246
GAUGE FIELDS AND STRINGS
each piece of which carries a ^-dimensional spin operator. Now, by a
Jordan-Wigner transformation this system is transformed into the
usual Majorana fermions on the world surface. On the other hand, since
each piece of the string carries spin 1/2, we can have two sectors,
depending on whether we have an even or an odd number of spins.
These are the Neveu-Schwarz and Ramond sectors respectively.
The spin operator, which interchanges these two sectors, can be
expressed by Jordan-Wigner transformations through the fermionic
field \¡/^. This representation, involving exponents of bi-linears of {¡/^ is
rather clumsy, but fortunately is not needed since we shall show how to
compute Sfl-correlations directly.
Consider first one fermionic field ij/ and let us compute the correlation function:
j r =
(9.412)
(This is precisely the case of the Ising model.)
It appears that the analytic properties of JT are sufficient for its
determination.
We know from (9.410) that it is an analytic function of and Z2 ,
which has square root branch points at z^ 2 = T- Also because of the
operator product relation
it must have a simple pole as z^ -► Z2 . Hence, we can write:
P(zi,Z2,x, y)
jT =
(^1 -
- y)(^2 - x){Z2 - y)Y'^
(9.413)
(9.414)
where P must be a polynomial in z^ and Z2 . This polynomial must be
arranged so that the residue at the pole Zj = Z2 is independent of z^
(because of the unit operator in (9.413). Hence:
P(zj, Z j, X, y) = e ( x , y)(zi - x ) ( Z j - y)
Also, because of
5(x)5(y)^|x-y|-^^/ + -.we have the requirement:
P(zi, Z2 , X, y) = (Z i - x X z 2 - x)
x-*y
(9.415)
(9.416)
GAUGE FIELDS AND STRINGS
each piece of which carries a ^-dimensional spin operator. Now, by a
Jordan-Wigner transformation this system is transformed into the
usual Majorana fermions on the world surface. On the other hand, since
each piece of the string carries spin 1/2, we can have two sectors,
depending on whether we have an even or an odd number of spins.
These are the Neveu-Schwarz and Ramond sectors respectively.
The spin operator, which interchanges these two sectors, can be
expressed by Jordan-Wigner transformations through the fermionic
field \¡/^. This representation, involving exponents of bi-linears of {¡/^ is
rather clumsy, but fortunately is not needed since we shall show how to
compute Sfl-correlations directly.
Consider first one fermionic field ij/ and let us compute the correlation function:
j r =
(9.412)
(This is precisely the case of the Ising model.)
It appears that the analytic properties of JT are sufficient for its
determination.
We know from (9.410) that it is an analytic function of and Z2 ,
which has square root branch points at z^ 2 = T- Also because of the
operator product relation
it must have a simple pole as z^ -► Z2 . Hence, we can write:
P(zi,Z2,x, y)
jT =
(^1 -
- y)(^2 - x){Z2 - y)Y'^
(9.413)
(9.414)
where P must be a polynomial in z^ and Z2 . This polynomial must be
arranged so that the residue at the pole Zj = Z2 is independent of z^
(because of the unit operator in (9.413). Hence:
P(zj, Z j, X, y) = e ( x , y)(zi - x ) ( Z j - y)
Also, because of
5(x)5(y)^|x-y|-^^/ + -.we have the requirement:
P(zi, Z2 , X, y) = (Z i - x X z 2 - x)
x-*y
(9.415)
(9.416)