Generalized Electric-Magnetic Duality and Holography

attractionlewdsterΗλεκτρονική - Συσκευές

18 Οκτ 2013 (πριν από 4 χρόνια και 8 μήνες)

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Generalized electric-magnetic duality and M-
theory: General Remarks.
Electric-Magnetic Duality in Linearized
Gravity: The 3+1 split formalism.
Electric-Magnetic duality and Holography -
the relevance of Chern-Simons.
Torsion Holography - a Janus solution.
HS ?
Standard Model of Holography
Q: What is String Theory?
A: It is the Holographic Geometrization of YM Gauge Theories
Non-Standard Model of Holography
[Klebanov & Polyakov (03)]
[Leigh & T.P. (03)]
Sezgin & Sundell (04)
HS ?
M-Theory ?
3-d CFT Models:
O(N), Gross-Neveu,
Conformal Scalars,

Gauge Fields,

Conformal Gravity
Q: What is the Holographic Geometrization of non-YM Theories?
A: M-Theory (?)
The Duality Conjecture: R.G. Leigh & T.C.P. (hep-th/0309177)
“Linearized higher-spin gauge fields on AdS4 possess a generalization of
electric-magnetic duality that is seen holographically in the boundary.”
Some recent works:
Spin-1 gauge fields: S. deHaro & G. Peng (hep-th/0701144), C. Herzog et. al. (hep-th/
0701036), G. Barnich & A. Gomberoff (arXiv.07082378)
Spin-2 (gravity): M. Henneaux & C. Teitelboim (gr-qc/0408101), R. G. Leigh & T. C. P.
arXiv 0704.0531

Weyl “shadow” symmetry of
Dualization & “double-trace” deformations
The Grand Picture (Speculative)
Quantum Duality of 3-d CFTs
(T.C.P. hep-th/9410093)
Holography - Boundary Conditions
E11, E10
Duality in the Hamiltonian formalism:
Canonical transformations leaving invariant the
of the Hamiltonian.
Example 1:
Example 2:
In Example 2, the 2nd order e.o.m. remain invariant.
Example 2 allows a modified duality transformation and “mixed” B.C..
Deser & Teitelboim (76)
Example 1: Electromagnetism
The Gauss Law maps to the Bianchi identity. Then, we can write:
The boundary modification is a Chern-Simons term.
See Glenn’s talk
Example 2: 4d Gravity
Consider the standard Hilbert-Palatini action.
The e.o.m. are:
The 3+1 split is the choice:
corresponds to AdS.
Henneaux & Teitelboim (04), Julia et. al. (05) Leigh & T.C.P. (07)
We split everything:
We define:
The action becomes:
The dynamics is carried by the terms:
with the definitions:
The electric field is a vector-valued one-form:
Define the magnetic field; vector valued one-form too:
The action becomes:
The q-constraints give:
and also:
We require that the latter transforms like a vector under SO(3) rotations of
the dreibein. The magnetic field term in an obstruction.
The choice:
and shows that the antisymmetric part of the magnetic field is a gauge d.o.f.
However, since the magnetic field does not appear in the kinetic term, its
variation gives an algebraic equation; the zero-torsion equation.
This is equivalent to choosing the de-Donder gauge:
Only the symmetric part of the magnetic field contributes to that.
Next use the shifted electric field:
For, symmetric electric and magnetic fields and zero torsion we get:
Linearize as:
Make an educated guess for a nice background i.e. the vacuum:
The action becomes:
The vanishing of the linear terms gives:
This is solved by (A)dS4:
Finally - the duality map:
The action dualizes to:
This differs from the initial action by
Nevertheless, this does not affect the second order e.o.m.
The constraints also dualize to:
Recall the linearized Bianchi identities:
The duality maps the constraints into the Bianchi identities.
Lastly, we notice that the modified duality transformations;
Leave the action invariant, up to additional terms in the constraints.
Using the relationship between the dual dreibein and the electric field;
we can show that the additional terms vanish. Hence, gravity
with a c.c. requires a modified duality transformation.
Duality implies that the switch from Dirichlet to Neumann B.C.s is a C.T.
In Example 2, the switch is from Dirichlet to mixed B.C.s
The “bulk” e.o.m. remain the same:
Duality implies that different “boundary” theories (i.e. corresponding to different
B.C.) are equivalent since they correspond to canonically equivalent “bulk” data.
Holography is equivalent to Hamiltonian dynamics.
Removal of divergences --> addition of “boundary” terms.
Addition of other “boundary” terms --> deformation of the “boundary” action.
with S. de Haro; to appear
The importance of self-dual configurations - Euclidean spaces
Fefferman-Graham holography:
We need to go on-shell and impose B.C. to find
in terms of
We can functionally integrate over
to the boundary generating functional
This is a non-local functional, hence it
be an effective action for
Gravity is non-dynamical in the boundary.
In the presence of a c.c. the e.o.m. and the Bianchi identities can be written as:
Hence, one can replace the e.o.m. by the self-duality condition:
We the find that the boundary e.m. tensor is the Cotton tensor:
This is produced by the variation of the gravitational Chern-Simons action.
The latter
be interpreted as an effective action for
See also: Compere & Marolf (08)
Similar to Lambda-Instantons: Julia et. al. (05)
The physical meaning of the self-dual configurations - consider Example 1:
The self-dual configurations yield a total derivative action:
The boundary “Chern-Simons” action describes all “bulk” states with zero energy.
The Gravitational Chern-Simons describes all zero energy states of 4d gravity.
Duality in a trivial automorphism of the Euclidean gravity Hilbert space.
The Nieh-Yan model
Consider a modified version of the D’Auria-Regge model.
Equivalent to gravity coupled to a pseudoscalar field.
Use the previous gauge fixing.
Look for domain-wall solutions.
The motivation is to give dynamics to the magnetic field.
with R. G. Leigh and N. NNguen; to appear
D’Auria & Regge (82)
The magnetic field becomes a proper dynamical variable.
Torsion is generally non-zero.
The e.o.m. are:
The solution is:
It is of the “Janus” type, since:
Quite possibly, there is only one boundary and the two limits take us to
different boundary patches.
The electric and magnetic fields are:
The boundary e.m. tensor vanishes.
Notice that the electric field diverges in both boundaries. In the one case, it
has a finite number of divergent terms; can be dealt with Holographic
In the other case, it has an infinite number of divergent terms; the resulting
boundary theory would seem non-renormalizable.
However, the two theories are very similar; they only differ in the exp.
value of a pseudoscalar dim=3 operator.
with D. Mansi & G. Tagliabue: to appear
The complete Holographic analysis of AdS4 gravity in the 3+1 split formalism.
Study of Black Holes, self-dual backgrounds and their Holography.
Bulk Electric-magnetic duality interchanges boundary energy-momentum density
with external geometry.
(In the same way that bulk electric-magnetic duality interchanges boundary
charge-density with external fields: interesting Quantum-Hall, superconductivity
Herzog, Horowitz, Hartnol, Sachdev et. al. (07-08)
Linearized gravity (and Higher-Spins)
possess a generalized electric-magnetic
The duality has important Holographic
The M-theory dualities can be studied