aa r X i v : . [ m a t h . QA ] D ec Geometric Vertex Algebras
Daniel Bruegmann ∗ A theorem of Huang [5] establishes the equivalence of vertex operator algebrasand geometric vertex operator algebras. We give a self-contained account ofthis theorem in the simplified setting of vertex algebras and geometric vertexalgebras, that is, without the infinitesimal conformal symmetries given by theVirasoro algebra.Both geometric vertex algebras and Z -graded vertex algebras have an un-derlying Z -graded vector space V over C . In a geometric vertex algebra, themultiplication maps µ take elements a , . . . , a n ∈ V placed at pairwise distinctpoint z , . . . , z n ∈ C , and the product µ ( a )( z ) is an element of V := Q k ∈ Z V k which contains V as a subspace. Since the result of the multiplication is aninfinite sequence of elements of V instead of a single element of V , associativityis not formulated in the usual way but instead involves an infinite sum. Further-more, µ ( a )( z ) holomorphically depends on z , and behaves meromorphically asthe z i come closer to each other. By contrast, the corresponding vertex algebracontains the Laurent expansions of the functions µ ( a, z, b, ∈ V of z ∈ C \ { } for all a, b ∈ V , whose coefficients turn out be elements of V . Theorem.
The set of Z -graded vertex algebra structures on a fixed Z -gradedvector space is in natural bijection with the set of geometric vertex algebra struc-tures. Related Work.
The meaning of the above theorem is very close to that ofTheorem 2.12 in Runkel’s lecture notes [6]. The holomorphic integral scalecovariant field theories of [6] have essentially the same data as geometric vertexalgebras. The axioms only differ in the kind of convergence in the infinitesum in the associativity property. The definition of geometric vertex algebrasin this article uses locally normal convergence in its associativity axiom, asopposed to pointwise convergence as in [6], seen to imply pointwise absoluteconvergence there. We show that the geometric vertex algebra constructedfrom a Z -graded vertex algebra always has locally normal convergence in itsassociativity property.The analogous theorem of Huang is Theorem 5.4.5 in [5]. Costello andGwilliam construct Z -graded vertex algebras from certain holomorphic factor-ization algebras on C . Our exposition of obtaining a vertex algebra from ageometric vertex algebra is modeled on their work. Acknowledgments.
The author would like to thank his advisors Peter Teich-ner and Andr´e Henriques for their support and suggestions. Thanks to BertramArnold, Achim Krause and Katrin Wendland for discussions. Thanks to Andr´e ∗ [email protected], Dec 17, 2020 Contents
Unless stated otherwise, all vector spaces are over the field C of complex num-bers. Geometric vertex algebras have an n -ary operation parametrized by z ∈ C n \ ∆ where∆ = ∆ n = { ( z , . . . , z n ) ∈ C n | z i = z j for some i, j ∈ { , . . . , j } with i = j } .for every natural number n ≥
0. We write ∆ instead of ∆ n when n is apparentfrom the context. Before giving the definition of a geometric vertex algebra, weexplain the relevant notions of holomorphicity and convergence of infinite sumsof holomorphic functions. Both are based on considering finite-dimensionalsubspaces of the V k for all k ∈ Z , a reflection of the rather algebraic nature ofvertex algebras. Definition 1.1.
Let U ⊆ C n and X be a vector space. A map f : U → X is holomorphic if f is locally a holomorphic function with values in a finite-dimensional subspace X . This means that every point p ∈ U has an open neigh-borhood V together with a finite-dimensional subspace Y ⊆ X with f ( V ) ⊆ Y and f | V : V → Y holomorphic. If V is a Z -graded vector space, then O ( U ; V )denotes the vector space of V -valued functions on U each of whose componentsis holomorphic.The condition that f | V be holomorphic is independent of the choice of Y . Itcan be shown that f ( U ) ⊆ Y for all Y as above if U is connected. In particular,a holomorphic function on a connected set like C n \ ∆ n globally takes values insome finite-dimensional subspace.For the associativity axiom of a geometric vertex algebra, we recall that aseries P i ∈ I f i of holomorphic functions on an open U ⊆ C n with values in afinite-dimensional normed space is called normal if P i ∈ I || f i || < ∞ where || || is the supremum norm. Such a series is called locally normal if every point inthe domain has an open neighborhood V s. t. P i ∈ I f i | V is normal, equivalentlyif P i ∈ I || f i || K < ∞ for each compact subset K of the domain. The notionsof normal and locally normal series are the same for every norm. Therefore, itmakes sense to say that a series of holomorphic functions with values in a vectorspace X is locally normal: Definition 1.2.
Let U ⊆ C n and X be a vector space. A series P i ∈ I f i of holomorphic X -valued functions on an open U ⊆ C n locally normal if, for2very p ∈ U , there is an open neighborhood V of p and a finite-dimensionalsubspace Y s. t. f ( V ) ⊆ Y and P i || f i | V || < ∞ , where the supremum norm isdefined with respect to a norm on Y . Definition 1.3. A geometric vertex algebra consists of: • A Z -graded vector space V = L k ∈ Z V k over C . • Linear maps µ : V ⊗ n → O ( C n \ ∆; V ) for n ≥ V = Q k ∈ Z V k . Forall n , we write µ ( a )( z ) = µ ( a, z ) = µ ( a , z , . . . , a n , z n )for the value of the function corresponding to a = a ⊗ . . . ⊗ a n ∈ V ⊗ n at z ∈ C n \ ∆.The axioms of a geometric vertex algebra are: • (permutation invariance) µ ( a σ , z σ ) = µ ( a, z )for a ∈ V n , z ∈ C n \ ∆ and every permutation σ ∈ Σ n . • (insertion at zero) µ ( a,
0) = a for all a ∈ V , where a is viewed as element of V via the embedding V = M n V n ֒ → Y n V n = V . • (associativity) For k ∈ Z let p k : V → V k denote the projection. For all a , . . . , a m ∈ V , b , . . . , b n ∈ V and z ∈ C m +1 \ ∆, w ∈ C n \ ∆ with max i | w i | < min ≤ j ≤ m | z j − z m +1 | we demandthat the sum X k ∈ Z µ ( a , z , . . . , a m , z m , p k µ ( b , w , . . . , b n , w n ) , z m +1 ) (1)converge in the sense that, for each l ∈ Z , the components of the sum-mands form a locally normal sum of V l -valued functions on C n \ ∆. Thislimit defines the left hand side of the equality µ ( a , z , . . . , a m , z m , µ ( b , w , . . . , b n , w n ) , z m +1 )= µ ( a , z , . . . , a m , z m , b , w + z m +1 , . . . , b n , w n + z m +1 ) (2)which we require to hold. Here m ≥ n ≥ • ( C × -equivariance) For all z ∈ C × , a , . . . a n ∈ V and w ∈ C n \ ∆ z.µ ( a , w , . . . , a n , w n ) = µ ( z.a , zw , . . . , z.a n , zw n )where z ∈ C × acts on V and V via multiplication by z l on V l for all l ∈ Z .3 (meromorphicity) For all a, b ∈ V , there exists an N s. t. the func-tion z N µ ( a, z, b,
0) of z ∈ C \ { } extends holomorphically to z ∈ C .In the associativity axiom of a geometric vertex algebra, the condition thatmax i | w i | < min ≤ j ≤ m | z j − z m +1 | is equivalent to saying that all w j + z m +1 , j = 1 , . . . , n , are contained in thelargest open ball around z m +1 not containing any of the z i for i = 1 , . . . , m .By permutation invariance the multiplication µ only depends on the set { ( a , z ) , . . . , ( a m , z m ) } for a , . . . , a m ∈ V and z ∈ C m \ ∆. The image µ ( ∅ ) of 1 ∈ C under themultiplication map C = V ⊗ → O (pt; V ) ∼ = V for n = 0 is called the vacuum vector | i of V , or unit. It is actually an elementof V ⊆ V because it is invariant under the action of C × . The case n = 0 of theassociativity axiom implies that µ ( a , z , . . . , a m , z m , | i , z m +1 ) = µ ( a , z , . . . , a m , z m )for a , . . . , a m ∈ V and z ∈ C m +1 \ ∆.The next proposition says that the action of C × on V extends to an actionof the group C × ⋉ C of affine transformations of C on the subspace V bb = { x ∈ V | ∃ C ∈ Z ∀ k < C : x k = 0 } of bounded below vectors. More generally, when constructing geometric vertexalgebras from vertex algebras, we will show that µ ( a , z , . . . , a m , z m ) vanishesin sufficiently low degree, i. e., is an element of V bb . We do not need the nextproposition to obtain a vertex algebra; rather it explains how the m = 0 caseof associativity can be thought of as translation invariance of µ by makingtranslations act on V bb by using µ . Proposition 1.4.
The vector space V bb is a representation of G = C × ⋉ C where w ∈ C acts as w.x = P k ∈ Z µ ( p k ( x ) , w ) and λ ∈ C × acts as ( λ.x ) k = λ k x k for x ∈ V bb .Proof. For w , w ∈ C and x ∈ V bb w . ( w .x ) = X k ∈ Z µ p k X k ∈ Z µ ( p k ( x ) , w ) ! , w ! = X k ∈ Z X k ∈ Z µ ( p k ( µ ( p k ( x ) , w )) , w )= X k ∈ Z X k ∈ Z µ ( p k ( µ ( p k ( x ) , w )) , w )= X k ∈ Z µ ( p k ( x ) , w + w ) (3)= ( w + w ) .x ,4here the sums are exchangeable because they are finite in each component of V and line (3) uses the associativity of V for m = 0. Furthermore0 .x = X k ∈ Z µ ( p k ( x ) ,
0) = X k ∈ Z p k ( x ) = x. These actions of C × and C assemble to an action of C × ⋉ C because λ. ( w. ( λ − .x )) = λ. X k ∈ Z µ ( p k ( λ.x ) , w )= X k ∈ Z λ.µ ( λ − .p k ( x ) , w )= X k ∈ Z µ ( p k ( x ) , λw )= ( λw ) .x because of C × -equivariance.If V is a Z -graded vector space and v ∈ V is homogeneous, then | v | denotesthe degree of v . Definition 1.5. A Z -graded vertex algebra consists of: • a Z -graded vector space V = L l ∈ Z V l over C , • a linear map Y : V −→
End V [[ x ± ]] a Y ( a, x ) = X k ∈ Z a ( k ) x − k − , where the k -th mode a ( k ) is a homogeneous endomorphism of V of de-gree | a | − k − a ∈ V , • a degree 1 endomorphism T of V , • a vector | i ∈ V called the vacuum ,such that: • (locality) For all a, b ∈ V , there exists a natural number N such that( x − y ) N [ Y ( a, x ) , Y ( b, y )] = 0in End( V )[[ x ± , y ± ]]. • (translation) T | i = 0 and [ T, Y ( a, x )] = ∂ x Y ( a, x ) for all a ∈ V . • (creation) Y ( a, x ) | i ∈ a + x V [[ x ]] for all a ∈ V . • (vacuum) Y ( | i , x ) = id V . In terms of the modes of a ∈ V , the translation axiom for a ∈ V is equivalentto demanding that [ T, a ( k ) ] = − ka ( k − for all k ∈ Z . The creation axiom for a is equivalent to the equation a ( − | i = a and a ( k ) | i = 0 for k ≥
0. The vacuumaxiom says that for all a ∈ V we have | i ( k ) a = 0 for k = − | i ( − a = a .5n a vertex algebra, the translation operator T and the vacuum | i areuniquely determined by the vertex operators Y ( a, x ). Proposition 1.7.
Let ( V , Y, T, | i ) be a vertex algebra. Then T a = a ( − | i forall a ∈ V . If a ∈ V satisfies Y ( a, x ) = id V , then a = | i .Proof. Let a ∈ V . It follows that T a = T a ( − | i = [ T, a ( − ] | i + aT | i = a ( − | i + 0by the creation axiom and the translation axiom. If Y ( a, x ) = id V , then a = Y ( a, x ) | i| x =0 = id( | i ) | x =0 = | i by the creation axiom. Proposition 1.8.
Let V be a geometric vertex algebra. Let a ∈ V and k ∈ Z .The k -th mode of a is a well-defined linear map a ( k ) : V −→ V b a ( k ) b := 12 πi Z S z k µ ( a, z ; b, dz. If a ∈ V is homogeneous, then a ( k ) is homogeneous of degree | a |− k − . A priori,the map a ( k ) is well-defined as a map V → V . Recall that we identify V = L k V k with its image in V = Q k V k under the natural injection.Proof. Let a, b ∈ V be homogeneous. Note that z. ( a ( k ) b ) = z | a |− k − | b | a ( k ) b forall z ∈ C × because2 πi z. ( a ( k ) b ) = z. Z S w k µ ( a, w ; b, dw = Z S w k z.µ ( a, w ; b, dw = Z S w k µ ( z.a, zw ; z.b, dw = z | a | + | b | Z S w k µ ( a, zw ; b, dw = z | a |− k − | b | Z zS w k µ ( a, w ; b, dw (substituted w/z )= z | a |− k − | b | Z S w k µ ( a, w ; b, dw (holomorphic)= 2 πi z | a |− k − | b | a ( k ) b. This implies that p l ( a ( k ) b ) is zero if l = | a | − k − | b | . Thus a ( k ) b ∈V | a |− k − | b | ⊆ V . Since every element of V is a finite sum of homogeneouselements, it follows that a ( k ) b ∈ V for all a, b ∈ V . Definition 1.9.
Let V be a Z -graded vector space over C . Let A ⊆ C be anannulus with center 0. If f : A → V is a holomorphic function, meaning inparticular that it takes values in a finite-dimensional subspace in each degree,then we define the Laurent expansion L ( f )( x ) ∈ Q k ∈ Z (cid:0) V k [[ x ± ]] (cid:1) of f on A componentwise. 6or V a Z -graded vector space, we identify V [[ x ± ]] with a subspace of Y k ∈ Z (cid:0) V k [[ x ± ]] (cid:1) = Y k ∈ Z V k ! [[ x ± ]] = V [[ x ± ]]via the injective linear map X l A l x l X l p k ( A l ) x l ! k ∈ Z . Proposition 1.10.
Let V be a geometric vertex algebra. For all a, b ∈ V , theLaurent expansion L [ z µ ( a, z, b, x ) is an element of V [[ x ± ]] and is equalto P l a ( l ) b x − l − .Proof. By the integral formula for the coefficients of the Laurent expansion L [ z µ ( a, z, b, x ) k = X l ∈ Z πi Z S z l p k ( µ ( a, z, b, dz · x − l − = X l ∈ Z p k (cid:18) πi Z S z l µ ( a, z, b, dz (cid:19) x − l − = X l ∈ Z p k ( a ( l ) b ) x − l − . For fixed l ∈ Z , the sum P k p k ( a ( l ) b ) is finite and equals a ( l ) b , so P l a ( l ) bx − l − is the desired preimage.We are now ready to state a more detailed version of the theorem. Theorem 1.11.
Let V be a Z -graded vector space over C . There is a bijectivemap from the set of geometric vertex algebra structures on V to the set of Z -graded vertex algebra structures on V defined by Y ( a, x ) b = L [ z µ ( a, z, b, x ) ( ∈ V [[ x ± ]]) T a = ∂ z µ ( a, z ) | z =0 | i = µ ( ∅ ) for all a, b ∈ V . The inverse to this bijection is uniquely determined by theequation µ ( a , z , . . . , a m , z m ) = X k ∈ Z ( a ) ( k ) z − k − . . . X k m ∈ Z ( a m ) ( k m ) z − k m − | i for a , . . . , a m ∈ V and z ∈ C m \ ∆ with | z | > . . . > | z m | . The r. h. s. above converges absolutely in each component of V by Proposi-tion 3.2. 7 From Geometric Vertex Algebras to VertexAlgebras
Given a geometric vertex algebra, we construct a Z -graded vertex algebra withthe same underlying vector space. Proposition 2.1.
Let V be a geometric vertex algebra. Let a , . . . , a m ∈ V and i, j ∈ { , . . . , m } with i < j . For all ( z , . . . , z i − , z i +1 , . . . , z m ) ∈ C m − \ ∆ and ε > with ε < | z l − z j | for all l = i, j , πi Z ∂B ε ( z j ) ( z i − z j ) k µ ( a, z ) dz i = µ ( a , z , . . . , \ a i , z i , . . . , a j − , z j − , a i ( k ) a j , z j , a j +1 , z j +1 , . . . , a m , z m ) . Proof.
Using permutation invariance and associativity µ ( a, z ) = X l ∈ Z µ ( . . . , \ a i , z i , . . . , p l µ ( a i , z i − z j , a j , , z j , . . . )and convergence is normal as functions of z i on ∂B ε ( z j ) in each component of V since it is locally normal by the associativity axiom and ∂B ε ( z j ) is compact. Itfollows that we can exchange integration over ∂B ε ( z j ) with summation to get12 πi Z ∂B ε ( z j ) ( z i − z j ) k µ ( a, z ) dz i = X l ∈ Z µ ( . . . , \ a i , z i , . . . , p l πi Z ∂B ε ( z j ) ( z i − z j ) k µ ( a i , z i − z j , a j , dz i , z j , . . . ) dz i . Here, we may move the integral into the argument of µ and p l because thesemaps are linear and the relevant functions take values in finite-dimensionalsubspaces. Shifting the contour integral to zero and noting that it does notmatter whether we integrate around a circle of radius ε or 1 as in the definitionof the modes, we get12 πi Z ∂B ε ( z j ) ( z i − z j ) k µ ( a i , z i − z j , a j , dz i = 12 πi Z ∂B ε (0) w k µ ( a i , w, a j , dw = a i ( k ) a j and thus12 πi Z ∂B ε ( z j ) ( z i − z j ) k µ ( a, z ) dz i = X l ∈ Z µ ( . . . , \ a i , z i , . . . , p l a i ( k ) a j , z j , . . . )= µ ( . . . , \ a i , z i , . . . , X l ∈ Z p l a i ( k ) a j , z j , . . . ) = µ ( . . . , \ a i , z i , . . . , p l a i ( k ) a j , z j , . . . )because the sum is finite.For i, j ∈ { , . . . , m } with i < j , let U ij be the set of z ∈ C m \ ∆ with | z i − z j | < min l = i,j | z l − z j | . The following proposition is the analogue of Proposi-tion 5.3.6 from [1, p. 167] for geometric vertex algebras. It expresses productsfor z i close to z j in terms of a series in z i − z j and other vertex algebra elementsinserted at z j and goes by the name of operator product expansion (OPE) .8 roposition 2.2. Let V be a geometric vertex algebra and let a , . . . , a m ∈ V .For z ∈ U ij µ ( a , z , . . . , a m , z m ) = X k ∈ Z µ ( . . . , \ a i , z i , . . . , a i ( k ) a j , z j , . . . )( z i − z j ) − k − with locally normal convergence.Proof. For fixed z , . . . , z i − , z i +1 , . . . , z m the l. h. s. is a holomorphic functionof z i such that z ∈ U ij . The integral formula for the coefficients of the locallynormal Laurent expansion and Proposition 2.1 imply µ ( a, z ) = X k ∈ Z πi Z ∂B ε ( z j ) ( w − z j ) k µ ( . . . , a i , w, . . . , a j , z j , . . . ) dw ( z i − z j ) − k − = X k ∈ Z µ ( . . . , \ a i , z i , . . . , a i ( k ) a j , z j , . . . )( z i − z j ) − k − . Proposition 2.3.
Assume that the underlying datum ( V , µ ) of a geometricvertex algebra satisfies all the axioms of a geometric vertex algebra except mero-morphicity. If V is bounded from below, then V is meromorphic.Proof. If V is bounded from below, then meromorphicity follows because µ ( b, w, c,
0) = X k ∈ Z µ ( b ( k ) c, w − k − = X k b ( k ) cw − k − and b ( k ) c is zero for k large enough for degree reasons.The map of Theorem 1.11 is well-defined: Proposition 2.4. If V is a geometric vertex algebra, then V forms a Z -gradedvertex algebra with the above Y , T and | i .Proof. Fix µ such that ( V , µ ) is a geometric vertex algebra. Our goal is to checkthat ( V , Y, T, | i ) as defined above is a vertex algebra. We determine the degreesof T and | i using the action of C × and the equivariance axiom similarly to howwe determined the degrees of the modes in Proposition 1.8: For all z ∈ C × , z. | i = z.µ ( ∅ ) = µ ( ∅ ) = | i so | i ∈ V ⊆ V since this shows that p l | i = 0 for l = 0. For all z ∈ C × and a ∈ V homogeneous, we have z.T a = z.∂ w µ ( a, w ) | w =0 = ∂ w z.µ ( a, w ) | w =0 = ∂ w µ ( z.a, zw ) | w =0 = ∂ w µ ( z | a | a, zw ) | w =0 = z | a | +1 ∂ w µ ( a, w ) | w =0 = z | a | +1 T a so T a is concentrated in degree | a | + 1 and T has degree 1. It follows that theimage of T is a subset of V . 9 ocality: Let a, b, c ∈ V . It suffices to treat the case of a, b, c homogeneous.Applying Proposition 2.2 twice, we find µ ( a, z, b, w, c,
0) = X l µ ( a, z, b ( l ) c w − l − ,
0) = X l X k a ( k ) b ( l ) c z − k − w − l − with locally normal convergence for | z | > > | w | >
0. Let A r,R = { z ∈ C | r < | z | < R } for 0 ≤ r, R ≤ ∞ . Using a similar notation for the Laurent expansion offunctions of several variables on products of annuli, we get L ( z,w ) ∈ A , × A , µ ( a, z, b, w, c, x, y ) = Y ( a, x ) Y ( b, y ) c and analogously L ( z,w ) ∈ A , × A , µ ( a, z, b, w, c, x, y ) = Y ( b, y ) Y ( a, x ) c. For all N ∈ N L ( z,w ) ∈ A , × A , ( z − w ) N µ ( a, z, b, w, c, x, y ) = ( x − y ) N Y ( a, x ) Y ( b, y ) cL ( z,w ) ∈ A , × A , ( z − w ) N µ ( a, z, b, w, c, x, y ) = ( x − y ) N Y ( b, y ) Y ( a, x ) c because Laurent expansion intertwines the action of polynomials as functionswith the action of polynomials as formal polynomials. For N large enough,the function ( z − w ) N µ ( a, z, b, w, c,
0) of ( z, w ) ∈ A , × A , \ ∆ extends to aholomorphic function of ( z, w ) ∈ A , × A , as a consequence of Proposition 2.2about the OPE and meromorphicity. If F ∈ O ( A , × A , ), then L A , × A , F = L A , × A , F in C [[ x ± , y ± ]], and therefore( x − y ) N Y ( a, x ) Y ( b, y ) c = ( x − y ) N Y ( b, y ) Y ( a, x ) c in V [[ x ± , y ± ]]. Vacuum:
Since µ ( | i , z, a,
0) = µ ( µ ( ∅ ) , z, a,
0) = µ ( a,
0) = a by the definition of | i , associativity, and insertion at zero, we have that | i ( k ) a = 12 πi Z S z k µ ( | i , z, a, dz = 12 πi Z S z k adz = δ k, − a for all a ∈ V and k ∈ Z . Creation:
Let a ∈ V . By associativity µ ( a, z, | i ,
0) = µ ( a, z ) and this is aholomorphic function of z ∈ C . Therefore its Laurent expansion has no negativepowers and its zeroth term is µ ( a,
0) = a because of the axiom about insertionat zero. 10 ranslation: Let T : V → V be defined by
T x = P k T p k x . This sum is finitein each degree, and T is linear. Let a, b ∈ V . The identity[ T, Y ( a, x )] b = T Y ( a, x ) b − Y ( a, x ) T b = ∂ x Y ( a, x ) b. is implied by T µ ( a, z, b, − µ ( a, z, T b,
0) = ∂ z µ ( a, z, b,
0) (4)because Laurent expansion is compatible with linear maps and with differentia-tion. Recall that
T c = ∂ w µ ( c, w ) | w =0 for c ∈ V . We may assume that w is closeto zero to compute the w -derivative. µ ( a, z, T b,
0) = µ ( a, z, X k p k T b,
0) (5)= X k µ ( a, z, p k ∂ w µ ( b, w ) | w =0 ,
0) (6)= X k ∂ w µ ( a, z, p k µ ( b, w ) , | w =0 (7)= " ∂ w X k µ ( a, z, p k µ ( b, w ) , w =0 (8)= ∂ w µ ( a, z, b, w ) | w =0 (9)Equation (9) follows from associativity which implies that the sum in (8) islocally normally convergent, and this implies that we can commute the sum andthe derivative in Equation (8), so the sum in (7) is locally normal. Equation (7)uses p k is linear and that µ is linear in each argument from V . The sums in (5)and (6) are finite. Similarly, T µ ( a, z, b,
0) = X k ∂ w µ ( p k µ ( a, z, b, , w ) | w =0 = ∂ w X k µ ( p k µ ( a, z, b, , w ) (cid:12)(cid:12) w =0 = ∂ w µ ( a, z + w, b, w ) | w =0 and thus T µ ( a, z, b, − µ ( a, z, T b,
0) = ∂ w µ ( a, z + w, b, w ) | w =0 − ∂ w µ ( a, z, b, w ) | w =0 = ∂ w µ ( a, z + w, b, | w =0 = ∂ z µ ( a, z, b, , which is Equation (4). Proposition 3.1.
Let V be a Z -graded vertex algebra. For all a, b ∈ V , thereexists a number N s. t. a ( n ) b = 0 if n ≥ N . µ . Proof.
Let a, b ∈ V . By the locality axiom, there exists an N such that( x − y ) N Y ( a, x ) Y ( b, y ) | i = ( x − y ) N Y ( b, y ) Y ( a, x ) | i . (10)By the creation axiom for b , we may set y = 0 on the l. h. s. and get x N Y ( a, x ) b = X n a ( n ) bx N − n − .Using Equation (10) and the creation axiom for a on the r. h. s., it follows thatthe l. h. s. contains no negative powers of x , even after setting y = 0. This meansthat a ( n ) b = 0 for N − n − <
0, equivalently n ≥ N . Proposition 3.2.
Let V be a Z -graded vertex algebra. Fix an integer m ≥ .For a , . . . , a m ∈ V and z ∈ C m \ ∆ the series Y ( a , z ) . . . Y ( a m , z m ) | i := X k ∈ Z ( a ) ( k ) z − k − . . . X k m ∈ Z ( a m ) ( k m ) z − k m − m | i converges locally normally in each component of V for z ∈ D m , where D m := { z ∈ C m | | z | > . . . > | z m |} ⊆ C m \ ∆ .As a function of z , the value of this series extends to a unique holomorphic V -valued function µ ( a, z ) = µ ( a , z , . . . , a m , z m ) of z ∈ C m \ ∆ . It satisfies µ ( a σ , z σ ) = µ ( a, z ) for every permutation σ ∈ Σ m .Furthermore, µ ( a ) is identically zero in sufficiently low degree for every a ∈V ⊗ m . This proposition and its proof are very similar to what is found in [2, 1.2and 4.5], [5, 5.3], [3, A.2], [4, 3.5.1], which treat the bounded below, degreewisefinite-dimensional case in these parts.
Proof.
We may assume that a , . . . , a m are homogeneous. Let f ( a, x ) ∈ V [[ x ± , . . . x ± m ]]denote the formal series corresponding to the sum in the claim. By the creationaxiom for a m , there are no negative powers of x m in f ( a, x ). The locality axiomfor a i and a j implies that there is a natural number N ij with( x i − x j ) N ij [ Y ( a i , x j ) , Y ( a j , x j )] = 0for 1 ≤ i < j ≤ m . Let g ( x ) = Y i The pair ( V , µ ) satisfies the axioms of a geometric vertexalgebra.Proof. Let m ≥ 0. Both the multilinearity and C × -equivariance may be checkedon D m by the uniqueness of analytic continuations from D m to C m \ ∆. On D m ,we can express µ in terms of Y and multilinearity follows from the fact that Y is linear and composition in End( V ) is bilinear. For C × -equivariance, it suffices14o consider homogeneous a , . . . , a m ∈ V : for all z ∈ C × and w ∈ D m , z.µ ( a, w )= X k ,...,k m ∈ Z z. (cid:16) ( a ) ( k ) w − k − . . . ( a m ) ( k m ) w − k m − m | i (cid:17) = X k ,...,k m ∈ Z z | a |− k − ... + | a m |− k m − ( a ) ( k ) w − k − . . . ( a m ) ( k m ) w − k m − m | i = X k ,...,k m ∈ Z ( z.a ) ( k ) ( zw ) − k − . . . ( z.a m ) ( k m ) ( zw m ) − k m − | i = µ ( z.a , zw , . . . , z.a m , zw m ) . The axiom about insertion at zero follows from the creation axiom of the vertexalgebra: for all a ∈ V , µ ( a, 0) = Y ( a, | i = a. To see that the meromorphicity axiom is satisfied, let N be the maximum ofthe N ij from the construction of µ , after fixing some elements of V . By con-struction, we then have the following strong version of meromorphicity: • For all a , . . . , a n ∈ V , there is a natural number N such that the function z Y i Let b , . . . , b n ∈ V and w ∈ C n \ ∆ . For all z ∈ C , e zT µ ( b , w , . . . , b n , w n ) = µ ( b , w + z, . . . , b n , w n + z ) . (15)In the case n = 1 and w = 0, the proposition together with insertion at zeroimplies that e zT b = Y ( b, z ) | i for b ∈ V and z ∈ C . This means that a ( k ) | i = − k − T − k − a for k ≤ − 1, which one could have also deduced directly fromthe translation axiom of the vertex algebra V . In particular T a = a ( − | i and µ ( a, w ) = a + T aw + . . . . Also, since e zT a = µ ( a, z ) for all a ∈ V and z ∈ C ,the proposition is the special case m = 0 of the associativity axiom for µ .15 roof. Both sides of the equation are holomorphic functions of ( z, w ) ∈ C × ( C n \ ∆). By the identity theorem we may assume | w | > . . . > | w n | , and having fixed a choice of w , we may assume z to be in the non-empty, open,and convex region B of those z such that | w + z | > . . . > | w n + z | . For z ∈ B , the equation is equivalent to e zT Y ( b , w ) . . . Y ( b n , w n ) | i = Y ( b , w + z ) . . . Y ( b n , w n + z ) | i .Let f ( z ) equal the l. h. s. and g ( z ) equal the r. h. s. for z ∈ B . The function f is the unique solution of the (holomorphic) initial value problem ∂ z ϕ ( z ) = T ϕ ( z ) ϕ (0) = Y ( b , w ) . . . Y ( b n , w n ) | i ϕ ∈ O ( B, V bb ) (16)as can be deduced from the fact that ϕ ( z ) = e zA x is the unique solution of ∂ z ϕ ( z ) = Aϕ ( z ) ϕ (0) = x ϕ ∈ O ( B, X ) . (17)We prove by induction on n that g is a solution of (16), too. The base case n = 0holds because then g ( z ) = | i . The induction hypothesis implies ∂ z Y ( b , w + z ) . . . Y ( b n , w n + z ) | i = T Y ( b , w + z ) . . . Y ( b n , w n + z ) | i .Proposition 3.2 implies g ( z ) = X k ∈ Z b k ) ( w + z ) − k − Y ( b , w + z ) . . . Y ( b m , w m + z ) | i and thus ∂ z g ( z ) = X k ∈ Z b k ) ( − k − w + z ) − k − Y ( b , w + z ) . . . Y ( b m , w m + z ) | i + X k ∈ Z b k ) ( w + z ) − k − ∂ z Y ( b , w + z ) . . . Y ( b m , w m + z ) | i = X k ∈ Z b k − ( − k )( w + z ) − k − Y ( b , w + z ) . . . Y ( b m , w m + z ) | i + X k ∈ Z b k ) ( w + z ) − k − T Y ( b , w + z ) . . . Y ( b m , w m + z ) | i after reindexing. The translation axiom implies that [ T, b k ) ] = − kb k − .Therefore ∂ z g ( z ) = X k ∈ Z T b k ) ( w + z ) − k − Y ( b , w + z ) . . . Y ( b m , w m + z ) | i = T Y ( b , w + z ) . . . Y ( b m , w m + z ) | i = T g ( z ) , and this concludes the induction step. 16 roposition 3.5. µ satisfies associativity.Proof. By Proposition 3.4, it suffices to show that X k µ ( a , z − y, . . . , a m , z m − y, p k ( µ ( b , w , . . . , b n , w n )) , z m +1 − y )= µ ( a , z − y, . . . , a m , z m − y, b , w + z m +1 − y, . . . , b n , w n + z m +1 − y )for some y ∈ C . Locally normal convergence is preserved by the applicationof e yT as this yields a finite sum in each component. We pick y so that | z m +1 − y | < | z m +1 − z i | and | w j + z m +1 − y | < | z i − y | (18)for all i = 1 , . . . , m and j = 1 , . . . , n and so that the | z i − y | are pairwisedistinct and the | w j + z m +1 − y | are pairwise distinct. The second conditionon the pairwise distinct norms holds for all y outside a finite union of lines.The first condition holds for all y close enough to z m +1 because max j | w j | < min ≤ i ≤ m | z i − z m +1 | is a prerequisite in the associativity axiom. It follows thatsuch a y exists.Permuting the z , . . . , z m and the w , . . . , w n and using the permutationinvariance of µ , we may assume that | z − y | > . . . > | z m − y | and | w + z m +1 − y | > . . . > | w n + z m +1 − y | , and we have | z m − y | > | w + z m +1 − y | by Equation 18 so | z − y | > . . . > | z m − y | > | w + z m +1 − y | > . . . > | w n + z m +1 − y | . Without loss of generality, we may assume y = 0 by redefining z i to z i − y and w j to w j − y so that | z | > . . . > | z m | > | w + z m +1 | > . . . > | w n + z m +1 | .Then, using the absolute convergence from Proposition 3.2, µ ( a , z , . . . , a m , z m , b , w + z m +1 , . . . , b n , w n + z m +1 )= X i ∈ Z m ,j ∈ Z n a i ) z − i − . . . a m ( i m ) z − i m − m b j ) ( w + z m +1 ) − j − . . . b n ( i n ) ( w n + z m +1 ) − j n − | i = X l X i ∈ Z m a i ) z − i − . . . a m ( i m ) z − i m − m p l X j ∈ Z n b j ) ( w + z m +1 ) − j − . . . b n ( i n ) ( w n + z m +1 ) − j n − | i = X l X i ∈ Z m a i ) z − i − . . . a m ( i m ) z − i m − m p l µ ( b , w + z m +1 , . . . , b n , w n + z m +1 )= X l X i ∈ Z m a i ) z − i − . . . a m ( i m ) z − i m − m p l µ ( b , w + z m +1 , . . . , b n , w n + z m +1 ) , µ ( b , w + z m +1 , . . . , b n , w n + z m +1 )= X k e z m +1 T p k µ ( b, w )= X k Y ( p k µ ( b, w ) , z m +1 ) | i = X k X i m +1 ( p k µ ( b, w )) ( i m +1 ) z − i m +1 − | i , by Proposition 3.4, the former equals X l X i ∈ Z m a i ) z − i − . . . a m ( i m ) z − i m − m p l X k X i m +1 ( p k µ ( b, w )) ( i m +1 ) z − i m +1 − | i = X l X i ∈ Z m +1 a i ) z − i − . . . a m ( i m ) z − i m − m p l X k ( p k µ ( b, w )) ( i m +1 ) z − i m +1 − | i = X k X l X i ∈ Z m +1 a i ) z − i − . . . a m ( i m ) z − i m − m p l ( p k µ ( b, w )) ( i m +1 ) z − i m +1 − | i = X k X i ∈ Z m +1 a i ) z − i − . . . a m ( i m ) z − i m − m ( p k µ ( b, w )) ( i m +1 ) z − i m +1 − | i = X k µ ( a , z , . . . , a m , z m , p k ( µ ( b , w , . . . , b n , w n )) , z m +1 ) , where convergence is locally normal to begin with since power series convergelocally normally and we may thus interchange the sums. Proof of Theorem 1.11. Let Φ denote the map from geometric vertex algebrastructures on V to vertex algebra structures on V as described in the statementof the theorem. Proposition 2.4 says that it is well-defined. Let Ψ denote themap from vertex algebra structures on V to geometric vertex algebra structuresdefined using Proposition 3.2 and Proposition 3.3. It is clear from the construc-tion that Φ ◦ Ψ = id because it suffices to consider Y to equate two vertexalgebra structures on the same Z -graded vector space (see Proposition 1.7). Tosee that Ψ ◦ Φ = id, we repeatedly apply Proposition 2.2 to deduce that µ ( a, z ) = Y ( a , z ) . . . Y ( a m , z m ) | i for all a , . . . , a m ∈ V and z ∈ C m \ ∆ with | z | > . . . > | z m | , where µ is a givengeometric vertex algebra structure and Y is part of Φ( µ ). Thus, for fixed a ,the functions µ ( a, z ) and Ψ(Φ( µ ))( a, z ) of z ∈ C m \ ∆ are analytic continua-tions of the same function, and therefore agree by the uniqueness statement inProposition 3.2, which characterizes Ψ(Φ( µ )). References [1] Kevin Costello and Owen Gwilliam. Factorization algebras in quantum fieldtheory. Vol. 1 , volume 31 of New Mathematical Monographs . CambridgeUniversity Press, Cambridge, 2017.182] Edward Frenkel and David Ben-Zvi. Vertex algebras and algebraic curves ,volume 88 of Mathematical Surveys and Monographs . American Mathemat-ical Society, Providence, RI, second edition, 2004.[3] Igor Frenkel, James Lepowsky, and Arne Meurman. Vertex operator algebrasand the Monster , volume 134 of Pure and Applied Mathematics . AcademicPress, Inc., Boston, MA, 1988.[4] Igor B. Frenkel, Yi-Zhi Huang, and James Lepowsky. On axiomatic ap-proaches to vertex operator algebras and modules. Mem. Amer. Math. Soc. ,104(494):viii+64, 1993.[5] Yi-Zhi Huang. 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