3.4 From Geometric Vertex Algebras to Prefactorization
Proof. The exterior product of functionals is compatible with pushforwards along complex-analytic embeddings. This implies that the multiplication maps for E are well-defined w. r. t. the permutation actions. Taking exterior prod-uct of functionals is a bounded linear map and the braiding of CBVS is as well, so the multiplication maps ofE are bounded linear maps. Unitality and associativity follow from the behavior of the exterior product of functionals.
Compatibility with the braiding is enforced by modding out the action of the permutation groups, letτ ∈Σm+n be the permutation moving the firstm elements past the lastnelements: We want to show that
βα(f) =β(y7→α(x7→f(y, x))) is equal to
τ∗(αβ)(f) =αβ(τ∗f) =α(x7→β(y7→τ∗f(x, y))) =α(x7→β(y7→f(y, x))) for
f ∈ O((UtV)m+n\∆)
α∈ O0(Um\∆), β∈ O0(Vn\∆) . The functionalsβαand τ∗(αβ) are the pushforwards of
β×α, τ∗(α×β)∈ O0((Vn\∆)×(Um\∆)) .
Thus, it suffices to check the desired equality on the dense subset of holomorphic functionsf which are a finite sum of functionsh(y)g(x) ofx∈Um\∆ andy∈ Vn\∆ forg andhholomorphic. Forf(y, x) =h(y)g(x),
β×α(f) =α(g)β(h) = (τ∗(α×β))(f) .
To identify the underlying precosheaf ofE with the precosheaf defined earlier, we note that multiplying with the unit in C ∼= O0(pt) = O0(∅0\∆0) is the pushforward of analytic functionals.
Now using the multiplication maps ofV, we define theevaluation map evU :EV(U)→ Vbb
forU ⊆Copen. Let α⊗a∈ O0(Um\∆)⊗ Vm. As part of the structure of a geometric vertex algebra, we have a holomorphic functionµ(a) :Um\∆→ V.
We would like to evaluateαon it to obtain an element ofV. Letk ∈Z. The holomorphic functionpk◦µ(a) takes values in a finite-dimensional subspaceF ⊆ Vk. The evaluation pairing tensored withF gives a map (X :=Um\∆)
O0(X)⊗ O(X;F)∼=O0(X)⊗ O(X)⊗F →C⊗F ∼=F
so we define the evaluation ofαonpk◦µ(a) to be the image ofα⊗(pk◦µ(a)) under this map to get an element ofVk for all k∈Z. Let α(µ(a)) denote the resulting element ofV. What we have just described more generally applies to any openX ⊆Cmand any holomorphic functionf ∈ O(X;V) andα∈ O0(X) and definesα(f)∈ V, whereV is aZ-graded vector space. It is clear thatα(f) depends linearly and boundedly on α⊗f, by the universal property of the
productV this means linearity and boundedness in each component. It follows thatα(µ(a)) depends linearly and boundedly onα⊗a, recall thatV⊗mis given the discrete bornology. If (ei)i is a basis of a finite-dimensional subspaceF in whichpk◦f takes values for somek∈Z, then
α(f)k=X
i
α(e∗i(pk◦f))ei.
The permutation invariance ofµ and the universal property of the direct sum give a well-defined and bounded linear map
evU :EV(U)−→ V [α⊗a]7−→α(µ(a)) .
The collection of the evU forU ⊆Copen defines a map of precosheaves fromE to the constant precosheaf constV.
Definition 3.4.4. The precosheaf R on C is defined as the kernel of ev : E(U)→constV, that is,
R(U) = ker evU ⊆E(U)
forU ⊆Copen. We call the elements ofR(U)relations onU.
Remark 3.4.5. Note thatR(U)⊆E(U) is b-closed as the kernel of a bounded linear map since such maps are continuous w. r. t. the b-topology.
The multiplication onEdoes not induce a multiplication onE/Rin general.
Example 3.4.6. We consider the case that there are a, b ∈ V s. t.a(m)b 6= 0 for somem ≥0. For example, this is the case for V the free boson,a=b the generator andm= 1. Let
U1=B2(0)\B1(0) U2=B1(0)
W =B2(0)
so that we haveU1∩U2=∅andU1, U2⊆W. Our goal is to show that there is no dotted arrow making
E(U1)⊗E(U2) E(W)
E(U1)/R(U1)⊗E(U2)/R(U2) E(W)/R(W)
commute, where the left map is the tensor product of the quotient maps and the right map is the quotient map.
Letα∈ O0(U1) andβ ∈ O0(U2) be the analytic functionals α:h∈ O(U1)7→
Z
3 2S1
zmh(z)dz
resp.
β :h∈ O(U2)7→h(0) .
Thenα⊗aandβ⊗brepresent classesx∈E(U1) resp.y∈E(U2). The image ofx⊗y inE(W) underMWU1,U2 is represented by αβ⊗a⊗b, where
αβ:h∈ O(W2\∆)7→α[z7→β[w7→h(z, w)]] = Z
3 2S1
zmh(z,0)dz. The image of this class in E(W)/R(W) is non-zero because this space maps injectively toV by construction and the image there is
evW(MWU1,U2([x⊗y])) = Z
3 2S1
zmµ(a, z, b,0)dz=a(m)b6= 0 .
If there were a dotted arrow making the diagram commute, then the lower composition would give zero since the image ofxinE(U1)/R(U1) is zero because
evU1(x) = Z
3 2S1
zmµ(a, z)dz= 0 since the integrand is defined onC and holomorphic.
Nevertheless, the multiplication maps ofE induce a multiplication onF = E/Rdisc; this is Proposition 3.4.9, whereRdiscare the relations visible on round, open discs. A(round, open) discis a subset D⊆C which is equal to
Br(z) ={w∈C| |z−w|< r}
for somez∈Cand some real numberr >0.
Definition 3.4.7. The multiplication maps ofE assemble to give a map M
d,W
R(d)⊗E(W)−→E(U)
where the direct sum runs over all discs d ⊆ U and open subsets W ⊆ U s. t.d∩W =∅. LetRdisc(U)⊆E(U) be the image of this map,
Rdisc(U) = im
M
d,W
R(d)⊗E(W)−→E(U)
.
We callRdisc the precosheaf of relations on discs. Let F be the precosheaf of complete bornological vector spaces defined by
F(U) =E(U)/Rdisc(U)
forU ⊆C open. If we want to make the dependence onV explicit, we use the notationFV forF.
Note thatF(U) is complete because E(U) is complete by Proposition 3.4.2 and because we are modding out by the b-closure ofRdisc(U). The author does not know ifRdisc(U) is b-closed or not. If U is a disc, thenRdisc(U) =R(U) by Lemma 3.5.1 further below, andR(U) is b-closed as the kernel of a bounded map. To prove thatFinherits multiplication maps fromE, we slightly rephrase
its definition so that it only refers to complete bornological spaces. LetU ⊆C open. The source of the map
M
d,W
R(d)⊗E(W)−→E(U)
as in the definition ofRdisc(U) is not complete, but Sdisc(U) :=M
d,W
R(d) ¯⊗E(W) ,
with d, W as above, is a completion. The space of relations R(U) ⊆E(U) is b-closed, because it is the intersection of the kernels of the components of evU, each of which is a bounded map, see its construction. SinceE(d) is complete, its b-closed subspaceR(d) is complete, too.
Proposition 3.4.8. The mapE(U)→F(U)is a cokernel of the map Sdisc(U)→E(U)
given by the sum of the multiplication maps.
Proof. This amounts toRdisc(U) =IwhereIis the image ofSdisc(U) inE(U).
Letd, W be as above. The algebraic tensor productR(d)⊗E(W) maps to its completionR(d) ¯⊗E(W), soRdisc(U)⊆I, henceRdisc(U)⊆I. Every algebraic tensor product has dense image in its completion, implying thatRdisc(U)⊇I andRdisc(U)⊇I.
Proposition 3.4.9. F is a prefactorization algebra onCwith its multiplication maps induced from those of E. The multiplication maps of E induce unique multiplication maps on F, also denoted M, such that the square
E(U) ¯⊗E(V) E(X)
F(U) ¯⊗F(V) F(X)
MXU,V
MXU,V
commutes for all X ⊆ C open and U, V ⊆ X open and disjoint, where the vertical maps are the completed tensor product of the quotient maps resp. the quotient map.
Proof. Let U and V be open and disjoint and contained in an open X ⊆ C.
Since cokernels in CBVS commute with ¯⊗in both variables separately, Propo-sition 3.4.8 implies that the mapE(U) ¯⊗E(V)→F(U) ¯⊗F(V) is the cokernel of the multiplication map
T :=Sdisc(U) ¯⊗E(V)⊕E(U) ¯⊗Sdisc(V)→E(U) ¯⊗E(V) .
For Y ⊆ C open, let qY : E(Y)→ F(Y) denote the quotient map. We want to see that T maps to zero under qX ◦MXU,V, as it then follows that F is a prefactorization algebra because E is a prefactorization algebra. To show that Sdisc(U) ¯⊗E(V) maps to zero, let d, W ⊆ U be as in the definition
of Sdisc(U). Using the associativity of E, we see that the map on the sum-mand (R(d) ¯⊗E(W)) ¯⊗E(V) factors through the summandR(d) ¯⊗E(WtV) ofSdisc(X). To show thatE(U) ¯⊗Sdisc(V) maps to zero, we also use the com-patibility of M with the braiding. This time d, W ⊆ V and the map on the summandE(U) ¯⊗(R(d) ¯⊗E(W)) factors through the summandR(d) ¯⊗E(UtW) ofSdisc(X).
To describe E as an affine-linearly invariant prefactorization algebra on C, letL0denote the grading operator ofV. This is the endomorphism ofV defined by L0a = |a|a for a homogeneous, so that λL0v = λ|a|a for a ∈ V homoge-neous andλ ∈C×. This defines an action ofC× on V and also C×nC be-cause (λ, w)7→λis a group homomorphism. ThereforeEV is an affine-linearly invariant prefactorization algebra where
σ(λ,w)U (α⊗a1⊗. . .⊗an) = (λ, w)∗(α)⊗λL0a1⊗. . .⊗λL0an
forα∈ O0(Un\∆), a1, . . . , an ∈ V, andU ⊆C open. Here, the pushforward of an analytic functionalα∈ O0(X) along a holomorphic mapg:X →Y is
g∗α:= [f ∈ O(Y)7→α(x7→f(g(x)))] .
We now proceed to show thatFV is affine-linearly invariant with the mapsσUg induced from those of EV. The evaluation maps are natural w. r. t. inclusions because they evaluate a function on Cn\∆ only depending on some vertex algebra elements which are not changed by the extension maps ofEV.
Proposition 3.4.10. LetV be a geometric vertex algebra. The evaluation maps areC×nC-equivariant in the sense that the square
EV(U) Vbb
EV(gU) Vbb σgU
evU
g evgU
commutes forU ⊆C open andg∈C×nC.
Recall that translations act on Vbb, the set of bounded-below vectors inV, by Proposition 2.1.4.
Proof. We check equivariance for translation and multiplication maps sepa-rately. Let
[α⊗a]∈(O0(Um\∆)⊗ Vm)Σm⊆EV(U) . Ifg∈C×nC, then
evgU(σUg([α⊗a])) = evgU([g∗α⊗(λ.a)]) = (g∗α)(µ(λ.a))
=α(z7→µ(g.a)(g.z)) . Ifg∈C×nC is multiplication with someλ∈C×, then
evgU(σgU([α⊗a]))
=α(z7→µ(λ.a)(λ.z))
=α(z7→λ.µ(a)(z)) (C×-equivariance)
=λ.α(µ(a)) (linearity)
=λ.evU([α⊗a]) ,
and thus
g.evU([α⊗a]) = evgU(σg([α⊗a])) . Ifg∈C×nCis translation by somew∈C, then
evgU(σgU([α⊗a]))
=α(z7→µ(a)(z+w))
=α(z7→g.µ(a)(z)) (associativity for m= 0)
=g.α(µ(a)) (linearity)
=g.evU([α⊗a]) .
Equivariance follows because C× nC is generated by the union of the two subgroups corresponding toC× andC.
Proposition 3.4.11. FV is an affine-linearly invariant prefactorization algebra onC with its invariance maps induced from those of EV.
Proof. We have to show that the maps σUg for E pass to the quotient, their properties then follow. The image of a disc under an affine-linear map is again a disc. The induced map on E preserves membership in R and hence Rdisc becauseR(U) is the kernel of evU and the evaluation maps are compatible with the actions of the affine-linear group by Proposition 3.4.10.
The rest of this subsection consists of a sequence of propositions proving that F is holomorphic, i. e., F is a holomorphic affine-linearly invariant pre-cosheaf on C. We first prove this for O0 using the Cauchy integral formula and the first of these propositions, then proceed to V, which is a G = C× -representation, thenE, and finallyF.
Proposition 3.4.12. Let D⊆Cbe a closed disc. The map D◦−→C0(∂D)
w7−→
z7→ 1 z−w
is holomorphic.
Proof. Letw0∈D◦ so ε:=dist(∂D, w)>0. Let δ=ε/2. Forw∈Bδ(w0)⊆ D◦ andz∈∂D,
q(z) :=
w−w0 z−w0
<1 and thus
1
z−w = 1
(z−w0)−(w−w0) = 1 z−w0
1 1−w−wz−w0
0
= 1
z−w0
∞
X
k=0
w−w0 z−w0
k
=
∞
X
k=0
1
(z−w0)k+1(w−w0)k.
We now prove uniform convergence of functions ofz∈∂D, that is,
z7→ 1 z−w
=
∞
X
k=0
z7→ 1
(z−w0)k+1
(w−w0)k inC0(∂D). The remainder term
1 z−w−
N
X
k=0
1
(z−w0)k+1(w−w0)k
=
∞
X
k=N+1
1
(z−w0)k+1(w−w0)k
≤ 1
|z−w0|
∞
X
k=N+1
w−w0
z−w0
k
≤ 1
|z−w0|
q(z)N+1 1−q(z)
converges to zero uniformly inz∈∂D because q(z) =
w−w0 z−w0
≤ dist(w, w0)
dist(∂D, w0) <1/2 .
Proposition 3.4.13. If X⊆Cn is open, then δ:X−→ O0(X)
x7−→δx is holomorphic.
Proof. Let x∈ X. There is a closed polydisc D = D1×. . .×Dn ⊆ X with center x. Let d = d1×. . .×dn be an open polydisc concentric with D s. t.
the radius of di is strictly than the radius of Di for i = 1, . . . , n. We claim that δ(d) ⊆ O0D(X) and that δ|d is holomorphic as a map to the Banach spaceO0D(X). Letz∈dandf ∈ O(X). We haveδz∈ OD0 (X) since
|δz(f)|=|f(z)| ≤ ||f||D. The map
F:d−→C0(∂D1×. . .×. . . ∂Dn) w7−→
"
z7→
n
Y
i=1
1 zi−wi
#
is holomorphic since the maps
d−→pri di−→C0(∂Di) w7−→
zi7→ 1 zi−wi
are holomorphic fori= 1, . . . , nby Proposition 3.4.12 and the pointwise exterior product of holomorphic maps is again holomorphic. The map
G:C0(∂D1×. . .×∂D2)−→ OD0 (X) f 7−→
g7→ 1 (2πi)n
Z
∂D1×...×...∂Dn
f(z)g(z)dz
is bounded because
1 (2πi)n
Z
∂D1×...×...∂Dn
f(z)g(z)dz
≤ ||f||||g||D.
The map δ|d is analytic because it is equal to G◦F by the Cauchy integral formula.
Proposition 3.4.14. Let G be a complex-analytic group manifold acting on a complex-analytic manifold X via an analytic map G×X → X. Then the precosheafO0of analytic functionals onXis a holomorphicG-invariant cosheaf.
Proof. LetU, V ⊆X be open. We wish to show that the action map DU,V −→BVS(O0(U),O0(V))
g7−→g∗
is holomorphic on the interior D◦U,V of the domain. For each g ∈ D◦U,V, the mapg∗ is the composite of
(u7→(g, u))∗:O0(U)→ O0(D◦U,V ×U) (24) and the bounded, linear pushforward map
O0(D◦U,V ×U)−→ O0(V) along the action map
D◦U,V ×U −→V .
The latter of these is independent ofg and pointwise composition with a fixed bounded linear map preserves holomorphicity, so it suffices to show that the map
I:D◦U,V −→BVS(O0(U),O0(DU,V◦ ×U)) g7−→(u7→(g, u))∗
is holomorphic. Proposition 3.4.13 forX =D◦U,V implies that D◦U,V −→ O0(D◦U,V)
g7−→δg
is holomorphic. It follows that
D◦U,V −→BVS(O0(U),O0(DU,V◦ )) ¯⊗ O0(U)) g7−→[α7→δg⊗¯ α]
is holomorphic because the map
O0(U)−→ O0(D◦U,V) ¯⊗ O0(U)) α7−→δg⊗¯ α
is bounded and linear. Postcomposing with the exterior product of functionals
×:O0(D◦U,V) ¯⊗ O0(U)−→ O0(D◦U,V ×U) preserves holomorphicity. Overall, the map
J :D◦U,V −→ O0(D◦U,V ×U) g7−→δg×α
is holomorphic. It remains to identify I with J. Let g ∈ D◦U,V. It suffices to show thatI(g) andJ(g) agree on the dense subset of product functionsp(z, w) = f(z)h(w),(z, w)∈ D◦U,V forf ∈ O(DU,V◦ ) andh∈ O(U):
J(g)(p) =δg(z7→α(w7→f(z)h(w))) =δg(f)α(h) =f(g)α(h) , which coincides with
I(g)(p) = ((u7→(g, u))∗(α)) (p) =α((u7→(g, u))∗p) =α(u7→p(g, u))
=α(u7→f(g)h(u)) =f(g)α(h) .
Proposition 3.4.15. IfAandB are holomorphic invariant precosheaves, then their pointwise completed tensor productA⊗¯B is a holomorphic invariant pre-cosheaf in a natural way.
Proof. The functionρAU,V⊗B¯ is holomorphic because
⊗¯ : BVS(A(U), A(V)) ¯⊗BVS(B(U), B(V))→BVS(A(U) ¯⊗B(U), A(V) ¯⊗B(V)) is a bounded linear map so postcomposing with it preserves holomorphicity.
Proposition 3.4.16. LetF1, F2 be affine-linearly invariant precosheaves onC of complete bornological vector spaces. IfF2 is a quotient ofF1, the invariance isomorphisms ofF1are induced from those ofF1, andF1is holomorphic, thenF1 is holomorphic.
Here, quotient means cokernel by some map of precosheaves of complete bornological spaces.
Proof. LetX, Y be complete bornological spaces andA⊆Xa sub vector space.
The map
{f ∈BVS(X, Y)|f|A= 0} −→BVS(X/A, Y) (25) sendingf to the map induced by f is a bounded linear map. Let i:G ,→F1 be a pointwise inclusion of precosheaves of which q : F1 → F2 is a cokernel.
ForU, V ⊆C open, we apply this to X =F1(U),A =R(U), and Y =F2(V) to conclude that the action map ρFU,V2 for F2 is holomorphic because it is the composite ofq∗◦ρFU,V1 with the bounded linear map from (25).
Proposition 3.4.17. Direct sums of holomorphic precosheaves are again holo-morphic.
Proof. This follows from the compatibility of direct sums with the enrichment of CBVS over itself and Proposition 3.1.15 that a product-valued function is holomorphic if and only if its components are.
The same argument proves that direct sums of holomorphic representations are holomorphic, so V is holomorphic as the direct sum of holomorphic one-dimensional representations. Thus the constant precosheaf assigningVto every open is holomorphic.
Proposition 3.4.18. The precosheavesE andF onC are holomorphic.
Proof. The preceding propositions imply that, forn≥0 the precosheaves given byO0(Un\∆)⊗ Vn onU⊆Care holomorphic. Proposition 3.4.16 and Propo-sition 3.4.17 about quotients resp. sums imply thatE is holomorphic. Proposi-tion 3.4.16 implies thatF is holomorphic.