Sharp Well-Posedness and Ill-Posedness for the Chern–Simons–Dirac System in One Dimension

We consider the Cauchy problem associated with the Chern–Simons–Dirac system in ℝ1+2. Using gauge invariance, we reduce the Chern–Simons–Dirac system to a Dirac equation and we uncover the null structure of this Dirac equation. Next, relying on null structure estimates, we establish that the Cauchy problem associated with this Dirac equation is locally-in-time well-posed in the Sobolev space Hs for all s > 1/4. Our proof uses modified L4-type estimates.

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Ill-posedness of the Cauchy problem for the Chern–Simons–Dirac system in one dimension

Journal of Differential Equations ◽

10.1016/j.jde.2014.10.020 ◽

2015 ◽

Vol 258 (4) ◽

pp. 1356-1394 ◽

Cited By ~ 6

Author(s):

Shuji Machihara ◽

Mamoru Okamoto

Keyword(s):

Cauchy Problem ◽

One Dimension ◽

Dirac System ◽

The Cauchy Problem ◽

Chern Simons

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Well-posedness for the dimension-reduced Chern–Simons–Dirac system

Journal of Evolution Equations ◽

10.1007/s00028-016-0371-1 ◽

2016 ◽

Vol 17 (3) ◽

pp. 1031-1048 ◽

Cited By ~ 1

Author(s):

Shuji Machihara ◽

Mamoru Okamoto

Keyword(s):

Well Posedness ◽

Dirac System ◽

Chern Simons

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LOW REGULARITY WELL-POSEDNESS OF THE DIRAC–KLEIN–GORDON EQUATIONS IN ONE SPACE DIMENSION

Communications in Contemporary Mathematics ◽

10.1142/s0219199708002740 ◽

2008 ◽

Vol 10 (02) ◽

pp. 181-194 ◽

Cited By ~ 15

Author(s):

SIGMUND SELBERG ◽

ACHENEF TESFAHUN

Keyword(s):

Dirac Operator ◽

Space Dimension ◽

One Dimension ◽

Well Posedness ◽

Low Regularity ◽

Klein Gordon ◽

One Space Dimension

We extend recent results of Machihara and Pecher on low regularity well-posedness of the Dirac–Klein–Gordon (DKG) system in one dimension. Our proof, like that of Pecher, relies on the null structure of DKG, recently completed by D'Ancona, Foschi and Selberg, but we show that in 1d the argument can be simplified by modifying the choice of projections for the Dirac operator. We also show that the result is best possible up to endpoint cases, if one iterates in Bourgain–Klainerman–Machedon spaces.

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On the well-posedness of Maxwell-Chern-Simons-Higgs system in the Lorenz gauge

Discrete and Continuous Dynamical Systems ◽

10.3934/dcds.2014.34.2389 ◽

2014 ◽

Vol 34 (5) ◽

pp. 2389-2403 ◽

Cited By ~ 3

Author(s):

Jianjun Yuan ◽

Keyword(s):

Well Posedness ◽

Lorenz Gauge ◽

Chern Simons

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Averaging over Narain moduli space

Journal of High Energy Physics ◽

10.1007/jhep10(2020)187 ◽

2020 ◽

Vol 2020 (10) ◽

Cited By ~ 2

Author(s):

Alexander Maloney ◽

Edward Witten

Keyword(s):

Einstein Gravity ◽

Two Dimensions ◽

Three Dimensions ◽

One Dimension ◽

Simons Theory ◽

Two Dimensional ◽

Dual Theory ◽

Recent Developments ◽

Chern Simons ◽

Chern Simons Theory

Abstract Recent developments involving JT gravity in two dimensions indicate that under some conditions, a gravitational path integral is dual to an average over an ensemble of boundary theories, rather than to a specific boundary theory. For an example in one dimension more, one would like to compare a random ensemble of two-dimensional CFT’s to Einstein gravity in three dimensions. But this is difficult. For a simpler problem, here we average over Narain’s family of two-dimensional CFT’s obtained by toroidal compactification. These theories are believed to be the most general ones with their central charges and abelian current algebra symmetries, so averaging over them means picking a random CFT with those properties. The average can be computed using the Siegel-Weil formula of number theory and has some properties suggestive of a bulk dual theory that would be an exotic theory of gravity in three dimensions. The bulk dual theory would be more like U(1)2D Chern-Simons theory than like Einstein gravity.

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