244
GAUGE FIELDS AND STRINGS
This is the standard form describing ^ /2 harmonic oscillators. The
Hamiltonian in this case is equal to zero, while the states are formed by
the “vacuum”
(/>J0> = 0
(9.405)
(9.406)
(9.407)
together with the “excited” states:
l/> = ;.•••
Since our operators are anticommuting,
the number of independent states in (9.406) is 2^'^ (each a can be either
empty or occupied). All these states have zero energy and describe a
single particle with 2^^^ states. This is just the degeneracy of the spinor
representation of the O(^) group, and in fact, what we have done is just
to construct such a representation. Of course, to complete the job, one
has to construct the O(^) generators out of (/>^, and show that the
representation is nondegenerate. This is not hard, but we stop here,
since our task was just to see the connection between ij/^ and spinors.
Returning to the string case, let us look at the ij/^ part of the string
action on a cylinder:
'n,R dt dcr
(9.408)
Two spinor structures correspond to {¡/^ being periodic or anti-periodic
as we go around the cylinder. In the periodic case (which is called the
Ramond sector)
has a tr-independent component, so that the action
(9.408) is reduced (for each
and
(9.403). We see that as we
integrate over the periodic case, the ground state of the string is a spacetime spinor. In the antiperiodic case we still have the states (9.406), but
now the Hamiltonian is nonzero (due to i\|; • 3^\|;) and these states are
not degenerate. Actually, in this case the ground state is bosonic (in the
space-time sense).
Now we have to return to our punctured sphere and the operator
algebra. This is achieved by a conformal transformation which makes
an annulus out of the cylinder. Then, the limit of an infinitely long
cylinder will correspond to the limit, when the internal circle of the
annulus shrinks to a point while the external one goes to infinity. That
is, we shall get a sphere with an omitted point. Our task is to find what
happens to different spinor structures under this transformation.
GAUGE FIELDS AND STRINGS
This is the standard form describing ^ /2 harmonic oscillators. The
Hamiltonian in this case is equal to zero, while the states are formed by
the “vacuum”
(/>J0> = 0
(9.405)
(9.406)
(9.407)
together with the “excited” states:
l/> = ;.•••
Since our operators are anticommuting,
the number of independent states in (9.406) is 2^'^ (each a can be either
empty or occupied). All these states have zero energy and describe a
single particle with 2^^^ states. This is just the degeneracy of the spinor
representation of the O(^) group, and in fact, what we have done is just
to construct such a representation. Of course, to complete the job, one
has to construct the O(^) generators out of (/>^, and show that the
representation is nondegenerate. This is not hard, but we stop here,
since our task was just to see the connection between ij/^ and spinors.
Returning to the string case, let us look at the ij/^ part of the string
action on a cylinder:
'n,R dt dcr
(9.408)
Two spinor structures correspond to {¡/^ being periodic or anti-periodic
as we go around the cylinder. In the periodic case (which is called the
Ramond sector)
has a tr-independent component, so that the action
(9.408) is reduced (for each
and
(9.403). We see that as we
integrate over the periodic case, the ground state of the string is a spacetime spinor. In the antiperiodic case we still have the states (9.406), but
now the Hamiltonian is nonzero (due to i\|; • 3^\|;) and these states are
not degenerate. Actually, in this case the ground state is bosonic (in the
space-time sense).
Now we have to return to our punctured sphere and the operator
algebra. This is achieved by a conformal transformation which makes
an annulus out of the cylinder. Then, the limit of an infinitely long
cylinder will correspond to the limit, when the internal circle of the
annulus shrinks to a point while the external one goes to infinity. That
is, we shall get a sphere with an omitted point. Our task is to find what
happens to different spinor structures under this transformation.