Generalized electricmagnetic duality and M
theory: General Remarks.
ElectricMagnetic Duality in Linearized
Gravity: The 3+1 split formalism.
ElectricMagnetic duality and Holography 
the relevance of ChernSimons.
Torsion Holography  a Janus solution.
Known
Unknown
Known
HOLOGRAPHY
Strings
SUGRA
HS ?
Gauge
Theory
Strongly
Coupled
Weakly
Coupled
Standard Model of Holography
Q: What is String Theory?
A: It is the Holographic Geometrization of YM Gauge Theories
Known
Unknown
Known
NonStandard Model of Holography
Known
Known
Unknown
[Klebanov & Polyakov (03)]
[Leigh & T.P. (03)]
Sezgin & Sundell (04)
HOLOGRAPHY
HS ?
MTheory ?
3d CFT Models:
O(N), GrossNeveu,
Thirring...
Conformal Scalars,
Gauge Fields,
Conformal Gravity
...
Strongly
Coupled
Weakly
Coupled
“DoubleTrace”
Deformations
Q: What is the Holographic Geometrization of nonYM Theories?
A: MTheory (?)
The Duality Conjecture: R.G. Leigh & T.C.P. (hepth/0309177)
“Linearized higherspin gauge ﬁelds on AdS4 possess a generalization of
electricmagnetic duality that is seen holographically in the boundary.”
Some recent works:
Spin1 gauge ﬁelds: S. deHaro & G. Peng (hepth/0701144), C. Herzog et. al. (hepth/
0701036), G. Barnich & A. Gomberoff (arXiv.07082378)
Spin2 (gravity): M. Henneaux & C. Teitelboim (grqc/0408101), R. G. Leigh & T. C. P.
arXiv 0704.0531
“doubletrace
deformation
Weyl “shadow” symmetry of
Dualization & “doubletrace” deformations
The Grand Picture (Speculative)
Mtheory

11d SUGRA
Quantum Duality of 3d CFTs
(T.C.P. hepth/9410093)
Holography  Boundary Conditions
E11, E10
dualities
M2branes
M5branes
BaggerLambert
Theory
Duality in the Hamiltonian formalism:
Canonical transformations leaving invariant the
form
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 ChernSimons term.
See Glenn’s talk
Example 2: 4d Gravity
Consider the standard HilbertPalatini 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 vectorvalued oneform:
Deﬁne the magnetic ﬁeld; vector valued oneform too:
The action becomes:
The qconstraints 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 zerotorsion equation.
This is equivalent to choosing the deDonder 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 selfdual conﬁgurations  Euclidean spaces
FeffermanGraham holography:
We need to go onshell and impose B.C. to ﬁnd
in terms of
We can functionally integrate over
to the boundary generating functional
This is a nonlocal functional, hence it
cannot
be an effective action for
Gravity is nondynamical 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 selfduality condition:
We the ﬁnd that the boundary e.m. tensor is the Cotton tensor:
This is produced by the variation of the gravitational ChernSimons action.
The latter
can
be interpreted as an effective action for
See also: Compere & Marolf (08)
Similar to LambdaInstantons: Julia et. al. (05)
The physical meaning of the selfdual conﬁgurations  consider Example 1:
The selfdual conﬁgurations yield a total derivative action:
The boundary “ChernSimons” action describes all “bulk” states with zero energy.
The Gravitational ChernSimons describes all zero energy states of 4d gravity.
Duality in a trivial automorphism of the Euclidean gravity Hilbert space.
The NiehYan model
Consider a modiﬁed version of the D’AuriaRegge model.
Equivalent to gravity coupled to a pseudoscalar ﬁeld.
Use the previous gauge ﬁxing.
Look for domainwall 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 nonzero.
Using:
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
Renormalization.
In the other case, it has an inﬁnite number of divergent terms; the resulting
boundary theory would seem nonrenormalizable.
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, selfdual backgrounds and their Holography.
Bulk Electricmagnetic duality interchanges boundary energymomentum density
with external geometry.
(In the same way that bulk electricmagnetic duality interchanges boundary
chargedensity with external ﬁelds: interesting QuantumHall, superconductivity
applications.
Herzog, Horowitz, Hartnol, Sachdev et. al. (0708)
Linearized gravity (and HigherSpins)
possess a generalized electricmagnetic
duality.
The duality has important Holographic
consequences.
The Mtheory dualities can be studied
Holographically.
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