Two-Loop Superstrings III, Slice Independence and Absence of Ambiguities

The chiral superstring measure constructed in the earlier papers of this series for general gravitino slices is examined in detail for slices supported at two points x_\alpha. In this case, the invariance of the measure under infinitesimal changes of gravitino slices established previously is strengthened to its most powerful form: the measure is shown, point by point on moduli space, to be locally and globally independent from the points x_\alpha, as well as from the superghost insertion points p_a, q_\alpha introduced earlier as computational devices. In particular, the measure is completely unambiguous. The limit x_\alpha = q_\alpha is then well defined. It is of special interest, since it elucidates some subtle issues in the construction of the picture-changing operator Y(z) central to the BRST formalism. The formula for the chiral superstring measure in this limit is derived explicitly.Comment: 20 pages, no figure

Calogero-Moser Systems in SU(N) Seiberg-Witten Theory

The Seiberg-Witten curve and differential for ${\cal N}=2$ supersymmetric SU(N) gauge theory, with a massive hypermultiplet in the adjoint representation of the gauge group, are analyzed in terms of the elliptic Calogero-Moser integrable system. A new parametrization for the Calogero-Moser spectral curves is found, which exhibits the classical vacuum expectation values of the scalar field of the gauge multiplet. The one-loop perturbative correction to the effective prepotential is evaluated explicitly, and found to agree with quantum field theory predictions. A renormalization group equation for the variation with respect to the coupling is derived for the effective prepotential, and may be evaluated in a weak coupling series using residue methods only. This gives a simple and efficient algorithm for the instanton corrections to the effective prepotential to any order. The 1- and 2- instanton corrections are derived explicitly. Finally, it is shown that certain decoupling limits yield ${\cal N}=2$ supersymmetric theories for simple gauge groups $SU(N_1)$ with hypermultiplets in the fundamental representation, while others yield theories for product gauge groups $SU(N_1) \times ...\times SU(N_p)$, with hypermultiplets in fundamental and bi-fundamental representations. The spectral curves obtained this way for these models agree with the ones proposed by Witten using D-branes and M-theory.Comment: 45 pages, Tex, no figure

Getting superstring amplitudes by degenerating Riemann surfaces

We explicitly show how the chiral superstring amplitudes can be obtained through factorisation of the higher genus chiral measure induced by suitable degenerations of Riemann surfaces. This powerful tool also allows to derive, at any genera, consistency relations involving the amplitudes and the measure. A key point concerns the choice of the local coordinate at the node on degenerate Riemann surfaces that greatly simplifies the computations. As a first application, starting from recent ansaetze for the chiral measure up to genus five, we compute the chiral two-point function for massless Neveu-Schwarz states at genus two, three and four. For genus higher than three, these computations include some new corrections to the conjectural formulae appeared so far in the literature. After GSO projection, the two-point function vanishes at genus two and three, as expected from space-time supersymmetry arguments, but not at genus four. This suggests that the ansatz for the superstring measure should be corrected for genus higher than four.Comment: 32 pages; v2: minor corrections, references adde

Two-Loop Superstrings V: Gauge Slice Independence of the N-Point Function

A systematic construction of superstring scattering amplitudes for $N$ massless NS bosons to two loop order is given, based on the projection of supermoduli space onto super period matrices used earlier for the superstring measure in the first four papers of this series. The one important new difficulty arising for the $N$-point amplitudes is the fact that the projection onto super period matrices introduces corrections to the chiral vertex operators for massless NS bosons which are not pure (1,0) differential forms. However, it is proved that the chiral amplitudes are closed differential forms, and transform by exact differentials on the worldsheet under changes of gauge slices. Holomorphic amplitudes and independence of left from right movers are recaptured after the extraction of terms which are Dolbeault exact in one insertion point, and de Rham closed in the remaining points. This allows a construction of GSO projected, integrated superstring scattering amplitudes which are independent of the choice of gauge slices and have only physical kinematical singularities.Comment: 33 pages, no figur

Two-Loop Superstrings VI: Non-Renormalization Theorems and the 4-Point Function

The N-point amplitudes for the Type II and Heterotic superstrings at two-loop order and for $N \leq 4$ massless NS bosons are evaluated explicitly from first principles, using the method of projection onto super period matrices introduced and developed in the first five papers of this series. The gauge-dependent corrections to the vertex operators, identified in paper V, are carefully taken into account, and the crucial counterterms which are Dolbeault exact in one insertion point and de Rham closed in the remaining points are constructed explicitly. This procedure maintains gauge slice independence at every stage of the evaluation. Analysis of the resulting amplitudes demonstrates, from first principles, that for $N\leq 3$, no two-loop corrections occur, while for N=4, no two-loop corrections to the low energy effective action occur for $R^4$ terms in the Type II superstrings, and for $F^4$, $F^2F^2$, $F^2R^2$, and $R^4$ terms in the Heterotic strings.Comment: 98 pages, no figur

Asyzygies, modular forms, and the superstring measure I

The goal of this paper and of a subsequent continuation is to find some viable ansatze for the three-loop superstring chiral measure. For this, two alternative formulas are derived for the two-loop superstring chiral measure. Unlike the original formula, both alternates admit modular covariant generalizations to higher genus. One of these two generalizations is analyzed in detail in the present paper, with the analysis of the other left to the next paper of the series.Comment: 30 page

Two-Loop Superstrings VII, Cohomology of Chiral Amplitudes

The relation between superholomorphicity and holomorphicity of chiral superstring N-point amplitudes for NS bosons on a genus 2 Riemann surface is shown to be encoded in a hybrid cohomology theory, incorporating elements of both de Rham and Dolbeault cohomologies. A constructive algorithm is provided which shows that, for arbitrary N and for each fixed even spin structure, the hybrid cohomology classes of the chiral amplitudes of the N-point function on a surface of genus 2 always admit a holomorphic representative. Three key ingredients in the derivation are a classification of all kinematic invariants for the N-point function, a new type of 3-point Green's function, and a recursive construction by monodromies of certain sections of vector bundles over the moduli space of Riemann surfaces, holomorphic in all but exactly one or two insertion points.Comment: 103 pages, 2 figure

On the Construction of Asymmetric Orbifold Models

Various asymmetric orbifold models based on chiral shifts and chiral reflections are investigated. Special attention is devoted to the consistency of the models with two fundamental principles for asymmetric orbifolds : modular invariance and the existence of a proper Hilbert space formulation for states and operators. The interplay between these two principles is non-trivial. It is shown, for example, that their simultaneous requirement forces the order of a chiral reflection to be 4, instead of the naive 2. A careful explicit construction is given of the associated one-loop partition functions. At higher loops, the partition functions of asymmetric orbifolds are built from the chiral blocks of associated symmetric orbifolds, whose pairings are determined by degenerations to one-loop.Comment: 40 pages, no figures, typos correcte

Exact Half-BPS Flux Solutions in M-theory III: Existence and rigidity of global solutions asymptotic to AdS4 x S7

The BPS equations in M-theory for solutions with 16 residual supersymmetries, $SO(2,2)\times SO(4)\times SO(4)$ symmetry, and $AdS_4 \times S^7$ asymptotics, were reduced in [arXiv:0806.0605] to a linear first order partial differential equation on a Riemann surface with boundary, subject to a non-trivial quadratic constraint. In the present paper, suitable regularity and boundary conditions are imposed for the existence of global solutions. We seek regular solutions with multiple distinct asymptotic $AdS_4 \times S^7$ regions, but find that, remarkably, such solutions invariably reduce to multiple covers of the M-Janus solution found by the authors in [arXiv:0904.3313], suggesting rigidity of the half-BPS M-Janus solution. In particular, we prove analytically that no other smooth deformations away from the M-Janus solution exist, as such deformations invariably violate the quadratic constraint. These rigidity results are contrasted to the existence of half-BPS solutions with non-trivial 4-form fluxes and charges asymptotic to $AdS_7 \times S^4$. The results are related to the possibility of M2-branes to end on M5-branes, but the impossibility of M5-branes to end on M2-branes, and to the non-existence of half-BPS solutions with simultaneous $AdS_4 \times S^7$ and $AdS_7 \times S^4$ asymptotic regions.Comment: 52 pages, 2 figures, pdf-latex. Minor change