From Prefactorization Algebras to Geometric Vertex Algebras

3.3 From Prefactorization Algebras to Geometric Vertex

We now introduce a little bit more notation to write down the explicit for-mulas for µ^z used to check the axioms of a geometric vertex algebra. These formulas differ from the above more abstract description in that specific radii of discs appear. As in [2] and [12] and other places, we consider a moduli space of a fixed number of discs sitting inside a larger disc. Forr1, . . . , rm>0, letD(r1, . . . , rm;R) denote the set ofz∈C^m with

Br₁(z1)t. . .tBr_m(zm)⊆BR(0) .

We also denote this set byD(r;R) and can often restrict attention to the case ofr1=. . .=rm=rfor some smallr >0. Let

Pr:VF=M

k∈Z

F(0)k −→F(Br(0)) be the map given by

F(0)_k = (lim_s>0F(B_s(0)))_k ^(π^r⁾^k F(B_r(0))_k ⊆F(B_r(0)) on thek-th summand fork∈Z. Let

L^R:F(B_R(0))−→ Y

k∈Z

F(0)_k=VF

be the map whosek-th component is given by (21). In terms of the mapsL^R,Pri, the definition ofµ^z is equivalent to the equation

µ(a₁, z₁, . . . , a_m, z_m) =µ^z(a)

=L^R(M_B^B^r¹^(z¹^),...,B^rm^(z^m⁾

R(0) (σ_(+z^B^r¹⁽⁰⁾

1) (P_r₁(a₁)), . . . , σ^B_(+z^rm⁽⁰⁾

m) (P_r_m(a_m))) (22) in V for a=a1⊗. . .⊗am∈ VF^⊗m and r1, . . . , rm, R >0 with z ∈ D(r;R).

This is becausePris the lower left composite in the commutative square

F(0)k F(0)

F(Br(0))k F(Br(0))

(π_r)_k π_r

in which the horizontal maps are the inclusions. Note that, for everyR >0, an elementx∈VF is uniquely determined by thePRpkxfor k∈Z. Considering one degreek∈Zat a time,

P_R(p_k(µ^z(a)))

=l^R_k(M_B^B^r¹^(z¹^),...,B^rm^(z^m⁾

R(0) (σ^B_(+z^r¹⁽⁰⁾

1) (Pr1(a1)), . . . , σ_(+z^B^rm⁽⁰⁾

m) (Prm(am)))) (23) inF(B_R(0))_k becauseP_Rp_kL^Ry=l^R_ky for ally∈F(B_R(0)).

So far, we know that the mapsµare linear maps from VF^⊗m to the set of functions onC^m\∆ with values inVF. Our next goal is holomorphicity.

Proposition 3.3.1. Let a ∈ V^⊗m. Then µ^z(a) = µ(a, z) is a holomorphic function ofz∈C^m\∆.

Proof. A function C^m \∆ → VF is holomorphic if and only if its compo-nents p_k ◦ f are holomorphic for all k ∈ Z. Since VF_k is assumed to be discrete, pk ◦f is holomorphic if it is locally a holomorphic map to a finite-dimensional subspace ofVFk. This notion of holomorphicity coincides with the notion of holomorphicity used in the definition of a geometric vertex algebra.

Let z ∈ C^m\∆. Pick R > 0 and r > 0 s. t. z ∈ D(r;R). Let s = r/2.

It suffices to find a neighborhoodU of z s. t.µ^z⁰(a) is a holomorphic function ofz⁰ ∈U for some given a=a1⊗. . .⊗am. We useU =Qm

i=1Bs(z) and note that it suffices to prove the holomorphicity of

N(z⁰) =M^B^s^(z

1),...,B_s(z⁰_m) B_R(0) (σ_(+z^B^s⁽⁰⁾0

1)(x₁), . . . , σ_(+z^B^s⁽⁰⁾0 m)(x_m))

for x_i ∈ F(B_s(0)) because we can then plug in x_i = P_sa_i and postcompose withL_R to getµ(a, z⁰). By the associativity ofM,

N(z⁰) =M_B^B^r^(z¹^),...B^r^(z^m⁾

R(0) (ρ₁(z⁰₁)(x₁), . . . , ρ_m(z_m⁰ )(x_m)) , where

ρ_i(z_i⁰) :F(B_s(0))−→F(B_r(z_i)) y7−→M^B^s^(z

0 i) Br(zi)(σ^B_(+z^s⁽⁰⁾0

i)(y)), soN is holomorphic as the composite of

• the map

U −→O^m

i=1BVS(F(Bs(0)), F(Br(zi))) (z⁰₁, . . . , z⁰_m)7−→ρ1(z₁⁰)⊗. . .⊗ρm(z⁰_m) ,

which is holomorphic because F is holomorphically translation-invariant, and the pointwise tensor product of holomorphic functions is holomorphic,

• the bounded evaluation-at-amap to Nm

i=1F(Br(zi)), and

• the bounded mapM_B^B^r^(z¹^),...B^r^(z^m⁾

R(0) .

Remark 3.3.2. If F is holomorphic in the sense of Definition 3.2.8, then it is holomorphic pointwise, meaning that ρU,V(z)(x) is a holomorphic function ofz ∈ DU,V for allx∈F(U). The proof of Proposition 3.3.1 works if we only assume thatF is holomorphic pointwise.

Proposition 3.3.3. (VF, µ)satisfies the insertion-at-zero axiom.

Proof. Leta∈VF. For allR >0 andk∈Z, P_Rp_kµ(a,0) =l_k^RM_B^B^r⁽⁰⁾

R(0)(P_ra) =l^R_kP_Ra=l_k^RP_RX

l∈Z

p_la

l∈Z

l^R_kP_Rp_la=P_Rp_ka,

where the last equality holds because PR isD^×-equivariant, soPRplahas de-greel, andl^R_k is the identity on the k-th weight space and zero on the others.

This impliesµ(a,0) =a.

Proposition 3.3.4. (VF, µ)isC^×-equivariant.

Proof. ForD^×instead ofC^×, this follows since the maps are equivariant w. r. t.

this semi-group in an appropriate sense: For all degrees k ∈ Z, it suffices to consider someR >0 withzi∈BR(zi) fori= 1, . . . , mwithr >0 small enough so thatz∈D(r, . . . , r;R), and to show equality after application ofPR◦pk.

PRpk(q.µ(a1, z1, . . . , am, zm))

=q.P_R(p_k(µ(a₁, z₁, . . . , a_m, z_m)))

=q.l_k^R(M_B^B^r^(z¹^),...,B^r^(z^m⁾

R(0) (σ_(+z^B^r⁽⁰⁾

1)(P_ra₁), . . . , σ_(+z^B^r⁽⁰⁾

m)(P_ra_m)))

=l^R_k(q.M_B^B^r^(z¹^),...,B^r^(z^m⁾

R(0) (σ_(+z^B^r⁽⁰⁾

1)(P_ra₁), . . . , σ_(+z^B^r⁽⁰⁾

m)(P_ra_m)))

=l^R_kM_B^B^qR⁽⁰⁾

R(0) σ^B_q^R⁽⁰⁾M_B^B^r^(z¹^),...,B^r^(z^m⁾

R(0) (σ^B_(+z^r⁽⁰⁾

1)(Pra1), . . . , σ_(+z^B^r⁽⁰⁾

m)(Pram))

=l^R_kM_B^B^qR⁽⁰⁾

R(0) M_B^B^qr^(qz¹^),...,B^qr^(qz^m⁾

qR(0) (σ_(q·)^B^r^(z¹⁾(σ^B_(+z^r⁽⁰⁾

1)(Pra1)), . . . , σ_(q·)^B^r^(z^m⁾(σ^B_(+z^r⁽⁰⁾

m)(P_ra_m)))

=l^R_kM_B^B^qR⁽⁰⁾

R(0) M_B^B^qr^(qz¹^),...,B^qr^(qz^m⁾

qR(0) (σ_(+qz^B^qr⁽⁰⁾

1)(σ_(q·)^B^r⁽⁰⁾(Pra1)), . . . , σ_(+qz^B^qr⁽⁰⁾

m)(σ_(q·)^B^r⁽⁰⁾(Pram)))

=l^R_kM_B^B^qR⁽⁰⁾

R(0) M_B^B^qr^(qz¹^),...,B^qr^(qz^m⁾

qR(0) (σ_(+qz^B^qr⁽⁰⁾

1)(σ_(q·)^B^r⁽⁰⁾(Pra1)), . . . , σ_(+qz^B^qr⁽⁰⁾

m)(σ_(q·)^B^r⁽⁰⁾(Pram)))

=l^R_kM_B^B^qr^(qz¹^),...,B^qr^(qz^m⁾

R(0) (σ_(+qz^B^qr⁽⁰⁾

1)(σ_(q·)^B^r⁽⁰⁾(P_ra₁)), . . . , σ_(+qz^B^qr⁽⁰⁾

m)(σ_(q·)^B^r⁽⁰⁾(P_ra_m)))

=l^R_kM_B^B^r^(qz¹^),...,B^r^(qz^m⁾

R(0) (M_B^B^qr^(qz¹⁾

r(qz₁) (σ^B_(+qz^qr⁽⁰⁾

1)(σ^B_(q·)^r⁽⁰⁾(Pra1))), . . . , M_B^B^qr^(qz^m⁾

r(qzm) (σ^B_(+qz^qr⁽⁰⁾

m)(σ^B_(q·)^r⁽⁰⁾(P_ra_m))))

=l^R_kM_B^B^r^(qz¹^),...,B^r^(qz^m⁾

R(0) (σ^B_(+qz^r⁽⁰⁾

1)(M_B^B^qr⁽⁰⁾

r(0) (σ_(q·)^B^r⁽⁰⁾(Pra1))), . . . , σ_(+qz^B^r⁽⁰⁾

m)(M_B^B^qr⁽⁰⁾

r(0) (σ_(q·)^B^r⁽⁰⁾(Pram))))

=l^R_kM_B^B^r^(qz¹^),...,B^r^(qz^m⁾

R(0) (σ^B_(+qz^r⁽⁰⁾

1)(q.Pra1), . . . , σ_(+qz^B^r⁽⁰⁾

m)(q.Pram))

=l^R_kM_B^B^r^(qz¹^),...,B^r^(qz^m⁾

R(0) (σ^B_(+qz^r⁽⁰⁾

1)(P_r(q.a₁)), . . . , σ^B_(+qz^r⁽⁰⁾

m)(P_r(q.a_m)))

=P_Rp_kµ(q.a₁, qz₁, . . . , q.a_m, qz_m)

Equivariance forC^× follows from the uniqueness of analytic continuation.

Proposition 3.3.5. µ satisfies the associativity axiom.

The map Φ in the following proof plays a very similar role as the map denoted by the same letter in Proposition 5.3.6 of [2].

Proof. Leta₁, . . . , a_m∈VF,b₁, . . . , b_n∈VF. Let A_m,n={(z, w)∈(C^m\∆)×(Cⁿ\∆)| max

1≤j≤n|w_j|< min

1≤i≤m|z_i−z_m+1|}

denote the set of tuples of points in C as in the associativity axiom of a ge-ometric vertex algebra. The summands in the associativity axiom, viewed as functions of (z, w)∈ Am,n, are elements ofO(Am,n;VF) and we prove absolute

convergence in a completely normable subobject. By the definition of the bornol-ogy on O(A_m,n;VF), this implies that, for every compact subset K ⊆ A_m,n and k ∈ Z, there is a completely normable subobject VK,k of VFk in which absolute convergence takes place. The discreteness of VFk implies that the bounded unit disc ofVK,k is contained in a finite-dimensional subspace ofVFk

soVK,k is finite-dimensional.

It suffices to prove convergence of functions on a neighborhood U of (z, w) for every (z, w)∈ Am,n. LetS > T2> T1>0 all be smaller than min_1≤i≤m|zi− zm+1|, but only slightly smaller, so that

w₁, . . . , w_n ∈B_T₁(0) (⊂B_T₂(0)⊂B_S(0)) .

Let r > 0 be small enough so that w ∈ D(r, . . . , r;T₁) and the B_r(z_i), i = 1, . . . , m, are pairwise disjoint and disjoint fromB_S(z_m+1). Let R >0 be big enough so that

z∈D(r, . . . , r, S;R) .

Lets=r/2 and

U =

i=1

B_s(z_i)×B_S−T₂(z_m+1)×

j=1

B_s(w_j)

which is a subset of Am,n as we now check. Let (z⁰, w⁰) ∈ U. For all i, j with 1≤i≤mand 1≤j≤n,

|w⁰_j|<|wj|+s=|wj|+r−s < T1−s

< S=S+r−2s≤ |zi−zm+1| −2s <|z_i⁰−z_m+1⁰ | so (z⁰, w⁰)∈ A_m,n.

We show convergence of holomorphic functions on U with values in VF by constructing a bounded linear map

Φ : BVS(F(BT1(0)), F(BT2(0)))−→ O(U;VF) with Φ(i^T_T¹

2) given by the r. h. s. of the associativity axiom, see (2), and Φ(i^T_T¹

2◦l^T_k¹) given by thek-th summand of the associativity axiom for allk∈Z, see (1). The existence of such a map implies the associativity axiom onU because bounded linear maps map absolutely convergent sums to absolutely convergent sums andi^T_T¹

2 =P

k∈Zi^T_T¹

2◦l^T_k¹ by Proposition 3.2.10. Let (Φ(f))(z⁰, w⁰)

=L^RM^B^s^(z

1),...,Bs(z⁰_m),BT2(z⁰_m+1) B_R(0) (σ_(+z^B^s⁽⁰⁾0

1)(Psa1), . . . , σ^B_(+z^s⁽⁰⁾0

m)(Psam), σ_(+z^B^T²0⁽⁰⁾

m+1)(f(M^B^s^(w

1),...,B_s(w⁰_n) B_T₁(0) (σ^B_(+w^s⁽⁰⁾0

1)(P_sb₁), . . . , σ_(+w^B^s⁽⁰⁾0

n)(P_sb_n)))) .

Forf =i^T_T¹

2, we use associativity, translation invariance, and associativity again:

(Φ(f))(z⁰, w⁰)

=L^RM^B^s^(z

1),...,Bs(z⁰_m),BT2(z_m+1⁰ ) B_R(0) (σ^B_(+z^s⁽⁰⁾0

1)(Psa1), . . . , σ^B_(+z^s⁽⁰⁾0

m)(Psam), σ_(+z^B^T²0⁽⁰⁾

m+1)(M^B^s^(w

1),...,B_s(w_n⁰) B_T₂(0) (σ_(+w^B^s⁽⁰⁾0

1)(Psb1), . . . , σ^B_(+w^s⁽⁰⁾0

n)(Psbn)))

=L^RM^B^s^(z

1),...,Bs(z⁰_m),BT2(z_m+1⁰ ) B_R(0) (σ^B_(+z^s⁽⁰⁾0

1)(Psa1), . . . , σ^B_(+z^s⁽⁰⁾0

m)(Psam), M^B^s^(w

1+z⁰_m+1),...,B_s(w⁰_n+z_m+1⁰ ) B_T₂(z_m+1⁰ ) (σ^B_(+w^s⁽⁰⁾0

1+z⁰_m+1)(P_sb₁), . . . , σ_(+w^B^s⁽⁰⁾0

n+z⁰_m+1)(P_sb_n)))

=L^RM^B^s^(z

1),...,Bs(z⁰_m),BT2(z_m+1⁰ ),Bs(w⁰₁+z⁰_m+1),...,Bs(w⁰_n+z_m+1⁰ )

B_R(0) (

σ_(+z^B^s⁽⁰⁾0

1)(P_sa₁), . . . , σ^B_(+z^s⁽⁰⁾0

m)(P_sa_m), σ^B_(+w^s⁽⁰⁾0

1+z⁰_m+1)(P_sb₁), . . . , σ_(+w^B^s⁽⁰⁾0

n+z⁰_m+1)(P_sb_n))

=µ(a1, z₁⁰, . . . , am, z⁰_m, b1, w₁⁰ +z_m+1⁰ , . . . , bn, w⁰_n+z_m+1⁰ ) Fork∈Zandf =i^T_T¹

2◦l^T_k¹, we consider the argument ofσ_(+z^B^T²0⁽⁰⁾

m+1)which is i^T_T¹

2◦l^T_k¹(M^B^s^(w

1),...,B_s(w_n⁰) B_T₁(0) (σ_(+w^B^s⁽⁰⁾0

1)(Psb1), . . . , σ^B_(+w^s⁽⁰⁾0

n)(Psbn))) and, because ofi^T_T¹

2◦l^T_k¹ =l^T_k²◦i^T_T¹

2 and associativity, equal to PT₂(pk(µ(b1, w⁰₁, . . . , bn, w_n⁰))) .

Plugging in, this implies that (Φ(f))(z⁰, w⁰)

=L^RM^B^s^(z

1),...,Bs(z⁰_m),BT2(z_m+1⁰ ) B_R(0) (σ^B_(+z^s⁽⁰⁾0

1)(Psa1), . . . , σ_(+z^B^s⁽⁰⁾0

m)(Psam), σ_(+z^B^T²0⁽⁰⁾

m+1)(PT₂(pk(µ(b1, w⁰₁, . . . , bn, w⁰_n)))))

=µ(a₁, z₁, . . . , a_m, z_m, p_k(µ(b₁, w⁰₁, . . . , b_n, w⁰_n)), z_m+1) which is thek-th summand in the associativity axiom.

Now, we check that Φ is a bounded linear map and actually maps to the space of holomorphic functions O(U;VF) by writing Φ as a composite of bounded linear maps. This argument is similar to the proof of Proposition 3.3.1 about holomorphicity. By associativity,

(Φ(f))(z⁰, w⁰)

=L^RM_B^B^r^(z¹^),...,B^r^(z^m^),B^S^(z^m+1⁾

R(0) (ρ1(z⁰₁)(Psa1), . . . , ρm(z_m⁰ )(Psam)), ρm+1(z_m+1⁰ )(f(M_B^B^r^(w¹^),...,B^r^(wⁿ⁾

T1(0) (τ1(w₁⁰)(Psb1), . . . , τn(w⁰_n)(Psbn)))) , where the map

ρ_i:B_s(z_i)−→BVS(F(B_s(0)), F(B_r(z_i))) z_i⁰7−→M_B^B^s^(z⁰ⁱ⁾

r(zi)◦σ_(+z^B^s⁽⁰⁾0 i)

is holomorphic for i= 1, . . . , m, and so are

ρ_m+1:B_S−T₂(z_m+1)−→BVS(F(B_T₂(0)), F(B_S(z_i))) z_m+1⁰ 7−→M^B^T²^(z

0 m+1)

B_S(z_m+1) ◦σ^B_(+z^T²0⁽⁰⁾ m+1)

and

τj :Bs(wj)−→BVS(F(Bs(0)), F(Br(wj))) w⁰_j7−→M^B^s^(w

0 j)

Br(wj)◦σ^B_(+w^s⁽⁰⁾0 j)

forj= 1, . . . , n. Thus the map

j=1

Bs(wj)−→Oⁿ

j=1F(Br(wj))

(w⁰₁, . . . , w_n⁰)7−→τ1(w₁⁰)(Psb1)⊗. . .⊗τn(w⁰_n)(Psbn) is holomorphic. Postcomposing withM_B^B^r^(w¹^),...,B^r^(wⁿ⁾

T1(0) andf defines a bounded linear map

BVS(F(B_T₁(0)), F(B_T₂(0))−→ O





j=1

B_s(w_j);F(B_T₂(0))



. Taking the exterior product with

ρm+1∈ O(B_S−T₂(zm+1); BVS(F(BT₂(0)), F(BS(zi)))) is a bounded linear map with target



B_S−T₂(z_m+1)×

j=1

B_s(w_j) ; BVS (F(B_T₂(0)), F(B_S(z_i))) ¯⊗F(B_T₂(0))





from which we can postcompose with the evaluation pairing to get an element of

O(B_S−T₂(z_m+1)×

j=1

B_s(w_j);F(B_S(z_i))) . Again using the holomorphicity ofF, the map

i=1

Bs(zi)−→O^m

i=1F(Br(zi))

(z⁰₁, . . . , z_m⁰ )7−→ρ1(z₁⁰)(Psb1)⊗. . .⊗ρm(z_m⁰ )(Psbm)

is holomorphic, and taking the exterior product with it is a bounded linear map with target

O(U;⊗^m_i=1F(B_r(z_i))⊗F(B_S(z_i))) . Postcomposing withL^R◦M_B^B^r^(z¹^),...,B^r^(z^m^),B^S^(z^m+1⁾

R(0) gives Φ(f)∈ O(U;VF).

Definition 3.3.6. LetF be a holomorphic affine-linearly invariant factorization algebra onC. We say thatF has discrete weight spaces ifVF_k is discrete for all k ∈ Z. If F has discrete weight spaces, we say that F has meromorphic operator product expansion (OPE)if (VF, µ) satisfies the meromorphicity axiom of a geometric vertex algebra, whereµ is the sequence of multiplication maps forVF constructed above.

We sometimes call holomorphic prefactorization algebras with meromorphic OPEmeromorphic prefactorization algebras, with the caveat that, in contrast to the language for functions, a meromorphic prefactorization algebra is a special kind of holomorphic prefactorization algebra.

Proposition 3.3.7. If F is a holomorphic prefactorization algebra onC with discrete weight spaces and meromorphic OPE, then(VF, µ)is a geometric vertex algebra.

Proof. Permutation invariance holds because the multiplication maps in a pref-actorization algebra are compatible with the braiding. We have already checked all the other axioms except meromorphicity which we now assume.

Recall from Proposition 2.2.3 thatVF is meromorphic if it is bounded from below. Being bounded from below is the assumption made in [2] when con-structing a vertex algebra from a prefactorization algebra. While less general, this assumption has the pleasing feature that it does not make reference to the multiplication maps of the prefactorization algebra.

3.4 From Geometric Vertex Algebras to Prefactorization

Im Dokument Vertex Algebras and Factorization Algebras (Seite 38-46)