Twisted Sector

theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...

, a twisted sector is a subspace of the full

Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...

closed string In physics, a string is a physical entity postulated in string theory and related subjects. Unlike elementary particles, which are zero-dimensional or point-like by definition, strings are one-dimensional extended entities. Researchers often ha ...

states in a particular theory over a (good)

orbifold In the mathematical disciplines of topology and geometry, an orbifold (for "orbit-manifold") is a generalization of a manifold. Roughly speaking, an orbifold is a topological space which is locally a finite group quotient of a Euclidean space. D ...

. In the first quantized formalism of

string theory In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and interac ...

(or in

two-dimensional conformal field theory A two-dimensional conformal field theory is a quantum field theory on a Euclidean two-dimensional space, that is invariant under local conformal transformations. In contrast to other types of conformal field theories, two-dimensional conformal fie ...

) the

target space Target may refer to: Physical items * Shooting target, used in marksmanship training and various shooting sports ** Bullseye (target), the goal one for which one aims in many of these sports ** Aiming point, in field artillery, fi ...

is an orbifold M/G if the observables of the string are only defined modulo G. Consequently, the value of the field after one cycle around the closed string need only be the same as its original value modulo some G transformation. i.e. there exists some

g\in G

such that :

X(\sigma+2\pi,\tau)=g (\sigma,\tau) /math>

For each

conjugacy class In mathematics, especially group theory, two elements a and b of a group are conjugate if there is an element g in the group such that b = gag^. This is an equivalence relation whose equivalence classes are called conjugacy classes. In other wor ...

of G, we have a different

superselection sector In quantum mechanics, superselection extends the concept of selection rules. Superselection rules are postulated rules forbidding the preparation of quantum states that exhibit coherence between eigenstates of certain observables. It was originall ...

(wrt the worldsheet). The conjugacy class consisting of the identity gives rise to the untwisted sector and all the other conjugacy classes give rise to twisted sectors. It's easy to see that since the observables are only modulo G, two different g's which are conjugate to each other give rise to the same sector. In the second quantized formalism, the different sectors give rise to different orbifold projections. String theory Conformal field theory {{string-theory-stub