The GSO projection (named after
Ferdinando Gliozzi
Ferdinando Gliozzi (; born 1940) is a string theorist at the Istituto Nazionale di Fisica Nucleare. Along with David Olive and Joël Scherk, he proposed the GSO projection
to map out the tachyon
A tachyon () or tachyonic particle is a hypo ...
,
Joël Scherk, and
David I. Olive
David Ian Olive ( ; 16 April 1937 – 7 November 2012) was a British theoretical physicist. Olive made fundamental contributions to string theory and Duality (mathematics), duality theory, he is particularly known for his work on the GSO projec ...
)
[ 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 model in
superstring
Superstring theory is an theory of everything, attempt to explain all of the Elementary particle, particles and fundamental forces of nature in one theory by modeling them as vibrations of tiny supersymmetry, supersymmetric String (physics), st ...
theory. The
projection is a selection of a subset of possible vertex operators in the
worldsheet conformal field theory (CFT)—usually those with specific worldsheet
fermion
In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks an ...
number and
periodicity 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 (OPE) of any two operators in ''A'' contains only operators which are in ''A''.
* Mutual locality — There are no
branch cut
In the mathematical field of complex analysis, a branch point of a multi-valued function (usually referred to as a "multifunction" in the context of complex analysis) is a point such that if the function is n-valued (has n values) at that point, a ...
s in the OPE of any two operators in the set ''A''.
* Modular invariance — The partition function on the
two-torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does not tou ...
of the theory containing only the operators in ''A'' respects
modular invariance.
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. 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 models), one typically chooses a GSO projection which eliminates the
tachyonic ground state of the string and preserves spacetime
supersymmetry
In a supersymmetric theory the equations for force and the equations for matter are identical. In theoretical and mathematical physics, any theory with this property has the principle of supersymmetry (SUSY). Dozens of supersymmetric theories e ...
.
Notes
References
*
Polchinski, Joseph (1998). ''String Theory'', Cambridge University Press. A modern textbook.
** Vol. 2: Superstring theory and beyond. .
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String theory