879 research outputs found
Spectra of quantum KdV Hamiltonians, Langlands duality, and affine opers
We prove a system of relations in the Grothendieck ring of the category O of
representations of the Borel subalgebra of an untwisted quantum affine algebra
U_q(g^) introduced in [HJ]. This system was discovered in [MRV1, MRV2], where
it was shown that solutions of this system can be attached to certain affine
opers for the Langlands dual affine Kac-Moody algebra of g^, introduced in
[FF5]. Together with the results of [BLZ3, BHK], which enable one to associate
quantum g^-KdV Hamiltonians to representations from the category O, this
provides strong evidence for the conjecture of [FF5] linking the spectra of
quantum g^-KdV Hamiltonians and affine opers for the Langlands dual affine
algebra. As a bonus, we obtain a direct and uniform proof of the Bethe Ansatz
equations for a large class of quantum integrable models associated to
arbitrary untwisted quantum affine algebras, under a mild genericity condition.
We also conjecture analogues of these results for the twisted quantum affine
algebras and elucidate the notion of opers for twisted affine algebras, making
a connection to twisted opers introduced in [FG].Comment: 54 pages (v3: some examples added; opers for twisted affine algebras
elucidated). Accepted for publication in Communications in Mathematical
Physic
Langlands duality for representations of quantum groups
We establish a correspondence (or duality) between the characters and the
crystal bases of finite-dimensional representations of quantum groups
associated to Langlands dual semi-simple Lie algebras. This duality may also be
stated purely in terms of semi-simple Lie algebras. To explain this duality, we
introduce an "interpolating quantum group" depending on two parameters which
interpolates between a quantum group and its Langlands dual. We construct
examples of its representations, depending on two parameters, which interpolate
between representations of two Langlands dual quantum groups.Comment: 37 pages. References added. Accepted for publication in Mathematische
Annale
Baxter's Relations and Spectra of Quantum Integrable Models
Generalized Baxter's relations on the transfer-matrices (also known as
Baxter's TQ relations) are constructed and proved for an arbitrary untwisted
quantum affine algebra. Moreover, we interpret them as relations in the
Grothendieck ring of the category O introduced by Jimbo and the second author
in arXiv:1104.1891 involving infinite-dimensional representations constructed
in arXiv:1104.1891, which we call here "prefundamental". We define the
transfer-matrices associated to the prefundamental representations and prove
that their eigenvalues on any finite-dimensional representation are polynomials
up to a universal factor. These polynomials are the analogues of the celebrated
Baxter polynomials. Combining these two results, we express the spectra of the
transfer-matrices in the general quantum integrable systems associated to an
arbitrary untwisted quantum affine algebra in terms of our generalized Baxter
polynomials. This proves a conjecture of Reshetikhin and the first author
formulated in 1998 (arXiv:math/9810055). We also obtain generalized Bethe
Ansatz equations for all untwisted quantum affine algebras.Comment: 41 pages (v3: New Section 5.6 added in which Bethe Ansatz equations
are written explicitly for all untwisted quantum affine algebras. New
examples, references, and historical comments added plus some minor edits.
v4: References added.
Spectral Curves, Opers and Integrable Systems
We establish a general link between integrable systems in algebraic geometry
(expressed as Jacobian flows on spectral curves) and soliton equations
(expressed as evolution equations on flat connections). Our main result is a
natural isomorphism between a moduli space of spectral data and a moduli space
of differential data, each equipped with an infinite collection of commuting
flows. The spectral data are principal G-bundles on an algebraic curve,
equipped with an abelian reduction near one point. The flows on the spectral
side come from the action of a Heisenberg subgroup of the loop group. The
differential data are flat connections known as opers. The flows on the
differential side come from a generalized Drinfeld-Sokolov hierarchy. Our
isomorphism between the two sides provides a geometric description of the
entire phase space of the Drinfeld-Sokolov hierarchy. It extends the Krichever
construction of special algebro-geometric solutions of the n-th KdV hierarchy,
corresponding to G=SL(n).
An interesting feature is the appearance of formal spectral curves, replacing
the projective spectral curves of the classical approach. The geometry of these
(usually singular) curves reflects the fine structure of loop groups, in
particular the detailed classification of their Cartan subgroups. To each such
curve corresponds a homogeneous space of the loop group and a soliton system.
Moreover the flows of the system have interpretations in terms of Jacobians of
formal curves.Comment: 64 pages, Latex, final version to appear in Publications IHE
Geometric Realization of the Segal--Sugawara Construction
We apply the technique of localization for vertex algebras to the
Segal-Sugawara construction of an ``internal'' action of the Virasoro algebra
on affine Kac-Moody algebras. The result is a lifting of twisted differential
operators from the moduli of curves to the moduli of curves with bundles, with
arbitrary decorations and complex twistings. This construction gives a uniform
approach to a collection of phenomena describing the geometry of the moduli
spaces of bundles over varying curves: the KZB equations and heat kernels on
non-abelian theta functions, their critical level limit giving the quadratic
parts of the Beilinson-Drinfeld quantization of the Hitchin system, and their
infinite level limit giving a Hamiltonian description of the isomonodromy
equations.Comment: References adde
MorphIC: A 65-nm 738k-Synapse/mm Quad-Core Binary-Weight Digital Neuromorphic Processor with Stochastic Spike-Driven Online Learning
Recent trends in the field of neural network accelerators investigate weight
quantization as a means to increase the resource- and power-efficiency of
hardware devices. As full on-chip weight storage is necessary to avoid the high
energy cost of off-chip memory accesses, memory reduction requirements for
weight storage pushed toward the use of binary weights, which were demonstrated
to have a limited accuracy reduction on many applications when
quantization-aware training techniques are used. In parallel, spiking neural
network (SNN) architectures are explored to further reduce power when
processing sparse event-based data streams, while on-chip spike-based online
learning appears as a key feature for applications constrained in power and
resources during the training phase. However, designing power- and
area-efficient spiking neural networks still requires the development of
specific techniques in order to leverage on-chip online learning on binary
weights without compromising the synapse density. In this work, we demonstrate
MorphIC, a quad-core binary-weight digital neuromorphic processor embedding a
stochastic version of the spike-driven synaptic plasticity (S-SDSP) learning
rule and a hierarchical routing fabric for large-scale chip interconnection.
The MorphIC SNN processor embeds a total of 2k leaky integrate-and-fire (LIF)
neurons and more than two million plastic synapses for an active silicon area
of 2.86mm in 65nm CMOS, achieving a high density of 738k synapses/mm.
MorphIC demonstrates an order-of-magnitude improvement in the area-accuracy
tradeoff on the MNIST classification task compared to previously-proposed SNNs,
while having no penalty in the energy-accuracy tradeoff.Comment: This document is the paper as accepted for publication in the IEEE
Transactions on Biomedical Circuits and Systems journal (2019), the
fully-edited paper is available at
https://ieeexplore.ieee.org/document/876400
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