Homotopy BV algebras, Courant algebroids and String Field Theory

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1 Yale University Department of Mathematics Homotopy BV algebras, Courant algebroids and String Field Theory Anton M. Zeitlin AMS Sectional Meeting Richmond, VA November 6-7, 2010

2 Introduction/Motivations Vertex/Courant algebroids with Calabi-Yau structure and associated homotopy BV algebras of Lian-Zuckerman Application: Einstein equations from the tensor product BV BV A -algebras. A (3) -algebra of Courant algebroid Yang-Mills equations from A (3) -algebra and their relation to Courant algebroid Open questions

3 Motivations from PHYSICS String Theory in background fields (perturbed 2D CFT) β-function: β(φ {v}, h) = i N hi β i (φ {v} ) Condition of conformal invariance: β(φ {v}, h) = 0 β 1 (φ {v} ) = 0 Classical Field Equations (Yang-Mills, Einstein) String Field Theory suggests an algebraic meaning of β-function β(φ {v}, h) = 0 QΦ+µ 2 (Φ, Φ)+µ 3 (Φ, Φ, Φ)+ = 0, Φ = Φ(φ {v}, h). µ i generate A - or L - algebras: homotopy generalizations of DGA and DGLA.

4 Motivations from MATHEMATICS Relations between vertex algebras and homotopy algebras. vertex algebroids vertex algebras (V = n 0 V n ) vertex algebroids homotopy BV with CY structure algebras vertex algebras? homotopy BV algebras Geometry: Vertex algebroids Courant algebroids 1) Homotopy algebras in generalized complex geometry 2) Relations between classical field equations and Courant algebroids

5 Vertex algebroids Def. (P. Bressler) Vertex A-algebroid when A is a commutative k-algebra is an A-module with pairing: A V V, f v f v, such that 1 v = v i) Leibniz k-algebra: [, ] : V k V V ii) a k-linear map of Leibniz algebras: π : V DerA(the anchor) iii) a symmetric k-bilinear pairing <, >: V k V A iv) a k-linear map : A V s.t. π = 0 wich satisfy 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) = fπ(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) We will le also interested in O X -vertex algebroid when A is replaced by O X and V is replaced by a sheaf of k-vector spaces.

6 Vertex algebroid Vertex algebra f ( 1) g = fg, f ( 1) v = f v, v ( 1) f = f v π(v)(f), v (0) f = f (0) v = π(v)(f), v (0) w = [v, w], v (1) w =< v, w > This gives a truncated vertex algebras. Then one can construct a vertex algebras Vert = n 0 V n, s.t. V 0 = A, V 1 = V The inverse statement is also true: for a given vertex algebra, s.t. V 0 = A, V 1 = V, one can construct a vertex algebroid. V. Gorbunov, F. Malikov, V. Schechtman, Invent. Math. 155 (2004), Calabi-Yau structure Def. Operator div : V A is called a Calabi- Yau structure on V if div = 0 div(f v) = f divv+ < f, v > div[v 1, v 2 ] = π(v 1 )( divv 2 ) π(v 2 )( divv 1 ) In the case of O X -vertex algebroids, on a manifold X with a volume form Φ, one can construct div div Φ π, where div Φ is a divergence operator for a vector field w.r.t. the volume form Φ. Example: Let Vert be VOA, then L 1 gives a Calabi-Yau structure.

7 Courant algebroids Def. (Z.-J.Liu, A. Weinstein, P. Xu; P. Bressler) Courant A-algebroid is an A-module Q equipped with i) a structure of Leibniz k-algebra [, ] : Q k Q Q ii) anchor map π : Q DerA iii) pairing <, >: Q A Q A iv) derivation : A Q They satisfy: π = 0 [q 1, fq 2 ] = f[q 1, q 2 ] + π(q 1 )(f)q 2 < [q, q 1 ], q 2 > + < q 1, [q, q 2 ] > = π(q) < q 1, q 2 > [q, (f)] = ( π(q)(f) ) < q, (f) > = π(q)(f) [q 1, q 2 ] + [q 2, q 1 ] = (< q 1, q 2 >) Courant algebroids Poisson vertex algebras

8 Classical limit Suppose V = Q[h], such that Q = V/hV is commutative, i.e. is commutative on Q, Leibniz bracket, pairing, anchor, and div take values in hv and Q Q. Then one can define the Courant A-algebroid structure on Q as follows: 1 π Q ( ) = lim h 0 h π V ( ), 1, Q = lim h 0 h, V, 1 [, ] Q = lim h 0 h [, ] V, 1 divq = lim h 0 h div V Later we will interpret this classical limit on the level of the homotopy BV algebras. Example. Vertex algebra, generated by the quantum fields p i (z) and X i (z) (i = 1,..., D) of conformal weights 1 and 0 correspondingly, and their OPE: p i (z)x k (w) hδk i z w Then, V 0 is given by the states are generated by F (X(z)), while V 1 is spanned by i : vi (X)p i : (z) (vector fields), k w k(x) X k (z) (1-forms). In fact, we have a VOA, where L(z) = 1 h i : p i X i : (z) + 2 φ(x)(z). The Calabi-Yau structure is given by the action of L 1 -operator.

9 Lian-Zuckerman operations Let Vert = n 0 V n be a VOA with Virasoro element L V (z). Consider the corresponding semi-infinite complex C = Vert Λ, where Λ is a VOA generated by b(z)c(w) 1 z w Theorem(I. Frenkel, H. Garland, G. Zuckerman) The operator D, such that D = (c(z)l(z)+ : c(z) c(z)b(z) :)dz is nilpotent on Vert Λ iff the central charge of the Virasoro algebra L V n is equal to 26. Proposition The operator D is nilpotent on the space C L0 = ker L 0, where L 0 = L V 0 + L Λ 0 for all values of central charge. C L0 is spanned by: u(z), c(z)a(z), c(z)a(z), c(z) c(z)ã(z), c(z) 2 c(z)ã(z), c(z) c(z) 2 c(z)ũ(z), where u, ũ, a, ã V 0 ; A, Ã V 1

10 Lian and Zuckerman introduced operations: µ(a 1, a 2 ) = Res a 1(z)a 2, {a 1, a 2 } = Res z (b 1 a 1 )(z)a 2 z B.H. Lian, G.J. Zuckerman Commun.Math.Phys.154 (1993) 613 These operations satisfy the relations of the homotopy BV algebra when central charge of {L V n } is equal to 26. Theorem (B.H. Lian-G.J. Zuckerman) i) The operation µ is homotopy commutative and homotopy associative: Dµ(a 1, a 2 ) = µ(da 1, a 2 ) + ( 1) a 1 µ(a 1, Da 2 ), µ(a 1, a 2 ) ( 1) a 1 a 2 µ(a 2, a 1 ) = = Dm(a 1, a 2 ) + m(da 1, a 2 ) + ( 1) a 1 m(a 1, Da 2 ), Dn(a 1, a 2, a 3 ) + n(da 1, a 2, a 3 ) + ( 1) a 1 n(a 1, Da 2, a 3 ) + +( 1) a 1 + a 2 n(a 1, a 2, Da 3 ) = = µ(µ(a 1, a 2 ), a 3 ) µ(a 1, µ(a 2, a 3 ))

11 ii) The operations µ and {, } are related in the following way: {a 1, a 2 } = b 0 µ(a 1, a 2 ) µ(b 0 a 1, a 2 ) ( 1) a 1 µ(a 1, b 0 a 2 ). iii) The operations µ and {, } satisfy the relations: {a 1, a 2 } + ( 1) ( a 1 1)( a 2 1) {a 2, a 1 } = ( 1) a 1 1 (Dm (a 1, a 2 ) m (Da 1, a 2 ) ( 1) a 2 m (a 1, Da 2 )), {a 1, µ(a 2, a 3 )} = µ({a 1, a 2 }, a 3 ) + ( 1) ( a 1 1) a 2 µ(a 2, {a 1, a 3 }), {µ(a 1, a 2 ), a 3 } µ(a 1, {a 2, a 3 }) ( 1) ( a 3 1) a 2 µ({a 1, a 3 }, a 2 ) = ( 1) a 1 + a 2 1 (Dn (a 1, a 2, a 3 ) n (Da 1, a 2, a 3 ) ( 1) a 1 n (a 1, Da 2, a 3 ) ( 1) a 1 + a 2 n (a 1, a 2, Da 3 ), {{a 1, a 2 }, a 3 } {a 1, {a 2, a 3 }} + ( 1) ( a 1 1)( a 2 1) {a 2, {a 1, a 3 }} = 0.

12 Proposition There is a structure of the homotopy BV algebra on C L0, such that the differential acts follows: V 0 V 1 V 1 V 0 id L 1 L 1 V 0 V 0 Theorem Let V be a vertex algebroid with a Calabi-Yau structure, then there exists a structure of a homotopy BV algebra on the complex F V : A V A id V div A div A Corollary The homotopy Lie algebra on the FV 1 subspace coincides with the homotopy Lie algebra of Roytenberg-Weinstein in the case of Courant algebroid. D. Roytenberg, A. Weinstein, Lett. Math. Phys. 46, 81-93

13 In the following we will call this homotopy algebra LZ(V ) or LZ(Vert). Let V = Q[h], and all the conditions of the classical limit are satisfied (see above). Then Q has a structure of a Courant algebroid. Question: Is it possible to describe classical limit LZ(V ) LZ(Q) in terms of LZ operations? One can find a subcomplex (F h,q, D V ) in (F V, D V ), which is isomorphic to (F Q, D Q ) on the level of k-vector spaces, such that on (F h,q, D V ) we have: D Q = lim h 0 D V, 1 {, } Q = lim h 0 h {, } V µ Q (, ) = lim h 0 µ V (, ), A.M.Z. Quasiclassical Lian-Zuckerman Homotopy Algebras, Courant Algebroid and Gauge Theory, Comm.Math.Phys., in press (arxiv: )

14 Equations of Field Theory and LZ algebra From now on we assume that all Courant algebroids we consider are Courant O X -algebroids. Closed strings Vert Vert Claim 1 Einstein equations (β 1 -function for closed strings) are equivalent to generalized Maurer-Cartan equations for the homotopy Lie algebra of LZ(Q Vert ) LZ(Q Vert ) Open strings (Vert) U(g) Claim 2 Yang-Mills equations (β 1 -function for open strings) are equivalent to generalized Maurer-Cartan equations for the homotopy associative algebra of LZ(Q Vert ) U(g)

15 Split Courant algebroids and Einstein equations Let Q be a Courant algebroid with Calabi- Yau structure and Q = T Ω, such that T is a Lie algebroid w.r.t. [, ], Im Ω. We refer to such Courant algebroid as split. Proposition The homotopy BV algebra LZ(Q) has a BV subalgebra on the subcomplex: C 0 T T div div C id A A Let T be the sheaf of the holomorphic sections of the tangent bundle. Let us denote the corresponding BV algebra BV h. Let BV a denote its antiholomorphic analogue. Let BV E BV h BV a. Consider the Maurer-Cartan equation: DΦ + {Φ, Φ} = 0, where Φ is an element of BV E of degree 2 Φ T T T T O X ŌX

16 The resulting equations (system of linear and bilinear equations) turn out to be Einstein Equations with dilaton and a B-field: R µν = 1 4 H µλρh λρ ν + 2 µ νφ, µ H µνρ 2( λ Φ)H λνρ = 0, 4( µ Φ) 2 4( µ µ Φ) + R H µνρh µνρ = 0, where {G µν } = {g i j} (hermitian), {B µν } = { g i j} (associated 2-form) and H = db. Tensor {g i j } comes from T T. These equations are very interesting on their own. In particular they imply that the corresponding manifold X is almost Calabi-Yau: Φ = log( ge Φ ) satisfies the equation i k Φ = 0, i.e. volume form defined by Φ is pluriharmonic. A.M.Z. Perturbed Beta-Gamma Systems and Complex Geometry, Nucl. Phys. B 794 (2008)

17 A -algebras Consider the chain complex (V, D). Consider multilinear operations µ i : V i V of degree 2 i such that µ 1 = D. Def. The space V is an A -algebra if the operations µ n satisfy the bilinear identity n 1 ( 1) i M i M n i+1 = 0 i=1 on V n, where M s acts on V m for any m s as follows: M s = n s ( 1) l(s+1) 1 l µ s 1 m s l l=0 Relations between D, µ 2, µ 3 : D 2 = 0, Dµ 2 (a 1, a 2 ) = µ 2 (Da 1, a 2 ) + ( 1) a 2 µ 2 (a 1, Da 2 ), Dµ 3 (a 1, a 2, a 3 ) + µ 3 (Da 1, a 2, a 3 ) + ( 1) a 2 µ 1 (a 1, Da 2, a 3 ) + ( 1) a 2 µ 3 (a 1, a 2, Da 3 ) = µ 2 ( µ2 (a 1, a 2 ), a 3 ) µ2 ( a1, µ 2 (a 2, a 3 ) ). For µ n = 0, n 3 it is just a DGA. A is called A (k) algebra if µ n = 0, n > k.

18 Generalized Maurer-Cartan equation Consider X V of degree 1. Then the equation DX + n 2 µ n(x,..., X) = 0 is called Generalized Maurer-Cartan equation. Symmetries: ( X X+ε Dα+ n 2,k α is an element of degree 0. ) ( 1) n k µ n (X,..., α,..., X),

19 Yang-Mills equations from A (3) -algebra of LZ(Q) η ij[ i, [ j, k ] ] = 0, i,j i = i + A i Two objects: {A i }, {η ij } A i comes from Maurer-Cartan element Where η ij comes from? Flat background deformation of A (3) : D D η = D + i,j η ij µ(s i {s j, }) where {s i, s j } = 0, such that s i F 1 and Ds i = 0 Theorem i) Operations D, µ, n generate A (3) - algebra on F Q. ii) There exists a deformation A η (3) (Q) of A (3)(Q), associated with D η, such that µ η = µ + ν η, n η = n. A.M.Z. Quasiclassical Lian-Zuckerman Homotopy Algebras, Courant Algebroid and Gauge Theory, Comm.Math.Phys., in press (arxiv: )

20 The corresponding GMC equation D η Φ + µ η (Φ, Φ) + n η (Φ, Φ, Φ) = 0 where Φ F 1 g will be called generalized Yang-Mills equation associated to Courant algebroid Q and Lie algebra g. Symmetries: Φ Φ + ε ( D η α + µ η (Φ, α) µ η (α, Φ) ) where α F 0 g Special case: V = T R n T R n η ij[ i, [ j, k ] ] = η ij[ ] [ k, φ i ], φ j i,j i,j η ij[ i, [ j, φ k ] ] = η ij[ φ i, [φ j, φ k ] ] i,j i,j Symmetries: A i ε( i u + [A i, u]), φ i φ i + ε[φ i, u] where φ i are matter fields.

21 A (3) algebra of pure YM theory D: 0 Ω 0 (R D ) d Ω 1 (R D ) d d Ω D 1 (R D ) d Ω D (R D ) 0 µ 2 (f 1, f 2 ): f 1 v A V a f 2 w vw Aw Vw aw B vb (A, B) B V 0 W vw A W 0 0 b vb f 1 takes values in {v, A, V, a} f 2 takes values in {w, B, W, b} u, v Ω 0 (R D ), A, B Ω 1 (R D ), V, W Ω D 1 (R D ), a, b Ω D (R D ) Trilinear operation µ 3 is defined to be nonzero only when all arguments belong to F 1. For A, B, C F 1 we have µ 3 (A, B, C) = A (B C) ( 1) D 2( (A B) ) C A.M.Z. Conformal Field Theory and Algebraic Structure of Gauge Theory, JHEP03(2010)056, arxiv:

22 Open questions Vertex algebras homotopy BV algebras What is the precise relation? Canonical string theory is not a vertex algebra: Closed : X µ (z, z)x ν (w, w) η µν log z w 2 (two series of oscillators) Open : X µ (t)x ν (s) η µν log t s 2 ; t, s R (one series of oscillators) Claim One can generalize the Lian-Zuckerman operations in such a way that closed strings will generate homotopy Lie algebra, while open strings will generate homotopy associative algebra.