Schwarzian Derivatives and Flows of Surfaces

This paper goes some way in explaining how to construct an integrable hierarchy of flows on the space of conformally immersed tori in n-space. These flows have first occured in mathematical physics -- the Novikov-Veselov and Davey-Stewartson hierarchies -- as kernel dimension preserving deformations of the Dirac operator. Later, using spinorial representations of surfaces, the same flows were interpreted as deformations of surfaces in 3- and 4-space preserving the Willmore energy. This last property suggest that the correct geometric setting for this theory is Moebius invariant surface geometry. We develop this view point in the first part of the paper where we derive the fundamental invariants -- the Schwarzian derivative, the Hopf differential and a normal connection -- of a conformal immersion into n-space together with their integrability equations. To demonstrate the effectivness of our approach we discuss and prove a variety of old and new results from conformal surface theory. In the the second part of the paper we derive the Novikov-Veselov and Davey-Stewartson flows on conformally immersed tori by Moebius invariant geometric deformations. We point out the analogy to a similar derivation of the KdV hierarchy as flows on Schwarzian's of meromorphic functions. Special surface classes, e.g. Willmore surfaces and isothermic surfaces, are preserved by the flows

A three-dimensional wavelet based multifractal method : about the need of revisiting the multifractal description of turbulence dissipation data

We generalize the wavelet transform modulus maxima (WTMM) method to multifractal analysis of 3D random fields. This method is calibrated on synthetic 3D monofractal fractional Brownian fields and on 3D multifractal singular cascade measures as well as their random function counterpart obtained by fractional integration. Then we apply the 3D WTMM method to the dissipation field issue from 3D isotropic turbulence simulations. We comment on the need to revisiting previous box-counting analysis which have failed to estimate correctly the corresponding multifractal spectra because of their intrinsic inability to master non-conservative singular cascade measures.Comment: 5 pages, 3figures, submitted to Phys. Rev. Let

Affine sl(2|1) and D(2|1;alpha) as Vertex Operator Extensions of Dual Affine sl(2) Algebras

We discover a realisation of the affine Lie superalgebra sl(2|1) and of the exceptional affine superalgebra D(2|1;alpha) as vertex operator extensions of two affine sl(2) algebras with dual levels (and an auxiliary level 1 sl(2) algebra). The duality relation between the levels is (k+1)(k'+1)=1. We construct the representation of sl(2|1) at level k' on a sum of tensor products of sl(2) at level k, sl(2) at level k' and sl(2) at level 1 modules and decompose it into a direct sum over the sl(2|1) spectral flow orbit. This decomposition gives rise to character identities, which we also derive. The extension of the construction to the affine D(2|1;k') at level k is traced to properties of sl(2)+sl(2)+sl(2) embeddings into D(2|1;alpha) and their relation with the dual sl(2) pairs. Conversely, we show how the level k' sl(2) representations are constructed from level k sl(2|1) representations.Comment: 50 pages, Latex2e, 2 figures, acknowledgements adde

Local inverses of shift maps along orbits of flows

Let $M$ be a smooth manifold and $F$ be a vector field on $M$. My article ["Smooth shifts along trajectories of flows", Topol. Appl. 130 (2003) 183-204, arXiv:math/0106199] concerning the homotopy types of the group of diffeomorphisms preserving orbits of $F$ contains two errors. They imply that the principal statement of that paper holds under additional assumptions on $F$. Unfortunately this result was essentially used in another paper of mine ["Homotopy types of stabilizers and orbits of Morse functions on surfaces" Ann. Glob. Anal. Geom., 29 no. 3, (2006) 241-285, arXiv:math/0310067]. The aim of this article is to expose the results of the first paper in a right way, extend them to a larger class of flows with degenerate singularities, and show that the results of the second paper remain true.Comment: 43 pages, 4 figures. V3. The class of admissible singularities for vector field is extende

Circuit complexity in interacting QFTs and RG flows

We consider circuit complexity in certain interacting scalar quantum field theories, mainly focusing on the $\phi^4$ theory. We work out the circuit complexity for evolving from a nearly Gaussian unentangled reference state to the entangled ground state of the theory. Our approach uses Nielsen's geometric method, which translates into working out the geodesic equation arising from a certain cost functional. We present a general method, making use of integral transforms, to do the required lattice sums analytically and give explicit expressions for the $d=2,3$ cases. Our method enables a study of circuit complexity in the epsilon expansion for the Wilson-Fisher fixed point. We find that with increasing dimensionality the circuit depth increases in the presence of the $\phi^4$ interaction eventually causing the perturbative calculation to breakdown. We discuss how circuit complexity relates with the renormalization group.Comment: 50 pages, 2 figures; references updated; version to appear in JHE