Gravity and Strings

17
The type-IIB superstring and type-II T duality
In the previous chapter we initiated the study of the 11- and ten-dimensional supergravity
theories which arise in the low-energy limits of the various string theories and M theory.
Our goal was to study the dualities that relate the various string theories and M theory using
effective-field-theory actions as we did in Section 15.2 with T duality in the effective action
of the common string sector. In the coming chapters we will study these dualities from the
point of view of their effect on classical solutions of the effective actions that represent the
classical long-range fields of perturbative and non-perturbative states of these theories, as
we did in Section 15.3 with the solutions associated with string and winding modes.
In this chapter we are going to study the N = 2B, d = 10 (chiral) supergravity theory, the
effective field theory of the type-IIB superstring, and how it is related to the N = 2A (nonchiral)
theory after compactification on a circle (type-II T duality). Furthermore, we are also
going to study the truncations to N = 1 theories, which are the effective-field theories of
the type-I and heterotic superstrings, finding the field-theory version of the type-I/heteroticstring
duality.
First, in Section 17.1, we will study the bosonic sector of the theory, giving a non-selfdual
action from which one can derive equations of motion that have to be supplemented
by the self-duality constraint of the 5-form field strength [111]. We will also give the supersymmetry
transformation rules to lowest order in fermions. Then, in Section 17.2 we will
study the S-duality symmetry of this theory, which becomes manifest only after several
redefinitions.
Next, in Section 17.3 we will perform the dimensional reduction to nine dimensions
of N = 2B, d = 10 supergravity compactified on a circle. As we will see, the ninedimensional
theory thus obtained is identical to the theory obtained by dimensional reduction
of the N = 2A, d = 10 theory, Eq. (16.84). This will allow us to establish a correspondence
between fields of the ten-dimensional N = 2A and N = 2B theories compactified on
circles. This correspondence is part of the T duality existing between the two corresponding
superstring theories and with our procedure we will have obtained the generalization
of the T-duality Buscher rules to type-II theories found in [125] and generalized in [691].
Finally, in Section 17.5 we will study various consistent truncations of the theory and
their relations to N = 1 theories and the corresponding full string theories. In particular,
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