Left-invariant Einstein metrics on S 3 × S 3 Alexander Haupt University of Hamburg MITP Workshop 2017 “Geometry, Gravity and Supersymmetry” Mainz, 26-Apr-2017 arXiv:1703.10512 w/ Florin Belgun, Vicente Cort´ es, David Lindemann
Outline 1 Introduction Motivation Some Definitions Previous Results 2 Methods for finding LI Einstein metrics on S 3 × S 3 =: G Reformulation as a Variational Problem Parameter Space System of Polynomial Equations 3 LI Einstein metrics invariant under finite subgroup Γ ⊂ Ad ( G ) Preliminaries & General Results Specific Results for Γ ∼ = Z 2 × Z 2 (Partial) Results for Γ ∼ = Z 2 4 Conclusions
Introduction Motivation Methods Some Definitions LI Einstein metrics inv. under finite subgrp. of Ad ( G ) Previous Results Conclusions Motivation: Mathematics Open problem: full classification of homogeneous compact Einstein manifolds in d = 6 Remaining open case : LI Einstein metrics on S 3 × S 3 Physics: string theory/supergravity Cp. Einstein manifolds play a role in AdS/CFT corresp. (string-/M-theory on AdS d × M ← → CFT on ( d − 1)-dim boundary of AdS d ) Flux compactifications of string theory from 10 to 4 dim: unbroken SUSY = ⇒ internal 6 d mfd. w/ SU (3)-structure Particular interest: SU (3)-structure nearly K¨ ahler or half-flat (Strict) nearly K¨ ahler = ⇒ 6 d mfd. is Einstein Left-invariant Einstein metrics on S 3 × S 3 Alexander Haupt (U. Hamburg)
Introduction Motivation Methods Some Definitions LI Einstein metrics inv. under finite subgrp. of Ad ( G ) Previous Results Conclusions Recall: Definition A (pseudo-)Riemannian mfd. ( M , g ) is called Einstein manifold if its Ricci tensor Ric g satisfies Ric g = λ g , for some constant λ ∈ R called Einstein constant . (Here: only consider the Riemannian case) Taking the trace yields: S = n λ , where S denotes the scalar curvature of g and n := dim M . Left-invariant Einstein metrics on S 3 × S 3 Alexander Haupt (U. Hamburg)
Introduction Motivation Methods Some Definitions LI Einstein metrics inv. under finite subgrp. of Ad ( G ) Previous Results Conclusions Later, we need concept of a symmetry of the metric . For LI Einstein metrics on S 3 × S 3 =: G [D’Atri, Ziller (1979)] : L G ⊂ Isom 0 ( G , g ) ⊂ L G · R G ∼ = ( G × G ) / { ( z , z ) | z ∈ Z ( G ) } (up to changing metric by an isometric LI metric). Notation: Isom 0 ( G , g ): conn. isometry grp. of some LI metric g on G L G ( R G ): group of left (right) translations Z ( G ) ∼ = Z 2 × Z 2 : center of G RHS contains group of inner automorphisms : Inn ( G ) = C G := { C a | a ∈ G } ⊂ L G · R G , where C a : G → G , x �→ axa − 1 (conjugation by a ) Hence, isotropy grp. of neutral el. e ∈ G in Isom 0 ( G , g ): Isom 0 ( G , g ) ∩ C G =: K 0 K 0 is the maximal connected subgroup of the Lie group Isom( G , g ) ∩ C G =: K Left-invariant Einstein metrics on S 3 × S 3 Alexander Haupt (U. Hamburg)
Introduction Motivation Methods Some Definitions LI Einstein metrics inv. under finite subgrp. of Ad ( G ) Previous Results Conclusions [Nikonorov, Rodionov (1999, 2003)] Partial classification : Theorem 1 (Nikonorov, Rodionov (1999, 2003)) A simply connected 6 d homogeneous cp. Einstein mfd. is either a symmetric space, or 1 isometric, up to multiplication of the metric by a constant, to one of 2 the following manifolds: CP 3 = Sp (2) Sp (1) × U (1) with squashed metric a the Wallach space SU (3) / T max with std or K¨ ahler metric b the Lie group SU (2) × SU (2) = S 3 × S 3 with some left-invariant c Einstein metric Classification of item (2c) is still open . Progress can be achieved by assuming additional symmetries of the metric... Left-invariant Einstein metrics on S 3 × S 3 Alexander Haupt (U. Hamburg)
Introduction Motivation Methods Some Definitions LI Einstein metrics inv. under finite subgrp. of Ad ( G ) Previous Results Conclusions [Nikonorov, Rodionov (2003)] Classification was achieved for the case that K = Isom( G , g ) ∩ C G contains a U (1) subgroup : Theorem 2 (Nikonorov, Rodionov (2003)) Let g be a LI Einstein metric on G := S 3 × S 3 . If K contains a U (1) subgroup, then ( G , g ) is homothetic to ( G , g can ) or ( G , g NK ). Here: g can = standard metric, g NK = nearly K¨ ahler metric These are the only known Einstein metrics on S 3 × S 3 (up to isometry and scale) These metrics are rigid [Kr¨ oncke (2015); Moroianu, Semmelmann (2007)] Left-invariant Einstein metrics on S 3 × S 3 Alexander Haupt (U. Hamburg)
Introduction Motivation Methods Some Definitions LI Einstein metrics inv. under finite subgrp. of Ad ( G ) Previous Results Conclusions Theorem 2 covers the case dim K ≥ 1 Remaining case: dim K = 0, i.e. K is a finite group Equivalently: require Isom 0 ( G , g ) = G i.e. the group of orientation preserving isometries is given by Isom + ( G , g ) = K ⋉ G , where K is a finite group of inner automorphisms of G → goal of rest of talk to analyze this case Left-invariant Einstein metrics on S 3 × S 3 Alexander Haupt (U. Hamburg)
Introduction Reformulation as a Variational Problem Methods Parameter Space LI Einstein metrics inv. under finite subgrp. of Ad ( G ) System of Polynomial Equations Conclusions Section 2 Methods for finding LI Einstein metrics on S 3 × S 3 =: G Left-invariant Einstein metrics on S 3 × S 3 Alexander Haupt (U. Hamburg)
Introduction Reformulation as a Variational Problem Methods Parameter Space LI Einstein metrics inv. under finite subgrp. of Ad ( G ) System of Polynomial Equations Conclusions [Jensen (1971); Wang, Ziller (1986); Besse (1987); ...] Finding Einstein metrics reformulated as variational problem : Theorem 3 A Riemannian metric g on a cp. orientable manifold M is Einstein iff it is a critical point of the Einstein-Hilbert functional � S EH [ g ] = S vol g , M � subject to the volume constraint V := M vol g = V 0 , where V 0 is a positive constant. Here, vol g is the metric volume form on ( M , g ). Left-invariant Einstein metrics on S 3 × S 3 Alexander Haupt (U. Hamburg)
Introduction Reformulation as a Variational Problem Methods Parameter Space LI Einstein metrics inv. under finite subgrp. of Ad ( G ) System of Polynomial Equations Conclusions Volume constr. incorp. by means of Lagrange multiplier Instead of S EH [ g ], we consider crit. pts. of ˜ S EH [ g , ν ] = S EH [ g ] − ν ( V − V 0 ) , where ν is a Lagrange multiplier . Simplifications occur when ( M , g ) = ( G , g ) is a cp. Lie group G with LI Riemannian metric g Then S = const. and hence, S EH [ g ] = S V Crit. pts. of ˜ S EH [ g , ν ] (i.e. Einstein metrics) satisfy 2 ν grad g S = 2 − n grad g V and V = V 0 Here, grad g = variation w.r.t. metric g . Assume n > 2. Left-invariant Einstein metrics on S 3 × S 3 Alexander Haupt (U. Hamburg)
Introduction Reformulation as a Variational Problem Methods Parameter Space LI Einstein metrics inv. under finite subgrp. of Ad ( G ) System of Polynomial Equations Conclusions LI Riemannian metric g on G ← → scalar product on Lie algebra g of G (also denoted by g ) Here, g = su (2) ⊕ su (2) w/ scal. prod. Q ( · , · ) = − 1 / 2 B ( · , · ) ( B ( X , Y ) = tr(ad( X ) ad( Y )) = Killing form of g ) Any other scalar product g : g ( · , · ) = Q ( L · , · ) ( L = Q -symmetric pos. def. endomorphism) Thus, space of LI Riemannian metrics ← → P ( g ) := { L ∈ End( g ) | L positive definite } Left-invariant Einstein metrics on S 3 × S 3 Alexander Haupt (U. Hamburg)
Introduction Reformulation as a Variational Problem Methods Parameter Space LI Einstein metrics inv. under finite subgrp. of Ad ( G ) System of Polynomial Equations Conclusions [Nikonorov, Rodionov (2003)] Parameterize P ( g ) by considering a change of basis : ( X , Y ) = ( E , F ) A T , A ∈ GL (6 , R ) Notation: Q -orthonormal basis ( E , F ) of g E := ( E 1 , E 2 , E 3 ), F := ( F 1 , F 2 , F 3 ) oriented ONBs of su (2) 1 , 2 g -orthonormal basis ( X , Y ) of g A satisfies A T A = L − 1 , i.e. g ( · , · ) = Q (( A T A ) − 1 · , · ) � D 0 � Can choose ( X , Y ) s.t. A = , where ˜ W D a 0 0 d 0 0 x u v , ˜ , W = D = 0 b 0 D = 0 e 0 α y w 0 0 0 0 β γ c f z a , . . . , f pos. params, components of W arbitrary real params Left-invariant Einstein metrics on S 3 × S 3 Alexander Haupt (U. Hamburg)
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