Binary Trees, Geometric Lie Algebras, and Operads

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1 Binary Trees, Geometric Lie Algebras, and Operads Ben West May 4, The moduli space of binary trees and the welding operation Let T be a binary tree. We put a total ordering on its nodes as follows. If p 1 and p 2 are two distinct nodes of T, then there are two unique paths γ 1 and γ 2 from the root r of T. We say p 1 < p 2 if p 1 is an ancestor of p 2, or if when γ 1 and γ 2 first diverge at a node q i, γ 1 goes right and γ 2 goes left. This total ordering is called the preorder on T. It induces a total ordering of the leaves of T. Intuitively, this ordering is the one that orders the leaves from right to left. Binary trees will now refer to binary trees with a total ordering on their leaves. Two binary trees are equivalent if there is an isomorphism preserving the ordering of their leaves. Let B n denote the moduli space of binary trees of size n, which is the set of equivalence classes of trees of size n. The moduli space of binary trees is denoted B = B n. n=1 The group S n+1 acts on trees of size n by permuting the ordering of its leaves, and this induces an action of S n+1 on B n. There exists σ T such that σ T (T ) has ordering of leaves induced by the preordering. Suppose T 1 and T 2 are trees of size n and m, respectively. For any i n + 1, we define a binary tree T 1 i T 2 of size n + m by welding the root of T 2 to the ith leaf of T 1. If the orderings of the leaves of T 1 and T 2 are p 1,..., p n+1 and q 1,..., q m+1, respectively, the ordering on the leaves of T 1 i T 2 is p 1,..., p i 1, q 1,..., q m+1, p i+1,..., p n+1. This induces a welding operation on B. We also have a cutting operation. Suppose p is an internal node of T other than the root. We can cut σ T (T ) at p to product two new trees Tp and T + P, where T p is the tree formed of p and its descendents, and T p + is the binary subtree of nodes which are not descendents of p. In particular, σ T (T ) = T p + i Tp. If T is a tree and p an internal node, denote by T p be T with the children of p exchanged. Essentially, T p is T with its chirality changed at p. Given an internal node p other than the root r of T, and two new nodes {p int, p ext }, we can construct a new tree T p (pint,pext) as follows. Let p be the parent of p. Replace it with p int so that p is the left child of p int, and p ext is the right child, and p ext is defined to be the last leaf. For r, define T r (pint,pext) with p int as its root, with r its left child, p ext the right child. We can think of the rod from p int to p ext in T p (pint,pext) coming out of the root p int in T (pint,pext) r. 2 Definition of geometric Lie algebras as a physical quantity coming in through p, or as Let V be a finite dimensional F -vector space, F a field of characteristic 0. Let F (n) denote the space of linear functionals on V V (n 1), and F = n=3 F (n). We have an action of S n on F n+1 by σ(f)(v v 1 v n ) = f(v v σ 1 (1) v σ 1 (n)). 1

2 Also, given a basis {e i } k i=1 of V with dual basis {e i }k i=1, and f F m, g F n, and i m 1, define the contraction f i g F m+n 2 by (f i g)(v v 1 v m+n 3 ) = k f(v v 1 v i 1 e l v n+i 1 v m+n 3 )g(e l v i v n+i 2 ). l=1 Definition 2.1. A geometric Lie algebra over F is a finite dimensional vector space V over F with a map ν : B F such that ν(b n ) F n+2 satisying 1. Chirality Axiom. For T B, ν( T p ) = ν(t ). 2. Conservation axiom. For T B n, 3. Permutation axiom. For σ S n+1, T B n, n+1 ν(t r (i,e) ) = ν(t p (i,e) i ). i=1 ν(σ(t )) = σ(ν(t )). 4. Welding axiom. For T 1, T 2 B, ν(t 1 i T 2 ) = ν(t 1 ) i ν(t 2 ). As for the naming of the conservation axiom, if T has size n, and x V V (n+1), we interpret ν(t r (i,e) )(x) as the amount of a physical quantity passing out through the root of T, and ν(t p (i,e) i )(x) as the amount of a quantity coming in through the p i leaf of T. The conservation axiom says these two quantities are the same. A homomorphism η : (V 1, ν 1 ) (V 2, ν 2 ) of geometric Lie algebras is a linear map such that ν 2 (T )(v η(v 1 ) η(v n+1 )) = ν 1 (T )(η (v ) v 1 v n+1 ). 3 The isomorphism between the category of geometric Lie algebras and category of finite dimensional Lie algebras Theorem 3.1. Let L and G be the categories of finite-dimensional Lie algebras and geometric Lie algebras, respectively. Then L and G are isomorphic. We construct a functor F : L G as follows. Let (V, [, ]) be a finite dimensional Lie algebra. Inductively define a map ν [, ] : B F as follows. Note B = {Λ, σ 12 Λ}, where Λ is the class of the tree of size 1 with order induced by the preordering. Define and From skew-symmetry of [, ], we have ν [, ] (Λ)(v v 1 v 2 ) = v, [v 1, v 2 ] ν [, ] (σ 12 (Λ))(v v 1 v 2 ) = v, [v 2, v 1 ]. ν [, ] (Λ) = ν [, ] (σ 12 (Λ)) = ν [, ] ( Λ λ ) so the Chirality axiom holds. Now if T is a tree, and p is any internal node, we have for some i, σ T (T ) = T + p i T p so T = σ 1 T (T + p i T p ). 2

3 Hence put ν [, ] (T ) = σ 1 T (ν [, ](T + p ) i ν [, ] (T p )). This is independent of the choice of p. Then (V, ν [, ] ) is a geometric Lie algebra. The permutation and welding axioms follow from the definition. (In the proof of independence, one shows ν [, ] (T ) = σ 1 T (ν [, ](σ T (T ))), so it s enough to prove the axioms when the ordering of leaves is induced from the preorder. If p is an internal node of T with p l and p r its left and right children, we have, for suitable i, σ T (T ) = T + p i (((Λ) 2 T p l ) 1 T p r ). Looking at the right side, we take a tree of size 1, weld the descents of the left child of p to the left, weld the descendents of the right child of p to the right, and then weld this tree into the tree of ancestors of p. From the welding axiom, we obtain, and similarly ν [, ] (T ) = ν [, ] (T + p ) i ((ν [, ] (Λ) 2 ν [, ] (T p l )) 1 ν [, ] (T p r )) ν [, ] ( T p ) = ν [, ] (T + p ) i ((ν [, ] ( Λ λ ) 2 ν [, ] (T p r )) 1 ν [, ] (T p l )) and these are negatives of each other since the Chirality axiom holds for trees of size 1. The conservation axiom is also proven by induction, by verifying it directly on B 1, and then proving it generally by breaking a tree into subtrees and using the welding axiom. The define F by the assignments F (V, [, ]) = (V, ν [, ] ) and F (f) = f for f a Lie algebra homomorphism. Also define a functor H : G L. Given (V, ν) a geometric Lie algebra, we define a bracket [, ] ν by v, [v 1, v 2 ] ν = ν(λ)(v v 1 v 2 ). The Chirality and Permutation axioms imply skew-symmetry. For instance, v, [v 1, v 2 ] ν = ν(λ)(v v 1 v 2 ) = ν( Λ λ )(v v 1 v 2 ) = ν(σ(12)(λ))(v v 1 v 2 ) = σ(12)(ν(λ))(v v 1 v 2 ) = ν(λ)(v v 2 v 1 ) = v, [v 2, v 1 ] ν so that [v 1, v 2 ] ν = [v 2, v 1 ] ν. Likewise, the Jacobi axiom follows the other three axioms. So one defines H by the assignment H(V, ν) = (V, [, ] ν ) and H(f) = f. Then one can directly verify that H F = 1 L and F H = 1 G. For example, and H F (V, [, ]) = H(V, ν [, ] ) = (V, [, ] ν[, ] ) v, [v 1, v 2 ] ν[, ] = ν [, ] (Λ)(v v 1 v 2 ) = v, [v 1, v 2 ] so that H F = 1 L. The other direction follows by induction. 4 Geometric Vertex Operator Algebras To tie it back to conformal theory, the geometric objects there are compact, connected, Riemann surfaces with ordered, oriented punctures with local (analytic) coordinates vanishing at these punctures. If p is an oriented puncture, and (ϕ, U) local coordinates centered at p. Then ϕ(u) contains somce disc D or radius r, and we have an equivalence relation on C by α β if α β 2iπZ. The classes with real part less than log r form an infinite half tube. We can view ϕ 1 (D \ {0}) as a half infinite tube with p a circle at the end. 3

4 In this way a compact, connected, Riemann surface with n punctures and vanishing local coordinates can be thought of as surface with n infinite half tubes, with circles attached to the ends. Given two Riemann surfaces with tubes, two punctures of opposite orientations can be glued together but cutting the tubes and identifying the boundaries. This is the Sewing operation, and induces an operation on the moduli space. We particularly focus on genus zero Riemann surfaces with tubes, or spheres with tubes, with one negatively oriented puncture. Two spheres are conformally equivalent if there is an analytic diffeomorphism all structure save maybe the local coordinate neighborhoods. Let K(n 1) denote the moduli space of spheres with n tubes. Given a sphere Q 1 in K(n) and Q 2 in K(m), we can sew the ith positive puncture of Q 1 to the negative puncture of Q 2, to get a sphere Q 1 i Q 2 in K(m + n 1). This should seem similar to the welding operation on the moduli space of binary trees, and motives the definition of a geometric vertex operator algebra. Let V = n Z V (n) be a Z-graded C-vector space such that dim V (n) <. Put the graded dual space, and let V = n Z V (n) V = n Z V (n) = V be the algebraic completion. Let H V (n) = Hom(V n, V ). Also, for m > 0 and n Z, and i m, we define a contraction map H V (m) H V (n) Hom(V m+n 1, V [[t, t 1 ]]) : (f, g) (f i g) t where (f i g) t (v 1 v n+m 1 ) is defined to be f(v 1 v i 1 P k g(v i v i+n 1 ) v i+n v m+n 1 )t k k Z where P k : V V ( k) is the projection. If the formal Laurent series v, (f i g) t (v 1 v n+m 1 ) is absolutely convergent for t = 1, then define f i g = (f i g) 1 H V (m + n 1). Definition 4.1. A geometric vertex operator algebra over C is a Z-graded C-vector space V = n Z V (n) such that dim V (n) <, and for any n N, a map ν n : K(n) H V (n) satisfying 1. Positive energy axiom: for sufficiently small n. 2. Grading axiom: For v V, v V (n) and a C, V (n) = 0 v, ν 1 (0, (a, 0))(v) = a n v, v. That is, V is graded according to the logarithms of eigenvalues of the operator corresponding to the rescaling of local coordinates. 3. Meromorphicity axiom: The (1 + n)-correlation function on K(n) Q v, ν n (Q)(v 1 v n is meromorphic, and if z i, z j, are the ith and jth punctures of Q, the for any v, v j V, there is a positive integer N(v i, v j ) such that the order of the pole z i = z j of v, ν n (Q)(v 1 v n ) for v k V, k i, j, is less than N(v i, v j ). That is, the correlation functions are meromorphic as functions of any of the punctures of Q, and the othe other punctures are the only possible poles. 4

5 4. Permuation axiom: For σ S n, σ(ν n (Q)) = ν n (σ(q)). That is, the ordering of the punctures is irrevelvant to the structure of the geometric vertex operator algebra. 5. Sewing axiom: If Q 1 K(m) and Q 2 K(n), and the ith tube of Q 1 can be sewn with the 0th tube of Q 2, then ν m (Q) i ν n (Q 2 ) exists, which is to say is absolutely convergent when t = 1, and v, (ν m (Q) i ν n (Q 2 )) t (v 1 v m+n 1 ) ν m+n 1 (Q 1 i Q 2 ) = (ν m (Q 1 ) i ν n (Q 2 )) exp( Γ(A (i), B (0), a (i) 0 )c) where c is the unique central charge. That is, the image of an element sewn in the moduli space is equal to the contraction of the images, up to a projective factor calculated up to second order in some particular variables. Theorem 4.2. Let V (c) and G(c) the categories of vertex operator algebras and geometric vertex operator algebras of central charge c, respectively. Then V (c) and G(c) are isomorphic as categories. 5 Operads It turns out, moduli spaces of spheres with tubes and the determinant line bundles can be given partial operad structures. Definition 5.1. An operad is a sequence of sets χ(j), j N with composition maps i : χ(k) χ(j) χ(k + j 1) : (a, b) a i b for any k > 0, j N, and 1 i k, and identity element I χ(1), and an action of S j on χ(j), satisfying the following axioms: Composition associativity: for k > 0, j, l N, 1 i 1 k, 1 i 2 k + j 1, and a χ(k), b χ(j), c χ(l), then (a i2 c) l+i1 1 b, i 2 < i 1, (a i1 b) i2 = a i1 (b i2 i 1+1 c), i 1 i 2 < i 1 + j, (a i2 j+1 c) i1 b, i 1 + j i 2. For k N, 1 i k, a χ(k), a i I = I 1 a = a. For k > 0, j N, 1 i k, and a χ(k), b χ(j), σ S k, τ S j, we have and σ(a) i b = σ(1,..., 1, j, 1,..., 1)(a }{{}}{{} σ(i) b) i 1 k i a i τ(b) = (1 1 τ 1 1)(a }{{}}{{} i b) i 1 k 1 where σ(j 1,..., j k ) denotes the permutation of k s=1 j s letters which permutes the k blocks of letters as σ permutes k letters. If the compositions i are only defined on subsets of χ(k) χ(j), and the axioms hold when both sides exist, we get a partial operad. If we drop the the composition-associativity axiom, we get a partial pseudooperad. 5

6 A morphism ψ : χ χ is a family of S j -equivariant maps ψ j : χ(j) χ (j) such that ψ 1 (I) = I, and χ(k) χ(j) i χ(k + j 1) χ (k) χ (j) i χ (k + j 1) commutes. Now let P be a partial operad. A subset G P(1) is a rescaling group if 1. I G 2. The partial compositions 1 : P(1) P(k) P(k) and i : P(k) P(1) P(k) are defined on G P(k) and P(k) G. 3. Composition-associativity holds in G sends G G G 5. Inverses exists with respect to 1 and I, then G is a group. Two elements c 1, c 2 P(j) are G-equivalent if there exists d G such that c 2 = d 1 c 1. Definition 5.2. A G-rescalable partial operad is a partial operad χ with rescaling group G such that for any k > 0, j N, 1 i j, and c P(k), d χ(j), there exists d P(j) that is G-equivalent to d such that c i d exists. In the case of the moduli space of spheres with tubes, to sew two given spheres, one cuts disks with reciprocal radii using the local coordinates, which contain no other punctures and then identifies the boundaries using the local coordinate maps and the inversion z 1 z. (This is required since Ĉ = C { } has an atlas {C, C { }}, with local coordinate map C { } C : w 1 w.) The ordering of positive punctures for the sewn sphere is obtained by inserting the ordering of the second sphere into that of the first, as in the binary tree case. The conformal class of spheres with n-tubes is denoted H(n 1), so the sewing operation gives maps, for each i m, i : K(m) K(m) K(m + n 1) and these satisfy the composition associativity axiom. The identity is the class of the standard sphere Ĉ in K(1) with its negative puncture, and 0 its positive puncture, and the standard local coordinates vanish at and 0. Also, S j acts on sphere with j positive punctures by permuting the punctures, and induces action on K(j). Sewing is only a partial operation since it s not always possible to have such disks. Rescaling the local coordinate map at the negative puncture makes it possible to satisfy the conditions, so K is a C -rescalable partial operad. 6