877 research outputs found

### 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

### 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

### 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

### Momumentum Analyticity and Finiteness of Compactified String Amplitudes, Part I: Tori

We generalize to the case of compactified superstrings a construction given
previously for critical superstrings of finite one loop amplitudes that are
well-defined for all external momenta. The novel issues that arise for
compactified strings are the appearance of infrared divergences from the
propagation of massless strings in four dimensions and, in the case of orbifold
schemes, the contribution of tachyons in partial amplitudes with given spin
structure and twist sectors. Methods are presented for the resolution of these
problems and expressions for finite amplitudes are given in terms of double and
single dispersion relations, with explicit spectral densities.Comment: 18 pages, te

### 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

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