#### Relating Berkovits and A ∞ superstring field theories; small Hilbert space perspective

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0 Arnold Sommerfeld Center, Ludwig-Maximilians University
In a previous paper it was shown that the recently constructed action for open superstring field theory based on A∞ algebras can be re-written in Wess-Zumino-Wittenlike form, thus establishing its relation to Berkovits' open superstring field theory. In this paper we explain the relation between these two theories from a different perspective which emphasizes the small Hilbert space, and in particular the relation between the A∞ structures on both sides.
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Berkovits and A
superstring field theories;
small Hilbert space perspective
Contents
1 Introduction
2 Cohomomorphisms and field redefinitions
Coderivations and cohomomorphisms
A∞ algebras
2.4 Symplectic form and cyclicity
3 Generating the A∞ theory from a free theory
Construction of multi-string products
Construction of improper field redefinition
4 Generating the reduced Berkovits theory from a free theory Action in small Hilbert space
Non-cyclic A∞ structure
Mapping the reduced Berkovits action into a free action
5 Equivalence of actions
6 Concluding remarks A The coproduct B Proof of (4.65) C Cyclic products of the reduced Berkovits theory
Introduction
emphasizes the small Hilbert space.
be written
motion take the form
Equivalently, we can write this as
gauge fixing as the reduced Berkovits theory.
With this substitution, the equations of
defined in the large Hilbert space, we will call it an improper field redefinition.
Now consider Berkovits’ superstring field theory. The equations of motion can be
written in the form
= 0,
= 0,
theories is given by equating the respective free fields:
As shown in [4] and as will be shown in the following, this turns out to be correct.
Berkovits theory. We end with some concluding remarks.
Cohomomorphisms and field redefinitions
discussion, see Kajiura [8].
The central objects in our discussion are multi-string products
the open string state space H into one copy:
Acting on tensor products of states, the operator cn is defined
cn(A1 ⊗ . . . . ⊗ An) ≡ cn(A1, . . . , An),
cn(A1, . . . , An),
cn : H⊗n → H.
string field A, denoted deg(A), is defined to be Grassmann parity ǫ(A) plus one:1
deg(A) = ǫ(A) + 1
(mod Z2).
unit of degree,
(mod Z2),
must incorporate a sign factor into the definition of multiplication
m2(A, B) ≡ (−1)deg(A)A ∗ B.
are reexpressed
Berkovits theory.
Often it is useful to express relations between multi-string products in the form of
operator equations. Consider, in general, a pair of linear maps
We define a tensor product map
bk,m : H⊗m → H⊗k,
cℓ,n : H⊗n → H⊗ℓ.
bk,m ⊗ cℓ,n : H⊗m+n → H⊗k+ℓ
as follows:
1The change of grading from Grassmann parity to degree is often referred to as a suspension [9, 10].
(anti)commuting cn,ℓ past the first m states.
With this we can rexpress the derivation
property of Q and associativity of the star product
Qm2 + m2(Q ⊗ I + I ⊗ Q) = 0,
m2(m2 ⊗ I + I ⊗ m2) = 0.
bk,m ⊗ I⊗0 = bk,m.
expressions (2.7) and (2.8).
Coderivations and cohomomorphisms
generated by taking tensor products of the open string state space:
T H = H⊗0 ⊕ H ⊕ H⊗2 ⊕ H⊗3 ⊕ . . . .
and so satisfies
1T H ⊗ V = V ⊗ 1T H = V
which follow. We give a more complete discussion in appendix A.
We seek a way to promote an n-string product Dn to a linear operator on the tensor
DnA1 ⊗ . . . ⊗ Am = 0
DnA1 ⊗ . . . ⊗ Am = Dn(A1, . . . , Am)
for m < n,
for m = n,
DnA1 ⊗ . . . ⊗ Am =
X I⊗m−1−ℓ ⊗ Dn ⊗ I⊗ℓ
A1 ⊗ . . . ⊗ Am
for m ≥ n,
I⊗k = I ⊗ . . . ⊗ I .
Then we can write this formula
X I⊗m−1−ℓ ⊗ Dn ⊗ I⊗ℓ
If IT H is the identity operator on the tensor algebra, we have
for m < n,
for m ≥ n.
IT H =
Acting Dn on both sides on this equation gives the formula:
k=0 ℓ=0
Dn = X
X I⊗k−ℓ ⊗ Dn ⊗ I⊗ℓ
Another useful relation is
|k−1{tzimes}
corresponding to a sequence of products3
D = D0 + D1 + D2 + D3 + . . .
D0, D1(A), D2(A, B), D3(A, B, C), . . . .
Then in general we have
D =
(I⊗ℓ ⊗ Dm ⊗ I⊗n)πℓ+m+n,
products Cm and Dn, we find:4
= X
k=0 ℓ=0
X I⊗k−ℓ ⊗ [Cm, Dn] ⊗ I⊗ℓ
3The zero string product can be seen as an operator which acts on 1T H and produces a string field. We
write D0 = D0(1T H).
4We always use [·, ·] to denote the commutator graded with respect to degree, except (mostly in the
[Cm, Dn] ≡ Cm
X I⊗m−1−ℓ ⊗ Dn ⊗ I⊗ℓ
− Dn
X I⊗n−1−ℓ ⊗ Cm ⊗ I⊗ℓ .
sequence of products
[C1, D0] + [C0, D1],
[C2, D0] + [C1, D1] + [C0, D2],
[C3, D0] + [C2, D1] + [C1, D2] + [C0, D3],
[C4, D0] + [C3, D1] + [C2, D2] + [C1, D3] + [C0, D4],
a sequence of degree even multi-string products
H0, H1(A), H2(A, B), H3(A, B, C), . . .
packaged into an operator on the tensor algebra in the following way:
ℓ=1 k1,k2,...,kℓ=0
are related to field redefinitions.
A∞ algebras
is a nilpotent coderivation on the tensor algebra:
D = D1 + D2 + D3 + . . .
[D, D] = 0.
called A∞ relations:
0 = [D1, D1],
0 = [D1, D2],
0 = [D1, D3] + [D2, D2], . . . .
0 = D1(D1(A)),
variation of a 3-product D3.
can be written using (2.29)
0 = [Q, Q],
0 = [Q, m2],
0 = [m2, m2],
0 = [Q, Q],
0 = [Q, m2],
0 = [m2, m2].
k=0 ℓ=0
which implies that the coderivations
k=0 ℓ=0
Q = X
X I⊗k−ℓ ⊗ Q ⊗ I⊗ℓ
m2 = X
X I⊗k−ℓ ⊗ m2 ⊗ I⊗ℓ
Q + m2 defines an A∞ algebra:
[Q + m2, Q + m2] = 0.
product would be expressed
m2(m2 ⊗ I − I ⊗ m2) = 0
(Grassmann grading).
Grassmann even.
corresponding object in the tensor algebra called a group-like element :
= 0.
= 0.
Therefore the equations of motion are equivalent to
used (2.24). Therefore the equations of motion can be written
Ψ⊗ℓ ⊗ Dm(Ψ, . . . , Ψ) ⊗ Ψ⊗n,
|m {tizmes}
X H[Ψ′]⊗ℓ,
= 1T H + X
= Hˆ
Now we want to see what happens when we make a field redefinition
the field redefinition
to the group-like element under field redefinition:
0 = DHˆ
Hˆ −1Hˆ = Hˆ Hˆ−1 = IT H.
D′ = Hˆ −1D Hˆ,
Hk1 (Ψ′, . . . , Ψ′) ⊗ . . . ⊗ Hkℓ+1 (Ψ′, . . . , Ψ′),
ℓ=1 k1,...,kℓ=0
| k1 {tizmes }
The inverse field redefinition is simply
as can be checked:
1 − H[Ψ′] = π1Hˆ −1Hˆ
The operator D′ is a coderivation (see appendix A). Moreover, it is nilpotent
on the zero-string component of the tensor algebra:
[D′, D′] = 0,
= Hˆ −1DHˆ 1T H,
= Hˆ −1D
1 − H0
product H0. Provided H0 shifts to a vacuum which satisfies the equations of motion, D0′
Symplectic form and cyclicity
respectively with a subscript L and a subscript S:
Hilbert spaces, respectively, with normalization
hA, BiL = hI ◦ A(0)B(0)iLUHP,
hA, BiS = hI ◦ A(0)B(0)iSUHP,
it satisfies
2-string vertex which maps two copies of the state space into a complex number:
A cyclic product cn then satisfies
ωk ◦ cn = ω ◦ (. . . ◦ (ω◦ cn) . . .).
In general, multi-string products are not cyclic.
We can define a cyclically permuted
permutation of a product
to itself, we have ωn+1 ◦ cn = cn.
out not to be the case in the reduced Berkovits theory.
left-hand side of the equation
field theory.
Hilbert space:
The action takes the form
or, factoring out A ⊗ B
SA =
corresponding vertex would vanish). For example, the 2-string product is defined
M2(A, B) =
Xm2(A, B) + m2(XA, B) + m2(A, XB)i,
M2 =
Xm2 + m2(X ⊗ I + I ⊗ X)i.
changing operator X is defined
satisfies two additional properties:
f (z) = − z2
f −
f (z) = go.
The action becomes
SA =
coupling constant:
hωL|I ⊗ µ 2 = −hωL|µ 2 ⊗ I.
where it is clear that we can identify go with the open string coupling constant.
Sfree =
and plug in the field redefinition
where µ 2 is a degree even 2-product of the form
µ 2 =
1 hξm2 − m2(ξ ⊗ I + I ⊗ ξ)i.
Hilbert space. Note that µ 2 is cyclic:
Sfree =
of the vertex. The product in the cubic term is BRST exact:
Qµ 2 − µ 2(Q ⊗ I + I ⊗ Q) = [Q, µ 2].
M2 = [Q, µ 2].
+higher orders,
The action can be written
Sfree =
field theory is generated from a free action by field redefinition.
coderivation M
which takes the form
M = Q + M2 + M3 + M4 + . . .
M = Gˆ−1Q Gˆ,
it is clear that M can be derived from free equations of motion
upon substituting
of multi-string products, which we refer to as follows:5
Mn+1 = products,
mn+2 = bare products
(degree odd, picture n),
(degree odd, picture n),
µ n+2 = gauge products
(degree even, picture n + 1).
defining generating functions
the differential equations
M(t) =
X tnMn+1,
m(t) =
µ n+2 =
|n+1{tzimes}
| n t{izmes }
|n+1{tzimes}
cyclic [1].
particular if
field, we have explicitly
Gˆ(t1, t2) = P exp
μ(t) = Gˆ(0, t)−1 d Gˆ(0, t),
M(t) = Gˆ(0, t)−1Q Gˆ(0, t),
m(t) = Gˆ(0, t)−1m2Gˆ (0, t).
Gˆ ≡ Gˆ(0, 1),
M = Gˆ−1Q Gˆ.
Consider the cohomomorphism6
the generating functions in the form
= =
6The fact that this is a cohomomorphism is explained in appendix A.
explicit, the gauge 3-product appearing above is
µ 3 =
ξm3 + m3(ξ ⊗ I ⊗ I + I ⊗ ξ ⊗ I + I ⊗ I ⊗ ξ) ,
where the bare 3-product m3 is given by
m3 = [m2, µ 2],
as computed in [1].
It is helpful to explain an analogy between the cohomomorphism Gˆ and a classical
cohomomorphism Gˆ satisfies
Let us verify that Gˆ is a solution. Following [4], we compute
or, equivalently,
dt Gˆ (0, t)m(t) Gˆ(t, 1),
dt Gˆ (0, t) Gˆ(0, t)−1m2 Gˆ(0, t) Gˆ(t, 1),
dt Gˆ(0, 1),
Under this transformation the products change as
[Q, Uˆ ] = 0,
M → M′ = Vˆ −1MVˆ .
a gauge transformation.
= 0,
= 0.
Multiplying these equations by Gˆ and making the substitution
= 0,
= 0,
the tensor algebra gives
= 0,
into (3.58) we obtain the equations of motion of Berkovits superstring field theory,
solution of the A∞ theory using
the role of these equations is reversed.
so that the field redefinition is explicitly (up to cubic order)
µ 2 and µ 3 are described in equations (3.18), (3.45), and (3.46).
the actions. That is, we should have
define the vertices:9
B˜n+1 = −
k0,...,km+1≥0
k0+...+km+1=n−m
and lives in the small Hilbert space
Computing (5.1), we find that Eˆ corresponds to a sequence of products
E0 = 0,
E1 = I,
E2 = F2 − µ 2,
9Equality of vertices only implies equality of the cyclic products up to terms which are antisymmetric
works for the 2-product. We should have
Plugging in E2 from (5.8) gives
where we use the fact that µ 2 is cyclic. Thus
is an extension of the computation given in appendix C, but is rather unwieldy.
be written as a free action
to introduce an auxiliary parameter t and write the action in the form
SA =
SA = X
1 − A
1 − A
1 − A
1 − A
SA =
product. Explicitly
with GˆM = Q Gˆ to find
Now note that
SA =
= t
= t
n=1 |
dt n=1
SA =
so the action further simplifies to
SA =
Therefore we find
SA =
Mapping the reduced Berkovits action into a free action
SB = − go2 0
1 Z 1
dt DξΨB, Q (ηetξΨB )e−tξΨB E
formulas, denote
Thus we can rewrite the action
SB = −
1 Z 1
ds DξΨB, Q gs ΨB g−s E .
Switching the order of the integration over s and t,
ds =
ds (1 − s) DξΨB, Q gs ΨB g−s E .
ds (1 − s) DξΨB, ηξQ gs ΨB g−s E .
Hilbert space BPZ inner product:
ds (1 − s) DΨB, ηξQ gs ΨB g−s E .
this as a free action.
SB =
ds (1 − s) ΨB, Q gs ΨB g−s
. (5.38)
Perform the following manipulations:
SB =
ds (1 − s) DΨB, Q gs ΨB g−s E
ds (1 − s) DΨB, Q gs ΨB g−s E
ds (1 − s) D[ξΨB, gs ΨB g−s], Q gt ΨB g−t E
dt (1 − s) D[ξΨB, gt ΨB g−t], Q gs ΨB g−s E .
SB = −
1 Z 1
SB = − g2
1 Z 1
SB =
Next note that
and in particular
d gs = gs(ξΨB) = (ξΨB)gs,
Then we have
SB =
ds (1 − s) DΨB, Q gs ΨB g−s E
ds (1 − s)
dt (1 − s)
d Dgs ΨB g−s, Q gt ΨB g−t E
d Dgt ΨB g−t, Q gs ΨB g−s E ,
ds Dgs ΨB g−s, Q gt ΨB g−t E
(1 − s) Dgs ΨB g−s, Q gt ΨB g−t E
ds (1 − s) DΨB, Q gs ΨB g−s E
ds (1 − s) Dgs ΨB g−s, Q gs ΨB g−s E
ds (1 − s) DΨB, Q gs ΨB g−s E .
(1 − s) Dgs ΨB g−s, Q gt ΨB g−t E
ds Dgs ΨB g−s, Q gt ΨB g−t E
dt (1 − t) Dgt ΨB g−t, Q gt ΨB g−t E ,
ds Dgs ΨB g−s, Q gt ΨB g−t E .
ds Dgs ΨB g−s, Q gt ΨB g−t E
ds Dgt ΨB g−t, Q gs ΨB g−s E ,
Dgs ΨB g−s, Q gt ΨB g−t E ,
SB =
1 Z 1
1 Z 1
derivative. Canceling terms and integrating the total d/ds derivative,
Symmeterizing the order of integration and relabeling s and t,
SB =
Dgs ΨB g−s, Q gt ΨB g−t E ,
E0 = 0,
1 + ω + ω2 ◦ E2 = 0,
1 + ω + ω2 + ω3 ◦ E3 = 0.
field redefinition given in (4.7)
field redefinition up to cubic order have no cyclic component:
Concluding remarks
through the improper field redefinition given in (3.44)
this correspondence holds to higher orders.
Acknowledgments
of excellence Origin and Structure of the Universe.
The coproduct
△ : T H → T H ⊗′ T H.
factorized states in the tensor algebra, the coproduct gives
△1T H = 1T H ⊗′ 1T H,
△A = 1T H ⊗′ A + A ⊗′ 1T H,
+(A ⊗ B) ⊗′ C + (A ⊗ B ⊗ C) ⊗′ 1T H.
More generally,
(△ ⊗′ IT H)△ = (IT H ⊗′ △)△.
of the tensor algebra into one copy:
This allows us to unambiguously define repeated products.
▽ : T H ⊗′ T H → T H.
▽(IT H ⊗′ ▽ ) = ▽(▽ ⊗′ IT H).
△D = (D ⊗′ IT H + IT H ⊗′ D)△ .
△ Hˆ = (Hˆ ⊗′ Hˆ )△ .
△V = V ⊗′ V.
from the definitions:
Property 1. The product of two cohomomorphisms is a cohomomorphism.
inverse operator is also a cohomomorphism.
exponential is a cohomomorphism.
To see property 1, note that if Hˆ 1 and Hˆ 2 are two cohomomorphisms then
the operator Gˆ (t1, t2) introduced in (3.38), which can be defined as a limit
Gˆ (t1, t2) = lim
1, the product of all of these factors is also a cohomomorphism.
We would like to prove the explicit formulas (2.27), (2.32) and (2.50) starting from
coproduct. For this we need two identities:
πk1+k2+...+kℓ = ▽ ℓ(πk1 ⊗′ πk2 ⊗′ . . . ⊗′ πkℓ )△ℓ,
and △ℓ denote ℓ-times repeated products and coproducts, respectively. This can
be defined, for example, recursively by the formulas
≡ ▽ ℓ(▽ ⊗′ IT H ⊗′ . . . ⊗′ IT H),
△ℓ+1 ≡ (△ ⊗′ IT H ⊗′ . . . ⊗′ IT H)△ℓ.
△ℓ+1D =
Projecting on the 2-string component we can use (A.12) to find
prove (2.27), (2.32) and (2.50).
△ℓ Hˆ =
Hˆ ⊗′ . . . ⊗′ Hˆ
Note the resemblance to (2.27) and (2.32).
Now we have the ingredients needed to
ℓ−1{tzimes
+ . . . + IT H ⊗′ . . . ⊗′ IT H ⊗′D △n,
the equation
V = X πnV = 1T H + X Ψ⊗n+1 =
in agreement with (2.50).
Now we can apply a similar procedure to find the general form of a coderivation D.
Therefore we can write
Compute the projection on the 2-string component,
and the sixth step we again used (A.12). Similarly computing
π3D = ▽ 2(π1 ⊗′ π1 ⊗′ π1)△2D,
= ▽(π1 ⊗′ π1)(D ⊗′ IT H + IT H ⊗′ D)△,
= X (Dm ⊗ I + I ⊗ Dm)πm+1.
In general, such a projection can take the form
= X ▽((dmπm) ⊗′ π1 + π0 ⊗′ (Dmπm))△,
= X h(dm ⊗ I)πm+1 + Dmπmi,
= X (dm ⊗ I)πm+1 + π1D.
∞ n
m=0 n=0 k=0
= X
X X(I⊗k ⊗ Dm ⊗ I⊗n−k)πm+n,
(I⊗ℓ ⊗ Dm ⊗ I⊗n)πℓ+m+n.
general form
which reproduces the expression (2.27).
Finally, let’s consider cohomomorphisms. We can take
onto the 2-string component
k1,k2,k3=0
general it can take the form
X (Hk1 ⊗ Hk2)▽(πk1 ⊗′ πk2)△,
X (Hk1 ⊗ Hk2)πk1+k2.
π3Hˆ = ▽ 2(π1 ⊗′ π1 ⊗′ π1)△2Hˆ
k1,k2=0
k1,k2=0
X (hk1 ⊗ hk2)▽(πk1 ⊗′ πk2)△,
X (hk1 ⊗ hk2)πk1+k2.
Comparing to (A.30) implies
hk = X hi ⊗ hk−i,
h0 = h0 ⊗ h0,
h1 = h0 ⊗ h1 + h1 ⊗ h0,
h2 = h0 ⊗ h2 + h1 ⊗ h1 + h2 ⊗ h0.
Similarly for the 3-string component we compute
Comparing to (A.30)
have h0 = I⊗0 and
Therefore a nonvanishing cohomomorphism takes the general form
ℓ=1 k1=0
kℓ=0
X . . . X (Hk1 ⊗ Hk2 ⊗ . . . ⊗ Hkℓ)πk1+k2+...+kℓ.
which reproduces the expression (2.32).
Proof of (4.65)
In this appendix we prove the identity (4.65):
It is convenient first to demonstrate a related formula satisfied by the products Fn+1:
[η, Fn+2] = go X m2(Fk+1 ⊗ Fn−k+1).
Hk = h0 ⊗ Hk.
that it holds for n → n + 1, as follows:
+ X m2(m2(Fk+1 ⊗ Fn−k+1) ⊗ ξ) ,
under m2:
m2(I ⊗ Fn+2 + Fn+2 ⊗ I)
+ X m2(m2(Fk+1 ⊗ ξ) ⊗ Fn−k+1))
+ X m2(Fk+1 ⊗ m2(ξ ⊗ Fn−k+1)) .
In the first step we use formula (4.46):
− X m2(Fk+1 ⊗ m2(Fn−k+1 ⊗ ξ)) .
Fm+2 = −
m + 2 m2(ξ ⊗ Fm+1 + Fm+1 ⊗ ξ).
+ X(n − k + 2)m2(Fk+1 ⊗ Fn−k+2) ,
terms cancel due to the associativity of m2, leaving
m2(I ⊗ Fn+2 + Fn+2 ⊗ I)
the upper and lower extremes of summation:
[η, Fn+3] = go X m2(Fk+1 ⊗ Fn−k+2),
to show that (B.2) holds for n = 0:
[η, F2] = − [η, m2(ξ ⊗ I + I ⊗ ξ)],
= gom2,
= gom2(F1 ⊗ F1),
which it does.
We can return to the proof of (4.65). We can write (4.65) in an equivalent form
algebra and project onto the 1-string component:
This vanishes as just demonstrated. We also have
can write
= ▽
phisms. This establishes (4.65).
SB = − go2 0
1 Z 1
dt DξΨB, Q (ηetξΨB )e−tξΨB E .
reduces to M = Gˆ−1Q Gˆ.
We start with the reduced Berkovits action
k=0 j=0
Cyclic products of the reduced Berkovits theory
In this appendix we show that the cyclic products B˜n+1 of the reduced Berkovits theory
theory to a free theory. Specifically, we prove the formula
k=0 j=0
SB =
go n=0 n + 2 ωS(ΨB, ηξQFn+1(ΨB, . . . , ΨB)).
| n+1{tzimes }
From this we can read off a sequence of degree odd non-cyclic products
B˜nn+on2cyclic =
products B˜n+2 are obtained by taking the cyclic projection:
B˜n+2 =
Plugging in,
B˜n+2 =
= QFn+2 + X ξQm2(Fk+1 ⊗ Fn−k+1).
+ X X ωj+1 ◦ (ξQm2(Fk+1 ⊗ Fn−k+1)) .
Computing the cyclic permutations we find
for the second term, and either
the number of cyclic permutations taken:
QFn+2 − X(ωj+1 ◦ Fn+2)(I⊗j ⊗ Q ⊗ In+1−j)
B˜n+2 =
k=0 j=0
n n−k
k=0 j=0
k=0 j=0
k=0 j=0
for 0 ≤ j ≤ n − k,
for 0 ≤ j ≤ k
n n−k
k=0 j=0
Relabeling indices allows us to combine the fourth and fifth terms:
B˜n+2 =
Then we find:
B˜n+2 =
QFn+2 + X(ωj+1 ◦ Fn+2)(I⊗j ⊗ Q ⊗ In+1−j)
B˜n+2 =
k=0 j=0
k=0 j=0
k=0 j=0
k=0 j=0
To simplify further we need to compute the cyclic permutations of equation (4.46):
Fm+2 = −
m + 2 m2(ξ ⊗ Fm+1 + Fm+1 ⊗ ξ).
(ωj+1 ◦ Fm+1)(I⊗j ⊗ m2(I ⊗ ξ) ⊗ I⊗m−j)
+(ωj ◦ Fm+1)(I⊗j−1 ⊗ m2(ξ ⊗ I) ⊗ Im−j) ,
(1 ≤ j ≤ m),
mismatch at the extremes of summation over j:
last 3 terms in (C.13)
Let us focus on the last three terms in (C.13):
last 3 terms in(C.13)
k=0 j=0
k=0 j=0
k=0 j=1
k=0 j=1
permutations of Fk+2, so we arrive at a simplification
k=0 j=1
k=0 j=0
k=0 j=0
k=0 j=0
k=0 j=0
Therefore (C.13) simplifies to
B˜n+2 =
QFn+2 + X(ωj+1 ◦ Fn+2)(I⊗j ⊗ Q ⊗ In+1−j)
In the last term relabel k → k − 1
− X
of summation over k:
k=0 j=0
B˜n+2 = − X
X(ωj+1 ◦ Fk+1)(I⊗j ⊗ QFn−k+2 ⊗ Ik−j),
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