Gravity duals of N = 2 superconformal field theories with no electrostatic description K. S IAMPOS , 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
P LAN OF THE TALK 1 G RAVITY DUALS OF N = 2 SCFT S 2 M ETHODS FOR SOLVING T ODA Known 11d solutions so far – Electrostatics Comments on electrostatics Beyond electrostatics 3 O UR 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 I N PROGRESS 5 D ISCUSSION & O UTLOOK K.Siampos (UMons) 4d instantons in 11d SUGRA 13 February 2014 2 / 19
S YNOPSIS 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
G RAVITY DUALS OF N = 2 SCFT S Solutions of 11d SUGRA which possess SO ( 2 , 4 ) × SO ( 3 ) × U ( 1 ) R isometry were constructed in Lin–Lunin–Maldacena (2004): � � 2 1 − z ∂ z Ψ ( d ϕ + ω ) 2 − ∂ z Ψ 4 d s 2 11 e 2 λ 4 d s 2 AdS 5 + z 2 e − 6 λ d Ω 2 γ ij d x i d x j 11 = κ 3 2 + , z ω x = 1 ω y = − 1 ω = ω x d x + ω y d y , 2 ∂ y Ψ , 2 ∂ x Ψ , ∂ z Ψ γ ij d x i d x j = d z 2 + e Ψ ( d x 2 + d y 2 ) , e − 6 λ = − z ( 1 − z ∂ z Ψ ) , G 4 = d C 3 = κ 11 F 2 ∧ d Ω 2 , 1 − z 2 e − 6 λ � z 3 e − 6 λ � d ω − ∂ z e Ψ d x ∧ d y . � � F 2 = 2 ( d ϕ + ω ) ∧ d + 2 z where Ψ ( x , y , z ) satisfies the continual Toda equation [continuum Lie algebras – Saveliev (1990)]: � � z e Ψ = 0 , ∂ 2 x + ∂ 2 Ψ + ∂ 2 y where z ∈ [ 0 , z c ] and z c : e Ψ = 0 . The boundary conditions for the 11d background regularity: e Ψ = finite � = 0 , z = 0 : ∂ z Ψ = 0 , ∂ z Ψ/ 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
M ETHODS FOR SOLVING T ODA Separable solutions They boil down to the Liouville equation | ∂ f | 2 q = 1 e Ψ = c 3 � − z 2 + c 1 z + c 2 � , 2 ( x + iy ) . ( 1 − c 3 | f | 2 ) 2 where c i ’s are constants and f = f ( q ) is a locally univalent meromorphic function. Example : Maldacena–N´ u˜ nez e Ψ = 4 N 2 − z 2 z ∈ [ 0 , N ] , r ∈ [ 0 , 1 ] . ( 1 − r 2 ) 2 , An extra U ( 1 ) symmetry ◮ The problem can be mapped to a Laplace equation – electrostatics Ward (1990). ◮ Examples: Maldacena–N˜ un´ ez and AdS 7 × S 4 : e Ψ = coth 2 ζ , r = sinh 2 ζ sin ϑ , z = cosh 2 ζ cos ϑ , ζ ∈ R ∗ ϑ ∈ [ 0 , π/ 2 ] . + , K.Siampos (UMons) 4d instantons in 11d SUGRA 13 February 2014 5 / 19
K NOWN 11 D SOLUTIONS SO FAR – E LECTROSTATICS Ward’s transformation: ρ = r e Ψ ( r , z ) / 2 , ln r = ∂ η Φ , z = ρ∂ ρ Φ , maps the Toda equation to a Laplace equation in cylindrical coordinates ( ρ, η ) 1 ⇒ 1 z e Ψ = 0 = r ∂ r ( r ∂ r Ψ ) + ∂ 2 ρ∂ ρ ( ρ∂ ρ Φ ) + ∂ 2 η Φ = 0 , on 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 � η , 0 � η � N , λ ( η ) = N , η � N . ◮ AdS 7 × S 4 � 2 η , 0 � η � 1 2 , λ ( η ) = η + 1 η � 1 2 , 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
C OMMENTS 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. a n η + q n , where a n ∈ Z . Kinks occur at integer values of η . ⇒ A k n − 1 singularity transverse to AdS 5 × S 2 . λ ( 0 ) = 0 and a n − 1 − a n = k n ∈ Z + = A class of 4d N = 2 SCFTs can be viewed as generalised quiver gauge theories Gaiotto (2009) ◮ ∃ SU ( λ n ) gauge group ∀ λ n = λ ( η ) η = n : λ n � λ n + 1 � N = ⇒ k n := 2 λ n − λ n − 1 − λ n + 1 < 0 . ◮ ∀ Kink η = n , ∃ k n = a n − 1 − a n fundamental hypermultiplets charged under the SU ( λ n ) = ⇒ A k n − 1 . � ◮ In total, this is a quiver with gauge group SU ( n ) described at strong coupling by supergravity. n K.Siampos (UMons) 4d instantons in 11d SUGRA 13 February 2014 7 / 19
B EYOND ELECTROSTATICS Charge distributions with A k − 1 singularities – irregular punctures; R 4 / Z k 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
G RAVITATIONAL INSTANTONS IN 4d AS A TOOL HyperK¨ ahler manifold, with a Killing vector ξ = ∂ ϕ d ϕ + ω i d x i � 2 + V − 1 d s 2 , d ℓ 2 = V d s 2 = γ ij d x i d x 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) Translational Killing vector 1 n � m i d V − 1 = ± ⋆ γ d ω , ∂ i ∂ i V − 1 = 0 , V − 1 = ε + γ ij = δ ij , x 0 i | . | � x − � i = 1 Rotational Killing vector – Toda frame 2 V − 1 = 1 ω x = 1 ω y = − 1 d s 2 = d z 2 + e Ψ ( d x 2 + d y 2 ) , 2 ∂ z Ψ , 2 ∂ y Ψ , 2 ∂ x Ψ , Self-duality of the Riemann tensor yields the continual Toda equation. Regularity requirement of the 4d (tool) geometry is opposite to the 11d one R κλµν R κλµν ∼ ( ∂ z Ψ ) − 6 . K.Siampos (UMons) 4d instantons in 11d SUGRA 13 February 2014 9 / 19
B IANCHI IX FOLIATIONS AND SELF – DUALITY Invariant metric under the left-action of the SU ( 2 ) algebra ξ i (right-invariant) fields d ℓ 2 = 1 4 Ω 1 Ω 2 Ω 3 d t 2 + Ω 2 Ω 3 1 + Ω 1 Ω 3 2 + Ω 1 Ω 2 σ 2 σ 2 σ 2 3 , Ω 1 Ω 2 Ω 3 d σ i = 1 σ 1 + i σ 2 = − e i ψ ( i d ϑ + sin ϑ d ϕ ) , σ 3 = d ψ + cos ϑ d ϕ , 2 ε ijk σ j ∧ σ k . Self-duality yields the 1 st order differential system Ω 1 = d Ω 1 ( t ) = 1 ˙ 2 ( Ω 2 Ω 3 − λ Ω 1 ( Ω 2 + Ω 3 )) , and cyclic . d t λ = 0 – ξ i 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 – ξ i 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
T ODA FRAME OF THE A TIYAH –H ITCHIN 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 3 � z = 1 q = 1 � � 1 − n 2 Ω i , 2 ( x + i y ) = q ( Ω i , ϑ, ψ via elliptic integrals ) , i 2 i = 1 3 3 � � Ω j Ω k e 2 Ψ = 4 � � n 2 i + n 2 j + n 2 i n 2 n 2 Ω i Ω j j − 1 − 2 ( 2 n i n j − 1 ) δ ij , V = , i Ω i i , j = 1 i = 1 where : n i = ( cos ψ sin ϑ, sin ψ sin ϑ, cos ϑ ) . We shall apply the b.c. for the 11d background regularity on Ω i and x i = ( t , ϑ, ψ ) parameters. K.Siampos (UMons) 4d instantons in 11d SUGRA 13 February 2014 11 / 19
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