We have constructed a generating function (and associated correlation functions) for string amplitudes in generic constant string backgrounds,GM N,BM N, Φ andU, onRD−1,1×
TDcr−D, so that also all K¨ahler and complex structure moduli (of the target space torus,
TDcr−D) contained inG
ab, Bab, background KK gauge fields,Aaµand Bµa, spacetime torsion,
Bµν and also spacetime metric, Gµν, are allowed to be turned on. In the process, we have
derived the chiral splitting theorem of D’Hoker and Phong [64] for string amplitudes, which we have generalised to the aforementioned background and with arbitrarily excited string vertex operator insertions (with generic KK and winding charges, as well as polarisation tensors associated to generic oscillators and spacetime indices46).
Our approach differs from that of D’Hoker and Phong [64] (and also Sen [67]), in that we did not make use of the “reverse engineering” approach (where the target spacetime embedding fields,xM(z,z), are first integrated out and only at a later stage of the compu-¯
tation is it noted that the result can be written as an integral whose integration variables get interpreted as AI-cycle loop momenta). As pointed out in a recent paper by Sen [67],
such a reverse engineering approach could potentially lead to ambiguities (because the same integrated-loop momenta amplitude can be written in more than one way as an integral over loop momenta [67]). Sen went on to explain that these ambiguities will not be visible in the final amplitudes after integrating out the loop momenta (while adopting an appro- priate analytic continuation for the loop momentum integral contours [65]), and that these ambiguities are therefore immaterial. However, as discussed in the Introduction above, it is sometimes desirable to not integrate out the loop momenta, and that this is also of interest for the computation of some physical observables, such as the spectrum of mass- less radiation associated to a decaying string. Therefore, the reverse engineering method could potentially lead to ambiguous results for observables. In our approach we have re- solved this potential ambiguity, in that we introduced loop momenta associated toAI-cycle
strings from the outset (by explicit momentum conserving delta function insertions into the original path integral where there is no room for this ambiguity), and have thus shown that the result of D’Hoker and Phong (that one can replace the target space fields, xM(z,z),¯ by a set of effective chiral fields, xM
+(z), xM−(¯z) for the left- and right-moving degrees of
freedom, appropriately modified so as to apply to generic backgrounds, RD−1,1 ×TDcr−D,
and vertex operators) is fully justified and leads to the correct un-ambiguous result for the fixed-loop momentum amplitudes.47
Let us now zoom in on the statement (5.100). Here it is crucial to note that the left-hand side denotes the usual path integral over matter,xM(z,z), and ghost fields,¯ b, c, whereas on
the right-hand side the matter and ghost fields have been integrated out, and the result has been written in terms of Wick contractions of effective (anti-)chiral fields,xM
+(z), (x−(¯z)),
whose correlation functions are determined from the chiral propagators (5.101), with the results given in (5.102) and (5.103). What we want to emphasise here is that on the left- hand side of (5.100) the target space embedding field appearing in vertex operators and the worldsheet action contains (generically) zero modes, instanton (or soliton) contributions, as well as quantum fluctuations, whereas the chiral fields on the right hand side are defined by their correlation functions, so that xM
± do not contain information about zero modes or instanton contributions. The latter have nevertheless been fully taken into account and appear in the overall delta function and loop momenta respectively. Therefore, using the chiral representation of amplitudes significantly simplifies amplitude computations.
Finally, we have also discussed how wave/particle (or rather wave/string) duality is manifested in string theory, and we have shown that the fixed-loop momenta representation can be thought of as the ‘wave picture’, the integrated loop momenta expression yielding the ‘string picture’. There are also hybrid formulations (or Routhians) whereby the compact and non-compact dimensions are in the wave or string picture, leading to four natural possibilities in total. In a forthcoming article [61] we will show that adopting a wave picture leads to significant simplifications and explicit analytic results (a string picture being much less tractable analytically).
The objective here has been to provide a working and efficient handle on computing string amplitudes involving HES vertex operators. In [59] we construct chiral HES coherent vertex operators (which is a very natural basis for excited strings) and discuss the notion of Euclidean adjoint vertex operators (which refines the rule of thumb of Polchinski [86, 99], a refinement that is necessary in order forallvertex operators to have positive norm48). These vertex operators are then [60] used to derive a generic expression for two-point amplitudes (where we keep the genus of the worldsheet generic in order to study generic properties),
47Note that there are still expected to be field-redefinition ambiguities that one expects from insight from
string field theory, see Sec. 4 in [67]. The authors thank Ashoke Sen for an extensive discussion of this point.
48The rule of thumb [86, 99] that to obtain the Euclidean adjoint of a vertex operator one is to conjugate
all explicit factors of ‘i=√−1’ is not sufficient when vertex operators have windingN−N¯ ∈2Z+ 1, in that there are some additional phases (hereN,N¯ are level numbers).
whose imaginary part at one loop [61] yields decay rates and power emitted into massless and massive radiation (including radiative backreaction and in particular α0 corrections), the real part giving mass shifts (relevant for black hole physics [32]). In [62] we discuss decay rates associated to gravitational radiation in particular and in [63] we make the connection to low energy effective field theory.