Boundary contributions to three loop superstring amplitudes

The super period matrix with Ramond punctures

Super Riemann surfaces are the arena for two-dimensional superconformal field theory. They can be regarded as smooth 212-dimensional supermanifolds equipped with a reduction of their structure group to the group of upper triangular 2 × 2 complex matrices. The integrability conditions for such a reduction turn out to be (most of) the famous torsion constraints of 2d supergravity. The other torsion constraints are merely conditions to fix some of the gauge freedom in this description, or to specify a particular connection on such a manifold, analogous to the Levi-Ci vita connection in Riemannian geometry. Unlike ordinary Riemann surfaces, a super Riemann surface \(\hat{X}\) cannot be regarded as having only one complex dimension; instead it has one commuting and one anticommuting complex coordinate. Nevertheless, in certain important aspects SRS behave as nicely as if they had only one dimension. In particular they possess an analog of the Cauchy-Riemann operator \(\bar{\partial }\) on an ordinary Riemann surface X. The latter is a first-order differential operator with values in a canonical line bundle ω º Ω1X over the surface; this bundle in turn is in a sense a square root of the bundle of volume forms. Similarly super Riemann surfaces have a canonically-defined first-order operator \(\hat{\bar{\partial }}\) taking values in the bundle \(\hat{\omega } \equiv BER{{\Omega }^{1}}\hat{X}\) of half-volume forms. This observation allows us to write a superstring world-sheet action depending only on a “superconformal structure” analogous to a conformal structure on an ordinary 2-surface. Furthermore, the operator \(\hat{\bar{\partial }}\) furnishes a short resolution of the structure sheaf of the super Riemann surface, making possible a Quillen theory of determinant line bundles. The material in this talk, and further discussion, can be found in the following papers: 1. S. Giddings and P. Nelson, “Torsion constraints and super Riemann surfaces,” Phys. Rev. Lett. 59, 2619 (1987). 2. P. Nelson, “Holomorphic coordinates for supermoduli space,” Comm. Math. Phys. 115, 167 (1988). 3. S. Giddings and P. Nelson, “The geometry of super Riemann surfaces,” Comm. Math. Phys. (1988)

We define super Riemann surfaces as smooth 2|2-dimensional supermanifolds equipped with a reduction of their structure group to the group of invertible upper triangular 2×2 complex matrices. The integrability conditions for such a reduction turn out to be (most of) the torsion constraints of 2d supergravity. We show that they are both necessary and sufficient for a frame to admit local superconformal coordinates. The other torsion constraints are merely conditions to fix some of the gauge freedom in this description, or to specify a particular connection on such a manifold, analogous to the Levi-Civita connection in Riemannian geometry. Unlike ordinary Riemann surfaces, a super Riemann surface cannot be regarded as having only one complex dimension. Nevertheless, in certain important aspects super Riemann surfaces behave as nicely as if they had only one dimension. In particular they posses an analog $$\widehat\partial $$ of the Cauchy-Riemann operator on ordinary Riemann surfaces, a differential operator taking values in the bundle of half-volume forms. This operator furnishes a short resolution of the structure sheaf, making possible a Quillen theory of determinant line bundles. Finally we show that the moduli space of super Riemann surfaces is embedded in the larger space of complex curves of dimension 1|1.

Multiloop superstring amplitudes are obtained upon integrating local amplitudes with respect to modular parameters of the Schottky supergroup and with respect to vertex coordinates in the nonsplit (1|1) complex supermanifold. Ambiguities in this integration are removed by the condition requiring that the local symmetries of the amplitudes under discussion be conserved. Local superstring amplitudes are calculated by summing local fermion-string amplitudes. The resulting superstring amplitudes are free from divergencies. The vacuum amplitude is zero, in just the same way as the one-, two-, and three-point massless-boson amplitudes are. The amplitudes in question vanish after integration of corresponding local amplitudes with respect to interaction-vertex coordinates and with respect to the limiting points of the Schottky supergroup, with the exception of those (3|2) points that are fixed by SL(2) symmetry.

The super Mumford form and Sato Grassmannian

We construct curves in moduli spaces of curves with prescribed intersection with boundary divisors. As applications, we obtain families of curves with maximal slope as well as extremal test curves for the weakly positive cone of the moduli space. MSC2010: 14D22.

We describe a general formalism based on the partial-wave decomposition to compute the iterative s-channel discontinuity of four-point amplitudes at any loop order. As an application, we focus on the low-energy expansions of type I and II superstring amplitudes. Besides providing new results for their leading and sub-leading logarithmic contributions beyond genus one, our approach elucidates the general structure of non-analytic threshold terms. In the case of open strings, the use of orthogonal colour projectors allows us to efficiently compute all contributions from different worldsheet topologies at a given loop order.

This work can be considered as a continuation of our previous one (J.Phys., 26 (1993) 313), in which an explicit form of coherent states (CS) for all SU(N) groups was constructed by means of representations on polynomials. Here we extend that approach to any SU(l,1) group and construct explicitly corresponding CS. The CS are parametrized by dots of a coset space, which is, in that particular case, the open complex ball $CD^{l}$. This space together with the projective space $CP^{l}$, which parametrizes CS of the SU(l+1) group, exhausts all complex spaces of constant curvature. Thus, both sets of CS provide a possibility for an explicit analysis of the quantization problem on all the spaces of constant curvature.

The paper continues our (in collaboration with A. Borisenko [J. Differential Geom. Appl. 20 p., to appear]) discovery of the new classes of \((k,\varepsilon)\)-saddle, \((k,\varepsilon)\)-parabolic, and \((k,\varepsilon)\)-convex submanifolds ( \(\,\varepsilon\ge0\)). These are defined in terms of the eigenvalues of the 2nd fundamental forms of each unit normal of the submanifold, extending the notion of k-saddle, k-parabolic, k-convex submanifolds ( \(\,\varepsilon=0\)). It follows that the definition of \((k,\varepsilon)\)-saddle submanifolds is equivalent to the existence of \(\varepsilon\)-asymptotic subspaces in the tangent space. We prove non-embedding theorems of \((k,\varepsilon)\)-saddle submanifolds, theorems about 1-connectedness and homology groups of these submanifolds in Riemannian spaces of positive (sectional or qth Ricci) curvature, in particular, spherical and projective spaces. We apply these results to submanifolds with ‘small’ normal curvature, \(k_n\le\varepsilon\), and for submanifolds with extrinsic curvature \(\le\varepsilon^2\) (resp., non-positive) and small codimension, and include some illustrative examples. The results of the paper generalize theorems about totally geodesic, minimal and k-saddle submanifolds by Frankel; Borisenko, Rabelo and Tenenblat; Kenmotsu and Xia; Mendonca and Zhou.

In the present paper, we study two different approaches of tensor decomposition. The first part aims to study some properties of tensors that result from the fact that some components are vanishing in certain coordinates. It is proven that these conditions allow tensor decomposition, especially (1, σ), σ=1,2,3 tensors. We apply the results for special tensors such as the Riemann, Ricci, Einstein, and Weyl tensors and the deformation tensors of affine connections. Thereby, we find new criteria for the Einstein spaces, spaces of constant curvature, and projective and conformal flat spaces. Further, the proof of the theorem of Mikeš and Moldobayev is repaired. It has been used in many works and it is a generalization of the criteria formulated by Schouten and Struik. The second part deals with the properties of a special differential operator with respect to the general decomposition of tensor fields on manifolds with affine connection. It is shown that the properties of special differential operators are transferred to the components of a given decomposition.

The homotopy type of spaces of rational curves on a toric variety

Given integers r>1, n>1 and q> n-2, we consider projective varieties X of dimension r+1 such that through n generic points of X passes a rational curve of degree q, contained in X. More precisely, we study the class X_{r+1,n}(q) of such varieties which moreover generate a projective space of the maximal dimension. We determine all varieties of a class X_{r+1,n}(q) when q is not equal to 2n-3. In particuliar, we show that there exists a variety X' in P^{r+n-1}, of minimal degree and a birational map F: X'---> X which sends a generic section of X' by a P^{n-1} onto a rational normal curve of degree q. Without hypothesis on q, we define a quasi-grassmannian structure on the space of the rational normal curves of degree q contained in a variety X of the class X_{r+1,n}(q). We prove that X is of the form described above if and only if this quasi-grassmannian structure is flat. We also give examples of varieties of the classes X_{r+1,3}(3) et X_{r+1,4}(5) which are not of this form.

The moduli space of stable rational curves with marked points has two distinguished families of maps: the forgetful maps, given by forgetting some of the markings, and the Kapranov maps, given by complete linear series of $\psi $-classes. The collection of all these maps embeds the moduli space into a product of projective spaces. We call the multidegrees of this embedding “Kapranov degrees,” which include as special cases the work of Witten, Silversmith, Gallet–Grasegger–Schicho, Castravet–Tevelev, Postnikov, Cavalieri–Gillespie–Monin, and Gillespie–Griffin–Levinson. We establish, in terms of a combinatorial matching condition, upper bounds for Kapranov degrees and a characterization of their positivity. The positivity characterization answers a question of Silversmith and gives a new proof of Laman’s theorem characterizing generically rigid graphs in the plane. We achieve this by proving a recursive formula for Kapranov degrees and by using tools from the theory of error correcting codes.

Simple formulas in terms of Chern classes are given for the volume of a tube about a Kähler submanifold of a space of constant holomorphic sectional curvature. A comparison theorem which generalizes these formulas is also given. Tubes about complete intersections in complex projective space are studied in detail.

In this paper, we show the first and second variational formulas of biharmonic maps and bi-Yang-Mills fields, and show the first variation formula of k-harmonic maps, and also give an overview of our recent results in (12), i.e., classifications of all biharmonic isopara- metric hypersurfaces in the unit sphere, and all biharmonic homogeneous real hypersurfaces in the complex or quaternionic projective spaces, answers in case of bounded geometry to Chen's conjecture or Caddeo, Montaldo and Piu's one on biharmonic maps into a space of non positive curvature and the isolation phenomena of bi-Yang-Mills fields.