Foliation of a Space Associated to Vertex Operator Algebra on A

Home , G2 (mathematics)

arXiv:2106.06539v2 [math.FA] 22 Sep 2021 endo t n,i atclr ti motn ocnie oitoso foliations consider to important is it verte particular, a of in structure and, spaces. algebraic it, the functions. on using correlation co manifold defined various of a obtain space of its can geometry and one the geometry, algebra the operator with vertex on parameters a Depending formal manifold. their a associating ve on of in side is geometric the vert functions m hand, of correlation other structure of the the nature On compute Algebraic and functions. understand correlation studies. to of helps field field this developing in rapidly a is surfaces algebra eu surfaces genus fsot aiod ntefae fLsksapoc [9]. approach Losik’s and of algebras frames operator the vertex in f a manifolds both for smooth important of function are correlation foliations of Such space surfaces. Riemann genus lower odfieflaino h pc soitdt etxoeao alg genus operator a vertex on a coordinate to local associated a space correlatio to the algebra associated of parameter operator foliation vertex define g a on to of c defined of Computations parameters space formal the with surfaces. for algebra foliation operator a vertex of for construction the introduce then we foliations. OITO FASAEASCAE OVRE OPERATOR VERTEX TO ASSOCIATED SPACE A OF FOLIATION nti ae eitouetefruafran for formula the introduce we paper this In h hoyo etxoeao lerscreainfntosco functions correlation algebras operator vertex of theory The e od n phrases. and words Key V ito ntesaeascae oteecreainfuncti correlation these to associated space s the genus Riemann general on on of liation considered computation on the algebras that operator vertex of tions ufc.Cranfrhrapiain ftedfiiinare definition the of applications further Certain surface. Abstract. nagenus a on S ( g 1 ) ecntutteflaino saeascae ocorrelat to associated aspace of foliation the construct We and g imn surface Riemann S etxoeao lers imn ufcs orlto f correlation surfaces; Riemann algebras; operator Vertex ( g 2 ) fgenera of 1. ALGEBRA Introduction .ZUEVSKY A. g 1 S 1 and g ( g orlto ucin eemnsafo- a determines functions correlation ) banda euto eigo lower of sewing of result a as obtained g 2 n , pitfnto o etxoperator vertex a for function -point g = g g 1 + imn ufc enfrom sewn surface Riemann proposed. g n enRiemann sewn a ons 2 sn hsformulation, this Using . rae.W prove We urfaces. nrlsw Riemann sewn eneral txoeao algebra operator rtex sdrdo Riemann on nsidered ucin lo us allow functions n n sal ostudy to able is One xoeao algebras operator ex osbyfrstudies for possibly reainfunctions orrelation prtralgebra operator x bawt formal with ebra oa coordinates local rsuiso the of studies or o func- ion tosapplied ethods sqecsfor nsequences associated f unctions; 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) associated with Möbius 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. Thus we assume that there exists non-degenerate bilinear form on V [4]. On the right hand side the form is present in the deﬁnition of the dual state u with respect to such form. FOLIATION OF A SPACE ASSOCIATED TO VERTEX OPERATOR ALGEBRA 5

There exists an algebraic procedure [23] relating n-point correlation functions to a sum of (n − 1)-point functions for vertex operator algebras on the torus. In [23] we ﬁnd that the genus one n-point correlation functions obey (1) ZV (v1,z1; ... ; vn,zn; τ) L(0) L(0) L(0)−c/24 = TrV (v1)wt(v1)−1Y (qz2 v2, qz2 ) ...Y (qzn vn, qzn )q n (1) + P1+j (z1 − zk, τ)ZV (v2,z2; ... ; v1[j]vk,zk; ... ; vn,zn; τ), Xk=2 Xj≥0 where Pl(z, τ) are Weierstrass functions [8], and square bracket on the right hand side denotes the deformed mode for vertex operator algebra on the torus [23]. A generalization [17] of the recursion procedure of [23] allows us to reduce a genus g = g1 + g2 n-point correlation function to the genus g zero-point correlation function (the partition function) on any higher genus Riemann surface formed from two lower genus g1, g2 Riemann surfaces in the ǫ sewing procedure [22]. Using explicit results of [17,18] for the Heisenberg vertex operator algebra in the Schottky formation of a genus g Riemann surface, we conjecture the following general form of the reduction formula for a general vertex operator algebra considered on a genus g Riemann surface obtained as a results of sewing of two lower genua g1 and g2 Riemann surfaces:

(g) (g) ZV v1,z1; ... ; vn,zn; T P(g) (g1) (g2) O(g) = k,n A1 , A2 · fV,k,n(A) · k,n, (3.3) Xk≥0

(g) (g1) (g1) where T = ǫ1,ǫ2,z1,z2, Ω , Ω , are parameters of the genus g n-point corre- P(g) (g1) (g2) lation function a sewn genus g Riemann surface, k,n A1 , A2 is a polynomial (g1) (g2) of special inﬁnite matrices A1 , A2 , [10, 17] associated to the period matrices (g1) (g2 ) P(g) O(g) for Riemann surfaces S and S , k,n, k,n are certain generalizations of clas- sical elliptic functions on higher genus Riemann surfaces depending on arguments (g1) (g2) (v1,z1; ... ; vn,zn), and fV,k,n is a function of the matrix A = I − A1 A2 . Note that, using results of [15, 18] for the multiple-sewn sphere case, and as it is shown in [20], the genus one case trace formulas can be obtain from the higher genus formulas (in particular, from genus two) in the complex parameter ρ-sewing procedure [22].

4. Construction of a foliation of the space of correlation functions over Riemann surfaces for a vertex operator algebra The construction of a vertex operator algebra n-point function of arbitrary genus gives us a hint how to deﬁne a special-type foliation of a space related to a vertex operator with the formal parameter associated to coordinates on a Riemann surface (formed from two lower genus surfaces) of genus g by means of vertex operator algebra correlation functions. 6 A. ZUEVSKY

A p-dimensional foliation F of an n-dimensional manifold M is a covering by a system of domains {Ui} of M together with maps n φi : Ui → C , such that for overlapping pairs Ui, Uj the transition functions n n ϕij : C → C , deﬁned by −1 ϕij = φj φi , take the form 1 2 ϕij (x, y)= ϕij (x), ϕij (x, y) , where x denotes the ﬁrst n − p coordinates, and y denotes the last p co-ordinates. That is, 1 Cn−p Cn−p ϕij : → , 2 Cp Cp ϕij : → . In this paper, we would like to foliate the space

⊗n (g) Mn = V × S , where V is a vertex operator algebra, and S(g) is a Riemann surface of genus g. Proposition 1. The correlation functions (3.3) determine a foliation on the space

M = n≥0 Mn. L Proof. We consider a system of charts {Um}, m ∈ N, covering the Riemann surface ⊗n part of the space Mn together with a filtration of the space V with respect to zi, i = 1,...,n belonging to {Um}. This defined an infinite-dimensional analog of (g1) charts for M. Suppose certain points xi, 1 ≤ i ≤ p ≤ n are situated on S -part (g2 ) (g) and other xi, p +1 ≤ i ≤ n are on S -part of the Riemann surface S . On the (g1 ) Riemann surface S of genus g1, let us choose a particular point xi, 1 ≤ i ≤ p ≤ n with associated local coordinate zi around xi which belongs to the domain Ui of the (g1) system {Um}. For another point xj , 1 ≤ j 6= i ≤ p on S -part we associate a domain Uj intersecting with Ui. It can be always done since we can move points xi, 1 ≤ i ≤ p around S(g1)-part of the Riemann surface S(g). (g1) Now let us compute a p ≤ n-point V -correlation functions Z (v1,z1; ... ; vp,zp) by means of the recursion procedure described above reducing p-point correlation (g1) functions to a one-point correlation function Z (vi,zi) associated to our chosen point xi in the domain Ui. Due to properties of vertex operators, the vertex algebra elements (v1,...,vp) can be chosen so that the expansion of the correlation func- (g1) tion Z (v1,z1; ... ; vp,zp) has a dimension p polynomial nature, [3]. The procedure (g1) of computation of Z (v1,z1; ... ; vp,zp) by the reduction to one-point correlation (g1) function Z (vi,zi) defines a map

p ⊗ (g1) p φi : V × S → C . FOLIATION OF A SPACE ASSOCIATED TO VERTEX OPERATOR ALGEBRA 7

(g1) The result of computation of Z (v1,z1; ... ; vp,zp) can re-written to reduce to an- (g1) other one-point function Z (vj ,zj ) associated to another coordinate zj around a point xj in the domain Uj. Thus, we can deﬁne the inverse map p −1 Cp ⊗ (g1) φj : → V × S .

(g1) For any set x of pairs of non-coinciding points among x1,...,xp on S we then deﬁne the function for the intersecting domains Ui and Uj on the Riemann surface S(g1) (g1) −1 ϕij (x)= φiφj (x). Exactly similar procedure is then performed for any set of y non-coinciding points (g2) (g2) among xp+1,...,xn on the Riemann surface S of genus g2 to deﬁne ϕij (y). Then the transition function is given by the map n n ϕij : C → C ,

(g1) (g2 ) ϕij (x, y)= ϕij (x), ϕij (y) . As a result, we have deﬁned a foliation of the space Mn by means of correlation functions for corresponding vertex operator algebras with the formal parameter associated to a coordinate on a Riemann surface. The total space M = n≥0 Mn is foliated similar to Mn. L The general situation is more complicated. Namely, we have to vary n, p, the set of vertex operator elements {vi}, i =1,...,n, and g1, g2 with g = g1 + g2 (note also, that another parameter can be provided by the type of grading of the vertex operator algebra).

5. Further directions The recursion procedure [23] plays a fundamental role in the theory of correlation functions for vertex operator algebras. It provides relations between n- and n−1-point correlation functions. In our foliation picture, the recursion procedure brings about relations among leaves of foliation. We work in the approach of formulation and computation of cohomologies of vertex operator algebras. Taking into account the above definitions and construction, we would like to develop a theory of characteristic classes ∞ ⊗n (g) for vertex operator algebras, and, in particular, for the space M = n=0 V × S . This can have a relation with Losik’s work [9] proposing a new framewoL rk for singular spaces and new kind of characteristic classes. Losik defines a smooth structure on the leaf space M/F of a foliation F of codimension n on a smooth manifold M that allows to apply to M/F the same techniques as to smooth manifolds. Losik defined the characteristic classes for a foliation as elements of the cohomology of certain bundles over the leaf space M/F. Similar to Losik’s theory, we use certain bundles (of correlation functions) over a foliated space. Relations to [1] on Reeb foliations modified Godbillon-Vey class and can be also considered. On the other hand, we would like to apply methodology of vertex algebras in order to complete Losik’s theory of characteristic classes [9]. One can mention a possibility to derive differential equations for correlation functions on separate leaves 8 A. ZUEVSKY of foliation. Such equations are derived for various genera and can be used in frames of Vinogradov theory [21]. The structure of foliation (in our sense) can be also studied from the automorphic function theory point of view. Since on separate leaves one proves automorphic properties of correlation functions, on can think about ”global” automorphic properties for the whole foliation. Since we consider multipoint correlation function for vertex algebras on Riemann surfaces of arbitrary genus, there exists also a connection to Krichever-Novikov type algebras [5, 6, 6, 11–13]. These are generalizations of the Witt, Virasoro, affine Lie algebras. In particular, one is able to use the structure of Krichever-Novikov type algebras (as higher-genus generalizations of algebras related to vertex algebras) to study foliated spaces of correlation functions introduced in this paper. The construction of n-point functions on genus g sewn Riemann surfaces can be used for introduction and computation of cohomology of Krichever-Novikov type algebras in appropriate setup. We plan to fully shed light on this subject in a future paper.

Acknowledgments The author would also like to thank a refereee provided comments to the ﬁrst version of the manuscript.

References

[1] Bazaikin, Y.V., Galaev, A.S., Gumenyuk, P. Non-diffeomorphic Reeb foliations and modified Godbillon-Vey class. Math. Z. (2021). https://doi.org/10.1007/s00209-021-02828-1 [2] Frenkel, E.; Ben-Zvi, D. Vertex algebras and algebraic curves. Mathematical Surveys and Mono- graphs, 88. American Mathematical Society, Providence, RI, 2001. xii+348 pp. [3] Francesco, Ph., Mathieu, P. and Senechal, D. Conformal field theory. Graduate Texts in Contemporary Physics. 1997. [4] Kac, V.: Vertex algebras for beginners, University Lecture Series 10, AMS, Providence 1998. [5] Krichever I.M., S. P. Novikov, Algebras of Virasoro type, Riemann surfaces and the structures of soliton theory (Russian), Funktsional. Anal, i Prilozhen. 21 (1987), no. 2, 46-63. [6] Krichever I.M., S. P. Novikov, Algebras of Virasoro type, Riemann surfaces and strings in Minkowski space (Russian), Funktsional. Anal, i Prilozhen. 21 (1987), no. 4, 47-61, 96, [7] Krichever I.M., S. P. Novikov, Algebras of Virasoro type, the energy-momentum tensor, and operator expansions on Riemann surfaces (Russian), Funktsional. Anal, i Prilozhen. 23 (1989), no. 1, 24-40, English transl., Funct. Anal. Appl. 23 (1989), no. 1, 19-33. [8] Lang, S.: Elliptic functions. With an appendix by J. Tate. Second edition. Graduate Texts in Mathematics, 112. New York: Springer-Verlag, 1987 [9] Losik, M. V.: Orbit spaces and leaf spaces of foliations as generalized manifolds, arXiv:1501.04993. [10] G. Mason, M. P. Tuite, On genus two Riemann surfaces formed from sewn tori, Communications in Mathematical Physics, v 270, 587 (2007). [11] Schlichenmaier, M. Krichever-Novikov type algebras. An introduction. Lie algebras, Lie super- algebras, vertex algebras and related topics, 181-220, Proc. Sympos. Pure Math., 92, Amer. Math. Soc., Providence, RI, 2016. [12] Schlichenmaier, M., Krichever-Novikov algebras for more than two points, Lett. Math. Phys. 19(1990), 15-165. [13] Krichever-Novikov algebras for more than two points: explicit generators, Lett. Math. Phys. 19(1990), 327-336. [14] Scheithauer, N. Vertex algebras, Lie algebras and superstrings, J.Alg. 200 (1998) 363–403. FOLIATION OF A SPACE ASSOCIATED TO VERTEX OPERATOR ALGEBRA 9

[15] M. P. Tuite, The Heisenberg generalized vertex operator algebra on a Riemann surface, arXiv:2006.01255. [16] M. P. Tuite, T. Gilroy, Genus two Zhu theory for vertex operator algebras, arXiv:1511.07664. [17] M. P. Tuite, M. Welby, General genus Zhu recursion for vertex operator algebras, arXiv:1911.06596 [18] M.P. Tuite, A. Zuevsky, The bosonic vertex operator algebra on a genus g Riemann surface RIMS Kokyuroko 1756 81-93 (2011). [19] M.P. Tuite, A. Zuevsky, Genus two partition and correlation functions for fermionic vertex operator sueralgebras I, Commun. Math. Phys. 306: 419–447, 2011. [20] M.P. Tuite, A. Zuevsky, Genus two partition and correlation functions for fermionic vertex operator sueralgebras II, arXiv:1308.2441. [21] Vinogradov, A. M. Cohomological analysis of partial diﬀerential equations and secondary cal- culus. Translations of Mathematical Monographs, 204. American Mathematical Society, Provi- dence, RI, 2001. xvi+247. [22] Yamada, A.: Precise variational formulas for abelian diﬀerentials. Kodai Math.J. 3 (1980) 114–143. [23] Y. Zhu: Modular invariance of characters of vertex operator algebras. J. Amer. Math. Soc. 9(1), 237–302.

Institute of Mathemtics, Czech Academy of Sciences, Zitna 25, Prague, Czech Republic Email address: [email protected]