Gravity duals of N = 2 superconformal field theories with no - - PowerPoint PPT Presentation

Gravity duals of N = 2 superconformal field theories with no electrostatic description

• K. SIAMPOS,

M´ ecanique et Gravitation, Universit´ e de Mons

partially based on JHEP 1311 (2013) 118 with P. M. Petropoulos and K. Sfetsos

Crete Center for Theoretical Physics

K.Siampos (UMons) 4d instantons in 11d SUGRA 13 February 2014 1 / 19

PLAN OF THE TALK

1 GRAVITY DUALS OF N = 2 SCFTS 2 METHODS FOR SOLVING TODA

Known 11d solutions so far – Electrostatics Comments on electrostatics Beyond electrostatics

3 OUR SOLUTION

Gravitational instantons in 4d as a tool Bianchi IX foliations and self–duality Toda frame of the Atiyah–Hitchin metric Appropriate boundary conditions

4 IN PROGRESS 5 DISCUSSION & OUTLOOK

K.Siampos (UMons) 4d instantons in 11d SUGRA 13 February 2014 2 / 19

SYNOPSIS

We constructed the first 11d supergravity solutions with SO(2, 4) × SO(3) × U(1)R isometry, which are regular and have no smearing. Absence of an extra U(1) symmetry, even asymptotically – No electrostatic description “Short motivation”: Explore the 11d landscape of qualitatively different solutions and potentially understand the dual SCFTs.

K.Siampos (UMons) 4d instantons in 11d SUGRA 13 February 2014 3 / 19

GRAVITY DUALS OF N = 2 SCFTS

Solutions of 11d SUGRA which possess SO(2, 4) × SO(3) × U(1)R isometry were constructed in Lin–Lunin–Maldacena (2004): ds2

11 = κ

2 3

11e2λ

• 4 ds2

AdS5 + z2e−6λdΩ2 2 +

4 1 − z ∂zΨ (dϕ + ω)2 − ∂zΨ z γijdx idx j

• ,

ω = ωxdx + ωydy , ωx = 1 2 ∂yΨ , ωy = − 1 2 ∂xΨ , γijdx idx j = dz2 + eΨ(dx2 + dy2) , e−6λ = − ∂zΨ z(1 − z ∂zΨ) , G4 = dC3 = κ11 F2 ∧ dΩ2 , F2 = 2(dϕ + ω) ∧ d

• z3e−6λ

+ 2z

• 1 − z2 e−6λ

dω − ∂zeΨdx ∧ dy . where Ψ(x, y, z) satisfies the continual Toda equation [continuum Lie algebras – Saveliev (1990)]:

• ∂2

x + ∂2 y

• Ψ + ∂2

z eΨ = 0 ,

where z ∈ [0, zc] and zc : eΨ = 0. The boundary conditions for the 11d background regularity: z = 0 : eΨ = finite = 0 , ∂zΨ = 0 , ∂zΨ/z = finite. Only known regular solutions so far involve separability or existence of an extra U(1) isometry.

K.Siampos (UMons) 4d instantons in 11d SUGRA 13 February 2014 4 / 19

METHODS FOR SOLVING TODA

Separable solutions They boil down to the Liouville equation eΨ = c3 |∂f|2 (1 − c3|f|2)2

• −z2 + c1 z + c2
• ,

q = 1 2 (x + iy). where ci’s are constants and f = f(q) is a locally univalent meromorphic function. Example: Maldacena–N´ u˜ nez eΨ = 4 N2 − z2 (1 − r2)2 , z ∈ [0, N] , r ∈ [0, 1] . An extra U(1) symmetry

◮ The problem can be mapped to a Laplace equation – electrostatics Ward (1990). ◮ Examples: Maldacena–N˜

un´ ez and AdS7 × S4 : eΨ = coth2 ζ , r = sinh2 ζ sin ϑ , z = cosh2 ζ cos ϑ , ζ ∈ R∗

+ ,

ϑ ∈ [0, π/2] .

K.Siampos (UMons) 4d instantons in 11d SUGRA 13 February 2014 5 / 19

KNOWN 11D SOLUTIONS SO FAR – ELECTROSTATICS

Ward’s transformation: ln r = ∂ηΦ , z = ρ∂ρΦ , ρ = r eΨ(r,z)/2 , maps the Toda equation to a Laplace equation in cylindrical coordinates (ρ, η) 1 r ∂r (r∂rΨ) + ∂2

zeΨ = 0 =

⇒ 1 ρ∂ρ (ρ∂ρΦ) + ∂2

ηΦ = 0 ,

• n an infinite conducting plane (

E = E⊥) with a charge density λ(η) along positive half-axis of η. Examples of line charge densities:

◮ Maldacena–N˜

un´ ez λ(η) = η , 0 η N , N , η N .

◮ AdS7 × S4

λ(η) = 2η , 0 η 1

2 ,

η + 1

2 ,

η 1

2 .

The line charge distribution over an infinite plane has an one to one correspondence with N = 2 quiver gauge theories Gaiotto–Maldacena (2012).

K.Siampos (UMons) 4d instantons in 11d SUGRA 13 February 2014 6 / 19

COMMENTS ON ELECTROSTATICS

Line charge density – related to the M5 sources Gaiotto–Maldacena, Reid-Edwards–Stefanski (2011), Donos–Simon (2011) & Aharony–Berdichevsky–Berkooz (2012)

◮ Extra U(1) isometry endows a smearing process with the typical validity limitations. An exception is

the Maldacena–N˜ un´ ez solution, i.e. no smearing and regular punctures.

◮ Regularity of spacetime imposes constraints on λ(η), arising from 4-flux quantisation on punctures.

λ(η) is continuous and piecewise segment, i.e. anη + qn, where an ∈ Z. Kinks occur at integer values of η. λ(0) = 0 and an−1 − an = kn ∈ Z+ = ⇒ Akn−1 singularity transverse to AdS5 × S2.

A class of 4d N = 2 SCFTs can be viewed as generalised quiver gauge theories Gaiotto (2009)

◮ ∃ SU(λn) gauge group ∀λn = λ(η)η=n : λn λn+1 N =

⇒ kn := 2λn − λn−1 − λn+1 < 0 .

◮ ∀ Kinkη=n, ∃kn = an−1 − an fundamental hypermultiplets charged under the SU(λn) =

⇒ Akn−1.

◮ In total, this is a quiver with gauge group

• n

SU(n) described at strong coupling by supergravity.

K.Siampos (UMons) 4d instantons in 11d SUGRA 13 February 2014 7 / 19

BEYOND ELECTROSTATICS

Charge distributions with Ak−1 singularities – irregular punctures; R4/Zk For k = 1 and large values of ρ: Electrostatics picture = ⇒ a non-U(1) Toda solution. Objective: Find Toda potentials which are not separable and depend on x, y, z. General solutions of the continual Toda equation are not known. Genuine solutions have been found in the framework of triaxial Bianchi-IX four-dimensional instantons.

◮ Riemann self-dual by Atiyah–Hitchin (1985). ◮ K¨

ahler and R = 0 (WASD) by Pedersen–Poon (1990).

◮ Weyl self-dual and Einstein by Tod (1994) & Hitchin (1995).

We move on with a revisit on gravitational instantons in 4d and the Atiyah–Hitchin solution.

K.Siampos (UMons) 4d instantons in 11d SUGRA 13 February 2014 8 / 19

GRAVITATIONAL INSTANTONS IN 4d AS A TOOL

HyperK¨ ahler manifold, with a Killing vector ξ = ∂ϕ dℓ2 = V

• dϕ + ωi dx i2 + V−1 ds2 ,

ds2 = γij dx i dx j , i = 1, 2, 3 Gibbons–Hawking (1979) Self-duality of Riemann tensor yields two district types of Killing vectors: Translational and Rotational. Boyer–Finley (1982) & Gegenberg–Das (1984)

1

Translational Killing vector dV−1 = ± ⋆γ dω , γij = δij , ∂i∂iV−1 = 0 , V−1 = ε +

n

• i=1

mi | x − x0i| .

2

Rotational Killing vector – Toda frame ds2 = dz2 + eΨ(dx2 + dy2) , V−1 = 1 2∂zΨ , ωx = 1 2∂yΨ , ωy = − 1 2 ∂xΨ , Self-duality of the Riemann tensor yields the continual Toda equation. Regularity requirement of the 4d (tool) geometry is opposite to the 11d one Rκλµν Rκλµν ∼ (∂zΨ)−6 .

K.Siampos (UMons) 4d instantons in 11d SUGRA 13 February 2014 9 / 19

BIANCHI IX FOLIATIONS AND SELF–DUALITY

Invariant metric under the left-action of the SU(2) algebra ξi (right-invariant) fields dℓ2 = 1 4 Ω1Ω2Ω3 dt2 + Ω2Ω3 Ω1 σ2

1 + Ω1Ω3

Ω2 σ2

2 + Ω1Ω2

Ω3 σ2

3 ,

σ1 + iσ2 = −ei ψ (i dϑ + sin ϑ dϕ) , σ3 = dψ + cos ϑ dϕ , dσi = 1 2 εijkσj ∧ σk . Self-duality yields the 1st order differential system ˙ Ω1 = dΩ1(t) dt = 1 2 (Ω2Ω3 − λ Ω1 (Ω2 + Ω3)) , and cyclic . λ = 0 – ξi translational – Lagrange system – Algebraically integrable Axisymmetric – Triaxial solutions found with a few months interval by Eguchi–Hanson (1978) & Belinsky–Gibbons–Page–Pope (1978). λ = 1 – ξi rotational – Darboux–Halphen system – Not always algebraically integrable The axisymmetric is the Taub–NUT, known since the 50s (Lorentzian) but revived as an instanton of SU(2) foliations by Gibbons–Hawking (1978) & Eguchi–Hanson (1979). The triaxial was found by: Atiyah–Hitchin (1985). General solution was known since the 19th century by Halphen’s (1881) works.

K.Siampos (UMons) 4d instantons in 11d SUGRA 13 February 2014 10 / 19

TODA FRAME OF THE ATIYAH–HITCHIN METRIC

The Toda frame was found by Olivier (1991) for the Atiyah–Hitchin metric and generalised by Finley–McIver (2010) for the general solution of the Halphen system z = 1 2

3

• i=1

Ωi

• 1 − n2

i

• ,

q = 1 2 (x + i y) = q(Ωi, ϑ, ψ via elliptic integrals) , e2Ψ = 4

3

• i, j=1

ΩiΩj

• n2

i + n2 j + n2 i n2 j − 1 − 2(2ninj − 1)δij

• ,

V =

3

• i=1

n2

i

ΩjΩk Ωi , where : ni = (cos ψ sin ϑ, sin ψ sin ϑ, cos ϑ) . We shall apply the b.c. for the 11d background regularity on Ωi and x i = (t, ϑ, ψ) parameters.

K.Siampos (UMons) 4d instantons in 11d SUGRA 13 February 2014 11 / 19

APPROPRIATE BOUNDARY CONDITIONS

Boundary conditions for 11d regularity are equivalent to: ∀(ϑ∗, ψ∗) = ⇒ ∃ t∗ : n∗

k = 0 ,

Ω∗

k = 0 ,

α Ω∗

i + β Ω∗ j = 0 ,

α + β = 2 , α, β ∈ (0, 2] , where (α, β) are expressed in terms of n∗

i = ni(ϑ∗, ψ∗).

They possess a smooth limit as z → 0 ∂zΨ z = 2 z V = − 2 Ω∗

i Ω∗ j n∗4 k

= finite = 0 . Comments on the solution:

◮ It is a genuine triaxial case. ◮ There are no punctures, i.e. eΨ = 0 – no additional non-compact branes. ◮ The axisymmetric limit (Taub–NUT) does not satisfy the b.c. – No electrostatic description.

Simple poles of Ωi – Singular four-dimensional instantons – Regular eleven-dimensional solutions.

K.Siampos (UMons) 4d instantons in 11d SUGRA 13 February 2014 12 / 19

THE HALPHEN SOLUTION

The original Halphen (Atiyah–Hitchin) regular solution Ωi,H(T), in terms of Theta functions and the quasimodular form of weight two or complete elliptic integrals, reads: Ω1,H = π 6

• E2(iT) − θ4

2(iT) − θ4 3(iT)

• = −K(κ)(E(κ) − κ′2K(κ)) ,

Ω2,H = π 6

• E2(iT) + θ4

3(iT) + θ4 4(iT)

• = −K(κ)E(κ) ,

Ω3,H = π 6

• E2(iT) + θ4

2(iT) − θ4 4(iT)

• = −K(κ)(E(κ) − K(κ)) ,

Where κ is the elliptic modulus t 2 = T = − 2K(κ′) π K(κ) , κ′2 = 1 − κ2 . This solution describes:

◮ A regular SU(2)-symmetric self-dual gravitational instanton. ◮ The configuration space of two slowly moving BPS SU(2) Yang–Mills–Higgs monopoles.

Manton (1982) & Gibbons–Manton (1986)

K.Siampos (UMons) 4d instantons in 11d SUGRA 13 February 2014 13 / 19

THE HALPHEN SOLUTION

Halphen original solution Ω1,H < 0 < Ω3,H < Ω2,H.

K.Siampos (UMons) 4d instantons in 11d SUGRA 13 February 2014 14 / 19

APPROPRIATE BOUNDARY CONDITIONS

The SL(2, R) covariance of the Darboux–Halphen system Ωi(T) = 1 (CT + D)2 Ωi,H AT + B CT + D

• +

C CT + D , A B C D

• ∈ SL(2, R) ,

T = t 2 . Generic SL(2, R) transformation, yield a singular 4d geometry. Note that z transforms like Ωi and the Toda potential as e2Ψ(T,ϑ,ψ) = 1 (CT + D)4 e2ΨH

• AT+B

CT+D ,ϑ,ψ

• .

Satisfying the boundary conditions We can satisfy the b.c. (Ω∗

3 = 0 & αΩ∗ 1 + βΩ∗ 2 = 0) if

T∗ = 1 πC2

• α K
• β/2
• − 2 E
• β/2
• K
• β/2
• − D

C , A C = − β K

• α/2
• − 2 E
• α/2
• α K
• β/2
• − 2 E
• β/2

. where T0 < T < T∞ , T0 = − B A , T∞ = − D C .

K.Siampos (UMons) 4d instantons in 11d SUGRA 13 February 2014 15 / 19

APPROPRIATE BOUNDARY CONDITIONS

Generic solution in the range T0 < T < T∞ and Ω1 < Ω3 < Ω2. Given value of α, there is a 2-parameter family due to unimodularity. Metric signature demands ∂zΨ

z

< 0 = ⇒ T ∈ [T0, T∗]. For the symmetric case, α = β = 1 = ⇒ T∗ = 1

2 (T0 + T∞) .

The Atiyah–Hitchin (non-singular) solution corresponds to T < T∞ < T0 < T and does not satisfy the b.c.

K.Siampos (UMons) 4d instantons in 11d SUGRA 13 February 2014 16 / 19

IN PROGRESS

K¨ ahler metric with symmetry and R = 0 – WASD (canonical orientation) LeBrun (1991) Once more : dℓ2 = V(dϕ + A)2 + V−1(dz2 + eΨ(dx2 + dy2)) , where the R = 0 and K¨ ahler conditions

• ∂2

x + ∂2 y

• Ψ + ∂2

z

• eΨ

= eΨ∇2Ψ = 0 ,

• ∂2

x + ∂2 y

• V−1 + ∂2

z

• V−1 eΨ

= 0 , A = ∂xV−1dy ∧ dz + ∂yV−1dz ∧ dx + ∂z

• V−1 eΨ

dx ∧ dy , with K¨ ahler form J = (dϕ + A) ∧ dz − V−1 eΨdx ∧ dy , dJ = 0 . Application: Non-supersymmetric 5d multi-center solutions – LeBrun metrics as base space. Bobev–Niehoff–Warner (2011), (2012) & Niehoff (2013) Known solutions involve methods of electrostatics.

K.Siampos (UMons) 4d instantons in 11d SUGRA 13 February 2014 17 / 19

PEDERSEN–POON METRICS

General diagonal K¨ ahler and R = 0 (WASD) metric with SU(2) isometry Pedersen–Poon (1990) Ω′

1 = Ω2Ω3 − αΩ1 ,

Ω′

2 = Ω1Ω3 − αΩ2 ,

Ω′

3 = Ω1Ω2 ,

T = t 2 . where α is a constant. For α = 0 – BGPP metric, i.e. RSD – ∂ϕ-translational. Its LeBrun frame Tod (1995) z = n3 Ω3 , x = eαT n2 Ω2 , y = eαT n1 Ω1 , eΨ = e−2αT , V =

3

• i=1

ΩjΩk Ωi n2

i ,

Ai dxi = V−1 Ω1Ω3 Ω2 − Ω2Ω3 Ω1

• sin ϑ sin ψ cos ψ dϑ + Ω1Ω2

Ω3 cos ϑ dψ

• ,

An application in 11d SUGRA:

◮ Triaxial solutions – regular 11d supegravities with SO(2, 4) × SO(3) × U(1)R. ◮ There is a regular puncture at eΨ = 0, ϑ = 0 =

⇒ α = N5.

◮ ∃ axisymmetric solutions – extra U(1) isometry – usual electrostatics-quiver description: α ∈ N.

K.Siampos (UMons) 4d instantons in 11d SUGRA 13 February 2014 18 / 19

DISCUSSION & OUTLOOK

◮ We constructed the first 11d supergravity solutions, which are regular, have no smearing and possess

• nly SO(2, 4) × SO(3) × U(1)R isometry.

◮ Use of an auxiliary problem: We retrieved the Toda potential for metrics with SU(2) rotational

symmetry.

◮ These are obtained for example by transforming the Atiyah–Hitchin instanton under SL(2, R). ◮ Absence of an extra U(1) symmetry, even asymptotically – no electrostatic description. ◮ Qualitatively different class of solutions dual to new 4d N = 2 SCFTs. Our next goal is to unravel

them... Other applications of the instantons tools:

1

Non-supersymmetric multi-center solutions – LeBrun metrics as base space. Bobev–Niehoff–Warner (2011) & (2012) & Niehoff (2013)

2

Scalar hypermultiplets in N = 2 supergravities – 4d quaternionic spaces. Bagger–Witten (1983)

K.Siampos (UMons) 4d instantons in 11d SUGRA 13 February 2014 19 / 19