Two-loop superstrings.

Abstract

In these lectures, recent progress on multiloop superstring perturbation theory is reviewed. A construction from first principles is given for an unambiguous and sliceindependent two-loop superstring measure on moduli space for even spin structure. A consistent choice of moduli, invariant under local worldsheet supersymmetry is made in terms of the super-period matrix. A variety of subtle new contributions arising from a careful gauge fixing procedure are taken into account.The superstring measure is computed explicitly in terms of genus two thetafunctions and reveals the importance of a new modular object of weight 6. For given even spin structure, the measure exhibits a behavior under degenerations of the worldsheet that is consistent with physical principles. The measure allows for a unique modular covariant GSO projection. Under this GSO projection, the cosmological constant, the 1-, 2-and 3-point functions of massless supergravitons vanish pointwise on moduli space. A certain disconnected part of the 4-point function is shown to be given by a convergent integral on moduli space. A general consistent formula is given for the two-loop cosmological constant in compactifications with central charge c = 15 and with N = 1 worldsheet supersymmetry. Finally, some comments are made on possible extensions of this work to higher loop order.

Superstring Scattering Amplitudes in Higher Genus

“…Showing that the cosmological constant vanishes is equivalent to showing that the sum ∆ Ξ (g) [∆] is identically zero. This has been verified for genus 2 in [8] and for genus 3 in [3]. In general notice that this sum, if non-zero, is a modular form with respect to the entire group Sp(2g, Z) of weight 8.…”

“…Following the earlier works, we try to find the superstring measure as the product of the bosonic measure (some formulas for which are known, but explicit point-independent expressions for which are apparently not known for high genus -see the discussion in [8]) and a function Ξ (g) [∆] depending on the characteristic ∆. As argued in [3], p.5 the factorization property can then be rewritten as a condition on Ξ (g) [∆].…”

“…We substitute the recursive expression (9) from the lemma into the left-hand-side of (8) where we used the fact that the i = −1 and I = n + m + 1 summands are zero in the last two sums, and the fact that formula (9) works for N's some of which are zero as well. Now we note that the two expressions in the last line are the same up to sign and renaming the variable from i to I, and thus they cancel, so that we finally use the inductive assumption that (8) Remark 25. This ansatz is a direct generalization of the formulas we obtained above for g ≤ 4 (and of course agrees with those).…”

Non-commutative Renormalization

“…Second the mathematical structure of string theory is complicated up to the point where it may become depressing. For instance great effort is needed to put the string theory at two loops on some rigorous footing [31], and three loops seem almost hopeless. Third, there was for some time the hope that string theory and the phenomenology at lower energies derived from it might be unique.…”

Factorization of the two loop four-particle amplitude in superstring theory revisited

“…To our knowledge the only explicitly known higher loop (≥ 2) non-vanishing amplitude is the four-particle amplitude in superstring theory, firstly obtained in [1] and later re-obtained in [2,3] in an explicitly gauge independent way, following the works of D' Hoker and Phong [4,5,6,7,8] on two loop measure of superstring theory. Recently D' Hoker and Phong [9,10] also gave a measure for three loop superstring theory.…”