TOPOLOGICAL FIELD THEORIES, STRING BACKGROUNDS AND HOMOTOPY ALGEBRAS 4 A. A. VORONOV∗ 9 Department of Mathematics 9 University of Pennsylvania 1 Philadelphia, PA 19104-6395 n USA∗∗ a J December 22, 1993 6 Abstract.Stringbackgroundsaredescribedaspurelygeometricobjectsrelatedtomoduli 1 spaces of Riemann surfaces, in the spirit of Segal’s definition of a conformal field theory. v Relations with conformal field theory,topological field theory and topological gravity are 3 studied. For each field theory, an algebraic counterpart, the (homotopy) algebra satisfied 2 by thetree level correlators, is constructed. 0 1 Keywords:Frobeniusalgebra–HomotopyLiealgebra–Homotopycommutativealgebra 0 – Gravity algebra – Moduli space – Topological field theory – Conformal field theory – 4 String theory – Stringbackground – Topological gravity 9 / h 1. Introduction t - p Theusualwayofdescribingastringbackgroundassomeconstructionontop e h of a conformal field theory involving the Virasoro operators, the antighost : fields and the BRST operator appears too eclectic to be seriously accepted v i by the general mathematical public. Here we make an attempt to include X stringtheoryintheframeworkofgeometric/topological fieldtheoriessuchas r a conformal field theory and topological field theory. Basically, we describe all two-dimensionalfieldtheoriesasvariationsonthethemeofSegal’sconformal field theories [9]. Our definition is in some sense dual to Segal’s definition of a string background, also known as a topological conformal field theory, via differential forms and operator formalism, see Segal [10] and Getzler [1]. In this paper, each geometric field theory is followed by a leitmotif, the structure of an algebra built on the state space of the theory. Whereas it is commonlyknownthattwo-dimensionalquantumfieldtheoriescomprisevery interestinggeometrical structures,related algebraicstructureshaveemerged ∗ Research supported in part byNSFgrant DMS-9108269.A03 ∗∗ AffiliatedtoDepartmentofMathematics,PrincetonUniversity,Princeton,NJ08544- 1000, USA 2 A.A.VORONOV very recently and are still experiencing a very active periodof growth. A list of references, perhaps, already outdated, can be found in the recent paper [5]. 2. Topological Field Theory and Frobenius Algebras Note. The field theories we are going to consider here will all have the total central charge zero. The general case can be done by involving the determi- nant line bundles over the moduli spaces. A topological field theory (TFT) is a complex vector space V, called the state space, together with a correspondence   m n 7−→ |Σi: V⊗m → V⊗n (1)   An oriented surface Σ A linear operator |Σi boundingm+n circles Here a surfaceis not necessarily connected. Its boundarycircles are enumer- ated and parameterized. The orientation of the first m ≥ 0 circles, called inputs, is opposite to the orientation of Σ and the orientation of the remain- ing n ≥ 0 circles, called outputs, is compatible with the orientation induced from the surface. The linear operator |Σi is called the state corresponding to the surface Σ. This correspondence should satisfy the following axioms. 1. Topological invariance: The linear mapping |Σi is invariant under orientation preserving diffeomorphisms of the surface Σ. 2. Permutation equivariance: The correspondence Σ 7→ |Σi commutes with the action of the symmetric groups S and S on surfaces and m n linear mappings by permutations of inputs and outputs. 3. Factorization property: Sewing along the parameterizations of the boundary corresponds to composing: TOPOLOGICALFIELDTHEORIES,STRINGBACKGROUNDS,ETC. 3 7−→ V⊗m → V⊗n → V⊗k The sewing of the outputs of The composition of the a surface with inputs of an- corresponding linear op- other surface erators 4. Normalization: 7−→ id :V → V A cylinder The identity operator These data and axioms can be formulated equivalently using functors. Withinthisapproach,aTFTisamultiplicative functorfroma“topological” tensor category Segal to a “linear” tensor category Hilbert. An object of the category Segal is a diffeomorphism class of parameterized one-dimensional compact manifolds, i.e., disjoint unions of circles. A morphism between two collections of circles is a diffeomorphism class of oriented surfaces bounding the circles, so that the induced orientation is against the parameterizations on the first collection and compatible with that on the second collection of circles. The identity morphism of an object is the cylinder over it. The operation of disjoint union of collections of circles introduces the structure of a tensor category on Segal. The other category Hilbert is the category of complex vector spaces (Hilbert in real examples), not necessarily finite dimensional, with the usual tensor product. Then the space V is the vector space corresponding to the single circle and it is easily checked that the functoriality plays the role of the factorization property and that the two definitions are equivalent. Any oriented surface can be cut into pants, caps and cylinders: 7−→ Thus, oriented surfaces have the following generators with respect the sewingoperation.Andrespectively,thespaceV isprovidedwithanalgebraic 4 A.A.VORONOV structure generated by the operations below with respect to composition of linear mappings. 7−→ V ⊗V → V 7−→ C → V ⊗V 7−→ V → V 7−→ V → C 7−→ C → V THEOREM 1 (Folklore). A TFT is equivalent to a Frobenius algebra, i.e., a commutative algebra V with a unity and a nondegenerate symmetric bilinear form h,i :V ⊗V → C which is invariant with respect to the multiplication: hab,ci = ha,bci and has an “adjoint” C → V ⊗V. An “adjoint” to a mapping φ : V ⊗V → C is a mapping ψ : C → V ⊗V, id⊗ψ φ12⊗id ψ⊗id such that the compositions V −→ V ⊗V ⊗V −→ V and V −→ V ⊗V ⊗ id⊗φ23 V −→ V are identities. When the space V is finite dimensional, an inner product establishes an isomorphism V → V∗, and an adjoint mapping gives a mapping V∗ → V, which is nothing but its inverse. Thus, in the finite dimensional case, a Frobenius algebra is just an algebra with an invaraint nondegenerate inner product. The theorem follows from the remark above about decomposing a surface into pants, caps and cylinders and the obvious fact that the symmetric form h,i in a Frobenius algebra V can be obtained from a linear functional f :V → C as ha,bi = f(ab). An important substructure is observed for a TFT at the tree level, when we restrict our attention to surfaces of genus zero and with exactly one TOPOLOGICALFIELDTHEORIES,STRINGBACKGROUNDS,ETC. 5 output:  n 7−→ V⊗n → V  7−→ C → V Topological invariance, permutation equivariance, the factorization and the normalization axioms make sense for such surfaces and are assumed. The following fact is worth mentioning, because we are aiming to study similar algebraic structures occurring in string theory at the tree level. COROLLARY 2. A TFT at the tree level is equivalent to a commutative algebra V with a unity. 3. Conformal Field Theory A conformal field theory (CFT) is a device very similar to a TFT, except that 1. the correspondence (1) is defined on Riemann surfaces bounding holo- morphic disks and the state |Σi depends smoothly on the Riemann sur- face Σ, 2. topological invariance is replaced by conformal invariance, 3. when two Riemann surfaces are sewn, the result is provided with a unique complex structure, and 4. normalization is slightly different: 7−→ id : H → H A cylinder of zero length The identity operator In other words, a CFT is a smooth mapping P → Hom(H⊗m,H⊗n), (2) m+n 6 A.A.VORONOV   m n 7−→ |Σi: H⊗m → H⊗n,   A point Σ in the A linear operator |Σi moduli space P m+n whereP is themodulispace of Riemann surfaces (one-dimensional com- m+n plex compact manifolds) boundingm negatively oriented holomorphic disks and n positively oriented disks. The surfaces can have arbitrary genera, the disksare holomorphicmappingsfrom theunitdisk to aclosed Riemann sur- face and they are enumerated. The mapping (2) must be equivariant with respect to permutations, transform sewing of Riemann surfaces into com- position of the corresponding linear operators and must be normalized as above. There is an evident reformulation of the CFT data as a functor from a suitable category Segal to the category Hilbert analogous to the one for TFT’s. 4. String Theory and Homotopy Lie Algebras 4.1. String Backgrounds Let H be a graded vector space with a differential Q, Q2 = 0, i.e., H be a complex. A string background is a correspondence C P → Hom(H⊗m,H⊗n), (3) • m+n 7−→ |Ci :H⊗m → H⊗n, Chains C in P Linear operators |Ci m+n TOPOLOGICALFIELDTHEORIES,STRINGBACKGROUNDS,ETC. 7 which satisfies the axioms below. [On the figure, the surface is nothing but a pair of pants (so m = 2, n = 1) and the chain is just a circle. The pants moving along the circle in the moduli space sweep out a “surface of revolution”,whichIattemptedtosketchabove.]Bychainsherewemeanthe (complex) vector space generated by oriented (sectionally) smooth chains. 1. Smoothness: The mapping (3) is smooth. 2. Equivariance: The mapping (3) is equivariant with respect to permu- tations of inputs and outputs. 3. Factorization: Thesewing of outputs of a chain with inputs of another chain (namely, outputs of each Riemann surface in the first chain are sewn with inputs of each Riemann surface in the second chain, each time producing a new Riemann surface) transforms under (3) into the composition of the corresponding linear operators. 4. Homogeneity and Q-∂-Invariance: The mapping (3) is a morphism of complexes. That means that it maps a chain of dimension k to a linear mapping of degree −k (with respect to the natural grading on the Hom) and that the boundary of a chain in P transforms into the m+n differential of the corresponding mapping, |∂Ci= Q|Ci, where Q acts on each of the m+n components H of Hom, as usual. 5. Normalization:Thepoint{Riemannspherewithtwounitdisksaround 0 and ∞ cut out} ∈ P maps to the identity operator id :H → H. m+n Thiscorrespondencecanalsobeaxiomatizedasafunctor,likeinthecases of TFT and CFT. The corresponding category Segal will still have disjoint unions of circles as objects, but its morphisms will be chains in the mod- uli spaces. In the category Hilbert, one has to consider graded spaces with differentials (i.e., complexes), but still all linear mappings as morphisms. The Virasoro semigroup of cylinders (including the group of diffeomor- phisms of the circle, represented by cylinders of zero width) acts on H via thedegree0statesexptT(v) = |exptvicorrespondingtocylinders,regarded as points in P : 1+1 7−→ exptT(v) : H → H, (4) exptv where v is the generating complex vector field on the circle. The so-called antighost operators b(v) on H can also be easily identified in our picture. They are the derivatives d b(v) = B(tv)| (5) t=0 dt of the operators B(tv) of degree −1 obtained when the same cylinder cor- responding to v is regarded as a one-chain in P . At time t, the cylinder 1+1 8 A.A.VORONOV exp(tv) is a point in P . When t changes, these points sweep out a path 1+1 in P . Note that 1+1 [T(v ),T(v )] = T([v ,v ]), 1 2 1 2 becausetheoperatorsexptT(v)definearepresentationoftheVirasorosemi- group, and {b(v ),b(v )} = 0, 1 2 because the two-chains exp(sv )× exp(tv ) and exp(tv )× exp(sv ) differ 1 2 2 1 only by orientation. In particular, b2(v) = 0. Moreover, {Q,b(v)} = T(v), because the boundary of the cylinder exp(tv) viewed as a one-chain is equal to the same cylinder viewed as apoint minus thetrivial zero-width cylinder. Stringtheories arealsoreferredtoastopological, becauseofthefollowing fact. THEOREM 3. The cohomology of the state space H of a string background with respect to the differential Q forms a TFT. Thus, the cohomology of H has a natural structure of a Frobenius algebra. Proof. Two Riemann surfaces Σ and Σ which are diffeomorphic can be 1 2 connected by a smooth path C in the moduli space. Hence, for the corre- sponding states we have |Σ i−|Σ i = |∂Ci = Q|Ci, 2 1 which means that their Q-cohomology classes are equal. 2 4.2. Higher Brackets The state |Ci is an operator from Hm to Hn, which for n = 1 may be thought of as an m-ary operation on the space H. By the factorization ax- iom,theoperation ofsewingofchainsC inthemodulispacescorrespondsto compositions of the corresponding operations on the space H. Respectively, any relation (involving compositions and boundaries) between chains in the moduli spaces produces an identity (involving compositions and the differ- ential Q) for the corresponding operations on H. At the tree level, when we consider Riemann surfaces of genus 0 only, this algebraic structure on H is rather tamable. This is because the topology of the finite dimensional moduli spaces M of isomorphism classes of m+1 punctured Riemann 0,m+1 spheres takes over the situation. Consider the following brackets: [x ,...,x ] = “|M i”(x ,...x ), m ≥ 2, (6) 1 m 0,m+1 1 m TOPOLOGICALFIELDTHEORIES,STRINGBACKGROUNDS,ETC. 9 where x ,...,x ∈ H are substituted on the right-hand side as arguments 1 m of the Hom(Hm,H), where the state |M i lives. The quotes are due to 0,m+1 the fact that the space M is not really a chain in P : there is no 0,m+1 m+1 natural mappings from M to P . A standard escape is to impose 0,m+1 m+1 these mappings as extra part of data. To preserve nice properties, this is achieved in the following two steps. Step 1. Push the correspondence C P → Hom(Hm,H) down to a map- • m+1 pingC P′ → Hom((Hrel)m,Hrel)fromchainsonthequotientspaceP′ • m+1 m+1 of P by rigid rotations of the holomorphic disks to the space of multilin- m+1 ear operators on Hrel. The latter is the subspace Hrel of vectors in H which are rotation-invariant, i.e., stable under the operators exp(tT(v)) of (4) and annihilated by the operators B(tv) of (5) corresponding to rigid rotations v ∈ S1. The pushdown is performed by pulling a chain C in P′ back m+1 to a chain C of the same dimension in P , restricting the operator |Ci m+1 to (Hrel)m aend projecting the value of the operator |Ci onto Hrel via thee mapping h 7→ b(∂/∂θ)h , where θ is the phase parameeter on the circle S1 0 and h is the rotation-invariant part of h (which exists provided the action 0 of S1 on H is diagonalizable). Step 2. Map the finite dimensional moduli spaces M to the infi- 0,m+1 nite dimensional quotient spaces P′ , so that gluing Riemann spheres in m+1 M ’satpuncturescorrespondstosewingofRiemannspheresinP′ ’s. 0,m+1 m+1 Sewing in P′ ’s can only be performed provided at least relative phases m+1 at sewn disks are given. The corresponding gluing operation should also be of this kind. Thus, the gluing operation takes us actually out of the spaces M to certain real compactifications of them, [5]. Such map- 0,m+1 pings M → P′ exist. Zwiebach’s string vertices [11] make up an 0,m+1 m+1 example of those. Here we thereby allow certain freedom of their choice. These additional data have been called a closed string-field theory in [5] after Zwiebach gave this title to the choice of his string vertices. After these modifications, we obtain brackets [x ,...,x ] defined on Hrel. 1 m THEOREM 4. These bracketsdefine the structure of ahomotopy Liealgebra (see next section) on the space Hrel. This result was obtained by Zwiebach in [11]. A mathematically rigorous proofof this theorem withthe useof operads was given in [5]. Thisalgebraic structuregeneralizesthetrivialoneofCorollary2tothecaseofstringtheory. 4.3. Homotopy Lie Algebras A homotopy Lie algebra is a graded vector space H, together with a differ- ential Q, Q2 = 0, of degree 1 and multilinear graded commutative brackets [x ,...,x ] of degree 3−2m for m ≥ 2 and x ,...,x ∈ H, satisfying the 1 m 1 m 10 A.A.VORONOV identities m Q[x ,...,x ]+ ǫ(i)[x ,...,Qx ,...,x ] 1 m 1 i m X i=1 = ǫ(σ)[[x ,...,x ],x ,...,x ], X X i1 ik j1 jl−1 k+l=m+1 unshufflesσ: k,l≥2 {1,2,...,m}=I1∪I2, I1={i1,...,ik}, I2={j1,...,jl−1} where ǫ(i) = (−1)|x1|+...+|xi−1| is the sign picked up by taking Q through x , 1 ...,x , |x| denoting the degree of x∈ H, ǫ(σ) is the sign picked up by the i−1 elements x passing through the x ’s during the unshuffle of x ,...,x , as i j 1 m usual in graded algebra. Note that for m=2, we have Q[x ,x ,x ] + (±[Qx ,x ,x ]±[x ,Qx ,x ]±[x ,x ,Qx ]) 1 2 3 1 2 3 1 2 3 1 2 3 = [[x ,x ],x ]±[[x ,x ],x ]±[[x ,x ],x ], 1 2 3 1 3 2 2 3 1 which means that the graded Jacobi identity is satisfied up to a null-homo- topy, the Q-exact term on the left-hand side. 4.4. Holomorphic String Backgrounds and Homotopy Commutative Algebras According to general ideology, cf. Kontsevich [7] and Ginzburg-Kapranov [4], there are three principal types of homotopy algebras: homotopy Lie, homotopy commutative and homotopy associative. The first two types are dual in certain sense, the third one is self-dual. It is remarkable that this duality is implemented in string theory by passing from left-right movers’ case to the chiral case, i.e., roughly speaking, from smooth “operator-valued differential forms” |Ci on the moduli spaces to holomorphic ones. More precisely, recall that the operators T(v) and b(v) of (4) and (5) are definedfor tangent vectors v to the Virasoro semigroup. Let us extend those operators to the complex tangent space to the Virasoro semigroup by C- linearity. Since the Virasoro semigroup is a complex manifold, the complex tangent space splits into the holomorphic and antiholomorphic parts Vir and Vir. Call a string background chiral, if T(v) = 0 and b(v) = 0 for v ∈ Vir. Note that the first equation implies that the correspondence C 7→ |Ci determines a holomorphic mapping C P → Hom(Hm,Hn). • m+n If we consider m−2-cycles (i.e., half-dimensional cycles relative to the boundary) in M instead of the fundamental cycle to define m-ary 0,m+1 products as in (6), the operad approach of [5] will lead to the structure of a homotopy commutative algebra [3, 4]. This is the matter of the forthcoming paper [6].