Gravity and Strings

432 The string effective action and T duality
to the dilaton in the string frame. They couple to the KR 2-form due to the definitions of
the field strengths and also to the presence of Chern–Simons (CS) topological terms in the
supergravity actions. These CS terms contain a great deal of information on the possible
intersections of extended objects of the theory [900].
Although the identification of the field theories on the basis of symmetry arguments is
correct, the identification of the fields with the string modes is ambiguous, since the supergravity
theories are unique up to field redefinitions. To establish completely the relation
between supergravity fields and string modes, it is necessary to have more information. For
instance, making use of the relations in Figure 14.1, the supergravity fields must be related
by dualities in the same way as the string modes are.
String effective actions also arise in a different way: string theories are usually quantized
in flat spacetime, but the string worldsheet action can be written in a curved background
as a non-linear σ -model, Eq. (14.5), and, furthermore, can be generalized to describe the
coupling to all background fields associated with the string massless modes. 1 The coupling
of the string to the Kalb–Ramond 2-form Bµν is represented by a WZ term that generalizes
the coupling of the Maxwell vector field to a charged point-particle, Eq. (8.53), i.e. it is the
integral of the pull-back of the 2-form over the two-dimensional worldsheet:
T
2
B, (15.4)
where B is given by
B = 1
2 Bijdξ i ∧ dξ j = d 2 ξɛ ij Bij = d 2 ξɛ ij ∂i X µ ∂ j X ν Bµν. (15.5)
Observe that the role of the electric charge is played here by the string tension. This coefficient
can be changed but we will take it as above, defining the normalization of Bµν that we
will use. This is the normalization that leads to the effective action Eq. (15.1). Observe also
that the WZ term, being topological (metric-independent), is automatically Weyl-invariant
and does not contribute to the γij equation of motion. Furthermore, the WZ term is invariant,
up to total derivatives, under gauge transformations of the 2-form, Eq. (15.3), which
means that strings are charged with respect to the KR 2-form and the charge is conserved. In
open-string worldsheets the non-vanishing boundary term is canceled out by the variation
of the term that represents the coupling of the open string to the 1-form Vµ:
V, (15.6)
provided that the vector transforms under the KR 2-form gauge symmetry Eq. (15.3),
∂
δVµ = T µ. (15.7)
Finally, this term is not parity-invariant and occurs only in oriented-string theories.
1 This is always true for the fields in the common sector for the bosonic and fermionic strings, although
worldsheet supersymmetry has to be studied case by case. The inclusion of RR massless superstring fields
in the σ -model is more complicated and how to do it is known only in certain cases.