Self-dual conformal gravity

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Self-dual conformal gravity Maciej Dunajski Department of Applied Mathematics and Theoretical Physics University of Cambridge MD, Paul Tod arxiv:1304.7772., Comm. Math. Phys. (2014). Dunajski (DAMTP, Cambridge) Self-dual conformal gravity June 2014 1 / 10

Conformal to Einstein (M, g) (pseudo) Riemannian n dimensional manifold with Levi Civita connection. Dunajski (DAMTP, Cambridge) Self-dual conformal gravity June 2014 2 / 10

Conformal to Einstein (M, g) (pseudo) Riemannian n dimensional manifold with Levi Civita connection. Curvature decomposition: Riemann=Weyl+Ricci+Scalar [ a, b ]V c = R ab c d V d R abcd = C abcd + 2 n 2 (g 2 a[cr d]b g b[c R d]a ) (n 1)(n 2) Rg a[cg d]b. Dunajski (DAMTP, Cambridge) Self-dual conformal gravity June 2014 2 / 10

Conformal to Einstein (M, g) (pseudo) Riemannian n dimensional manifold with Levi Civita connection. Curvature decomposition: Riemann=Weyl+Ricci+Scalar [ a, b ]V c = R ab c d V d R abcd = C abcd + 2 n 2 (g 2 a[cr d]b g b[c R d]a ) (n 1)(n 2) Rg a[cg d]b. (M, g) is 1 Einstein, if R ab = 1 n Rg ab. 2 Ricci flat if R ab = 0. 3 Conformal to Einstein (Ricci flat) if there exists Ω : M R + such that ĝ = Ω 2 g is Einstein (Ricci flat). Dunajski (DAMTP, Cambridge) Self-dual conformal gravity June 2014 2 / 10

Two problems in conformal geometry (M, g) Lorentzian four-manifold. Does there exist Ω : M R +, and a local coordinate system (x, y, z, t) such that ĝ = Ω 2 g = dt 2 + dx 2 + dy 2 + dz 2. Dunajski (DAMTP, Cambridge) Self-dual conformal gravity June 2014 3 / 10

Two problems in conformal geometry (M, g) Lorentzian four-manifold. Does there exist Ω : M R +, and a local coordinate system (x, y, z, t) such that ĝ = Ω 2 g = dt 2 + dx 2 + dy 2 + dz 2. Answer. Need C ab c d = 0 (Conformally invariant, as Ĉ = C). Dunajski (DAMTP, Cambridge) Self-dual conformal gravity June 2014 3 / 10

Two problems in conformal geometry (M, g) Lorentzian four-manifold. Does there exist Ω : M R +, and a local coordinate system (x, y, z, t) such that ĝ = Ω 2 g = dt 2 + dx 2 + dy 2 + dz 2. Answer. Need C ab c d = 0 (Conformally invariant, as Ĉ = C). Given an n dimensional manifold (M, g), does there exist Ω : M R + such that Ω 2 g is Einstein? Dunajski (DAMTP, Cambridge) Self-dual conformal gravity June 2014 3 / 10

Two problems in conformal geometry (M, g) Lorentzian four-manifold. Does there exist Ω : M R +, and a local coordinate system (x, y, z, t) such that ĝ = Ω 2 g = dt 2 + dx 2 + dy 2 + dz 2. Answer. Need C ab c d = 0 (Conformally invariant, as Ĉ = C). Given an n dimensional manifold (M, g), does there exist Ω : M R + such that Ω 2 g is Einstein? 1 n = 2. All metrics are Einstein. Dunajski (DAMTP, Cambridge) Self-dual conformal gravity June 2014 3 / 10

Two problems in conformal geometry (M, g) Lorentzian four-manifold. Does there exist Ω : M R +, and a local coordinate system (x, y, z, t) such that ĝ = Ω 2 g = dt 2 + dx 2 + dy 2 + dz 2. Answer. Need C ab c d = 0 (Conformally invariant, as Ĉ = C). Given an n dimensional manifold (M, g), does there exist Ω : M R + such that Ω 2 g is Einstein? 1 n = 2. All metrics are Einstein. 2 n = 3. Einstein=constant curvature. Need Y abc = 0 (Cotton tensor). Dunajski (DAMTP, Cambridge) Self-dual conformal gravity June 2014 3 / 10

Two problems in conformal geometry (M, g) Lorentzian four-manifold. Does there exist Ω : M R +, and a local coordinate system (x, y, z, t) such that ĝ = Ω 2 g = dt 2 + dx 2 + dy 2 + dz 2. Answer. Need C ab c d = 0 (Conformally invariant, as Ĉ = C). Given an n dimensional manifold (M, g), does there exist Ω : M R + such that Ω 2 g is Einstein? 1 n = 2. All metrics are Einstein. 2 n = 3. Einstein=constant curvature. Need Y abc = 0 (Cotton tensor). 3 n = 4?? Brinkman 1920s, Szekeres 1963, Kozameh Newman Tod 1985. Dunajski (DAMTP, Cambridge) Self-dual conformal gravity June 2014 3 / 10

Conformal gravity B bc := ( a d 1 2 Rad )C abcd, Bach E abc := C ef cb d ( C) afed ( C) ef cb d (C) afed, Eastwood-Dighton. Dunajski (DAMTP, Cambridge) Self-dual conformal gravity June 2014 4 / 10

Conformal gravity B bc := ( a d 1 2 Rad )C abcd, Bach E abc := C ef cb d ( C) afed ( C) ef cb d (C) afed, Eastwood-Dighton. Necessary conditions: B ab = 0, E abc = 0. Dunajski (DAMTP, Cambridge) Self-dual conformal gravity June 2014 4 / 10

Conformal gravity B bc := ( a d 1 2 Rad )C abcd, Bach E abc := C ef cb d ( C) afed ( C) ef cb d (C) afed, Eastwood-Dighton. Necessary conditions: B ab = 0, E abc = 0. Sufficient conditions (locally): B ab = 0, E abc = 0 + genericity on Weyl. Dunajski (DAMTP, Cambridge) Self-dual conformal gravity June 2014 4 / 10

Conformal gravity B bc := ( a d 1 2 Rad )C abcd, Bach E abc := C ef cb d ( C) afed ( C) ef cb d (C) afed, Eastwood-Dighton. Necessary conditions: B ab = 0, E abc = 0. Sufficient conditions (locally): B ab = 0, E abc = 0 + genericity on Weyl. Conformal gravity: C 2 vol g Euler-Lagrange B ab = 0. M Dunajski (DAMTP, Cambridge) Self-dual conformal gravity June 2014 4 / 10

Anti-self-duality (genericity fails) An oriented 4-manifold (M, g) is anti-self-dual (ASD) if C abcd = 1 2 ɛ ab ef C cdef. Dunajski (DAMTP, Cambridge) Self-dual conformal gravity June 2014 5 / 10

Anti-self-duality (genericity fails) An oriented 4-manifold (M, g) is anti-self-dual (ASD) if B ab = 0, E abc = 0. C abcd = 1 2 ɛ ab ef C cdef. Dunajski (DAMTP, Cambridge) Self-dual conformal gravity June 2014 5 / 10

Anti-self-duality (genericity fails) An oriented 4-manifold (M, g) is anti-self-dual (ASD) if C abcd = 1 2 ɛ ab ef C cdef. B ab = 0, E abc = 0.... and yet (in the real analytic category) ASD depends on 6 functions of 3 variables. R ab = 0+ASD depends o 2 functions of 3 variables. Dunajski (DAMTP, Cambridge) Self-dual conformal gravity June 2014 5 / 10

Anti-self-duality (genericity fails) An oriented 4-manifold (M, g) is anti-self-dual (ASD) if C abcd = 1 2 ɛ ab ef C cdef. B ab = 0, E abc = 0.... and yet (in the real analytic category) ASD depends on 6 functions of 3 variables. R ab = 0+ASD depends o 2 functions of 3 variables. Signature (4, 0) or (2, 2). (Lorentzian+ASD=conformal flatness). Dunajski (DAMTP, Cambridge) Self-dual conformal gravity June 2014 5 / 10

Anti-self-duality (genericity fails) An oriented 4-manifold (M, g) is anti-self-dual (ASD) if C abcd = 1 2 ɛ ab ef C cdef. B ab = 0, E abc = 0.... and yet (in the real analytic category) ASD depends on 6 functions of 3 variables. R ab = 0+ASD depends o 2 functions of 3 variables. Signature (4, 0) or (2, 2). (Lorentzian+ASD=conformal flatness). Question: Given a 4-manifold (M, g) with ASD Weyl tensor, how to determine whether g is conformal to a Ricci flat metric? Dunajski (DAMTP, Cambridge) Self-dual conformal gravity June 2014 5 / 10

Twistor equation C T M = S S, where (S, ε), (S, ε ) are rank two complex symplectic vector bundles (spin bundles) over M. Dunajski (DAMTP, Cambridge) Self-dual conformal gravity June 2014 6 / 10

Twistor equation C T M = S S, where (S, ε), (S, ε ) are rank two complex symplectic vector bundles (spin bundles) over M. Spinor indices (love it or hate it): a = AA, where A, B, = 0, 1, and A, B, = 0, 1. Metric g = ε ε. Dunajski (DAMTP, Cambridge) Self-dual conformal gravity June 2014 6 / 10

Twistor equation C T M = S S, where (S, ε), (S, ε ) are rank two complex symplectic vector bundles (spin bundles) over M. Spinor indices (love it or hate it): a = AA, where A, B, = 0, 1, and A, B, = 0, 1. Metric g = ε ε. Twistor operator. D : Γ(S ) Γ(T M S ). π A A(A π B ) Dunajski (DAMTP, Cambridge) Self-dual conformal gravity June 2014 6 / 10

Twistor equation C T M = S S, where (S, ε), (S, ε ) are rank two complex symplectic vector bundles (spin bundles) over M. Spinor indices (love it or hate it): a = AA, where A, B, = 0, 1, and A, B, = 0, 1. Metric g = ε ε. Twistor operator. D : Γ(S ) Γ(T M S ). Conformally invariant if π A A(A π B ) g Ω 2 g, ε Ω ε, ε Ω ε, π Ω π. Dunajski (DAMTP, Cambridge) Self-dual conformal gravity June 2014 6 / 10

Prolongation of the twistor equation Proposition. An ASD metric g is conformal to a Ricci flat metric if and only if dim(ker D) = 2. Dunajski (DAMTP, Cambridge) Self-dual conformal gravity June 2014 7 / 10

Prolongation of the twistor equation Proposition. An ASD metric g is conformal to a Ricci flat metric if and only if dim(ker D) = 2. Prolongation π α ɛ = 0, α + π P = 0, where α Γ(S), and P ab = (1/12)Rg ab (1/2)R ab. Dunajski (DAMTP, Cambridge) Self-dual conformal gravity June 2014 7 / 10

Prolongation of the twistor equation Proposition. An ASD metric g is conformal to a Ricci flat metric if and only if dim(ker D) = 2. Prolongation π α ɛ = 0, α + π P = 0, where α Γ(S), and P ab = (1/12)Rg ab (1/2)R ab. Connection D on a rank-four complex vector bundle E = S S. Dunajski (DAMTP, Cambridge) Self-dual conformal gravity June 2014 7 / 10

Prolongation of the twistor equation Proposition. An ASD metric g is conformal to a Ricci flat metric if and only if dim(ker D) = 2. Prolongation π α ɛ = 0, α + π P = 0, where α Γ(S), and P ab = (1/12)Rg ab (1/2)R ab. Connection D on a rank-four complex vector bundle E = S S. Spinor conjugation in Riemannian signature: σ : S S. π = (p, q), σ(π) = ( q, p), so π Ker D σ(π) Ker D. Dunajski (DAMTP, Cambridge) Self-dual conformal gravity June 2014 7 / 10

Prolongation of the twistor equation Proposition. An ASD metric g is conformal to a Ricci flat metric if and only if dim(ker D) = 2. Prolongation π α ɛ = 0, α + π P = 0, where α Γ(S), and P ab = (1/12)Rg ab (1/2)R ab. Connection D on a rank-four complex vector bundle E = S S. Spinor conjugation in Riemannian signature: σ : S S. π = (p, q), σ(π) = ( q, p), so π Ker D σ(π) Ker D. Theorem 1. There is a one-to-one correspondence between parallel sections of (E, D) and Ricci flat metrics in an ASD conformal class. Dunajski (DAMTP, Cambridge) Self-dual conformal gravity June 2014 7 / 10

Tensor Obstructions Integrability conditions for parallel sections: RΨ = 0 where R = [D, D] is the curvature (4 by 4 matrix), and Ψ = (π, α) T. Dunajski (DAMTP, Cambridge) Self-dual conformal gravity June 2014 8 / 10

Tensor Obstructions Integrability conditions for parallel sections: RΨ = 0 where R = [D, D] is the curvature (4 by 4 matrix), and Ψ = (π, α) T. If R = 0 then g is conformally flat. Otherwise differentiate: (DR)Ψ = 0, (DDR)Ψ = 0,... Dunajski (DAMTP, Cambridge) Self-dual conformal gravity June 2014 8 / 10

Tensor Obstructions Integrability conditions for parallel sections: RΨ = 0 where R = [D, D] is the curvature (4 by 4 matrix), and Ψ = (π, α) T. If R = 0 then g is conformally flat. Otherwise differentiate: (DR)Ψ = 0, (DDR)Ψ = 0,... Necessary conditions: rank (R) = 2. Dunajski (DAMTP, Cambridge) Self-dual conformal gravity June 2014 8 / 10

Tensor Obstructions Integrability conditions for parallel sections: RΨ = 0 where R = [D, D] is the curvature (4 by 4 matrix), and Ψ = (π, α) T. If R = 0 then g is conformally flat. Otherwise differentiate: (DR)Ψ = 0, (DDR)Ψ = 0,... Necessary conditions: rank (R) = 2. Sufficient conditions: rank (R, DR) = 2. Dunajski (DAMTP, Cambridge) Self-dual conformal gravity June 2014 8 / 10

Tensor Obstructions Integrability conditions for parallel sections: RΨ = 0 where R = [D, D] is the curvature (4 by 4 matrix), and Ψ = (π, α) T. If R = 0 then g is conformally flat. Otherwise differentiate: (DR)Ψ = 0, (DDR)Ψ = 0,... Necessary conditions: rank (R) = 2. Sufficient conditions: rank (R, DR) = 2. Set V a = 4 C 2 C bcd a e C bcde. Theorem 2. An ASD Riemannian metric g is conformal to a Ricci flat metric if and only if det(r) := 4 e C bcde f C bcdf V 2 C 2 = 0, and T ab := P ab + a V b + V a V b 1 2 V 2 g ab = 0. Dunajski (DAMTP, Cambridge) Self-dual conformal gravity June 2014 8 / 10

Example Two parameter family of Riemannian metrics (LeBrun 1988) g = f 1 dr 2 + 1 4 r2 (σ1 2 + σ2 2 + fσ3), 2 where f = 1 + A r 2 + B r 4, where σ j are 1-forms on SU(2) such that dσ 1 = σ 2 σ 3 etc. Dunajski (DAMTP, Cambridge) Self-dual conformal gravity June 2014 9 / 10

Example Two parameter family of Riemannian metrics (LeBrun 1988) g = f 1 dr 2 + 1 4 r2 (σ 2 1 + σ 2 2 + fσ 2 3), where f = 1 + A r 2 + B r 4, where σ j are 1-forms on SU(2) such that dσ 1 = σ 2 σ 3 etc. Kahler and vanishing Ricci scalar, so ASD. Dunajski (DAMTP, Cambridge) Self-dual conformal gravity June 2014 9 / 10

Example Two parameter family of Riemannian metrics (LeBrun 1988) g = f 1 dr 2 + 1 4 r2 (σ 2 1 + σ 2 2 + fσ 2 3), where f = 1 + A r 2 + B r 4, where σ j are 1-forms on SU(2) such that dσ 1 = σ 2 σ 3 etc. Kahler and vanishing Ricci scalar, so ASD. Conformal to Ricci flat? det(r) = 0, T ab = A(4B A2 ) (Ar 2 + 2B) 2 g ab. Dunajski (DAMTP, Cambridge) Self-dual conformal gravity June 2014 9 / 10

Example Two parameter family of Riemannian metrics (LeBrun 1988) g = f 1 dr 2 + 1 4 r2 (σ 2 1 + σ 2 2 + fσ 2 3), where f = 1 + A r 2 + B r 4, where σ j are 1-forms on SU(2) such that dσ 1 = σ 2 σ 3 etc. Kahler and vanishing Ricci scalar, so ASD. Conformal to Ricci flat? det(r) = 0, T ab = A(4B A2 ) (Ar 2 + 2B) 2 g ab. If A = 0 then g is Ricci flat (Eguchi Hanson). Dunajski (DAMTP, Cambridge) Self-dual conformal gravity June 2014 9 / 10

Example Two parameter family of Riemannian metrics (LeBrun 1988) g = f 1 dr 2 + 1 4 r2 (σ 2 1 + σ 2 2 + fσ 2 3), where f = 1 + A r 2 + B r 4, where σ j are 1-forms on SU(2) such that dσ 1 = σ 2 σ 3 etc. Kahler and vanishing Ricci scalar, so ASD. Conformal to Ricci flat? det(r) = 0, T ab = A(4B A2 ) (Ar 2 + 2B) 2 g ab. If A = 0 then g is Ricci flat (Eguchi Hanson). If B = A 2 /4 then ĝ = (2r 2 + A) 1 g is Ricci flat. Dunajski (DAMTP, Cambridge) Self-dual conformal gravity June 2014 9 / 10

Example Two parameter family of Riemannian metrics (LeBrun 1988) g = f 1 dr 2 + 1 4 r2 (σ 2 1 + σ 2 2 + fσ 2 3), where f = 1 + A r 2 + B r 4, where σ j are 1-forms on SU(2) such that dσ 1 = σ 2 σ 3 etc. Kahler and vanishing Ricci scalar, so ASD. Conformal to Ricci flat? det(r) = 0, T ab = A(4B A2 ) (Ar 2 + 2B) 2 g ab. If A = 0 then g is Ricci flat (Eguchi Hanson). If B = A 2 /4 then ĝ = (2r 2 + A) 1 g is Ricci flat. ASD Taub NUT ( ) ĝ = U dr 2 + R 2 (σ1 2 + σ2) 2 + U 1 σ3, 2 where U = 1 R 4A. Dunajski (DAMTP, Cambridge) Self-dual conformal gravity June 2014 9 / 10

Example Two parameter family of Riemannian metrics (LeBrun 1988) g = f 1 dr 2 + 1 4 r2 (σ 2 1 + σ 2 2 + fσ 2 3), where f = 1 + A r 2 + B r 4, where σ j are 1-forms on SU(2) such that dσ 1 = σ 2 σ 3 etc. Kahler and vanishing Ricci scalar, so ASD. Conformal to Ricci flat? det(r) = 0, T ab = A(4B A2 ) (Ar 2 + 2B) 2 g ab. If A = 0 then g is Ricci flat (Eguchi Hanson). If B = A 2 /4 then ĝ = (2r 2 + A) 1 g is Ricci flat. ASD Taub NUT ( ) ĝ = U dr 2 + R 2 (σ1 2 + σ2) 2 + U 1 σ3, 2 where U = 1 R 4A. g is ASD Kahler, Ω 2 g is hyper-kahler! Dunajski (DAMTP, Cambridge) Self-dual conformal gravity June 2014 9 / 10

Example Two parameter family of Riemannian metrics (LeBrun 1988) g = f 1 dr 2 + 1 4 r2 (σ 2 1 + σ 2 2 + fσ 2 3), where f = 1 + A r 2 + B r 4, where σ j are 1-forms on SU(2) such that dσ 1 = σ 2 σ 3 etc. Kahler and vanishing Ricci scalar, so ASD. Conformal to Ricci flat? det(r) = 0, T ab = A(4B A2 ) (Ar 2 + 2B) 2 g ab. If A = 0 then g is Ricci flat (Eguchi Hanson). If B = A 2 /4 then ĝ = (2r 2 + A) 1 g is Ricci flat. ASD Taub NUT ( ) ĝ = U dr 2 + R 2 (σ1 2 + σ2) 2 + U 1 σ3, 2 where U = 1 R 4A. g is ASD Kahler, Ω 2 g is hyper-kahler! Are there more such examples?? (Clue: The Kahler form for g becomes a conformal Killing Yano tensor for ĝ). Dunajski (DAMTP, Cambridge) Self-dual conformal gravity June 2014 9 / 10

Thank You! Dunajski (DAMTP, Cambridge) Self-dual conformal gravity June 2014 10 / 10

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