Equivariant localization and completion in cyclic homology and derived loop spaces

We prove an equivariant localization theorem over an algebraically closed field of characteristic zero for smooth quotient stacks by reductive groups $X/G$ in the setting of derived loop spaces as well as Hochschild homology and its cyclic variants. We show that the derived loop spaces of the stack $X/G$ and its classical $z$-fixed point stack $\pi_0(X^z)/G^z$ become equivalent after completion along a semisimple parameter $[z] \in G//G$, implying the analogous statement for Hochschild and cyclic homology of the dg category of perfect complexes $\text{Perf}(X/G)$. We then prove an analogue of the Atiyah-Segal completion theorem in the setting of periodic cyclic homology, where the completion of the periodic cyclic homology of $\text{Perf}(X/G)$ at the identity $[e] \in G//G$ is identified with a 2-periodic version of the derived de Rham cohomology of $X/G$. Together, these results identify the completed periodic cyclic homology of a stack $X/G$ over a parameter $[z] \in G//G$ with the 2-periodic derived de Rham cohomology of its $z$-fixed points.Comment: Pre-publication version. 42 pages. Comments welcom

CKM and Tri-bimaximal MNS Matrices in a SU(5) x (d)T Model

We propose a model based on SU(5) x {}^{(d)}T which successfully gives rise to near tri-bimaximal leptonic mixing as well as realistic CKM matrix elements for the quarks. The Georgi-Jarlskog relations for three generations are also obtained. Due to the {}^{(d)}T transformation property of the matter fields, the b-quark mass can be generated only when the {}^{(d)}T symmetry is broken, giving a dynamical origin for the hierarchy between m_{b} and m_{t}. There are only nine operators allowed in the Yukawa sector up to at least mass dimension seven due to an additional Z_{12} x Z'_{12} symmetry, which also forbids, up to some high orders, operators that lead to proton decay. The resulting model has a total of nine parameters in the charged fermion and neutrino sectors, and hence is very predictive. In addition to the prediction for \theta_{13} \simeq \theta_{c}/3 \sqrt{2}, the model gives rise to a sum rule, \tan^{2}\theta_{\odot} \simeq \tan^{2} \theta_{\odot, \mathrm{TBM}} - {1/2} \theta_{c} \cos\beta, which is a consequence of the Georgi-Jarlskog relations in the quark sector. This deviation could account for the difference between the experimental best fit value for the solar mixing angle and the value predicted by the tri-bimaximal mixing matrix.Comment: 11 pages; v2: additional references added; minor modifications made; conclusion unchanged; v3: version to appear in Phys. Lett.

Categorical cyclic homology and filtered D\mathcal{D}-modules on stacks: Koszul duality

Motivated by applications to the categorical and geometric local Langlands correspondences, we establish an equivalence between the category of filtered $\mathcal{D}$-modules on a smooth stack $X$ and the category of $S^1$-equivariant ind-coherent sheaves on its formal loop space $\widehat{\mathcal{L}} X$, exchanging compact $\mathcal{D}$-modules with coherent sheaves, and coherent $\mathcal{D}$-modules with continuous ind-coherent sheaves. The equivalence yields a sheaf of categories over $\mathbb{A}^1/\mathbb{G}_m$ whose special fiber is a category of coherent sheaves on stacks appearing in categorical traces, and whose generic fiber is a category of equivariant constructible sheaves.Comment: Improved main result (now functorial for all pullbacks and proper pushforward), slightly changed title, 42 pages, comments welcom

Tri-Bimaximal Neutrino Mixing, A4 and the Modular Symmetry

We formulate and discuss a 4-dimensional SUSY version of an A4 model for tri-bimaximal neutrino mixing which is completely natural. We also study the next-to-the-leading corrections and show that they are small, once the ratios of A4 breaking VEVs to the cutoff are fixed in a specified interval. We also point out an interesting way of presenting the A4 group starting from the modular group. In this approach, which could be interesting in itself as an indication on a possible origin of A4, the lagrangian basis where the symmetry is formulated coincides with the basis where the charged leptons are diagonal. If the same classification structure in A4 is extended from leptons to quarks, the CKM matrix coincides with the unit matrix in leading order and a study of non leading corrections shows that the departures from unity of the CKM matrix are far too small to accomodate the observed mixing angles.Comment: 21 pages, 1 figure; added section on a see-saw realization; version to appear on Nucl. Phys.

Constraints on the average magnetic field strength of relic radio sources 0917+75 and 1401-33 from XMM-Newton observations

We observed two relic radio sources, 0917+75 and 1401-33, with the XMM-Newton X-ray observatory. We did not detect any X-ray emission, thermal or non-thermal, in excess of the local background level from either target. This imposes new upper limits on the X-ray flux due to inverse Compton scattering of photons from the cosmic microwave background by relativistic electrons in the relic sources, and new lower limits on the magnetic field strength from the relative strength of the radio and X-ray emission. The combination of radio and X-ray observations provides a measure of the magnetic field independent of equipartition or minimum energy assumptions. Due to increasing sensitivity of radio observations, the known population of cluster relics has been growing; however, studies of non-thermal X-ray emission from relics remain scarce. Our study adds to the small sample of relics studied in X-rays. In both relics, our field strength lower limits are slightly larger than estimates of the equipartition magnetic field.Comment: 11 pages, 5 figures. Accepted by MNRA

T′T^{'} Predictions of PMNS and CKM Angles

Generalizing a previous model to accommodate the third quark family and CP violation, we present a $T^{'}$ model which predicts tribimaximal neutrino (PMNS) mixings while the central predictions for quark mixings are $|V_{td}/V_{ts}| = 0.245$ and $|V_{ub}/V_{cb}| = 0.237$ with a predicted CP violating KM phase $\delta_{KM} = 65.8^{0}$. All these are acceptably close to experiment, including the KM phase for which the allowed values are $63^{0} < \delta_{KM} < 72^{0}$, and depend only on use of symmetry $T^{'} \times Z_2$ to define the model and no additional parameters.Comment: Simplified presentatio