Enumeration of N-rooted maps using quantum field theory

A one-to-one correspondence is proved between the N-rooted ribbon graphs, or maps, with e edges and the (e-N+1)-loop Feynman diagrams of a certain quantum field theory. This result is used to obtain explicit expressions and relations for the generating functions of N-rooted maps and for the numbers of N-rooted maps with a given number of edges using the path integral approach applied to the corresponding quantum field theory.Comment: 27 pages, 7 figure

Magnetic rock properties of the gabbros from the ODP Drill Hole 1105 A of the Atlantis Bank, Southwest Indian Ridge

Laboratory studies of 30 samples from 158 m long drill core of the Hole 1105 A (ODP Leg 179) of the Atlantis Bank, Southwest Indian Ridge have revealed magnetic properties of the gabbros, olivine gabbros, oxide gabbros and olivine oxide gabbros down the core. Comparison of modal proportions of the oxides, grain sizes and magnetization parameters of the rocks has confirmed that most coarse-grained oxide mineral bearing rocks record low Koenigsberger ratio (2 to 5) and median destructive fields (5 to 7 mT). Average natural remanent magnetization (Jnrm) and stable remanent magnetization (Jst) of the core samples are 5.8 A/m and 1.9 A/m, respectively. Their mean stable magnetic inclination is 66° ± 4°, about 14° steeper than the expected dipole inclination of the area similar to the one reported at Hole 735 B. The excess inclination perhaps marks a tectonic block rotation of the reversely magnetized rocks of the bank. We interpret that gabbros and serpentinites devoid of basaltic carapace significantly contribute to seafloor spreading anomalies of the bank

Feynman diagrams, ribbon graphs, and topological recursion of Eynard-Orantin

We consider two seemingly unrelated problems, the calculation of the WKB expansion of the harmonic oscillator wave functions and the counting the number of Feynman diagrams in QED or in many-body physics and show that their solutions are both encoded in a single enumerative problem, the calculation of the number of certain types of ribbon graphs. In turn, the numbers of such ribbon graphs as a function of the number of their vertices and edges can be determined recursively through the application of the topological recursion of Eynard-Orantin to the algebraic curve encoded in the Schr\"odinger equation of the harmonic oscillator. We show how the numbers of these ribbon graphs can be written down in closed form for any given number of vertices and edges. We use these numbers to obtain a formula for the number of N-rooted ribbon graphs with e edges, which is the same as the number of Feynman diagrams for 2N-point function with e+1-N loops.Comment: 29 pages, 7 figure

Nature of the crust in the Laxmi Basin (14°-20°N), western continental margin of India

The nature of the crust in the Laxmi Basin, western margin of India, is an uncertain issue; more importantly, this has implications on paleogeographic reconstructions of the western Indian Ocean. We have analyzed three geophysical data sets and modeled gravity and magnetic anomalies for determining nature of the crust. Basement of the Laxmi Basin includes numerous highs, which make the basement uneven and shallower compared to the Western Basin. The Laxmi Basin is characterized by a broad gravity high and a narrower prominent gravity low within it, while within the basin the broad anomaly gradually increases toward north. The Panikkar Ridge is associated with the gravity low, which is comparable, at least in sign, to known negative gravity anomaly of the Laxmi Ridge. Intrusive structures mapped in the Laxmi Basin coincide with significant magnetic anomalies, which were earlier interpreted as seafloor-spreading anomalies. Model studies reveal that the Laxmi Basin consists of ∼14 km thick stretched continental crust, in which magmatic bodies have been emplaced, whereas the Panikkar Ridge remains less altered stretched continental crust. The crust of the Laxmi Basin is mostly thinner than crust under the Laxmi Ridge and continental margin. In addition to the rift-drift-related stretching of the continental margin, the Laxmi Basin possibly has undergone extra stretching in E-W direction during the pre-Tertiary period. At ~68 Ma Deccan volcanism on western India may have disrupted the initial conditions that were leading to onset of spreading in the basin. Subsequently the Reunion hot spot had emplaced the volcanic material within the stretched thinned continental crust. We interpret the Laxmi Basin as a failed rift, undergone stretching following intraplate kinematics prior to Deccan volcanism

BKM superalgebras from counting dyons in N=4 supersymmetric type II compactifications

We study the degeneracy of quarter BPS dyons in N =4 type II compactifications of string theory. We find that the genus-two Siegel modular forms generating the degeneracies of the quarter BPS dyons in the type II theories can be expressed in terms of the genus-two Siegel modular forms generating the degeneracies of quarter BPS dyons in the CHL theories and the heterotic string. This helps us in understanding the algebra structure underlying the degeneracy of the quarter BPS states. The Conway group, Co_1, plays a role similar to Mathieu group, M_{24}, in the CHL models with eta quotients appearing in the place of eta products. We construct BKM Lie superalgebra structures corresponding to Z_N (for N=2,3,4) orbifolds of the type II string compactified on a six-torus.Comment: 21 page

Formation of diapiric structure in the deformation zone, central Indian Ocean: a model from gravity and seismic reflection data

Analyses of bathymetry, gravity and seismic reflection data of the diffusive plate boundary in the central Indian Ocean reveal a new kind of deformed structure besides the well-reported structures of long-wavelength anticlinal basement rises and high-angle reverse faults. The structure (basement trough) has a length of about 150 km and deepens by up to 1 km from its regional trend (northward dipping). The basement trough includes a rise at its center with a height of about 1.5km. The rise is about 10 km wide with rounded upper surface and bounded by vertical faults. A broad freeair gravity low of about 20 mGal and a local high of 8 mGal in its center are associated with the identified basement trough and rise structure respectively. Seismic results reveal that the horizontal crustal compression prevailing in the diffusive plate boundary might have formed the basement trough possibly in early Pliocene time. Differential loading stresses have been generated from unequal crust/sediment thickness on lower crustal and upper mantle rocks. A thin semi-ductile serpentinite layer existing near the base of the crust that is interpreted to have been formed at mid-ocean ridge and become part of the lithosphere, may have responded to the downward loading stresses generated by the sediments and crustal rocks to inject the serpentinites into the overlying strata to form a classic diapiric structure

Enumeration of multi-rooted plane trees

We give closed form expressions for the numbers of multi-rooted plane trees with specified degrees of root vertices. This results in an infinite number of integer sequences some of which are known to have an alternative interpretation. We also propose recursion relations for numbers of such trees as well as for the corresponding generating functions. Explicit expressions for the generating functions corresponding to plane trees having two and three roots are derived. As a by-product, we obtain a new binomial identity and a conjecture relating hypergeometric functions.Comment: 28 pages, 3 figure