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Foliation of a Space Associated to Vertex Operator Algebra on A FOLIATION OF A SPACE ASSOCIATED TO VERTEX OPERATOR ALGEBRA A. ZUEVSKY Abstract. We construct the foliation of aspace associated to correlation func- tions of vertex operator algebras on considered on Riemann surfaces. We prove that the computation of general genus g correlation functions determines a fo- liation on the space associated to these correlation functions a sewn Riemann surface. Certain further applications of the definition are proposed. 1. Introduction The theory of vertex operator algebras correlation functions considered on Riemann surfaces is a rapidly developing field of studies. Algebraic nature of methods applied in this field helps to understand and compute the structure of vertex operator algebras correlation functions. On the other hand, the geometric side of vertex operator algebra correlation functions is in associating their formal parameters with local coordinates on a manifold. Depending on the geometry, one can obtain various consequences for a vertex operator algebra and its space of correlation functions. One is able to study the geometry of a manifold using the algebraic structure of a vertex operator algebra defined on it, and, in particular, it is important to consider foliations of associated spaces. In this paper we introduce the formula for an n-point function for a vertex operator algebra V on a genus g Riemann surface S(g) obtained as a result of sewing of lower (g1) (g2) genus surfaces S and S of genera g1 and g2, g = g1 +g2. Using this formulation, we then introduce the construction of a foliation for the space of correlation functions for vertex operator algebra with formal parameters defined on general sewn Riemann surfaces. Computations of a vertex operator algebra correlation functions allow us to define foliation of the space associated to a vertex operator algebra with formal arXiv:2106.06539v2 [math.FA] 22 Sep 2021 parameter associated to a local coordinate on a genus g Riemann surface sewn from lower genus Riemann surfaces. Such foliations are important both for studies of the space of correlation function for a vertex operator algebras and possibly for studies of smooth manifolds in the frames of Losik’s approach [9]. Key words and phrases. Vertex operator algebras; Riemann surfaces; correlation functions; foliations. 1 2 A. ZUEVSKY 2. Vertex operator algebras First, we recall the definitions of the standard formal series x δ = xny−n, (2.1) y nX∈Z κ (x + y)κ = xκ−mym, (2.2) m mX≥0 κ κ(κ−1)...(κ−m+1) for any formal variables x,y,κ where m = m! . A vertex operator algebra [2,4] (V, Y, 1,ω) consists of a Z-graded complex vec- Z tor space V = n∈Z V(n) where dim V(n) < ∞ for each n ∈ , a linear map Y : V → End(VL)[[z,z−1]] for a formal parameter z and pair of distinguished vec- tors: the vacuum 1 ∈ V(0) and the conformal vector ω ∈ V(2). For each v ∈ V , the image under the map Y is the vertex operator Y (v,z)= v(n)z−n−1, nX∈Z with modes v(n) ∈ End(V ), where Y (v,z)1 = v + O(z). The linear operators (modes) u(n): V → V satisfy creativity Y (u,z)1 = u + O(z) (2.3) and lower truncation u(n)v =0, (2.4) for each u, v ∈ V and n ≫ 0. Each vertex operator satisfies the translation property Y (L(−1)u,z)= ∂zY (u,z). (2.5) Finally, the vertex operators satisfy the Jacobi identity −1 z1 − z2 −1 z2 − z1 z0 δ Y (u,z1)Y (v,z2) − z0 δ Y (v,z2)Y (u,z1) z0 −z0 −1 z1 − z0 = z2 δ Y (Y (u,z0)v,z2) . (2.6) z2 Vertex operators satisfy locality, i.e., for all u, v ∈ V there exists an integer k ≥ 0 such that k (z1 − z2) [Y (u,z1), Y (v,z2)]=0. The vertex operator of the conformal vector ω is Y (ω,z)= L(n)z−n−2, nX∈Z where the modes L(n) satisfy the Virasoro algebra with central charge c m3 − m [L(m),L(n)] = (m − n)L(m + n)+ δ c Id . 12 m,−n V We define the homogeneous space of weight k to be V(k) = {v ∈ V |L(0)v = kv}, FOLIATION OF A SPACE ASSOCIATED TO VERTEX OPERATOR ALGEBRA 3 and we write (v) = k for v ∈ V(k). Amongst other properties, these axioms imply locality, associativity, commutativity and skew-symmetry: m m (z1 − z2) Y (u,z1)Y (v,z2) = (z1 − z2) Y (v,z2)Y (u,z1), (2.7) n n (z0 + z2) Y (u,z0 + z2)Y (v,z2)w = (z0 + z2) Y (Y (u,z0)v,z2)w, (2.8) k u(k)Y (v,z) − Y (v,z)u(k) = Y (u(j)v,z)zk−j , (2.9) j Xj≥0 Y (u,z)v = ezL(−1)Y (v, −z)u, (2.10) for u, v, w ∈ V and integers m, n ≫ 0 [4]. In [23] Zhu introduced a second ”square-bracket” vertex operator algebra (V, Y [, ], 1, ω˜) associated to a given vertex operator algebra (V, Y (, ), 1,ω). The new square bracket vertex operators are defined by a change of parameters, namely −n−1 L(0) Y [v,z]= v[n]z = Y (qz v, qz − 1), (2.11) nX∈Z c with qz = exp(z), while the new conformal vector isω ˜ = ω − 24 1. For v of L(0) weight wt(v) ∈ R and m ≥ 0, v[m] = m! c(wt(v),i,m)v(i), (2.12) iX≥m i wt(v) − 1+ x c(wt(v),i,m)xm = . (2.13) i mX=0 wt(v)−1 In particular we note that v[0] = i v(i). iP≥0 2.1. The invariant form. The subalgebra {L(−1),L(0),L(1)} =∼ SL(2, C) associ- ated with M¨obius transformations on z naturally acts on a vertex algebra [4]. In particular, 0 1 1 γ = , z 7→ w = − , (2.14) −1 0 z is generated by T = exp(L(−1))exp(L(1)) exp (L(−1)) , where T Y (u,z) T −1 = Y (exp(−z)L(1)) (−z)−2L(0) u, −z−1 . (2.15) Following [14], we therefore define Y †(u,z)= u†(n)z−n−1 = T Y (u,z) T −1. (2.16) Xn For a quasi-primary vector u (i.e., satisfying the condition L(1)u = 0) of weight wt(u), we have u†(n) = (−1)n+1u(2wt(u) − n − 2), (2.17) and L†(n) = (−1)nL(−n). 4 A. ZUEVSKY Definition 1. We call a bilinear form h.,.i on V invariant if for all a, b, u ∈ V , [14] hY (u,z)a,bi = ha, Y †(u,z)bi, (2.18) i.e., hu(n)a,bi = ha,u†(n)bi. Thus it follows that hL(0)a,bi = ha,L(0)bi, so that ha,bi = 0 if wt(a) 6= wt(b) for homogeneous a, b. One also finds ha,bi = hb,ai [14]. The form h.,.i is unique up to normalization if L(1)V1 = V0. We choose the normalization h1V , 1V i = 1. It is non-degenerate if and only if V is simple [8]. Given α β α β αβ any V basis {u } we define the dual V basis {u } where hu , u iλ = δ . 3. Construction of an n-point function at a genus g Riemann surface Recall that a conformal field theory defined on a Riemann surface S(g) of genus (g) (g) g [3, 10, 16, 17, 19, 20] is determined by the set ZV v1,z1; ... ; vn,zn;Ω of all n (g) o correlation functions for all n, and all choices of points zi ∈ S , and all choices of elements vi of corresponding vertex operator algebra V . In this section, extending the genus two results of [16, 19, 20], we introduce the definition of an n-point correlation function for a vertex operator algebra V on a genus g Riemann surface S(g) obtained as a result of sewing of lower genus surfaces S(g1) (g2 ) and S of genera g1 and g2, g = g1 + g2. The genus g n-point correlation function (g1 ) for a1,...,aL ∈ V and b1,...,bR ∈ V , L + R = n inserted at x1,...,xL ∈ S and (g1) y1,...,yR ∈ S , respectively, can be defined by (g) (g1) (g2) ZV a1, x1; ... ; aL, xL|b1,y1; ... ; bR,yR;Ω1 , Ω2 ,ǫ,z1,z2 n (g1) (g1) = n≥0 u∈V ǫ ZV Y (a1, x1) ...Y (aL, xL)u;Ω1 ,z1 (3.1) P P (g2) (g2) ·ZV Y (bR,yR) ...Y (b1,y1)u;Ω2 ,z2 . (3.2) Note that this construction of the correlation functions depends on the choice of in- (g1 ) (g2) sertion points z1 ∈ S and z2 ∈ S of the ǫ-sewing construction [22]. Here ǫ is the (gi) sewing complex parameter [22] and Ωi , i = 1, 2 are period matrices for Riemann surfaces S(gi). To avoid misunderstanding, we say here, that the lower genus cor- (g1) (g1) (g2) relation functions ZV Y (a1, x1) ...Y (aL, xL)u; Ω1 ,z1 and ZV (Y (bR,yR) ... (g2) Y (b1,y1)u; Ω2 ,z2 are supposed to be know for given fixed g1, g2, or obtained via (3.1) recursively, from known lower genus correlation functions (e.g., starting from the partition functions, or torus correlation functions). The explicit dependence on Ω(gi), i = 1, 2 is assumed. Not all vertex operator algebras admit the notion of dual states.
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