Gravity and Strings

14.2 Quantum theories of strings 417
This action, whose covariant quantization poses many problems, has very interesting
features. First, it can be generalized to the other string theories with N = 2 spacetime supersymmetry
[137, 481] and other higher-dimensional objects of Table 14.1, always with a
WZ term (see also [78, 79]). This was done for the super 3-brane in d = 6in[574] and for
the super 2-brane (supermembrane,nowM2-brane)ofd = 11 in [138, 139]. The M2-brane
is related to the ten-dimensional string by double dimensional reduction [14, 335] (the last
two members of the NM = 32 series).
These generalizations include the coupling of the supersymmetric extended objects to supergravity
fields: since GS actions are manifestly spacetime-supersymmetric, the coupling
to gravity implies the coupling to all the background fields of a supergravity theory. 11 The
fact that it can and must be coupled to supergravity explains in part the necessity of the WZ
term: supergravity theories contain potentials that are (p + 1)-forms in superspace that can
naturally couple to p-dimensional objects through a WZ term (the integral of the pull-back
of the (p + 1)-form over the (p + 1)-dimensional worldvolume), just like particles couple
to the Maxwell 1-form Eq. (8.53) (see also Eq. (15.4)). The expansion of the (p + 1)-form
fields in components contains terms that do not vanish in flat spacetime and 2 above is
one such term. The full WZ term for the superstring coupled to supergravity also contains
the purely bosonic term Eq. (15.4).
κ-symmetry imposes constraints on the supergravity fields [137–9, 335, 481, 961]. In
particular [138, 139], the action for the 11-dimensional supermembrane coupled to the
fields of 11-dimensional supergravity can be κ-invariant only if certain constraints are
solved by the equations of motion of that theory. These constraints coincide with the superspace
constraints of d = 11 supergravity [262]. Thus worldvolume κ-invariance implies the
equations of motion of the spacetime supergravity fields, a highly non-trivial fact. Something
similar happens in string theory coupled to background fields: by requiring invariance
under worldsheet Weyl transformations in the quantum theory one obtains the equations of
motion of the spacetime fields (see Section 15.1).
Although there is no clear motivation at this point, we could also include other supermultiplets
in the super-p-brane actions [339]. The new supersymmetric extended objects
include the Dp-branes we will study in more detail later [16, 17, 81, 140, 223, 569, 899].
14.2 Quantum theories of strings
In this section we are going to overview the quantization in Minkowski spacetime of the
bosonic and fermionic string actions that we have introduced in the previous section. We
will focus on the definition of quantum string theory (in particular on string interactions)
and on the results: the critical dimensions, mode expansions, and massless spectra of the
simplest consistent string theories.
14.2.1 Quantization of free-bosonic-string theories
Free strings can translate, rotate, and vibrate. The various allowed vibrational modes are
seen as different particle states in spacetime. These particle states must fit into Poincaré
11 Only recently has it been learned how to couple superstrings to RR backgrounds (to be defined later) [695,
694, 696].