arXiv subjects

C. Pech

Publications and source records attributed to C. Pech.

4 recordsLinked to original sources

arXiv2013

A comparison of Landau-Ginzburg models for odd dimensional Quadrics

In [Rie08], the second author defined a Landau-Ginzburg model for homogeneous spaces G/P. In this paper, we reformulate this LG model in the case of the odd-dimensional quadric X=Q_{2m-1}. Namely we introduce a regular function Wcan on a variety Xcan x C*, where Xcan is the complement of a particular anticanonical divisor in the the projective space CP^{2m-1}=P(H*(X,C)*). Firstly we prove that the Jacobi ring associated to Wcan is isomorphic to the quantum cohomology ring of the quadric, and that this isomorphism is compatible with the identification of homogeneous coordinates on Xcan with elements of H*(X,C). Secondly we find a very natural Laurent polynomial formula for Wcan by restricting it to a `Lusztig torus' in Xcan. Thirdly we show that the Dubrovin connection on H*(X,C[q]) embeds into the Gauss-Manin system associated to Wcan and deduce a flat section formula in terms of oscillating integrals. Finally, we compare (Xcan,Wcan) with previous Landau-Ginzburg models defined for odd quadrics. Namely, we prove that it is a partial compactification of Givental's original LG model [Giv96]. We show that our LG model is isomorphic to the Lie-theoretic LG model from [Rie08]. Moreover it is birationally equivalent to an LG model introduced by Gorbounov and Smirnov [GS13], and it is algebraically isomorphic to Gorbounov and Smirnov's mirror for Q_3, implying a tameness property in that case.

math.AG↗

arXiv2013

A Landau-Ginzburg model for Lagrangian Grassmannians, Langlands duality and relations in quantum cohomology

In [Rie08], the second author defined a Landau-Ginzburg model for homogeneous spaces G/P, as a regular function on an affine subvariety of the Langlands dual group. In this paper, we reformulate this LG-model (X^,W_t) in the case of the Lagrangian Grassmannian LG(m) as a rational function on a Langlands dual orthogonal Grassmannian, in the spirit of work by R. Marsh and the second author [MR12] for type A Grassmannians. This LG model has some very interesting features, which are not visible in the type A case, to do with the non-triviality of Langlands duality. We also formulate a conjecture relating our superpotential with the quantum differential equations of LG(m). Finally, our expression for W_t also leads us to conjecture new formulas in the quantum Schubert calculus of LG(m).

math.AG↗

arXiv2009

Laurent polynomial moment problem: a case study

In recent years, the so-called polynomial moment problem, motivated by the classical Poincare center-focus problem, was thoroughly studied, and the answers to the main questions have been found. The study of a similar problem for rational functions is still at its very beginning. In this paper, we make certain progress in this direction; namely, we construct an example of a Laurent polynomial for which the solutions of the corresponding moment problem behave in a significantly more complicated way than it would be possible for a polynomial.

math.CV↗

arXiv2001

The aromatic infrared bands as seen by ISO-SWS: probing the PAH model

We discuss the Aromatic Infrared Band (AIB) profiles observed by ISO-SWS towards a number of bright interstellar regions where dense molecular gas is illuminated by stellar radiation. Our sample spans a broad range of excitation conditions (exciting radiation fields with effective temperature, Teff, ranging from 23,000 to 45,000 K). The SWS spectra are decomposed coherently in our sample into Lorentz profiles and a broadband continuum. We find that the individual profiles of the main AIBs at 3.3, 6.2, 8.6 and 11.3 microns are well represented with at most two lorentzians. Furthermore, we show that the positions and widths of these AIBs are remarkably stable (within a few cm-1). We then extract the profiles of individual AIBs from the data and compare them to a model of Polycyclic Aromatic Hydrocarbon (PAH) cation emission which includes the temperature dependence of the AIB profiles. The present similarity of the AIB profiles requires that the PAH temperature distribution remains roughly the same whatever the radiation field hardness. Deriving the temperature distribution of interstellar PAHs, we show that its hot tail, which controls the AIB spectrum, sensitively depends on Nmin (the number of C-atoms in the smallest PAH) and Teff. Comparing the observed profiles of the individual AIBs to our model results, we can match most of the AIB profiles if Nmin is increased with Teff. We then discuss our results in the broader context of ISO observations of fainter interstellar regions where PAHs are expected to be in neutral form.

astro-ph↗