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

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 II, The Chiral Measure on Moduli Space

A detailed derivation from first principles is given for the unambiguous and slice-independent formula for the two-loop superstring chiral measure which was announced in the first paper of this series. Supergeometries are projected onto their super period matrices, and the integration over odd supermoduli is performed by integrating over the fibers of this projection. The subtleties associated with this procedure are identified. They require the inclusion of some new finite-dimensional Jacobian superdeterminants, a deformation of the worldsheet correlation functions using the stress tensor, and perhaps paradoxically, another additional gauge choice, ``slice \hat\mu choice'', whose independence also has to be established. This is done using an important correspondence between superholomorphic notions with respect to a supergeometry and holomorphic notions with respect to its super period matrix. Altogether, the subtleties produce precisely the corrective terms which restore the independence of the resulting gauge-fixed formula under infinitesimal changes of gauge-slice. This independence is a key criterion for any gauge-fixed formula and hence is verified in detail.Comment: 64 pages, no figure

The Box Graph In Superstring Theory

In theories of closed oriented superstrings, the one loop amplitude is given by a single diagram, with the topology of a torus. Its interpretation had remained obscure, because it was formally real, converged only for purely imaginary values of the Mandelstam variables, and had to account for the singularities of both the box graph and the one particle reducible graphs in field theories. We present in detail an analytic continuation method which resolves all these difficulties. It is based on a reduction to certain minimal amplitudes which can themselves be expressed in terms of double and single dispersion relations, with explicit spectral densities. The minimal amplitudes correspond formally to an infinite superposition of box graphs on $\phi ^3$ like field theories, whose divergence is responsible for the poles in the string amplitudes. This paper is a considerable simplification and generalization of our earlier proposal published in Phys. Rev. Lett. 70 (1993) p 3692.Comment: Plain TeX, 67 pp. and 9 figures, Columbia/UCLA/94/TEP/3

Transport coefficients in high temperature gauge theories: (II) Beyond leading log

Results are presented of a full leading-order evaluation of the shear viscosity, flavor diffusion constants, and electrical conductivity in high temperature QCD and QED. The presence of Coulomb logarithms associated with gauge interactions imply that the leading-order results for transport coefficients may themselves be expanded in an infinite series in powers of 1/log(1/g); the utility of this expansion is also examined. A next-to-leading-log approximation is found to approximate the full leading-order result quite well as long as the Debye mass is less than the temperature.Comment: 38 pages, 6 figure

HI in the Outskirts of Nearby Galaxies

The HI in disk galaxies frequently extends beyond the optical image, and can trace the dark matter there. I briefly highlight the history of high spatial resolution HI imaging, the contribution it made to the dark matter problem, and the current tension between several dynamical methods to break the disk-halo degeneracy. I then turn to the flaring problem, which could in principle probe the shape of the dark halo. Instead, however, a lot of attention is now devoted to understanding the role of gas accretion via galactic fountains. The current $\rm \Lambda$ cold dark matter theory has problems on galactic scales, such as the core-cusp problem, which can be addressed with HI observations of dwarf galaxies. For a similar range in rotation velocities, galaxies of type Sd have thin disks, while those of type Im are much thicker. After a few comments on modified Newtonian dynamics and on irregular galaxies, I close with statistics on the HI extent of galaxies.Comment: 38 pages, 17 figures, invited review, book chapter in "Outskirts of Galaxies", Eds. J. H. Knapen, J. C. Lee and A. Gil de Paz, Astrophysics and Space Science Library, Springer, in pres

Challenges of open innovation: the paradox of firm investment in open-source software

Open innovation is a powerful framework encompassing the generation, capture, and employment of intellectual property at the firm level. We identify three fundamental challenges for firms in applying the concept of open innovation: finding creative ways to exploit internal innovation, incorporating external innovation into internal development, and motivating outsiders to supply an ongoing stream of external innovations. This latter challenge involves a paradox, why would firms spend money on R&D efforts if the results of these efforts are available to rival firms? To explore these challenges, we examine the activity of firms in opensource software to support their innovation strategies. Firms involved in open-source software often make investments that will be shared with real and potential rivals. We identify four strategies firms employ – pooled R&D/product development, spinouts, selling complements and attracting donated complements – and discuss how they address the three key challenges of open innovation. We conclude with suggestions for how similar strategies may apply in other industries and offer some possible avenues for future research on open innovation

Geometric effects on T-breaking in p+ip and d+id superconductors

Superconducting order parameters that change phase around the Fermi surface modify Josephson tunneling behavior, as in the phase-sensitive measurements that confirmed $d$ order in the cuprates. This paper studies Josephson coupling when the individual grains break time-reversal symmetry; the specific cases considered are $p \pm ip$ and $d \pm id$, which may appear in Sr$_2$RuO$_4$ and Na$_x$CoO$_2 \cdot$(H$_2$O)$_y$ respectively. $T$-breaking order parameters lead to frustrating phases when not all grains have the same sign of time-reversal symmetry breaking, and the effects of these frustrating phases depend sensitively on geometry for 2D arrays of coupled grains. These systems can show perfect superconducting order with or without macroscopic $T$-breaking. The honeycomb lattice of superconducting grains has a superconducting phase with no spontaneous breaking of $T$ but instead power-law correlations. The superconducting transition in this case is driven by binding of fractional vortices, and the zero-temperature criticality realizes a generalization of Baxter's three-color model.Comment: 8 page

Two-Loop Superstrings IV, The Cosmological Constant and Modular Forms

The slice-independent gauge-fixed superstring chiral measure in genus 2 derived in the earlier papers of this series for each spin structure is evaluated explicitly in terms of theta-constants. The slice-independence allows an arbitrary choice of superghost insertion points q_1, q_2 in the explicit evaluation, and the most effective one turns out to be the split gauge defined by S_{\delta}(q_1,q_2)=0. This results in expressions involving bilinear theta-constants M. The final formula in terms of only theta-constants follows from new identities between M and theta-constants which may be interesting in their own right. The action of the modular group Sp(4,Z) is worked out explicitly for the contribution of each spin structure to the superstring chiral measure. It is found that there is a unique choice of relative phases which insures the modular invariance of the full chiral superstring measure, and hence a unique way of implementing the GSO projection for even spin structure. The resulting cosmological constant vanishes, not by a Riemann identity, but rather by the genus 2 identity expressing any modular form of weight 8 as the square of a modular form of weight 4. The degeneration limits for the contribution of each spin structure are determined, and the divergences, before the GSO projection, are found to be the ones expected on physical grounds.Comment: 58 pages, no figure

The Solution Space of the Unitary Matrix Model String Equation and the Sato Grassmannian

The space of all solutions to the string equation of the symmetric unitary one-matrix model is determined. It is shown that the string equation is equivalent to simple conditions on points $V_1$ and $V_2$ in the big cell \Gr of the Sato Grassmannian $Gr$. This is a consequence of a well-defined continuum limit in which the string equation has the simple form \lb \cp ,\cq_- \rb =\hbox{\rm 1}, with \cp and \cq_- $2\times 2$ matrices of differential operators. These conditions on $V_1$ and $V_2$ yield a simple system of first order differential equations whose analysis determines the space of all solutions to the string equation. This geometric formulation leads directly to the Virasoro constraints \L_n\,(n\geq 0), where \L_n annihilate the two modified-KdV \t-functions whose product gives the partition function of the Unitary Matrix Model.Comment: 21 page