Two new results on spanning sets for vertex operator algebras and

Two new results on spanning sets for vertex operatoralgebras and their quasimodules

Geoffrey Buhl

California State University Channel Islands

July 9, 2008

Geoffrey Buhl (CSUCI) Spanning Set Results July 9, 2008 1 / 19

Outline

1 A whirlwind tour of spanning sets

2 Spanning sets for quasimodules

3 Spanning sets for vertex operator algebras

4 Theorem ingredients for widget cranking


Notation, always with the notation . . .

V - a vertex operator algebra with Vn = 0 for n < 0

modules are N-gradable a.k.a. admissible

CN(V ) = {u−Nv : u, v ∈ V } for N ≥ 2

C1(V ) = span{u−1v , L(−1)u : u, v ∈⊕

n>0 Vn}

XN a set of homogeneous representatives of V /CN(V )

wtun = wt(u)− n − 1 for u homogeneous


A whirlwind tour of spanning sets

Minimal spanning set

Theorem (Karel-Li)

For a vertex operator algebra V , let X1 be a set of homogeneousrepresentatives of a spanning set for the quotient space V /C1(V ), V isspanned by the elements of the form

x1n1· · · x r

nr1,

where r ∈ N; x1, . . . , x r ∈ X1; n1, . . . , nr ∈ Z; andwt(x1

n1) ≥ · · · ≥ wt(x r

nr) > 0.

another equivalent ordering restriction is: n1 ≤ · · · ≤ nr < 0



No-repetition spanning set

Theorem (Gaberdiel-Neitzke 2003)

For a vertex operator algebra V , let X2 be a set of homogeneousrepresentatives of a spanning set for the quotient space V /C2(V ), V isspanned by the elements of the form

x1n1· · · x r

nr1,

where r ∈ N; x1, . . . , x (r) ∈ X2; n1, . . . , nr ∈ Z; and n1 < · · · < nr < 0.



Module spanning sets

No-repetition spanning sets for:

Modules for C2-cofinite vertex operator algebras with V0 = C1 (B2002)

Modules for vertex operator algebras and modern proof (Miyamoto2004)

Twisted modules for vertex operator algebras (Yamauchi 2004)


Spanning sets for quasimodules

Which quasi is this?

Locality axiom for vertex operator algebras:

For any Y (u, x1) and Y (v , x2), there exists a non-negative integer k suchthat

(x1 − x2)k [Y (u, x1),Y (v , x2)] = 0.

Quasilocality axiom for vertex operator algebras:

For any Y (u, x1) and Y (v , x2), there exists a non-zero f (x1, x2) ∈ C[x1, x2]such that

f (x1, x2)[Y (u, x1),Y (v , x2)] = 0.



Which quasi is this?

Locality axiom for vertex operator algebras:

For any Y (u, x1) and Y (v , x2), there exists a non-negative integer k suchthat

(x1 − x2)k [Y (u, x1),Y (v , x2)] = 0.

Quasilocality axiom for vertex operator algebras:

For any Y (u, x1) and Y (v , x2), there exists a non-zero f (x1, x2) ∈ C[x1, x2]such that

f (x1, x2)[Y (u, x1),Y (v , x2)] = 0.



. . . and the definition is . . .

Definition

A quasimodule for a vertex operator V is a vector space W satisfies thenormal module axioms with the Jacobi identity replaced by thequasi-Jacobi identity: for u, v ∈ V there exists a non-zero polynomialf (x1, x2) ∈ C[x1, x2] such that,

x−10 δ

(x1−x2

x0

)f (x1, x2)YW (u, x1)YW (v , x2)

−x−10 δ

(x2−x1−x0

)f (x1, x2)YW (v , x2)YW (u, x1)

= x−12 δ

(x1−x0

x2

)f (x1, x2)YW (Y (u, x0)v , x2).



Cold war doctrine? No, mathematical definition.

Definition

A vector w of a module W for a vertex operator algebra V is uniformlyannihilated by a set of vectors X ⊆ V if there exists T ∈ N such thatxnw = 0 for any x ∈ X , n ≥ T . The smallest such T for a given X iscalled the order of uniform annihilation of w by X .



Difference- one spanning set for quasimodules

Theorem (B)

For a vertex operator algebra V , X2 a set of homogeneous representativesof a basis for V /C2(V ), and a quasimodule module W generated by avector w that is uniformly annihilated by X2, W is spanned by elements ofthe form

x1n1· · · x r

nrw (1)

with n1 < · · · < nr < T where r ∈ N; x1, . . . , x r ∈ X2; n1, . . . , nr ∈ Z; andT is order of uniform annihilation of w by X2.


Spanning sets for vertex operator algebras

Revisit algebra spanning sets

u1n1

u2n2· · · ur

nr1

Spanning Set Generating space Ordering

Karel-Li X1 n1 ≤ · · · ≤ nr < 0Gaberdiel-Neitzke X2 n1 < · · · < nr < 0

......

XN+1 difference-N




u1n1

u2n2· · · ur

nr1


Karel-Li X1 difference-zero conditionGaberdiel-Neitzke X2 difference-one condition

......

XN+1 difference-N




u1n1

u2n2· · · ur

nr1


Karel-Li X1 difference-zero conditionGaberdiel-Neitzke X2 difference-one condition

......

XN+1 difference-N



Difference-N spanning set for VOAs

Theorem (B-Karaali)

For a vertex operator algebra V , let XN+1 be a set of homogeneousrepresentatives of a spanning set for the quotient space V /CN+1(V ), V isspanned by the elements of the form

x1n1

x2n2· · · x r

nr1 (2)

where r ∈ N; x1, . . . , x r ∈ XN+1; n1, . . . , nr ∈ Z−; andn1 < n2 < · · · < nr < 0 with ni − ni+1 ≥ N for each 1 ≥ i ≥ r − 1.


Theorem ingredients for widget cranking

Ingredients needed to crank the theorem widget

u1n1

u2n2· · · ur

nr1

Proof ingredients:

A nice filtration - framework for induction argument of proof

Commutativity identity - reordering modes

Associativity identity - replacement and straightening modes



Difference-one filtration

V (0) ⊂ V (1) ⊂ V (2) ⊂ · · · ⊂ V ,

where

V (s) = span{u1n1

u2n2· · · ur

nr1 :

r∑i=1

wtui ≤ s}



Commutativity identities

For u, v ∈ V a vertex algebra and m, n ∈ Z:

Commutativity identity:

[um, vn] =∑j≥0

(mj

)(ujv)m+n−j

Compatible with filtration giving a reordering property: Givenu1n1

u2n2· · · uk

nk1 ∈ V (s), and a permutation σ ∈ Sk , we have:

u1n1

u2n2· · · uk

nk1 = u

σ(1)nσ(1)

uσ(2)nσ(2)· · · uσ(k)

nσ(k)1 + R

where R ∈ V (s−1).



Associativity identities

For u, v ∈ V a vertex algebra and m, n ∈ Z:

Associativity identity:

(umv)n =∑i≥0

(mi

)(−1)i (um−ivn+i − (−1)mvm+n−iui )

Compatible with filtration giving a replacement property:

Given u1n1

u2n2· · · uk

nk1 ∈ V (s),

u1n1

u2n2· · · uk

nk1 = x1

n1x2n2· · · xk

nk1 + R,

where x i is a representative of ui + C2(V ) and R ∈ V (s−1).



Straightening identities

Use one instance of the associativity identity to remove repeated modes:

unvn = (u−1v)2n+1 −∑

i≥0,i 6=−n

u−1−iv2n+i −∑i≥0

v2n−iui .

Use two instances of the associativity identity to remove adjacent modeswith indices that differ by one:

un−1vn = (n + 1)((u−1v)2n − A0,n,1(u, v)− B0,n(u, v))

+ ((u−2v)2n+1 − A1,n,1(u, v) + B1,n(u, v))



Straightening identities

Use one instance of the associativity identity to remove repeated modes:

unvn = (u−1v)2n+1 −∑

i≥0,i 6=−n

u−1−iv2n+i −∑i≥0

v2n−iui .

Use two instances of the associativity identity to remove adjacent modeswith indices that differ by one:

un−1vn = (n + 1)((u−1v)2n − A0,n,1(u, v)− B0,n(u, v))

+ ((u−2v)2n+1 − A1,n,1(u, v) + B1,n(u, v))



Quasi-versions of associativity and commutativity

Associativity identity for quasimodules:∑L≥i ,j≥0

∑k≥0

aij

(i

k

)(um+kv)n+i+j−k

=∑

L≥i ,j≥0

∑k≥0

(−1)k

(m

k

)aijum+i−kvn+j+k

−∑

L≥i ,j≥0

∑k≥0

(−1)k+m

(m

k

)aijvm+n+j−kui+k

Commutativity identity for quasimodules:∑L≥i ,j≥0

aij [um+i , vn+j ] =∑

L≥i ,j≥0

∑k≥0

aij

(m + i

k

)(ukv)m+n+i+j−k

where f (x1, x2) =∑

L≥i ,j≥0 aijxi1x

j2



Conclusion

Difference-one (or higher difference) spanning sets should exist for vertexoperator algebra-like objects whenever you have some formulation ofassociativity and commutativity for those objects.


Documents

Two new results on spanning sets for vertex operator algebras and