Self-Dual metrics on non-simply connected 4-manifolds

We construct self-dual(SD) but not locally conformally flat(LCF) metrics on families of non-simply connected 4-manifolds with small signature. We construct various sequences with bounded or unbounded Betti numbers and Euler characteristic. These metrics have negative scalar curvature. As an application, this addresses the Remark 4.79 of Besse.Comment: 8 pages, 1 figure. Minor revisio

Recursive Construction of Higgs-Plus-Multiparton Loop Amplitudes: The Last of the Phi-nite Loop Amplitudes

We consider a scalar field, such as the Higgs boson H, coupled to gluons via the effective operator H tr G_{mu nu} G^{mu nu} induced by a heavy-quark loop. We treat H as the real part of a complex field phi which couples to the self-dual part of the gluon field-strength, via the operator phi tr G_{SD mu nu} G_{SD}^{mu nu}, whereas the conjugate field phi^dagger couples to the anti-self-dual part. There are three infinite sequences of amplitudes coupling phi to quarks and gluons that vanish at tree level, and hence are finite at one loop, in the QCD coupling. Using on-shell recursion relations, we find compact expressions for these three sequences of amplitudes and discuss their analytic properties.Comment: 63 pages, 7 figures; v2 references added; v3 minor typos corrected and note added; v4 fixed error in eq. (7.11) (lower limit of sum should be l=2, not l=3), also affecting eqs. (7.14), (8.20), (8.21), (8.27) and (8.28

Restrictions on infinite sequences of type IIB vacua

Ashok and Douglas have shown that infinite sequences of type IIB flux vacua with imaginary self-dual flux can only occur in so-called D-limits, corresponding to singular points in complex structure moduli space. In this work we refine this no-go result by demonstrating that there are no infinite sequences accumulating to the large complex structure point of a certain class of one-parameter Calabi-Yau manifolds. We perform a similar analysis for conifold points and for the decoupling limit, obtaining identical results. Furthermore, we establish the absence of infinite sequences in a D-limit corresponding to the large complex structure limit of a two-parameter Calabi-Yau. In particular, our results demonstrate analytically that the series of vacua recently discovered by Ahlqvist et al., seemingly accumulating to the large complex structure point, are finite. We perform a numerical study of these series close to the large complex structure point using appropriate approximations for the period functions. This analysis reveals that the series bounce out from the large complex structure point, and that the flux eventually ceases to be imaginary self-dual. Finally, we study D-limits for F-theory compactifications on K3\times K3 for which the finiteness of supersymmetric vacua is already established. We do find infinite sequences of flux vacua which are, however, identified by automorphisms of K3.Comment: 35 pages. v2. Typos corrected, ref. added. Matches published versio

On self-duality and unigraphicity for 33-polytopes

Recent literature posed the problem of characterising the graph degree sequences with exactly one $3$-polytopal (i.e. planar, $3$-connected) realisation. This seems to be a difficult problem in full generality. In this paper, we characterise the sequences with exactly one self-dual $3$-polytopal realisation. An algorithm in the literature constructs a self-dual $3$-polytope for any admissible degree sequence. To do so, it performs operations on the radial graph, so that the corresponding $3$-polytope and its dual are modified in exactly the same way. To settle our question and construct the relevant graphs, we apply this algorithm, we introduce some modifications of it, and we also devise new ones. The speed of these algorithms is linear in the graph order

Brst Cohomology and Invariants of 4D Gravity in Ashtekar Variables

We discuss the BRST cohomologies of the invariants associated with the description of classical and quantum gravity in four dimensions, using the Ashtekar variables. These invariants are constructed from several BRST cohomology sequences. They provide a systematic and clear characterization of non-local observables in general relativity with unbroken diffeomorphism invariance, and could yield further differential invariants for four-manifolds. The theory includes fluctuations of the vierbein fields, but there exits a non-trivial phase which can be expressed in terms of Witten's topological quantum field theory. In this phase, the descent sequences are degenerate, and the corresponding classical solutions can be identified with the conformally self-dual sector of Einstein manifolds. The full theory includes fluctuations which bring the system out of this sector while preserving diffeomorphism invariance.Comment: 15 page

Spectral Orbits and Peak-to-Average Power Ratio of Boolean Functions with respect to the {I,H,N}^n Transform

We enumerate the inequivalent self-dual additive codes over GF(4) of blocklength n, thereby extending the sequence A090899 in The On-Line Encyclopedia of Integer Sequences from n = 9 to n = 12. These codes have a well-known interpretation as quantum codes. They can also be represented by graphs, where a simple graph operation generates the orbits of equivalent codes. We highlight the regularity and structure of some graphs that correspond to codes with high distance. The codes can also be interpreted as quadratic Boolean functions, where inequivalence takes on a spectral meaning. In this context we define PAR_IHN, peak-to-average power ratio with respect to the {I,H,N}^n transform set. We prove that PAR_IHN of a Boolean function is equivalent to the the size of the maximum independent set over the associated orbit of graphs. Finally we propose a construction technique to generate Boolean functions with low PAR_IHN and algebraic degree higher than 2.Comment: Presented at Sequences and Their Applications, SETA'04, Seoul, South Korea, October 2004. 17 pages, 10 figure

Multifractal properties of power-law time sequences; application to ricepiles

We study the properties of time sequences extracted from a self-organized critical system, within the framework of the mathematical multifractal analysis. To this end, we propose a fixed-mass algorithm, well suited to deal with highly inhomogeneous one dimensional multifractal measures. We find that the fixed mass (dual) spectrum of generalized dimensions depends on both the system size L and the length N of the sequence considered, being however stable when these two parameters are kept fixed. A finite-size scaling relation is proposed, allowing us to define a renormalized spectrum, independent of size effects.We interpret our results as an evidence of extremely long-range correlations induced in the sequence by the criticality of the systemComment: 12 pages, RevTex, includes 9 PS figures, Phys. Rev. E (in press