Gerstenhaber algebras

Transcription

1 Anton M. Zeitlin Columbia University, Department of Mathematics Institute des Hautes Études Scientifiques 30th International Colloqium on Group Theoretical Methods in Physics Ghent, Belgium July 13 - July 19, 2014

2 G -algebra and from G -algebras via differential

3 Sigma-model: S so = 1 4πh d 2 z(g µν + B µν) X µ X ν + γr (2) (γ)φ(x ) Σ Symmetries: Diff symmetry and B B + dλ. Conformal invariance : µ d dµ Gµν = βg µν(g, B, Φ, h) = 0, µ d dµ Bµν = βb µν(g, B, Φ, h) = 0, µ d dµ Φ = βφ (G, B, Φ, h) = 0 at the level h 0 turn out to be : R µν = 1 4 Hµλρ Hλρ ν 2 µ ν Φ, µh µνρ 2( λ Φ)H λνρ = 0, 4( µφ) 2 4 µ µ Φ + R HµνρHµνρ = 0.

4 Early days of string theory: linearized and their symmetries: (G µν = η µν + s µν): Q η Ψ s = 0, Ψ s Ψ s + QΛ in a BRST complex associated to certain Virasoro module. It was conjectured (Sen, Zwiebach,...) that Einstein equations with h-corrections and their symmetries are Generalized Maurer-Cartan (GMC) Equations and their symmetries: QΨ [Ψ, Ψ] h + 1 3! [Ψ, Ψ, Ψ] h +... = 0 Ψ Ψ + QΛ + [Ψ, Λ] h [Ψ, Ψ, Λ] h +..., where [,,..., ] h operations generate L -algebra. We show that for a proper background, there is a richer structure, namely G -algebra, as well as a well-defined classical limit of the L -subalgebra, so that GMC equations are equivalent to Einstein Equations.

5 First order version of sigma-model action We start from free action: S 0 = 1 ( p 2πih X p X ), Σ where p X (Ω (1,0) (M)) Ω (1,0) (Σ), p X (Ω (0,1) (M)) Ω (0,1) (Σ). Symmetries: X i X i v i (X ), X ī X ī v ī( X ), p i p i + i v k p k, p ī p ī + ī v k p k p i p i X k ( k ω i i ω k ), Not invariant under general diffeomorphisms, i.e. p ī p ī X k ( kω ī ī ω k). δs 0 = 1 ( 2πih v, p X + v, p X ). It is necessary to add extra terms: δs µ = 1 ( µ, p 2πih X + µ, X p ), where µ Γ(T (1,0) M T (0,1) (M)), µ Γ(T (0,1) M T (1,0) (M)), so that: µ µ v +..., µ µ v +....

6 Continuing the procedure: where µ ī j S = 1 ( p 2πih X p X Σ µ, p X µ, X p b, X X ), µ ī j jv i + v k k µ ī j + v k kµ ī j + µī k jv k µ k j kv i + µ ī l µk j kv l, b i j b i j + v k k b i j + v k kb i j + b i k jv k + b l j i v l + b i kµ k j kv k + b l j µ k i kv l, so that: X i X i v i (X, X ), p i p i + p k i v k p k µ k l iv l b j k i v k X j, X ī X ī v ī(x, X ), p ī p ī + p k ī v k p k µ k l i v l b jk ī v k X j.

7 Similarly, the 1-form transformation: such that b i j b i j + jω i i ω j + µ ī j ( iω k k ω i ) + µ s i ( jω s sω j) + µījµ s k ( sωī ī ω s) p i p i X k ( k ω i i ω k ) r ω i X r µ s k i ω s X k, p ī p ī X k ( kω ī ī ω k) r ω ī X r µ s k iω s X k.

8 For simplicity: E = TM T M and E = T (1,0) M T (1,0) M, Ē = T (0,1) M T (0,1) M, i.e. E = E Ē. Let M Γ(E Ē), such that ( ) 0 µ M =. µ b Introduce α Γ(E), i.e. α = (v, v, ω, ω). Let D : Γ(E) Γ(E Ē), such that ( ) 0 v Dα =. v ω ω Then the transformation of M can be expressed: M M Dα + φ 1(α, M) + φ 2(α, M, M). Let us describe φ 1, φ 2 algebraically. In order to do that we need to pass to jet bundles, i.e. α J (O M ) J (Ō(Ē)) J (O(E)) J (ŌM), M J (O(E)) J (Ō(Ē))

9 Then: α = J f J b J + K b K f K, M = I a I ā I, where a I, b J J (O(E)), f I J (O M ) and ā I, b J J (Ō(Ē)), f I J (ŌM). Then φ 1(α, M) = I,J [b J, a I ] D f J ā I + I,K f K a I [ b K, ā I ] D, where [, ] D is a Dorfman bracket: Similarly: [v 1, v 2] D = [v 1, v 2] Lie, [v, ω] D = L v ω, [ω, v] D = i v dω, [ω 1, ω 2] D = 0. φ 2(α, M, M) = M Dα M 1 b I, a K a J ā J ( f I )ā K + 1 a J (f I )a K b I, ā K ā J. 2 2 I,J,K I,J,K

10 Relation to standard second order sigma-model: Let us fill in 0 in M: ( ) g µ M =. µ b S fo = 1 ( p 2πih X + p X Σ g, p p µ, p X µ, p X b, X X ). Same formulas express symmetries. If {g i j } is nondegenerate, then : S so = 1 4πh d 2 z(g µν + B µν) X µ X ν, G s k = g īj µīsµ j k + g s k b s k, B s k = g īj µīsµ j k g s k b s k G si = g i j µ j s g s j µ j i, G sī = g sj µ j ī g īj µ j s B si = g s j µ j i g i j µ j s, B sī = g īj µ j s g sj µ j ī, Symmetries: infinitesimal diffeomorphism transformations and the 2-form B symmetry G G L vg, B B 2dω if α = (v, ω), so that v Γ(TM), ω Ω 1 (M). B B L vb

11 Vertex algebroids The CFT corresponding to the chiral part of the free first order action S 0 = 1 p 2πih X R (2) (γ)φ (X ) is locally described via VOA: with Virasoro element Σ X i (z)p j (w) hδi j z w T (z) = 1 h : p(z) X (z) : + 2 φ (X (z)). The corresponding space of states V = + n=0 Vn, V 0 O M C[h] = O h M, V 1 V = O(E) C[h] O(E) h The structure of vertex algebra imposes algebraic relations on V 0 V 1 giving it a structure of a vertex algebroid.

12 A vertex O M -algebroid is a sheaf of C-vector spaces V with a pairing O M C[h] V V, i.e. f v f v such that 1 v = v, equipped with a structure of a Leibniz C[h]-algebra [, ] : V C[h] V V, a C[h]-linear map of Leibniz algebras π : V Γ(TM) usually referred to as an anchor, a symmetric C-bilinear pairing, : V C[h] V O h M a C-linear map : O M V such that π = 0, which satisfy the relations f (g v) (fg) v = π(v)(f ) (g) + π(v)(g) (f ), [v 1, f v 2] = π(v 1)(f ) v 2 + f [v 1, v 2], [v 1, v 2] + [v 2, v 1] = ( v 1, v 2 ), π(f v) = f π(v), f v 1, v 2 = f v 1, v 2 π(v 1)(π(v 2)(f )), π(v)( v 1, v 2 ) = [v, v 1], v 2 + v 1, [v, v 2], (fg) = f (g) + g (f ), [v, (f )] = (π(v)(f )), v, (f ) = π(v)(f ), where v, v 1, v 2 V, f, g O h M.

13 f = df, π(v)f = hv(f ), π(ω) = 0, f v = fv + hdx i i j fv j, f ω = f ω, [v 1, v 2] = h[v 1, v 2] D h 2 dx i i k v s 1 sv k 2, [v, ω] = h[v, ω] D, [ω, v] = h[ω, v] D, [ω 1, ω 2] = 0, v, ω = h v, ω s, v 1, v 2 = h 2 i v j 1 jv i 2, ω 1.ω 2 = 0, Given a holomorphic volume form on open neighborhood Uof M, one can associate a homotopy Gerstenhaber algebra to the vertex algebroid on U. Consider the light modes of the corresponding BRST complex, (i.e. L 0 = 0). The resulting complex (F h, Q) is: O h M V V 1 2 hdiv 1 hdiv 2 O h M O h M. O h M id div stands for divergence operator with respect to the nonvanishing volume form applied to sections of Γ(U, T (1,0) (M)).

14 The homotopy Gerstenhaber algebra of Lian and Zuckerman The homotopy associative and homotopy commutative product of Lian and Zuckerman: (A, B) = Res z A(z)B z Q(a 1, a 2) h = (Qa 1, a 2) h + ( 1) a 1 (a 1, Qa 2) h, (a 1, a 2) h ( 1) a 1 a 2 (a 2, a 1) h = Qm(a 1, a 2) + m(qa 1, a 2) + ( 1) a 1 m(a 1, Qa 2), Q(a 1, a 2, a 3) h + (Qa 1, a 2, a 3) h + ( 1) a 1 (a 1, Qa 2, a 3) h + ( 1) a 1 + a 2 (a 1, a 2, Qa 3) h = ((a 1, a 2) h, a 3) h (a 1, (a 2, a 3) h ) h Operator b of degree -1 on (F h, Q) which anticommutes with Q: V id V O h id M O h M O h id M O h M

15 One can define a bracket: ( 1) a1 {a 1, a 2} h = b(a 1, a 2) h (ba 1, a 2) h ( 1) a1 (a 1ba 2) h, {a 1, a 2} + ( 1) ( a 1 1)( a 2 1) {a 2, a 1} = ( 1) a1 1 (Qm h(a 1, a 2) m h(qa 1, a 2) ( 1) a2 m h(a 1, Qa 2)), {a 1, (a 2, a 3) h } h = ({a 1, a 2} h, a 3) h + ( 1) ( a 1 1) a 2 (a 2, {a 1, a 3} h ) h, {(a 1, a 2) h, a 3} h (a 1, {a 2, a 3} h ) h ( 1) ( a 3 1) a 2 ({a 1, a 3} h, a 2) h = ( 1) a 1 + a 2 1 (Qn h(a 1, a 2, a 3) n h(qa 1, a 2, a 3) ( 1) a1 n h(a 1, Qa 2, a 3) ( 1) a 1 + a 2 n h(a 1, a 2, Qa 3), {{a 1, a 2} h, a 3} h {a 1, {a 2, a 3} h } h + ( 1) ( a 1 1)( a 2 1) {a 2, {a 1, a 3} h } h = 0. The conjecture of Lian and Zuckerman, which was later proven by series of papers (Kimura, Zuckerman, Voronov; Huang, Zhao; Voronov) says that the symmetrized product and bracket of homotopy Gerstenhaber algebra constructed above can be lifted to G -algebra.

16 Quasiclassical limit of LZ G algebra Let V h=0 = V 0. One can see that (F, Q) = (F 1, Q) is a subcomplex of (F h, Q), which is: O M V 0 hv hdiv 1 hdiv 2 ho M ho M h 2 O M id Then (, ) h : F i F j F i+j [h], {, } : F i F j hf i+j 1 [h], b : F i hf i 1 [h], so that (, ) 0 = lim h 0 (, ) h, are well defined. {, } 0 = lim h 0 h 1 {, } h, b 0 = lim h 0 h 1 b

17 The symmetrized operations (, ) 0, {, } 0 satisfy the relations of homotopy Gerstenhaber algebra. The resulting C and L algebras are reduced to C 3 and L 3 algebras. Conjecture: This G -algebra is G 3-algebra (no higher homotopies). This is very close to the classical limit procedure for vertex algebroid: 1 [, ] 0 = lim h 0 [, ], h π0 = lim h 0 1 π,, h 0 = 1,. h As a result we get a Courant algebroid: A Courant O M -algebroid is an O M -module Q equipped with a structure of a Leibniz C-algebra [, ] 0 : Q C Q Q, an O M -linear map of Leibniz algebras (the anchor map) π 0 : Q Γ(TM), a symmetric O M -bilinear pairing, : Q OM Q O M, a derivation : O M Q which satisfy π = 0, [q 1, fq 2] 0 = f [q 1, q 2] 0 + π 0(q 1)(f )q 2 [q, q 1], q 2 + q 1, [q, q 2] = π 0(q)( q 1, q 2 0), [q, (f )] 0 = (π 0(q)(f )) q, (f ) = π 0(q)(f ) [q 1, q 2] 0 + [q 2, q 1] 0 = ( q 1, q 2 0) for f O M and q, q 1, q 2 Q. In our case Q = O(E), π 0 is just a projection on O(TM) [q 1, q 2] 0 = [q 1, q 2] D, q 1, q 2 0 = q 1, q 2 s, = d. Question: Is there a direct path (avoiding vertex algebra) from Courant algebroid to G 3-algebra?

18 Baby version: Gerstenhaber algebra Subcomplex (F sm, Q): O(T (1,0) M) O(T (1,0) M) div 1 div 2 C C O M i O M The G algebra degenerates to G-algebra. Moreover, due to b 0 it is a BV-algebra. Combine chiral and antichiral part: F sm = F sm F sm ( 1) a 1 {a 1, a 2} = b (a 1, a 2) (b a 1, a 2) ( 1) a 1 (a 1b a 2), where b = b b.

19 Maurer-Cartan elements, closed under b : Γ(T (1,0) (M) T (0,1) (M)) O(T (0,1) (M) O(T (1,0) (M) O M ŌM Components:(g, v, v, φ, φ). The Maurer-Cartan equation is equivalent to: 1). Vector field div Ω g, where Ω = Ω e 2φ+2 φ is determined by f 2Φ 0 = 2(Φ 0 + φ φ) and i jφ 0 = 0, is such that its Γ(T (1,0) M), Γ(T (0,1) M) components are correspondingly holomorphic and antiholomorphic. 2). Bivector field g Γ(T (1,0) M T (0,1) M) obeys the following equation: [[g, g]] + L divω (g)g = 0, where L divω (g) is a Lie derivative with respect to the corresponding vector fields and [[g, h]] k l (g i j i jh k l + h i j i jg k l i g k j jh i l i h k j jg i l ) 3). div Ω div Ω (g) = 0. The infinitesimal symmetries of the Maurer-Cartan equation coincide with the holomorphic coordinate transformations of the volume form and tensor {g i j }.

20 These are Einstein equations with the following constraints: Physically: G i k = g i k, B i k = g i k, Φ = log g + Φ 0, G ik = G ī k = G ik = G ī k = 0, [dp][d p][dx ][d X ]e 2πih 1 Σ ( p X p X g,p p )+ Σ R(2) (γ)φ 0 (X ) = [dx ][d X ]e 1 4πh d 2 z(gµν +B µν ) X µ X ν + R (2) (γ)(φ 0 (X )+ g)

21 Main Conjecture Consider F b = F F b =0 with the L -algebra structure given by Lian-Zuckerman construction. One can explicitly check that GMC symmetry reproduces Ψ Ψ + QΛ + [Ψ, Λ] h [Ψ, Ψ, Λ] h +..., M M Dα + φ 1(α, M) + φ 2(α, M, M). up to the second order in M. Conjecture: The corresponding Maurer-Cartan equation gives Einstein equations on G, B, Φ expressed in terms of differential. The symmetries of the Maurer-Cartan equation reproduce mentioned above symmetries of Einstein equations.

22 Thank you!