String Theory and M-Theory

Homework Problems 295
where the coordinates x, y, z each have period 2π. Suppose there is also a
nonvanishing three-form Hxyz = N, where N is an integer. For example,
Bxy = Nz.
(i) Using the T-duality rules for background fields derived in Chapter 6,
carry out a T-duality transformation in the x direction followed by
another one in the y direction. What is the form of the resulting
metric and B fields?
(ii) One can regard the T 3 as a T 2 , parametrized by x and y, fibered over
the z-circle. Going once around the z-circle is trivial in the original
background. What happens when we go once around the z-circle
after the two T-dualities are performed?
(iii) The background after the T-dualities has been called nongeometrical.
Explain why. Hint: use the results of the preceding problem.
PROBLEM 7.14
Consider the compactification of each of the two supersymmetric heterotic
string theories on a circle of radius R. As discussed in Section 7.4, the
moduli space is 17-dimensional and at generic points the left-moving gauge
symmetry is U(1) 17 . However, at special points there are enhanced symmetries.
Assume that the gauge fields in the compact dimensions, that is,
the Wilson lines, are chosen in each case to give SO(16) × SO(16) × U(1)
left-moving gauge symmetry. Show that the two resulting nine-dimensional
theories are related by a T-duality transformation that inverts the radius of
the circle. This is very similar to the T-duality relating the type IIA and
IIB superstring theories compactified on a circle.
PROBLEM 7.15
(i) Compactifying the E8 × E8 heterotic string on a six-torus to four
dimensions leads to a theory with N = 4 supersymmetry in four
dimensions. Verify this statement and assemble the resulting massless
spectrum into four-dimensional supermultiplets.
(ii) Repeat the analysis for the type IIA or type IIB superstring. What is
the amount of supersymmetry in four dimensions in this case? What
is the massless supermultiplet structure in this case?