A Large k Asymptotics of Witten's Invariant of Seifert Manifolds

We calculate a large $k$ asymptotic expansion of the exact surgery formula for Witten's $SU(2)$ invariant of Seifert manifolds. The contributions of all flat connections are identified. An agreement with the 1-loop formula is checked. A contribution of the irreducible connections appears to contain only a finite number of terms in the asymptotic series. A 2-loop correction to the contribution of the trivial connection is found to be proportional to Casson's invariant.Comment: 51 pages (Some changes are made to the Discussion section. A surgery formula for perturbative corrections to the contribution of the trivial connection is suggested.

Residue Formulas for the Large k Asymptotics of Witten's Invariants of Seifert Manifolds. The Case of SU(2)

We derive the large k asymptotics of the surgery formula for SU(2) Witten's invariants of general Seifert manifolds. The contributions of connected components of the moduli space of flat connections are identified. The contributions of irreducible connections are presented in a residue form. This form is similar to the one used by A. Szenes, L. Jeffrey and F. Kirwan. This similarity allows us to express the contributions of irreducible connections in terms of intersection numbers on their moduli spaces.Comment: 39 pages, no figures, LaTe

Witten's Invariants of Rational Homology Spheres at Prime Values of KK and Trivial Connection Contribution

We establish a relation between the coefficients of asymptotic expansion of trivial connection contribution to Witten's invariant of rational homology spheres and the invariants that T.~Ohtsuki extracted from Witten's invariant at prime values of $K$. We also rederive the properties of prime $K$ invariants discovered by H.~Murakami and T.~Ohtsuki. We do this by using the bounds on Taylor series expansion of the Jones polynomial of algebraically split links, studied in our previous paper. These bounds are enough to prove that Ohtsuki's invariants are of finite type. The relation between Ohtsuki's invariants and trivial connection contribution is verified explicitly for lens spaces and Seifert manifolds.Comment: 32 pages, no figures, LaTe

Rare first rib pseudoarthrosis with thoracic outlet syndrome in pediatric gymnast: A case report

Background: This case study evaluates the diagnosis and treatment of a 12 year old Caucasian male gymnast who had several diagnoses including an isolated first rib fracture, resultant pseudoarthrosis of the first rib, and the development of symptomatic thoracic outlet syndrome. We discuss the causes, prevalence, and suggestions for prompt diagnosis and treatment of these conditions in pediatric patients. Although all three conditions are rare in a child, this case highlights the importance of having a high clinical index of suspicion in recurrent pain in pre-pubertal athletes. Case presentation: A 10 year old Caucasian male presented with a two to three month history of worsening left shoulder pain. He was a competitive gymnast who practiced approximately ten hours per week. His shoulder pain was accompanied by a tic type movement consisting of hyperextension of the left shoulder multiple times per day. The patient was seen by a pediatric orthopedic surgeon who diagnosed the patient with overuse syndrome and prescribed physical therapy. Within one to two months, the patient\u27s shoulder pain and tightness returned. For two years, the patient continued the cycle multiple times of two to three months of physical therapy two to three times a week, relative rest, then returned to activity. He continued to be diagnosed with “overuse syndrome”. At the age of 12, the patient\u27s mother noticed atrophy to the left upper scapula region and vague weakness of the left upper extremity. Cervical MRI showed “unusual nodular mass at the apex of the left hemithorax involving the antero-lateral aspect of the left first rib.” 3D reconstructed CT images were done showing first rib pseudoarthrosis as well as demonstrating a non-displaced fracture through the left second rib. The patient underwent a left first rib resection without complication. He recovered well post operatively; the pain, “tic”, and atrophy drastically improved, and he returned to his baseline activity level. Conclusions: Children involved in high impact sports are subject to fractures due to the muscles pulling on the bone. Our patient not only had a first rib fracture, but also had incorrect healing of the fracture leading to pseudoarthrosis and eventual thoracic outlet syndrome. With the continued failure of conservative treatment for pain, more imaging studies should be ordered to evaluate for any missed pathologies. Removal of the first rib is a definitive treatment and should be considered if the patient’s thoracic outlet syndrome symptoms do not improve with conservative measures such as lifestyle modifications or physical therapy

Arguments for F-theory

After a brief review of string and $M$-Theory we point out some deficiencies. Partly to cure them, we present several arguments for ``$F$-Theory'', enlarging spacetime to $(2, 10)$ signature, following the original suggestion of C. Vafa. We introduce a suggestive Supersymmetric 27-plet of particles, associated to the exceptional symmetric hermitian space $E_{6}/Spin^{c}(10)$. Several possible future directions, including using projective rather than metric geometry, are mentioned. We should emphasize that $F$-Theory is yet just a very provisional attempt, lacking clear dynamical principles.Comment: To appear in early 2006 in Mod. Phys. Lett. A as Brief Revie

Renormalization Ambiguities in Chern-Simons Theory

We introduce a new family of gauge invariant regularizations of Chern-Simons theories which generate one-loop renormalizations of the coupling constant of the form $k\to k+2 s c_v$ where $s$ can take any arbitrary integer value. In the particular case $s=0$ we get an explicit example of a gauge invariant regularization which does not generate radiative corrections to the bare coupling constant. This ambiguity in the radiative corrections to $k$ is reminiscent of the Coste-L\"uscher results for the parity anomaly in (2+1) fermionic effective actions.Comment: 10 pages, harvmac, no changes, 1 Postscript figure (now included

The partition bundle of type A_{N-1} (2, 0) theory

Six-dimensional (2, 0) theory can be defined on a large class of six-manifolds endowed with some additional topological and geometric data (i.e. an orientation, a spin structure, a conformal structure, and an R-symmetry bundle with connection). We discuss the nature of the object that generalizes the partition function of a more conventional quantum theory. This object takes its values in a certain complex vector space, which fits together into the total space of a complex vector bundle (the `partition bundle') as the data on the six-manifold is varied in its infinite-dimensional parameter space. In this context, an important role is played by the middle-dimensional intermediate Jacobian of the six-manifold endowed with some additional data (i.e. a symplectic structure, a quadratic form, and a complex structure). We define a certain hermitian vector bundle over this finite-dimensional parameter space. The partition bundle is then given by the pullback of the latter bundle by the map from the parameter space related to the six-manifold to the parameter space related to the intermediate Jacobian.Comment: 15 pages. Minor changes, added reference

Charge Lattices and Consistency of 6D Supergravity

We extend the known consistency conditions on the low-energy theory of six-dimensional N = 1 supergravity. We review some facts about the theory of two-form gauge fields and conclude that the charge lattice Gamma for such a theory has to be self-dual. The Green-Schwarz anomaly cancellation conditions in the supergravity theory determine a sublattice of Gamma. The condition that this sublattice can be extended to a self-dual lattice Gamma leads to a strong constraint on theories that otherwise appear to be self-consistent.Comment: 15 pages. v2: minor changes; references, additional example added; v3: minor corrections and clarifications added, JHEP versio