Power series and quantum fields

In this section, we are going to study different kinds of power series: polynomials, formal power series, Laurent polynomials, Laurent series and, finally, a notion of “formal power series” which can be infinite “in both directions”. Each of these kinds of power series will later be used in our work; it is important to know the properties and the shortcomings of each of them.

3.3.1. Definitions

Parts of the following definition should sound familiar to the reader (indeed, we have already been working with polynomials, formal power series and Laurent polynomials), although maybe not in this generality.

Definition 3.3.1. For every vector space B and symbol z, we make the following definitions:

(a) We denote by B[z] the vector space of all sequences (b_n)_n∈

N∈ B^N such that only finitely many n ∈ N satisfy b_n 6= 0. Such a sequence (b_n)_n∈

N is denoted by P

n∈N

b_nzⁿ. The elements of B[z] are called polynomials in the indeterminate z over B (even when B is not a ring).

(b) We denote by B[[z]] the vector space of all sequences (b_n)_n∈

N ∈ B^N. Such a sequence (bn)_n∈

N is denoted by P

n∈N

bnzⁿ. The elements of B[[z]] are called formal power series in the indeterminate z over B (even when B is not a ring).

(c)We denote by B[z, z⁻¹] the vector space of all two-sided sequences (b_n)_n∈

Z ∈ B^Zsuch that only finitely manyn∈Zsatisfyb_n 6= 0. (Atwo-sided sequence means a sequence indexed by integers, not just nonnegative integers.) Such a sequence (b_n)_n∈

is denoted by P Z n∈Z

b_nzⁿ. The elements ofB[z, z⁻¹] are called Laurent polynomials in the indeterminate z over B (even whenB is not a ring).

(d)We denote byB((z)) the vector space of all two-sided sequences (b_n)_n∈

Z ∈B^Z such that only finitely many among the negative n ∈Zsatisfy b_n 6= 0. (A two-sided sequence means a sequence indexed by integers, not just nonnegative integers.) Such a sequence (b_n)_n∈

Z is denoted by P

n∈Z

b_nzⁿ. Sometimes, B((z)) is also denoted by B[[z, z⁻¹]. The elements of B((z)) are called formal Laurent series in the indeter-minate z over B (even when B is not a ring).

(e)We denote byB[[z, z⁻¹]] the vector space of all two-sided sequences (bn)_n∈

Z ∈ B^Z. Such a sequence (b_n)_n∈

Z is denoted by P

n∈Z

b_nzⁿ.

All five of these spaces B[z], B[[z]], B[z, z⁻¹], B((z)) and B[[z, z⁻¹]] are C [z]-modules. (Here, the C[z]-module structure on B[[z, z⁻¹]] is given by

X

n∈N

c_nzⁿ

!

· X

n∈Z

b_nzⁿ

!

=X

n∈Z

X

m∈N

c_m·bn−m

!

zⁿ (108)

for all P

n∈Z

b_nzⁿ ∈B[[z, z⁻¹]] and P

n∈N

c_nzⁿ∈C[z], and theC[z]-module structures on the other four spaces are defined similarly.) Besides, B[[z]] and B((z)) are C [[z]]-modules (defined in a similar way to (108)). Also, B((z)) is a C((z))-module (in a similar way). Besides, B[z, z⁻¹], B((z)) and B[[z, z⁻¹]] are C[z, z⁻¹]-modules (defined analogously to (108)).

Of course, ifB is aC-algebra, then the above-defined spacesB[z],B[z, z⁻¹],B[[z]]

and B((z)) are C-algebras themselves (with the multiplication defined similarly to (108)), and in fact B[z] is the algebra of polynomials in the variable z over B, and B[z, z⁻¹] is the algebra of Laurent polynomials in the variablez overB, andB[[z]]

is the algebra of formal power series in the variable z over B.

It should be noticed thatB[z]∼=B⊗C[z] and B[z, z⁻¹]∼=B⊗C[z, z⁻¹] canon-ically, but such isomorphisms do not hold for B[[z]], B((z)) and B[[z, z⁻¹]] unless B is finite-dimensional.

We regard the obvious injections B[z] → B[z, z⁻¹], B[z⁻¹] → B[z, z⁻¹] (this is the map sending z⁻¹ ∈ B[z⁻¹] to z⁻¹ ∈ B[z, z⁻¹]), B[z] → B[[z]], B[z⁻¹] → B[[z⁻¹]], B[[z]]→B((z)),B[[z⁻¹]]→B((z⁻¹)), B[z, z⁻¹] →B((z)), B[z, z⁻¹]→ B((z⁻¹)),B((z))→B[[z, z⁻¹]] andB((z⁻¹))→B[[z, z⁻¹]] as inclusions.

Clearly, all five spaces B[z], B[[z]], B[z, z⁻¹], B((z)) and B[[z, z⁻¹]] depend functorially on B.

Before we do anything further with these notions, let us give three warnings:

1) Given Definition 3.3.1, one might expect B[[z, z⁻¹]] to canonically become a C[[z, z⁻¹]]-algebra. But this is not true even for B =C(because there is no reasonable way to define a product of two elements of C[[z, z⁻¹]] ⁹⁶). This also answers why B[[z, z⁻¹]] does not become a ring whenB is aC-algebra. Nor isB[[z, z⁻¹]], in general, aB[[z]]-module.

2) The C[z, z⁻¹]-module B[[z, z⁻¹]] usually has torsion. For example, (1−z) · P

n∈Z

zⁿ = 0 in C[[z, z⁻¹]] despite P

n∈Z

zⁿ 6= 0. As a consequence, working in B[[z, z⁻¹]]

requires extra care.

3) Despite the suggestive notation B((z)), it is of course not true that B((z)) is a field whenever B is a commutative ring. However, B((z)) is a field whenever B is a field.

Convention 3.3.2. LetB be a vector space, and z a symbol. By analogy with the notationsB[z],B[[z]] andB((z)) introduced in Definition 3.3.1, we will occasionally

96If we would try the natural way, we would get nonsense results. For instance, if we tried to compute the coefficient of

P

n∈Z

1zⁿ

·

P

n∈Z

1zⁿ

before z⁰, we would get P

(n,m)∈Z²; n+m=0

1·1, which is not a convergent series.

also use the notations B[z⁻¹], B[[z⁻¹]] and B((z⁻¹)). For example, B[z⁻¹] will mean the vector space of all “reverse sequences” (b_n)_n∈−

N such that only finitely many n ∈ −N satisfy b_n 6= 0 ⁹⁷. Of course, B[z] ∼= B[z⁻¹] as vector spaces, but B[z] andB[z⁻¹] are two different subspaces ofB[z, z⁻¹], so it is useful to distinguish between B[z] and B[z⁻¹].

Now, let us extend Definition 3.3.1 to several variables. The reader is advised to only skim through the following definition, as there is nothing unexpected in it:

Definition 3.3.3. Let m ∈ N. Let z₁, z₂, ..., z_m be m symbols. For every vector space B, we make the following definitions:

(a) We denote by B[z1, z2, ..., zm] the vector space of all families b_{(n1,n2,...,nm)}

(n1,n2,...,nm)∈N^m ∈B^Nm such that only finitely many (n₁, n₂, ..., n_m)∈N^m satisfy b_{(n1,n2,...,nm)} 6= 0. Such a family b_{(n1,n2,...,nm)}

(n1,n2,...,nm)∈N^m is denoted by P

(n1,n2,...,nm)∈N^m

b_{(n1,n2,...,nm)}z₁ⁿ¹z₂ⁿ²...z_m^nm. The elements of B[z₁, z₂, ..., z_m] are called polynomials in the indeterminates z₁, z₂, ..., z_m over B (even whenB is not a ring).

(b) We denote by B[[z₁, z₂, ..., z_m]] the vector space of all families b_{(n1,n2,...,nm)}

(n1,n2,...,nm)∈N^m ∈ B^Nm. Such a family b_{(n1,n2,...,nm)}

(n1,n2,...,nm)∈N^m is

denoted by P

(n1,n2,...,nm)∈N^m

b_{(n1,n2,...,nm)}z₁ⁿ¹zⁿ₂²...z_m^nm. The elements ofB[[z₁, z₂, ..., z_m]]

are calledformal power series in the indeterminates z1, z2, ..., zm over B (even when B is not a ring).

(c) We denote by B

z₁, z₁⁻¹, z₂, z₂⁻¹, ..., z_m, z_m⁻¹

the vector space of all families b_{(n1,n2,...,nm)}

(n1,n2,...,nm)∈Z^m ∈B^Zm such that only finitely many (n1, n2, ..., nm)∈Z^m satisfy b_{(n1,n2,...,nm)} 6= 0. Such a family b_{(n1,n2,...,nm)}

(n1,n2,...,nm)∈Z^m is denoted by P

(n1,n2,...,nm)∈Z^m

b_{(n1,n2,...,nm)}z₁ⁿ¹z₂ⁿ²...zⁿ_m^m. The elements ofB

z₁, z₁⁻¹, z₂, z₂⁻¹, ..., z_m, z_m⁻¹ are calledLaurent polynomials in the indeterminates z₁, z₂, ..., z_mover B (even when B is not a ring).

(d) We denote by B((z1, z2, ..., zm)) the vector space of all families b_{(n1,n2,...,nm)}

(n1,n2,...,nm)∈Z^m ∈B^Zm for which there exists an N ∈Z such that every (n1, n2, ..., nm)∈Z^m\ {N, N + 1, N + 2, . . .}^m satisfiesb_{(n1,n2,...,nm)} = 0. Such a fam-ily b_{(n1,n2,...,nm)}

(n1,n2,...,nm)∈Z^m is denoted by P

(n1,n2,...,nm)∈Z^m

b_{(n1,n2,...,nm)}z₁ⁿ¹zⁿ₂²...z_m^nm. The elements ofB((z₁, z₂, ..., z_m)) are called formal Laurent series in the indetermi-nates z₁, z₂, ..., z_m over B (even when B is not a ring).

(e) We denote by B

z₁, z₁⁻¹, z₂, z⁻¹₂ , ..., z_m, z_m⁻¹

the vector space of all families b_{(n1,n2,...,nm)}

(n1,n2,...,nm)∈Z^m ∈ B^Zm. Such a family b_{(n1,n2,...,nm)}

(n1,n2,...,nm)∈Z^m is

denoted by P

(n1,n2,...,nm)∈Z^m

b_{(n1,n2,...,nm)}z₁ⁿ¹z₂ⁿ²...z_m^nm.

All five of these spaces B[z1, z2, ..., zm], B[[z1, z2, ..., zm]], B

z₁, z₁⁻¹, z₂, z₂⁻¹, ..., z_m, z_m⁻¹

,B((z₁, z₂, ..., z_m)) andB

z₁, z⁻¹₁ , z₂, z⁻¹₂ , ..., z_m, z_m⁻¹ are C[z₁, z₂, ..., z_m]-modules. (Here, the C[z₁, z₂, ..., z_m]-module structure on

97Here, −N denotes the set {0,−1,−2,−3, ...}, and a “reverse sequence” is a family indexed by elements of−N.

B ]-module structures on the other four spaces are defined similarly.) Besides, B[[z₁, z₂, ..., z_m]] and B((z₁, z₂, ..., z_m)) are C[[z₁, z₂, ..., z_m]]-modules (defined in a similar fashion to (109)). Also,B((z₁, z₂, ..., z_m)) is aC((z₁, z₂, ..., z_m))-module (de-fined in analogy to (109)). Besides,B

z₁, z₁⁻¹, z₂, z₂⁻¹, ..., z_m, z⁻¹_m themselves (with multiplication defined by a formula analogous to (109) again), and in factB[z₁, z₂, ..., z_m] is the algebra of polynomials in the variablesz₁, z₂, ..., z_m over B, and B

z₁, z₁⁻¹, z₂, z₂⁻¹, ..., z_m, z_m⁻¹

is the algebra of Laurent polynomials in the variables z₁, z₂, ..., z_m over B, and B[[z₁, z₂, ..., z_m]] is the algebra of formal power

unless B is finite-dimensional or m= 0.

There are several obvious injections (analogous to the ones listed in Definition 3.3.1) which we regard as inclusions. For example, one of these is the injection B[z₁, z₂, ..., z_m]→B[[z₁, z₂, ..., z_m]]; we won’t list the others here.

Clearly, when m= 1, Definition 3.3.3 is equivalent to Definition 3.3.1.

Definition 3.3.3 can be extended to infinitely many indeterminates; this is left to the reader.

Our definition ofB((z₁, z₂, ..., z_m)) is rather intricate. The reader might gain a better understanding from the following equivalent definition: The setB((z1, z2, ..., zm)) is the subset ofB B[[z₁, z₂, ..., z_m]] at the multiplicatively closed subset consisting of all monomials.

The reader should be warned that if B is a field, m is an integer > 1, and z₁, z₂,

..., z_m are m symbols, then the ring B((z₁, z₂, ..., z_m)) isnot a field (unlike in the case m = 1); for example, it does not contain an inverse to z₁ −z₂. This is potentially confusing and I would not be surprised if some texts define B((z₁, z₂, ..., z_m)) to mean a different ring which actually is a field.

When B is a vector space and z is a symbol, there is an operator we can define on each of the five spacesB[z],B[[z]],B[z, z⁻¹],B((z)) and B[[z, z⁻¹]]: derivation with respect toz:

Definition 3.3.4. For every vector space B and symbol z, we make the following definitions:

Define a linear map d

dz :B[z]→B[z] by the formula d

dz X

n∈N

b_nzⁿ

!

=X

n∈N

(n+ 1)b_n+1zⁿ (110)

for every X

n∈N

b_nzⁿ∈B[z].

Define a linear map d

dz :B[[z]]→B[[z]] by the very same formula, and define linear maps d

dz : B[z, z⁻¹] → B[z, z⁻¹], d

dz : B((z)) → B((z)) and d

dz : B[[z, z⁻¹]] → B[[z, z⁻¹]] by analogous formulas (more precisely, by formulas which differ from (110) only in that the sums range over Z instead of over N).

For every f ∈ B[[z, z⁻¹]], the image d

dzf of f under the linear map d

dz will be denoted by df

dz or by f⁰ and called thez-derivative of f (or, briefly, the derivative of f). The operator d

dz itself (on any of the five vector spaces B[z], B[[z]],B[z, z⁻¹], B((z)) and B[[z, z⁻¹]]) will be called thedifferentiation with respect to z.

An analogous definition can be made for several variables:

Definition 3.3.5. Letm∈N. Let z1, z2, ..., zm be m symbols. Let i∈ {1,2, ..., m}.

For every vector space B, we make the following definitions:

Define a linear map ∂

∂z_i :B[z₁, z₂, ..., z_m]→B[z₁, z₂, ..., z_m] by the formula

∂

∂z_i





X

(n1,n2,...,nm)∈N^m

b(n1,n2,...,nm)z₁ⁿ¹z₂ⁿ²...z_m^nm





= X

(n1,n2,...,nm)∈N^m

(n_i+ 1)b_{(n1,n2,...,ni−1,ni+1,ni+1,ni+2,...,nm)}z₁ⁿ¹z₂ⁿ²...z_m^nm (111)

for every X

(n1,n2,...,nm)∈N^m

b(n1,n2,...,nm)z₁ⁿ¹z₂ⁿ²...z_m^nm ∈B[z1, z2, ..., zm]. Define a linear map ∂

∂z_i : B[[z₁, z₂, ..., z_m]] → B[[z₁, z₂, ..., z_m]] by the very same formula, and define linear maps ∂

∂z_i : B

z₁, z⁻¹₁ , z₂, z⁻¹₂ , ..., z_m, z_m⁻¹

→

B

z₁, z₁⁻¹, z₂, z₂⁻¹, ..., z_m, z_m⁻¹ , ∂

∂z_i : B((z₁, z₂, ..., z_m)) → B((z₁, z₂, ..., z_m)) and

∂

∂z_i : B

z₁, z⁻¹₁ , z₂, z₂⁻¹, ..., z_m, z_m⁻¹

→ B

z₁, z₁⁻¹, z₂, z₂⁻¹, ..., z_m, z_m⁻¹

by analo-gous formulas (more precisely, by formulas which differ from (111) only in that the sums range over Z^m instead of over N^m).

For every f ∈ B

z₁, z₁⁻¹, z₂, z₂⁻¹, ..., z_m, z_m⁻¹

, the image ∂

∂zi

f of f under the linear map ∂

∂z_i will be denoted by ∂f

∂z_i and called the z_i-derivative of f (or the partial derivative of f with respect to z_i). The operator ∂

∂z_i itself (on any of the five vector spaces B[z₁, z₂, ..., z_m], B[[z₁, z₂, ..., z_m]], B

z₁, z⁻¹₁ , z₂, z₂⁻¹, ..., z_m, z_m⁻¹ , B((z₁, z₂, ..., z_m)) and B

z₁, z₁⁻¹, z₂, z₂⁻¹, ..., z_m, z⁻¹_m

) will be called the differenti-ation with respect to zi.

Again, it is straightforward (and left to the reader) to extend this definition to infinitely many indeterminates.

3.3.2. Quantum fields

Formal power series which are infinite “in both directions” might seem like a perverse and artificial notion; their failure to form a ring certainly does not suggest them to be useful. Nevertheless, they prove very suitable when studying infinite-dimensional Lie algebras. Let us explain how.

For us, when we study Lie algebras, we are mainly concerned with their elements, usually basis elements (e. g., the an inA). For physicists, instead, certain generating functions built of these objects are objects of primary concern, since they are closer to what they observe. They are called quantum fields.

Now, what are quantum fields?

For example, in A, let us set a(z) = P

n∈Z

a_nz⁻ⁿ⁻¹, where z is a formal variable.

This sum P

n∈Z

a_nz⁻ⁿ⁻¹ is a formal sum which is infinite in both directions, so it is not an element of any of the rings U(A) [[z]] or U(A) ((z)), but only an element of U(A) [[z, z⁻¹]].

As we said, the vector space U(A) [[z, z⁻¹]] is not a ring (even though U(A) is a C-algebra), so we cannot multiply two “sums” like a(z) in general. However, in the following, we are going to learn about several things that wecando with such “sums”.

One first thing that we notice about our concrete “sum”a(z) = P

n∈Z

a_nz⁻ⁿ⁻¹ is that if we apply a(z) to some vectorv inF_µ(by evaluating the term (a(z))v componentwise⁹⁸), then we get a sum P

n∈Z

z⁻ⁿ⁻¹a_nv which evaluates to an element of F_µ((z)) (because every sufficiently large n∈Z satisfies z⁻ⁿ⁻¹ a_nv

|{z}

=0

= 0). As a consequence, a(z) “acts”

98By “evaluating” a term like (a(z))v at a vector v “componentwise”, we mean evaluating P

n∈Z

a_nz⁻ⁿ⁻¹

(v). Here, the variable z is decreed to commute with everything else, so that anz⁻ⁿ⁻¹

(v) meansz⁻ⁿ⁻¹anv.

onF_µ. I am saying “acts” in quotation marks, since this “action” is not a mapF_µ→F_µ but a mapF_µ →F_µ((z)), and sincea(z) does not lie in a ring (as I said,U(A) [[z, z⁻¹]]

is not a ring).

Physicists call a(z) a quantum field (more precisely, a free bosonic field).

While we cannot take the square (a(z))² of our “sum”a(z) (sinceU(A) [[z, z⁻¹]] is not a ring), we can multiply two sums “with different variables”; e. g., we can multiply a(z) anda(w), wherezandware two distinct formal variables. The producta(z)a(w) is defined as the formal sum P

(n,m)∈Z²

a_na_mz⁻ⁿ⁻¹w^−m−1 ∈U(A) [[z, z⁻¹]] [[w, w⁻¹]]. Note that elements ofU(A) [[z, z⁻¹]] [[w, w⁻¹]] are two-sided sequences of two-sided sequences of elements of U(A); of course, we can interpret them as maps Z² →U(A).

It is easy to see that [a(z), a(w)] = P

n∈Z

nz⁻ⁿ⁻¹wⁿ⁻¹. This identity, in the first place, holds on the level of formal sums (where P

n∈Z

nz⁻ⁿ⁻¹wⁿ⁻¹ is a shorthand notation for a particular sequence of sequences: namely, the one whose j-th element is the sequence whose i-th element is δ_i+j+2,0(j+ 1)), but if we evaluate it on an element v of F_µ, then we get an identity [a(z), a(w)]v = P

n∈Z

nz⁻ⁿ⁻¹wⁿ⁻¹v which holds in the space F_µ((z)) ((w)).

We can obtain the “series” [a(z), a(w)] = P

n∈Z

nz⁻ⁿ⁻¹wⁿ⁻¹ by differentiating a more basic “series”:

δ(w−z) := X

n∈Z

z⁻ⁿ⁻¹wⁿ.

This, again, is a formal series infinite in both directions. Why do we call itδ(w−z) ? Because in analysis, the delta-“function” (actually a distribution) satisfies the formula R δ(x−y)f(y)dy = f(x) for every function f, whereas our series δ(w−z) satisfies a remarkably similar property⁹⁹. And now, [a(z), a(w)] = P

n∈Z

nz⁻ⁿ⁻¹wⁿ⁻¹ becomes [a(z), a(w)] = ∂_wδ(w−z) =: δ⁰(w−z).

Something more interesting comes out for the Witt algebra: SetT (z) = P

n∈Z

L_nz⁻ⁿ⁻²

99Namely, if we define the “formal residue” 1 2πi

H

|z|=1

q(z)dzof an elementq(z)∈B((z)) (forB being some vector space) to be the coefficient ofq(z) beforez⁻¹, then everyf = P

n∈Z

fnzⁿ(withfn∈B) satisfies 1

2πi H

|z|=1

z⁻ⁿ⁻¹f(z)dz=f_n, and thus 1 2πi

H

|z|=1

δ(w−z)f(z)dz=f(w).

in the Witt algebra. Then, we have [T (z), T (w)]

= X

(n,m)∈Z²

(n−m)L_n+mz⁻ⁿ⁻²w^−m−2 = X

(k,m)∈Z²

L_k (k−2m)

| {z }

=(k+2)+2(−m−1)

z^m−k−2w^−m−2

= X

k∈Z

L_k(k+ 2)z^−k−3

!

| {z }

=−T⁰(z)

X

m∈Z

z^m+1w^−m−2

!

| {z }

=δ(w−z)

+ 2 X

k∈Z

L_kz^−k−2

!

| {z }

=T(z)

X

m∈Z

(−m−1)z^mw^−m−2

!

| {z }

=δ⁰(w−z)

=−T⁰(z)δ(w−z) + 2T (z)δ⁰(w−z).

Note that this formula uniquely determines the Lie bracket of the Witt algebra. This is how physicists would define the Witt algebra.

Now, let us set T (z) = P

n∈Z

Lnz⁻ⁿ⁻² in the Virasoro algebra. (This power se-ries T looks exactly like the one before, but note that the L_n now mean elements of the Virasoro algebra rather than the Witt algebra.) Then, our previous computa-tion of [T (z), T(w)] must be modified by adding a term of P

n∈Z

n³−n

12 Cz⁻ⁿ⁻²wⁿ⁻² = C

12δ⁰⁰⁰(w−z). So we get

[T(z), T (w)] = −T⁰(z)δ(w−z) + 2T(z)δ⁰(w−z) + C

12δ⁰⁰⁰(w−z).

Exercise: Check that, if we interpret L_n and a_m as the actions of L_n ∈ Vir and a_m ∈ A on the VirnA-module F_µ, then the following identity between maps F_µ → Fµ((z)) ((w)) holds:

[T (z), a(w)] = a(z)δ⁰(w−z). Recall

:a_ma_n : =

a_ma_n, if m≤n;

a_na_m, if m > n .

So we can reasonably define the “normal ordered” product :a(z)a(w) : to be X

(n,m)∈Z²

:a_na_m :z⁻ⁿ⁻¹w^−m−1 ∈U(A)

z, z⁻¹ w, w⁻¹ .

This definition of :a(z)a(w) : is equivalent to the definition given in Problem 2 of Problem Set 3.

That :a(z)a(w) : is well-defined is not a surprise: the variablesz andware distinct, so there are no terms to collect in the sum P

(n,m)∈Z²

:anam :z⁻ⁿ⁻¹w^−m−1, and thus there is no danger of obtaining an infinite sum which makes no sense (like what we would

get if we would try to define a(z)²). ¹⁰⁰ But it is more interesting that (although we cannot define a(z)²) we can define a “normal ordered” square :a(z)² : (or, what is the same, :a(z)a(z) : ), although it will not be an element of U(A) [[z, z⁻¹]] but rather of a suitable completion. We are not going to do elaborate on how to choose this completion here; but for us it will be enough to notice that, if we reinterpret thea_n as endomorphisms ofF_µ(using the action ofAonF_µ) rather than elements ofU(A), then the “normal ordered” square :a(z)² : is a well-defined element of (EndF_µ) [[z, z⁻¹]].

this is how power series are always multiplied; but we don’t yet know that the sum P

(n,m)∈Z²; n+m=k

:a_na_m : makes sense for allk (although we will see in a few lines that it does)



here, we substituted k by n in the first sum, and we substituted m by −m in the second sum

,

and the sums P

m∈Z

:a_−ma_n+m : are well-defined for all n ∈ Z (by Lemma 3.2.10 (c)).

We can simplify this result if we also reinterpret the Ln ∈Vir as endomorphisms ofFµ

(using the action of Vir on F_µ that was introduced in Proposition 3.2.13) rather than elements of U(Vir). In fact, the “series” T (z) = P

Remark 3.3.6. In Definition 3.2.4, we have defined the normal ordered product :a_ma_n : in the universal enveloping algebra of the Heisenberg algebra. This is not the only situation in which we can define a normal ordered product, but in other situations the definition can happen to be different. For example, in Proposition 3.4.4, we will define a normal ordered product (on a different algebra) which will not be commutative, and not even “super-commutative”. There is no general rule to define normal ordered products; it is done on a case-by-case basis.

100For the same reason, the producta(z)a(w) (without normal ordering) is well-defined.

However, the definition of the normal ordered product of two quantum fields given in Problem 2 of Problem Set 3 is general, i. e., it is defined not only for quantum fields over U(A).

Exercise 1. For any β ∈ C, the formula T (z) = 1

2 :a(z)² : +βa⁰(z) defines a representation of Vir on F_µ with c= 1−12β².

Exercise 2. For any β ∈ C, there is a homomorphism ϕβ : Vir → VirnA (a splitting of the projection VirnA →Vir) given by

ϕ_β(L_n) =L_n+βa_n, n 6= 0;

ϕ_β(L₀) =L₀+βa₀+ β² 2 K, ϕ_β(C) =C.

Exercise 3. If we twist the action of Exercise 1 by this map, we recover the action of problem 1 of Homework 2 for β=iλ.

3.3.3. Recognizing exponential series

Here is a simple property of power series (actually, an algebraic analogue of the well-known fact from analysis that the solutions of the differential equation f⁰ = αf are scalar multiples of the functionx7→exp (αx)):

Proposition 3.3.7. Let R be a commutative Q-algebra. Let U be an R-module.

Let (α1, α2, α3, ...) be a sequence of elements of R. Let P ∈ U[[x1, x2, x3, ...]] is a formal power series with coefficients in U (where x₁, x₂, x₃, ... are symbols) such that every i > 0 satisfies ∂P

∂x_i = α_iP. Then, there exists some f ∈ U such that P =f·exp P

j>0

xjαj

! .

The proof of Proposition 3.3.7 is easy (just let f be the constant term of the power series P, and prove by induction that every monomial of P equals the corresponding monomial of f·exp P

j>0

x_jα_j

! ).

3.3.4. Homogeneous maps and equigraded series

The discussion we will be doing now is only vaguely related to power series (let alone quantum fields); it is meant as a preparation for a later proof (namely, that of Theorem 3.11.2), where it will provide “convergence” assertions (in a certain sense).

A well-known nuisance in the theory of Z-graded vector spaces is the fact that the endomorphism ring of a Z-graded vector space is not (in general) Z-graded. It does, however, contain a Z-graded subring, which we will introduce now:

Definition 3.3.8. (a) Let V and W be two Z-graded vector spaces, with gradings (V [n])_n∈

Z and (W[n])_n∈

Z, respectively. Letf :V →W be a linear map. Letm∈Z. Then, f is said to be a homogeneous linear map of degree m if every n∈Z satisfies f(V [n])⊆W[n+m].

(It is important not to confuse this notion of “homogeneous linear maps of de-gree m” with the notion of “homogeneous polynomial maps of degree n” defined in Definition 2.6.16 (a); the former of these notions is not a particular case of the latter.)

Note that the homogeneous linear maps of degree 0 are exactly the graded linear maps.

(b) Let V and W be two Z-graded vector spaces. For every m ∈ Z, let Hom_hg=m(V, W) denote the vector space of all homogeneous linear mapsV →W of degreem. This Hom_hg=m(V, W) is a vector subspace of Hom (V, W) for everym∈Z. Moreover, L

m∈Z

Homhg=m(V, W) is a well-defined internal direct sum, and will be de-noted by Hom_hg(V, W). This Hom_hg(V, W) is a vector subspace of Hom (V, W), and is canonically a Z-graded vector space, with its m-th graded component being Hom_hg=m(V, W).

(c) Let V be a Z-graded vector space. Then, let End_hgV denote the Z-graded vector subspace Homhg(V, V) of Hom (V, V) = EndV. Then, EndhgV is a subalge-bra of EndV, and a Z-graded algebra. Moreover, the canonical action of End_hgV on V (obtained by restricting the action of EndV on V to End_hgV) makes V into a Z-graded EndhgV-module.

We next need a relatively simple notion for a special kind of power series. I (Darij) call them “equigraded power series”, though noone else seems to use this nomenclature.

Definition 3.3.9. Let B be a Z-graded vector space, and z a symbol. An element P

n∈Z

b_nzⁿ ofB[[z, z⁻¹]] (withb_n∈B for everyn ∈Z) is said to beequigraded if every n ∈Zsatisfiesb_n ∈B[n] (where (B[m])_m∈

Z denotes the grading onB). SinceB[[z]]

and B((z)) are vector subspaces of B[[z, z⁻¹]], it clearly makes sense to speak of equigraded elements of B[[z]] or of B((z)). We will denote by B[[z, z⁻¹]]_equi the set of all equigraded elements of B[[z, z⁻¹]]. It is easy to see that B[[z, z⁻¹]]_equi is a vector subspace of B[[z, z⁻¹]].

Elementary properties of equigraded elements are:

Proposition 3.3.10. (a)LetB be aZ-graded vector space, andz a symbol. Then, {f ∈B[z] | f is equigraded},

f ∈B z, z⁻¹

| f is equigraded , {f ∈B[[z]] | f is equigraded}, {f ∈B((z)) | f is equigraded}, f ∈B

z, z⁻¹

| f is equigraded =B

z, z⁻¹

equi

are vector spaces.

(b) Let B be a Z-graded algebra. Then, {f ∈B[[z]] | f is equigraded} is a subalgebra of B[[z]] and closed with respect to the usual topology on B[[z]].

(c)Let B be aZ-graded algebra. If f ∈B[[z]] is an equigraded power series and invertible in the ring B[[z]], thenf⁻¹ also is an equigraded power series.

We will only use parts (a) and (b) of this proposition, and these are completely straightforward to prove. (Part (c)is less straightforward but still an easy exercise.)

Equigradedness of power series sometimes makes their actions on modules more manageable. Here is an example:

Proposition 3.3.11. Let A be a Z-graded algebra, and let M be a Z-graded A-module. Assume thatM is concentrated in nonnegative degrees. Letube a symbol.

(a) It is clear that for anyf ∈A[[u, u⁻¹]] and anyx∈M[u, u⁻¹], the productf x is a well-defined element of M[[u, u⁻¹]].

(b) For any equigraded f ∈A[[u, u⁻¹]] and any x∈M[u, u⁻¹], the productf x is a well-defined element of M((u)) (and not only of M[[u, u⁻¹]]).

(c) For any equigraded f ∈A[[u⁻¹]] and any x ∈M[u⁻¹], the productf x is a well-defined element of M[u⁻¹] (and not only ofM[[u⁻¹]]).

The proof of this proposition is quick and straightforward. (The only idea is that for any fixed x ∈ M[u, u⁻¹], any sufficiently low-degree element of A annihilates x due to the “concentrated in nonnegative degrees” assumption, but sufficiently low-degree monomials in f come with sufficiently low-degree coefficients due to f being equigraded.)

Im Dokument Pavel Etingof: 18.747: Infinite-dimensional Lie algebras (Spring term 2012 at MIT) (Seite 173-184)