diffeomorphisms of the torus. The name comes from the classical name modular group of this group, as in modular form theory. In string theory ...

. Starting from the same worldsheet CFT, different choices in the GSO projection will lead to string theories with different physical particles and properties in

spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...

. For example, the Type II and Type 0 string theories result from different GSO projections on the same worldsheet theory. Furthermore, the two distinct Type II theories, IIA and IIB, differ in their GSO projections. In building models of realistic string vacua (as opposed to

toy model In the modeling of physics, a toy model is a deliberately simplistic model with many details removed so that it can be used to explain a mechanism concisely. It is also useful in a description of the fuller model. * In "toy" mathematical models ...

s), one typically chooses a GSO projection which eliminates the

tachyon A tachyon () or tachyonic particle is a hypothetical particle that always travels faster than light. Physicists believe that faster-than-light particles cannot exist because they are not consistent with the known laws of physics. If such partic ...

ic ground state of the string and preserves spacetime supersymmetry.

Notes

References

* Polchinski, Joseph (1998). ''String Theory'', Cambridge University Press. A modern textbook. ** Vol. 2: Superstring theory and beyond. . {{string-theory-stub String theory