Transmutations between Singular and Subsingular Vectors of the N=2 Superconformal Algebras

We present subsingular vectors of the N=2 superconformal algebras other than the ones which become singular in chiral Verma modules, reported recently by Gato-Rivera and Rosado. We show that two large classes of singular vectors of the Topological algebra become subsingular vectors of the Antiperiodic NS algebra under the topological untwistings. These classes consist of BRST- invariant singular vectors with relative charges $q=-2,-1$ and zero conformal weight, and no-label singular vectors with $q=0,-1$. In turn the resulting NS subsingular vectors are transformed by the spectral flows into subsingular and singular vectors of the Periodic R algebra. We write down these singular and subsingular vectors starting from the topological singular vectors at levels 1 and 2.Comment: 21 pages, Latex. Minor improvements. Very similar to the version published in Nucl. Phys.

Spectral Flows and Twisted Topological Theories

We analyze the action of the spectral flows on N=2 twisted topological theories. We show that they provide a useful mapping between the two twisted topological theories associated to a given N=2 superconformal theory. This mapping can also be viewed as a topological algebra automorphism. In particular null vectors are mapped into null vectors, considerably simplifying their computation. We give the level 2 results. Finally we discuss the spectral flow mapping in the case of the DDK and KM realizations of the topological algebra.Comment: The presentation of the results has been very much improved. Some references have been adde

Topological Descendants: DDK and KM Realizations

The "minimal matter + scalar" system can be embedded into the twisted N=2 topological algebra in two ways: a la DDK or a la KM. Here we present some results concerning the topological descendants and their DDK and KM realizations. In particular, we prove four "no-ghost" theorems (two for null states) regarding the reduction of the topological descendants into secon- daries of the "minimal matter + scalar" conformal field theory. We write down the relevant expressions for the case of level 2 descendants.Comment: 10 pgs, Late

The Even and the Odd Spectral Flows on the N=2 Superconformal Algebras

There are two different spectral flows on the N=2 superconformal algebras (four in the case of the Topological algebra). The usual spectral flow, first considered by Schwimmer and Seiberg, is an even transformation, whereas the spectral flow previously considered by the author and Rosado is an odd transformation. We show that the even spectral flow is generated by the odd spectral flow, and therefore only the latter is fundamental. We also analyze thoroughly the four ``topological'' spectral flows, writing two of them here for the first time. Whereas the even and the odd spectral flows have quasi-mirrored properties acting on the Antiperiodic or the Periodic algebras, the topological even and odd spectral flows have drastically different properties acting on the Topological algebra. The other two topological spectral flows have mixed even and odd properties. We show that the even and the even-odd topological spectral flows are generated by the odd and the odd-even topological spectral flows, and therefore only the latter are fundamental.Comment: 15 pages, Latex. Minor improvements in the last paragraph of the conclusions. Numbering of references has change

Half-quadratic transportation problems

We present a primal--dual memory efficient algorithm for solving a relaxed version of the general transportation problem. Our approach approximates the original cost function with a differentiable one that is solved as a sequence of weighted quadratic transportation problems. The new formulation allows us to solve differentiable, non-- convex transportation problems

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Hole probability for nodal sets of the cut-off Gaussian Free Field

Let ($\Sigma$, g) be a closed connected surface equipped with a riemannian metric. Let ($\lambda$ n) n$\in$N and ($\psi$ n) n$\in$N be the increasing sequence of eigenvalues and the sequence of corresponding L 2-normalized eigenfunctions of the laplacian on $\Sigma$. For each L \textgreater{} 0, we consider $\phi$ L = 0\textless{}$\lambda$n$\le$L $\xi$n \sqrt $\lambda$n $\psi$ n where the $\xi$ n are i.i.d centered gaussians with variance 1. As L $\rightarrow$ $\infty$, $\phi$ L converges a.s. to the Gaussian Free Field on $\Sigma$ in the sense of distributions. We first compute the asymptotic behavior of the covariance function for this family of fields as L $\rightarrow$ $\infty$. We then use this result to obtain the asymptotics of the probability that $\phi$ L is positive on a given open proper subset with smooth boundary. In doing so, we also prove the concentration of the supremum of $\phi$ L around 1 \sqrt 2$\pi$ ln L

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