ATTEMPT AT A SYNTHESIS
273
identity on Zj and on t , restricting the t integral to the fundamental
region. If the partition function violates modular invariance the term
d/dx will give a nonzero contribution, coming from the boundaries.
Thus a modular anomaly on the world sheet will manifest itself as a
gauge and gravitational anomaly in space-time.
The conclusion is that in heterotic string theories one has to check
modular invariance, which severely limits the possibilities. On the other
hand, in ordinary closed strings modular invariance is automatic, since
non-chiral determinants can be regularized in a manifestly covariant
way. Such theories, even when they contain chiral fermions in spacetime, are safe from anomalies.
Let us summarize. We see that we now have a consistent description
of quantum gravity and supergravity, and we have many options for
including chiral fermions and other particles into the scheme. This
approach explains the enigma of generations, the strange copying
which started with the muon. The number of copies now is determined
by the number of zero modes of the generalized Dirac operators, which
in turn is fixed by the topology of the background manifold. Actually it
is hard to obtain a small enough number in this approach.
At the same time, we cannot explain at the moment why compactification took place, and most importantly, why the dimensionality of our
world is equal to four. Until these problems are solved, we cannot be
quite sure that we are on the right track. Still, there are many favourable
signs.
10.3 The Three-Dimensional Ising Modelf
It is well known that most of the phase transitions of the second kind
occurring in Nature are equivalent, as far as critical behaviour is
concerned, to the three dimensional Ising model. After Onsager’s
remarkable success with the two dimensional version of this model,
many people have tried to find its solution in 3d (private communication). No such solution has been found and we have a strong feeling
now that the model is not exactly solvable. At the same time, the total
solution is not actually needed, because only the critical behaviour of
this theory presents a universal interest. By this we mean that the
t Some of the results of this section have been obtained in collaboration with V.
Dotsenko (1978).
273
identity on Zj and on t , restricting the t integral to the fundamental
region. If the partition function violates modular invariance the term
d/dx will give a nonzero contribution, coming from the boundaries.
Thus a modular anomaly on the world sheet will manifest itself as a
gauge and gravitational anomaly in space-time.
The conclusion is that in heterotic string theories one has to check
modular invariance, which severely limits the possibilities. On the other
hand, in ordinary closed strings modular invariance is automatic, since
non-chiral determinants can be regularized in a manifestly covariant
way. Such theories, even when they contain chiral fermions in spacetime, are safe from anomalies.
Let us summarize. We see that we now have a consistent description
of quantum gravity and supergravity, and we have many options for
including chiral fermions and other particles into the scheme. This
approach explains the enigma of generations, the strange copying
which started with the muon. The number of copies now is determined
by the number of zero modes of the generalized Dirac operators, which
in turn is fixed by the topology of the background manifold. Actually it
is hard to obtain a small enough number in this approach.
At the same time, we cannot explain at the moment why compactification took place, and most importantly, why the dimensionality of our
world is equal to four. Until these problems are solved, we cannot be
quite sure that we are on the right track. Still, there are many favourable
signs.
10.3 The Three-Dimensional Ising Modelf
It is well known that most of the phase transitions of the second kind
occurring in Nature are equivalent, as far as critical behaviour is
concerned, to the three dimensional Ising model. After Onsager’s
remarkable success with the two dimensional version of this model,
many people have tried to find its solution in 3d (private communication). No such solution has been found and we have a strong feeling
now that the model is not exactly solvable. At the same time, the total
solution is not actually needed, because only the critical behaviour of
this theory presents a universal interest. By this we mean that the
t Some of the results of this section have been obtained in collaboration with V.
Dotsenko (1978).