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| Wei Zhang |
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| Zhiwei Yun |
Shtukas and the Taylor expansion of L-functions
Abstract. We define the Heegner–Drinfeld cycle on the moduli stack of Drinfeld Shtukas of rank two with r-modifications for an even integer r. We prove an identity between
(1) The r-th central derivative of the quadratic base change L-function associated to an
everywhere unramified cuspidal automorphic representation π of PGL2;
(2) The self-intersection number of the π-isotypic component of the Heegner–Drinfeld cycle. This identity can be viewed as a function-field analog of the Waldspurger and Gross–Zagier formula for higher derivatives of L-functions.
http://arxiv.org/abs/1512.02683
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| Steven Weinberg |
Leptons interact only with photons, and with the intermediate bosons that presumably mediate weak interactions. What could be more natural than to unite these spin-one bosons into a multiplet of gauge fields? Standing in the way of this synthesis are the obvious differences in the masses of the photon and intermediate meson, and in their couplings. We might hope to understand these differences by imagining that the symmetries relating the weak and electromagnetic interactions are exact symmetries of the Lagrangian but are broken by the vacuum. However, this raises the specter of unwanted massless Goldstone bosons. This note will describe a model in which the symmetry between the electromagnetic and weak interactions is spontaneously broken, but in which the Goldstone bosons are avoided by introducing the photon and the intermediate-boson fields as gauge fields. The model may be renormalizable.
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| Juan Martín Maldacena |
We show that the large N limit of certain conformal field theories in various dimen- sions include in their Hilbert space a sector describing supergravity on the product of Anti-deSitter spacetimes, spheres and other compact manifolds. This is shown by taking some branes in the full M/string theory and then taking a low energy limit where the field theory on the brane decouples from the bulk. We observe that, in this limit, we can still trust the near horizon geometry for large N. The enhanced supersymmetries of the near horizon geometry correspond to the extra supersymmetry generators present in the super- conformal group (as opposed to just the super-Poincare group). The ’t Hooft limit of 3+1 N = 4 super-Yang-Mills at the conformal point is shown to contain strings: they are IIB strings. We conjecture that compactifications of M/string theory on various Anti-deSitter spacetimes is dual to various conformal field theories. This leads to a new proposal for a definition of M-theory which could be extended to include five non-compact dimensions.
- Conformal field theory
- Hilbert space
- Supergravity
- Anti-deSitter spacetime
- Brane
- IIB string
- M theory
- String theory
- Super-conformal group
- Super-Poincare group
- t Hooft limit
- Super-Yang-Mills
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| André Füzfa |
The curved space-time around current loops and solenoids carrying arbitrarily large steady electric currents is obtained from the numerical resolution of the coupled Einstein-Maxwell equations in cylindrical symmetry. The artificial gravitational field associated to the generation of a magnetic field produces gravitational redshift of photons and deviation of light. Null geodesics in the curved space-time of current loops and solenoids are also presented. We finally propose an experimental setup, achievable with current technology of superconducting coils, that produces a phase shift of light of the same order of magnitude than astrophysical signals in ground-based gravitational wave observatories.
- Solenoids (ソレノイド)
- Cylindrical symmetry (円筒対称性)
- Deviation
- Null geodesics
