PDF Publication Title:
Text from PDF Page: 034
R. M. Kaufmann, D. Li & B. Wehefritz-Kaufmann index by the eta invariant where b a μ a ( f ) : = 1 b n [ L ∗2 Ω − L ∗1 Ω ] + 2 η ̃ ( D ( L 1 , L 2 ) ) | ba 1 n ( 4 . 7 ) μa := η(D(L1, L2)). (4.5) We stress that this analytical Maslov index is real-valued. For example, consider two typical Lagrangians R and L(θ) = R{eiθ} in (R2,ω), the path connecting them can be taken to be φ(t) = ei(lπ+θ)t, for some l ∈ Z where the integer l is added since the Lagrangian R{eiθ} is invariant by multiplying a minus sign. Since D(R,L(θ))φ=−idφ =(lπ+θ)φ dt the eta function of D(R, L(θ)) is η(D(R, L(θ)), s) = 1 [ζH (s, θ/π) − ζH (s, 1 − θ/π)] πs where ζH (s, x) is the Hurwitz zeta function ζH (s, x) = ∞n=0(n+x)−s for 0 < x < 1. The Hurwitz zeta function has a regular point at s = 0, indeed ζH (0, x) = 1 − x. 2 Hence the real-valued Maslov index is easily obtained by direct computation μa = η(D(R,L(θ)),0) = 1 − 2θ. (4.6) π In order to be compatible with the integral Lagrangian intersection number, i.e. the geometric Maslov index, the integer-valued analytical Maslov index is defined as follows. Define the canonical 1-form on Λ(1) by pulling back the standard 1-form on S1, dθ 2π which is the generator of the cohomology group H1(Λ(1), Z) ≃ H1(S1, Z) ≃ Z. Now the integer-valued analytical Maslov index is defined for a pair of continuous and piecewise smooth Lagrangians f(t) = (L1(t),L2(t)), for some interval a ≤ t ≤ b, Ω := (Sq)∗ b1 a L∗sΩ=π In (R2,ω), let L1 = R,L2 = R{eiθ}, then it is easy to compute the eta invariant dθsi=π [θsi(b)−θsi(a)],s=1,2. i=1 a i=1 η(D(L1, L2)) = 1 − 2θ for 0 < θ ≤ π π 4 0 for θ = 0 1630003-34 −1−2θ for−π≤θ<0 π 4 Rev. Math. Phys. 2016.28. Downloaded from www.worldscientific.com by PURDUE UNIVERSITY on 08/11/17. For personal use only.PDF Image | Notes on topological insulators
PDF Search Title:
Notes on topological insulatorsOriginal File Name Searched:
RMP.pdfDIY PDF Search: Google It | Yahoo | Bing
Sulfur Deposition on Carbon Nanofibers using Supercritical CO2 Sulfur Deposition on Carbon Nanofibers using Supercritical CO2. Gamma sulfur also known as mother of pearl sulfur and nacreous sulfur... More Info
CO2 Organic Rankine Cycle Experimenter Platform The supercritical CO2 phase change system is both a heat pump and organic rankine cycle which can be used for those purposes and as a supercritical extractor for advanced subcritical and supercritical extraction technology. Uses include producing nanoparticles, precious metal CO2 extraction, lithium battery recycling, and other applications... More Info
CONTACT TEL: 608-238-6001 Email: greg@infinityturbine.com (Standard Web Page)