Generalized Electric-Magnetic Duality and Holography

Generalized Electric-Magnetic 

Duality and Holography 

T a s s o s C . P e t k o u 

U n i v e r s i t y o f C r e t e 

B r u s s e l s 4 J u n e 2 0 0 8

Plan of the talk 

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.

Electric-Magnetic Generalized Electric-Magnetic Duality in Linearized Duality and Gravity: M-theory: The 3+1 General split formalism Remarks 

K 

Q: What is String Theory 

A: It is the Holographic Geometrization of YM Gauge Theories 

Standard Model of Holography 

Known 

α ′ → 0 

SUGRA 

Strongly 

Coupled 

Unknown 

λ = g 2 Y MN → ∞ 

Strings 

HOLOGRAPHY 

Gauge 

Theory 

α ′ → ∞ 

Unknown 

HS 

Weakly 

Coupled 

Known 

λ = g 2 Y MN → 0


Q: What is the Holographic Geometrization of non-YM Theories 

A: M-Theory () 

Non-Standard Model of Holography 

d = 4 d = 3 

Conformal Scalars, 

Strongly 

Coupled 

Known 

K 

HS 

M-Theory 

Gauge Fields, 

Conformal Gravity 

“Double-Trace” 

Deformations 

3-d CFT Models: 

O(N), Gross-Neveu, 

Thirring... 

Unknown 

... 

Weakly 

Coupled 

HOLOGRAPHY 

Known 

[Klebanov & Polyakov (03)] 

[Leigh & T.P. (03)] 

Sezgin & Sundell (04) 

γ s =→ s→∞ 2γ 1 

[ 

1 − 3 2s + · · · ]


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 

DIMENSION 

Weyl “shadow” symmetry of O(4, 1) 

UNITARITY BOUND 

2 

“double-trace 

deformation 

1 

0 

Dualization & “double-trace” deformations 

1 

2 

SPIN

Generalized Electric-Magnetic Duality and M-theory: General Remarks 

The Grand Picture (Speculative) 

Bagger-Lambert 

Theory 

M2-branes 

Holography - Boundary Conditions 

M-theory 

- 

11d SUGRA 

E11, E10 

dualities 

M5-branes 

Quantum Duality of 3-d CFTs 

(T.C.P. hep-th/9410093)

Electric-Magnetic Duality in Linearized Gravity: The 3+1 split formalism 

Duality in the Hamiltonian formalism: 

Canonical transformations leaving invariant the form of the Hamiltonian. 

p ↦→ q , q ↦→ −p , I = 

∫ b 

a 

[p ˙q − H(p, q)] ↦→ I D = 

∫ b 

a 

Deser & Teitelboim (76) 

[−qṗ − H(q, −p)] 

H(q, −p) = H(q, p) ⇒ I D = I − pq∣ b δq∣ a⇒ b 0 ↦→ δp∣ a= b 0 

a= 

Example 1: H(p, q) = 1 2 (p2 + q 2 ) 

Example 2: 

H(p, q; λ) = 1 2 (p2 + q 2 + 2λpq) = H(q, −p; −λ) 

In Example 2, the 2nd order e.o.m. remain invariant. 

Example 2 allows a modified duality transformation and “mixed” B.C.. 

p ↦→ q , q ↦→ −p − 2λq ⇒ I(λ) ↦→ I(λ) − (pq + λq 2 ∣ 

) , ⇒ δ(p + 2λq) ∣ b 0 

a= 

∣ b a


I = 

∫ 

dt ∧ 

Example 1: Electromagnetism 

{ 

A˙ 

∧ ∗ 3 E − 1 } 

2 (E ∧ ∗ 3E + B ∧ ∗B) − A 0 ˜d ∗3 E 

See Glenn’s talk 

, B = ∗ 3 ˜dA 

E ↦→ −B , B = ∗ 3 ˜dA ↦→ ∗3 E 

I ↦→ 

∫ 

dt ∧ 

{ 

−ȦD ∧ ∗ 3 B − 1 } 

2 (E ∧ ∗ 3E + B ∧ ∗B) + A 0 ˜d ∗3 B 

, ˜dA D = ∗ 3 E 

The Gauss Law maps to the Bianchi identity. Then, we can write: 

I ↦→ I D = I − 

∫ 

A D ∧ ˜dA 

The boundary modification is a Chern-Simons term.


Example 2: 4d Gravity 

Henneaux & Teitelboim (04), Julia et. al. (05) Leigh & T.C.P. (07) 

Consider the standard Hilbert-Palatini action. 

∫ 

I = 

(e a ∧ e b ∧ R cd + Λ ) 

2 ea ∧ e b ∧ e c ∧ e d 

ɛ abcd 

R a b = dω a b + ω a c ∧ ω c b = 1 2 Ra bcde c ∧ e d 

The e.o.m. are: 

Λ > 0 

corresponds to AdS. 

R ab + Λe a ∧ e b = 0 , T a = de a + ω a b ∧ e b = 0 

The 3+1 split is the choice: 

e 0 = σ ⊥ n = Ndt , e α = ẽ α + N α dt


We split everything: 

ω a b = q a bdt + ˜ω a b , R a b = ˜R a b + dt ∧ r a b 

We define: 

r a b = ˙˜ω a b = ˜dq a b − ˜ω a bq c b + q a c ˜ω c b 

The action becomes: 

I HP = 2ɛ αβγ 

∫ 

dt ∧ 

{ 

N 

( ˜Rαβ + Λẽ α ∧ ẽ β) ∧ ẽ γ − 2N α ˜R0β ∧ ẽ γ 

+r 0α ∧ ẽ β ∧ ẽ γ}


The dynamics is carried by the terms: 

2ɛ αβγ r 0α ∧ ẽ β ∧ ẽ γ = Π α ∧ ˙ẽ α + 4q 0α ɛ αβγ ˜T β ∧ ẽ γ + 2σ ⊥ q αδ ɛ αβγ K δ ∧ ẽ β ∧ ẽ γ 

with the definitions: 

Π α = −4σ ⊥ ɛ αβγ K β ∧ ẽ γ , 

˜T α = dẽ α ∧ ˜ω α β ∧ ẽ β 

The electric field is a vector-valued one-form: 

K α = σ ⊥ ˜ω 0 α = K αβ ẽ β , 

˜Rα β = (3) R α β − σ ⊥ K α ∧ K β 

˜R 0 α = σ ⊥ ( ˜dK α + K β ∧ ˜ω β α) = σ ⊥ ( ˜DK) α


Define the magnetic field; vector valued one-form too: 

B α = 1 2 σ ⊥ɛ αβγ ˜ω βγ , 

˜ω αβ = −ɛ αβγ B γ 


I HP = 

∫ 

dt ∧ 

{ 

˙ẽ α ∧ Π α − 4σ ⊥ N α ɛ αβγ ( ˜DK) β ∧ ẽ γ 

+2σ ⊥ N 

( 

2 ˜dB γ + ɛ αβγ B α ∧ B β − ɛ αβγ K α ∧ K β +σ ⊥ Λɛ αβγ ẽ α ∧ ẽ β) ∧ẽ γ 

+4q 0α ɛ αβγ ˜T β ∧ ẽ γ + 2q αβ σ ⊥ ɛ ασρ K β ∧ ẽ σ ∧ ẽ ρ}


The q-constraints give: 

and also: 

q αβ ⇒ K [α,β] = 0 

ɛ αβγ ˜T β ∧ ẽ γ = ɛ αβγ ˜dẽ β ∧ ẽ γ − σ ⊥ B β ∧ ẽ β ∧ ẽ α = 0 

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: 

B [α,β] = ɛ γ 

αβ V γ , V = V α ẽ α , ẽ α ↦→ Λ α βẽ β 

( 

Λ 

−1 ) γ 

β ˜dΛ α γ = σ ⊥ V δ α β 

⇒ B [α,β] = 0 

and shows that the antisymmetric part of the magnetic field is a gauge d.o.f.


This is equivalent to choosing the de-Donder gauge: 

ɛ αβγ ˜dẽ α ∧ ẽ γ = 0 

However, since the magnetic field does not appear in the kinetic term, its 

variation gives an algebraic equation; the zero-torsion equation. 

˜T α = ˜dẽ α + ɛ αβγ B β ∧ ẽ γ = 0 

Only the symmetric part of the magnetic field contributes to that. 

Next use the shifted electric field: 

ˆK α = K α = ρẽ α 

, ρ 2 = σ ⊥ Λ 

For, symmetric electric and magnetic fields and zero torsion we get: 

∫ { 

I HP = dt ∧ ˙ẽ α ∧ ˆΠ α − 4σ ⊥ N α ɛ αβγ ( ˜D ˆK) β ∧ ẽ γ 

( 

−2σ ⊥ Nɛ αβγ B α ∧ B β + ˆK α ∧ ˆK β + 2ρ ˆK α ∧ ẽ β) ∧ ẽ γ]}


Linearize as: 

ẽ α = ẽ α + E α , N = 1 + n , N α = n α 

B α = B α + b α , 

ˆKα = ˆK α + k α 

Make an educated guess for a nice background i.e. the vacuum: 

B α = 0 = ˆK α 


∫ { 

I HP = dt ∧ (Ėα + ρE α ∧ p α − 2σ ⊥ ɛ αβγ (b α ∧ b β + k α ∧ k β ) ∧ ẽ γ 

−4σ ⊥ η α ɛ αβγ ˜dk β ∧ ẽ γ + n(4σ ⊥ ˜dbγ + ρp γ ) ∧ ẽ γ} 

p α = −4σ ⊥ ɛ αβγ k β ∧ ẽ γ


The vanishing of the linear terms gives: 

˙ẽ α + ρẽ α = 0 

This is solved by (A)dS4: 

ẽ 0 = dt , ẽ α = e −ρt dx α , K α = ρẽ α 

Ric ab = − 3σ ⊥ 

L 2 η ab 

, R = − 12σ ⊥ 

L 2 

Finally - the duality map: 

k α ↦→ −b α , b α ↦→ k α E ↦→ E , p ↦→ −p D 

ɛ αβγ b β ∧ ẽ γ + ˜dE α ↦→ ɛ αβγ k β ∧ ẽ γ + ˜dE α = 0


The action dualizes to: 

∫ { 

I ↦→ I D = dt ∧ −E ˙α 

∧ p D,α − ρẼα ∧ p D,α 

−2σ ⊥ ɛ αβγ (b α ∧ b β + k α ∧ k β ) ∧ ẽ γ 

+4σ ⊥ n α ɛ αβγ ˜db β ∧ ẽ γ + n(4σ ⊥ ˜dkγ + ρp D,α ) ∧ ẽ γ} 

This differs from the initial action by 

ρ ↦→ −ρ 

Nevertheless, this does not affect the second order e.o.m. 

The constraints also dualize to: 

C α ≡ ɛ αβγ ˜dk β ∧ e γ ↦→ −ɛ αβγ ˜db β ∧ ẽ γ 

C 0 ≡ −σ ⊥ ( ˜db γ − ρɛ αβγ k α ∧ ẽ β ) ∧ ẽ γ ↦→ −σ ⊥ ( ˜dk γ + ρɛ αβγ b α ∧ ẽ β ) ∧ ẽ γ


Recall the linearized Bianchi identities: 

B α T = −ɛ αβγ ˜db β ∧ ẽ γ + · · · 

B 0 T = −σ ⊥ ( ˜dk α + ρɛ αβγ b β ∧ ẽ γ ) ∧ ẽ α + · · · = 0 

The duality maps the constraints into the Bianchi identities. 

C α ↦→ B T,α , C 0 ↦→ B 0 T B T,α ↦→ −C α , B 0 T ↦→ −C 0


Lastly, we notice that the modified duality transformations; 

k α ↦→ −b α − 2ρE α , b α ↦→ k α 

Leave the action invariant, up to additional terms in the constraints. 

−8ρn α k β ∧ ẽ α ∧ ẽ β 

8nΛɛ αβγ E β ∧ ẽ γ ∧ ẽ α 

Using the relationship between the dual dreibein and the electric field; 

E α β = 1 

∂ 2 ɛα δγ∂ γ k δ β 

we can show that the additional terms vanish. Hence, gravity 

with a c.c. requires a modified duality transformation.

Electric-Magnetic Duality and Holography: the relevance of Chern-Simons 

with S. de Haro; to appear 

Holography 

∫ 

is equivalent to Hamiltonian dynamics. 

δW [q] 

δI = pδq + {e.o.m.} , p ≡ = 〈O〉 q 

δq 

∂M 

Removal of divergences --> addition of “boundary” ∫ terms. 

I ↦→ I Ren = I + f ct (q) 

Addition of other “boundary” terms --> deformation of the “boundary” action. 

∫ 

I Ren ↦→ I def = I Ren + f def (p) , q + f def ′ (p) ∣ = 0 

∂M 

∂M 

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. 

∂M


The importance of self-dual configurations - Euclidean spaces 

Fefferman-Graham holography: 

ds 2 = l ( 

dr 2 

r 2 + g ij (r, ¯x)dx i dx j) , g ij (r¯x) = g (0) 

ij (¯x) + ... + r3 g (3) 

ij (¯x) + ... 

δ(I EH + I GH + I ct ) = 1 2 

∫ 

d 3 x √ g (0) g (3) 

ij δg(0),ij , 〈T ij (¯x)〉 = 3l2 

16πG N 

g (3) 

ij (¯x) 

We need to go on-shell and impose B.C. to find g (3) 

ij (¯x) 

in terms of 

g(0) ij (¯x) 

We can functionally integrate over g (0) 

ij (¯x) to the boundary generating functional 

This is a non-local functional, hence it cannot be an effective action for g (0) 

ij (¯x) 

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: 

˜W ab ∧ e c = 0 , W a b ∧ e b = 0 

W ab = R ab + e a ∧ e b , 

˜W ab = 1 2 ɛab bcW bc 

Hence, one can replace the e.o.m. by the self-duality condition: 

W ab = ± ˜W ab ⇒ C µνρσ = ± 1 2 ɛ kl 

µν C klρσ 

We the find that the boundary e.m. tensor is the Cotton tensor: 

= C ij = ɛi kl ∇ l 

(R jk − 1 ) 

4 g jkR 

g (3) 

ij 

Similar to Lambda-Instantons: Julia et. al. (05) 

This is produced by the variation of the gravitational Chern-Simons action. 

The latter can be interpreted as an effective action for g (0) 

ij (¯x) 

See also: Compere & Marolf (08)


The physical meaning of the self-dual configurations - consider Example 1: 

p ↦→ q , q ↦→ −p : ⇒ H(p, q) = 1 2 (p2 − q 2 ) ↦→ −H(p, q) 

The self-dual configurations yield a total derivative action: 

p = ±q 

⇒ H(p, q) = 0 , ⇒ I = ± 1 2 q2 ∣ 

∣∣∂M 

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.

Torsion Holography - a Janus solution 

with R. G. Leigh and N. NNguen; to appear 

The Nieh-Yan model 

NY = 2R a b ∧ e a ∧ e b − T a ∧ T a = −d(T a ∧ e a ) 

Consider a modified version of the D’Auria-Regge model. 

Equivalent to gravity coupled to a pseudoscalar field. 

∫ 

I NY = I HP − 2 F d(T a ∧ e a ) , F = F (t) 

Use the previous gauge fixing. 

K α ∧ ẽ α = 0 , B α ∧ ẽ α = 0 , ɛ αβγ ˜dẽ β ∧ ẽ γ = 0 

Look for domain-wall solutions. 

N = 1 , N α = 0 , ˜dẽ α = 0 

The motivation is to give dynamics to the magnetic field. 

D’Auria & Regge (82)

Torsion Holography: - An a Janus exact solution 

I NY = 

∫ 

dt ∧ 

{ 

˙ẽ α ∧ (−4σ ⊥ ɛ αβγ K β ∧ ẽ γ ) − 2σ ˙ F ɛ αβγ B α ∧ ẽ β ∧ ẽ γ 

+2σ ⊥ 

( 

ɛαβγ 

[ 

σB α ∧ B β − K α ∧ K β] ∧ ẽ γ + σ ⊥ Λɛ αβγ ẽ α ∧ ẽ β ∧ ẽ γ) } 

The magnetic field becomes a proper dynamical variable. 

Torsion is generally non-zero. 

˜T α = −σɛ αβγ B β ∧ ẽ γ = − 1 2 σ 3 ˙ F ɛ αβγ ẽ β ∧ ẽ γ 

Using: 

The e.o.m. are: 

σ ⊥ = σ = 1 

Ä(t) + 3 ˙ A(t) 2 − 3σ ⊥ Λ = 0 

ẽ α = e A(t) dx α 

¨F (t) + 3 ˙ F (t) ˙ A(t) = 0 

Ȧ(t) 2 − σ ⊥ Λ = 1 4 σ ˙ F (t) 2

Torsion Holography: - An a Janus exact solution 

The solution is: 

ds 2 = dt 2 + e −2Λt (1 + e 6Λt ) 2 3 dx α dx α , F (t) = ± 4 3 arctan(e3Λt ) 

It is of the “Janus” type, since: 

F or t → ±∞ ⇒ ds 2 → AdS , F (t) → ( 2π 3 , 0) 

Quite possibly, there is only one boundary and the two limits take us to 

different boundary patches. 

The electric and magnetic fields are: 

K α = Λe −Λt 

B α = √ 6Λ 

1 − e 6Λt 

( 

(1 + e 6Λt ) 2/3 dxα = Λe −Λt 1 − 5 ) 

3 e6Λt + ... 

e 2Λt 

(1 + e 6Λt ) 2/3 dxα = √ 6Λe 2Λt (1 + ..)dx α 

dx α

Torsion Holography - a Janus solution 

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 non-renormalizable. 

However, the two theories are very similar; they only differ in the exp. 

value of a pseudoscalar dim=3 operator. 

δI NY = −2 

∫ 

F (t) ∼ e 3Λt + ... , as t → −∞ 

∣ ∣∣∣∣ 

∞ ∫ 

δF (t)ɛ αβγ B α ∧ ẽ β ∧ ẽ γ = d 3 xδF (t)〈[O + (x)〉 − 〈O − (x)〉] 

−∞ 

〈[O + (x)〉 − 〈O − (x)〉] = 2 √ 6Λ

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 

applications. 

Herzog, Horowitz, Hartnol, Sachdev et. al. (07-08)

Conclusions 

Linearized gravity (and Higher-Spins) 

possess a generalized electric-magnetic 

duality. 

The duality has important Holographic 

consequences. 

The M-theory dualities can be studied 

Holographically.

Generalized Electric-Magnetic Duality and Holography

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