Domenico Fiorenza
Vertex algebras avant Borcherds



If you open your favourite book in conformal field theory, within the first 10 pages you will almost surely find an expression like this:

T(z)T(w)D/2(zw) 4+2(zw) 2T(w)+1(zw)T(w) T(z)T(w)\sim \frac{D/2}{(z-w)^4}+\frac{2}{(z-w)^2}T(w)+\frac{1}{(z-w)}\partial T(w)

or equivalently, in the D’Andrea-Kac notation

[T λT]=D12λ 3+2λT+T, [T_\lambda T]= \frac{D}{12}{\lambda^3}+2\lambda T+\partial T,

or even, directly in terms of Laurent coefficients,

[L m,L n]=(mn)L m+n+Dm 3m12δ m,n. [L_m,L_n]= (m-n)L_{m+n}+D\frac{m^3-m}{12}\delta_{m,-n}.

Following Borcherds, expressions like the above are at the basis of the modern axiomatization of vertex algebras. It’s worth recalling how they historically arose from path-integral heuristics.

The basic rules

The players

Notation: Let VV be a (topological) vector space; if f:MVf\colon M\to V is integrable w.r.t. the measure e Sdvol Me^{-S}d vol_M, we write

f= Mfe Sdvol M \langle f\rangle =\int_M f\, e^{-S}d vol_M

nn-point functions

To a pair (A,X)(A,X) consisting of a field and a global section, we can associate a distribution A X𝒟(Σ)A_X\in \mathcal{D}'(\Sigma) as follows:

A X:f Σf(p)A(X,p)drmvol Σ A_X\colon f\mapsto \int_\Sigma f(p)A(X,p)d{\rm vol}_\Sigma

Distributions are an algebra (w.r.t. tensor product)

𝒟(Σ m)𝒟(Σ n)𝒟(Σ m+n) \mathcal{D}'(\Sigma^m)\otimes \mathcal{D}'(\Sigma^n)\to \mathcal{ D}'(\Sigma^{m+n})


φψ:f Σ m+nφ(p 1)ψ(p 2)f(p 1,p 2)drmvol Σ m+n \varphi\otimes\psi\colon f\mapsto \int_{\Sigma^{m+n}}\varphi(\vec{p}_1)\psi(\vec{p}_2)f(\vec{p}_1,\vec{p}_2)d{\rm vol}_{\Sigma^{m+n}}

so we get a map n×M𝒟(Σ n)\mathcal{F}^n\times M \to \mathcal{D}'(\Sigma^n)

(A 1,,A n;X)A 1,XA n,X (A_1,\dots,A_n;X)\mapsto A_{1,X}\otimes\cdots \otimes A_{n,X}

which we can look at as a map

MrmHom( n,𝒟(Σ n)). M\to{\rm Hom}(\mathcal{F}^{\otimes n},\mathcal{D}'(\Sigma^n)).

Integrating over MM we get the nn-point function

n: n𝒟(Σ n) \langle\,\rangle_n\colon\mathcal{F}^{\otimes n}\to\mathcal {D}'(\Sigma^n)

Explicitly, for A 1,,A nA_1,\dots,A_n in \mathcal{F}, the nn-point function A 1A n n\langle A_1\cdots A_n\rangle_n is the distribution on Σ n\Sigma^n whose density at (p 1,,p n)(p_1,\dots, p_n) is

A 1(X,p 1)A n(X,p n) n= MA 1(X,p 1)A n(X,p n)e S(X)dvol M(X) \langle A_1(X,p_1)\cdots A_n(X,p_n)\rangle_n= \int_M A_1(X,p_1)\cdots A_n(X,p_n) e^{-S(X)}d vol_M(X)

We can also look at the map n×M𝒟(Σ n)\mathcal{F}^n\times M \to \mathcal{ D}'(\Sigma^n) as to a map nC (M,𝒟(Σ n)\mathcal{F}^n\to C^\infty(M, \mathcal {D}'(\Sigma^n); we denote denote by A 1A nA_1\cdots A_n the image of (A 1,,A n)(A_1,\dots,A_n) by this map.

Off-diagonal regularity

Let Δ nΣ n\Delta_n\hookrightarrow \Sigma^n be the big diagonal

Δ n={non-injective maps{1,,n}Σ}} \Delta_n=\bigl\{\text{non-injective maps} \{1,\dots,n\}\to \Sigma\}\bigr\}

Consider the restrictions 𝒟(Σ n)𝒟(Σ nΔ n)\mathcal{D}'(\Sigma^n)\to \mathcal {D}'(\Sigma^n\setminus \Delta_n). We ask the restrictions of the nn-point functions to be regular distributions on Σ nΔ n\Sigma^n\setminus \Delta_n, i.e., for fixed A 1,,A nA_1,\dots,A_n, the density A 1(p 1)A n(p n) n\langle A_1(p_1)\cdots A_n(p_n)\rangle_n is a smooth function on on Σ nΔ n\Sigma^n\setminus \Delta_n. Singularities may appear when p i,p jpp_i,p_j\to p for iji\neq j. In particular, 11-point functions are smooth.


We further assume to have an operator ::\colon\,\,\colon,

nC (M×Σ n;)\mathcal{F}^{\otimes n}\to C^\infty(M\times \Sigma^n;\mathbb{C})
A 1A n:A 1A n: A_1\otimes \cdots\otimes A_n\mapsto :A_1\cdots A_n:

such that

A 1A n n=:A 1A n:+singular distribution, \langle A_1\cdots A_n\rangle_n=\langle:A_1\cdots A_n:\rangle+\text{singular distribution},

where the \langle\,\,\rangle on the right-hand side is integration over MM with respect to the measure e Sdrmvol Me^{-S}d{\rm vol}_M:

:C (M×Σ n;)C (Σ n;). \langle\,\,\rangle: C^\infty(M\times \Sigma^n;\mathbb{C})\to C^\infty( \Sigma^n;\mathbb{C}).

Via the embedding C (Σ n;)𝒟(Σ n;)C^\infty(\Sigma^n;\mathbb{C})\hookrightarrow \mathcal{D}'(\Sigma^n;\mathbb{C}) induced by a choice of a volume form on Σ\Sigma, we can look at both A 1A nA_1\cdots A_n and :A 1A n::A_1\cdots A_n: as 𝒟(Σ n;)\mathcal{D}'(\Sigma^n;\mathbb{C})-valed functions on MM. With this identification, the difference A 1A n:A 1A n:A_1\cdots A_n - :A_1\cdots A_n: is an 𝒟(Σ n;)\mathcal{D}'(\Sigma^n;\mathbb{C})-valed functions on MM whose integral over MM is the singular part of A 1A n n\langle A_1\cdots A_n\rangle_n. Moreover, we also require :A:=A 1\langle:A:\rangle=\langle A\rangle_1 for 11-point functions. This rules out the trivial regularization given by :A 1A n:=0:A_1\cdots A_n: = 0 for any A 1,,A nA_1,\dots, A_n. For the 2-point functions regularization gives

AB=:AB:+φ AB A B = : A B: +\,\varphi_{A B}

for a suitable function φ AB:M𝒟(Σ 2;)\varphi_{A B}:M \to \mathcal{D}'(\Sigma^2;\mathbb{C}). One the 2-point regularizations have been chosen, we can define higher order regularizations iteratively, as follows:


:AB:=ABφ AB:A B:=A B-\varphi_{A B}

:ABC:=ABCφ ABCφ BCAφ ACB:A B C:= A B C -\varphi_{A B} C-\varphi_{B C} A-\varphi_{A C} B

:ABCD:=ABCDφ ABCDφ ACBDφ CDAB+φ ABφ CD+φ ACφ BD+φ ADφ BC:A B C D:=A B C D-\varphi_{A B} C D-\varphi_{A C} B D-\cdots- \varphi_{C D} A B +\varphi_{A B}\varphi_{C D}+\varphi_{A C}\varphi_{B D}+\varphi_{A D}\varphi_{B C}

and so on. Therefore, in general, if 𝒜=A 1A n\mathcal{A}=A_1\cdots A_n, then

:𝒜:=𝒜contractions {:}\mathcal{A}:=\mathcal{A}-\text{contractions}

Remark: If 𝒜=A 1A n\mathcal{A}=A_1\cdots A_n and =B 1B m\mathcal{B}=B_1\cdots B_m, then

:𝒜:=𝒜allcontractions {:}\mathcal{A}\mathcal{B}:=\mathcal{A}\mathcal{B}-all contractions
:𝒜:=𝒜𝒜contractions {:}\mathcal{A}:=\mathcal{A}-\mathcal{A}contractions
::=contractions {:}\mathcal{B}:=\mathcal{B}-\mathcal{B}contractions

and so

:𝒜:::=:𝒜:+𝒜contractions. {:}\mathcal{A}::\mathcal{B}:=:\mathcal{A}\mathcal {B}:+\mathcal{ A}\mathcal{B}contractions.

Therefore singularities of :𝒜:::\langle:\mathcal{A}::\mathcal{B}:\rangle come entirely from the 𝒜\mathcal{A}\mathcal{B}-contractions of 𝒜\mathcal{A}\mathcal{B}.

Example: :A 1A 2::B 1B 2:=φ A 1B 1φ A 2B 2+φ A 1B 2φ A 2B 1+φ A 1B 1:A 2B 2:+φ A 1B 2:A 2B 1:+φ A 2B 1:A 1B 2:+φ A 2B 2:A 1B 1:+:A 1A 2B 1B 2::A_1 A_2: :B_1 B_2:=\varphi_{A_1 B_1}\varphi_{A_2 B_2}+\varphi_{A_1 B_2}\varphi_{A_2 B_1}+\varphi_{A_1 B_1}:A_2 B_2:+\varphi_{A_1 B_2}:A_2 B_1:+\varphi_{A_2 B_1}:A_1 B_2:+\varphi_{A_2 B_2}:A_1 B_1:+:A_1 A_2 B_1 B_2:

Notation: One defines

R AB(z)=:AB:(z,z); R_{A B}(z)=:A B:(z,z);

it is a smooth function on Σ\Sigma. In physicists’ notation, one writes :A(z)B(w)::A(z)B(w): for :AB:(z,w):AB:(z,w), and so

R AB(z)=lim wz:A(z)B(w): R_{AB}(z)=\lim_{w\to z}:A(z)B(w):

Moreover, multilinearity of regularization gives

z:A(z)B(w):=: zA(z)B(w): \partial_z:A(z)B(w):=:\partial_zA(z)B(w):

and so

zR AB(z)=R zAB(z)+R A zB(z) \partial_z R_{AB}(z)=R_{\partial_z A\, B}(z)+R_{ A\partial_z B}(z)

The game

The rules of the game

Promote each element AA of \mathcal{F} to an operator A^\hat{A}, its quantization. By this we mean that A^\hat{A} is an element of some associative algebra. More precisely, consider the free associative algebra

T()= n0 n T(\mathcal{F})=\bigoplus_{n\geq 0}\mathcal{F}^{\otimes n}

generated by \mathcal{F}, modulo the following relations:

A^ 1A^ n=0iffA 1A nB 1B m m+n| Σ m+nΔ m+n n=0 \hat{A}_1\cdots \hat{A}_n=0 \qquad iff\qquad \langle A_1\cdots A_n\, B_1\cdots B_m\rangle_{m+n}\biggr\vert_{\Sigma^{m+n}\setminus \Delta^{\ge n}_{m+n}}=0

for any B 1,B mB_1,\cdots B_m, where

Δ n k={(p 1,,p n)|p i=p jforsomeijwithjk} \Delta_n^{\geq k}=\{(p_1,\dots,p_n)\,|\, p_i=p_j for some i\neq j with j\geq k\}

We will adopt the following shorthand notation for the above relations:


Advanced tricks

Lie derivatives

Assume a linear map

i:C 0 (Σ)H 0(M;TM) i\colon C^\infty_0(\Sigma)\to H^0(M;T M)

is given, where TMT M denotes the tangent bundle of MM. Then each 11-form ω\omega on MM gives a distribution (ω|i)(\omega|i) on Σ\Sigma by

f(ω|i f)= M(ω|i f)e Sdvol M. f\mapsto\langle (\omega|i_f)\rangle=\int_M(\omega|i_f) e^{-S}d vol_M.

For any vector field vv on MM, the Lie derivative v\mathcal{L}_v satisfies

(dS|v)+div(v)= M v(e Sdvol M)=0. \langle (d S|v)\rangle+\langle div(v)\rangle=\int_M \mathcal{L}_v(e^{-S}d vol_M)=0.

Hence, if the vector fields i fi_f are divergence-free, the 11-point function (dS|i)(dS|i) is zero. If ff is supported away from q 1,,q mq_1,\dots,q_m, then, for any B 1,,B mB_1,\dots,B_m,

i f(B 1(X,q 1)B m(X,q m))=0 \mathcal{L}_{i_f}(B_1(X,q_1)\cdots B_m(X,q_m))=0


(dS|i)=0 \langle (d S|i)\cdots\rangle=0

so we recover within this formalism a version of Ehrenfest’s theorem.

Noether’s theorem

Let vv be a symmetry of the action, i.e., a vector field such (dS|v)=0(d S|v)=0, and assume furthermore that div(v)=0div(v)=0, so that

div S(v)=0, div_S(v)=0,


div S(v)= v(e Sdvol M). div_S(v)=\mathcal {L}_{v}(e^{-S}d vol_M).

Then, for any XX in MM, the map C (M;)C^\infty(M;\mathbb{C})\to \mathbb{C} given by

ρdiv S(ρv) X \rho\mapsto div_S(\rho v)_X

is a distribution on Σ\Sigma which is zero on constant functions. From the exact sequence

0C (Σ)d{Exact1formsonΣ}0, 0\to \mathbb{C}\to C^\infty(\Sigma)\stackrel{d}{\to}\{Exact\, 1-forms on \Sigma\}\to 0,

there exist

j v,X:{Exact1formsonΣ} j_{v,X}:\{Exact\, 1-forms on \Sigma\}\to \mathbb{C}

such that

div S(ρv) X=(j v,X|dρ). div_S(\rho v)_X=(j_{v,X}|d\rho).

Assume j v,Xj_{v,X} extends to a 1-current j v,X:{1formsonΣ}j_{v,X}: \{1-forms on \Sigma\}\to \mathbb{C}. Then

0= M ρv(e Sdvol M)=(j v,X|dρ)=(j v|ρ) 0=\int_M\mathcal{L}_{\rho v}(e^{-S}d vol_M)= \langle(j_{v,X}|d\rho)\rangle= \langle(\partial j_{v}|\rho)\rangle

Hence j v\langle \partial j_{v}\rangle is the zero distribution and, more in general, j v=0,i.e.j v^=0\langle \partial j_{v}\cdots\rangle=0, i.e. \partial\hat{j_v}=0. Identify j v,Xj_{v,X} with a 1-form via the canonical pairing of 1-forms on Σ\Sigma:

(j v,X|ω)= Σj v,Xω (j_{v,X}|\omega)=\int_\Sigma j_{v,X} \wedge \omega


dj v=0. \langle d j_{v}\cdots\rangle=0.

Ward identities

Now add a field AA. Then

0= M ρvA(p)(e Sdvol M)=(j X|ρ)A(p)+ ρvA(p). 0=\int_M\mathcal{L}_{\rho v}A(p)(e^{-S}d vol_M)= \langle(\partial j_X|\rho)A(p)\rangle+\langle\mathcal{L}_{\rho v}A(p)\rangle.

If ρ\rho is a bump function at pp, then ρvA(p)= vA(p)\mathcal{L}_{\rho v}A(p)=\mathcal {L}_{v}A(p) and so

vA(p)=A(p) Σρdj v=A(p) B pdj v= B pA(p)j v, \langle\mathcal{L}_v A(p)\rangle=\langle A(p)\int_\Sigma \rho d j_{v}\rangle=\langle A(p)\int_{B_p} d j_{v}\rangle =\int_{\partial B_p}\langle A(p) j_{v} \rangle,

where B pB_p denotes a little disk centered at pp (the support of the bump function ρ\rho). Let

Q v,p= B pj v Q_{v,p}=\int_{\partial B_p}\langle j_v\cdots \rangle

be the charge of vv at pp. Then

v| p=Q v,p \langle\mathcal{L}_v\cdots\rangle\bigr\vert_p=Q_{v,p}

The holomorphic case

If A(p)j v\langle A(p) j_v \rangle is holomorphic in B ppB_p\setminus{p}, then

B pA(p)j v=Res zpA(p)j v(z)dz \int_{\partial B_p}\langle A(p) j_v \rangle=Res_{z\to p}\langle A(p) j_v(z) \rangle dz

And we obtain

vA(p)=Res zpA(p)j v, \langle\mathcal{L}_v A(p)\rangle=Res_{z\to p}\langle A(p) j_v \rangle,

That is

Q v,p=Res zpj v(z)dz. Q_{v,p}= Res_{z\to p}\langle j_v(z)\cdots\rangle dz.


Assume A(z)B(w)\langle A(z)B(w)\cdots \rangle is a holomorphic function of zz for zwz\neq w.

We write

(1)A(z)B(w) k1C k(w)1(zw) k A(z)B(w)\sim \sum_{k\geq 1} C_k(w)\frac{1}{(z-w)^k}

to mean

A(z)B(w)= k1C k(w)1(zw) k+:A(z)B(w): \langle A(z)B(w)\cdots \rangle= \sum_{k\geq 1} \langle C_k(w)\cdots \rangle \frac{1}{(z-w)^k}+\langle:A(z)B(w):\cdots \rangle

The expression in equation (1) is called operator product exapansion of AA and BB (ABA B OPE for short). Note that :A(z)B(w)::A(z)B(w): is a holomorphic function of zz also at z=wz=w.

Since Res wz:A(z)j v(w):dw=0Res_{w\to z}:A(z)j_v(w):dw=0, to compute vA(z)\langle\mathcal {L}_v A(z)\rangle one only needs the Aj vA j_v OPE

A(z)j v(w) k1C k(w)1(zw) k. A(z)j_v(w)\sim\sum_{k\geq 1} C_k(w)\frac{1}{(z-w)^k}.

The algebra of currents

Assume two conserved currents j 1j_1 and j 2j_2 are given, and let Q 1,p,Q 2,pQ_{1,p}, Q_{2,p} be the associated charges at pp. Then the commutator [Q 1,p,Q 2,p][Q_{1,p}, Q_{2,p}] acts as

Res zpRes wpj 1(z)j 2(w)Res wpRes zpj 1(z)j 2(w)=Res wpRes zwj 1(z)j 2(w))=Res wp(Res zwj 1(z)j 2(w))). Res_{z\to p}Res_{w\to p}\langle j_1(z)j_2(w)\cdots\rangle-Res_{w\to p}Res_{z\to p}\langle j_1(z)j_2(w)\cdots\rangle= Res_{w\to p}Res_{z\to w}\langle j_1(z)j_2(w))\cdots\rangle =Res_{w\to p}\langle (Res_{z\to w} j_1(z)j_2(w)))\cdots\rangle.

In other words

[j 1,j 2](w)=Res zwj 1(z)j 2(w) [j_1,j_2](w)=Res_{z\to w} j_1(z)j_2(w)

and the Lie bracket is completely determined by the OPE of j 1(z)j 2(w)j_1(z)j_2(w).

Let’s play

Now we specialize the above general setup to conformal field theory on the complex plane. So our Σ\Sigma will be the complex plane \mathbb{C}, the sheaf \mathcal{M} will be the sheaf of smooth fuctions on \mathbb{C} with values in D\mathbb{R}^D, and the action will be the Polyakov action for the standard Euclidean metric both on the source \mathbb{C} and on the target D\mathbb{R}^D, i.e.,

S[X]=12π ΣX μ¯X μ S[X]=\frac{1}{2\pi}\int_\Sigma \partial X^\mu\overline{\partial}X_\mu

The tangent space of M=C (; D)M=C^\infty(\mathbb{C};\mathbb{R}^D) at each point XX is identified with the subspace C 0 (; D)C^\infty_0(\mathbb{C};\mathbb{R}^D) of compactly supported functions. For any μ=1,,D\mu=1,\dots, D, a linear map i μ:C 0 (,)H 0(M,TM)i_\mu: C^\infty_0(\mathbb{C},\mathbb{R})\to H^0(M, T M) is given by i μ,f:X νX ν+δ μ νϵfi_{\mu,f}\colon X^\nu\mapsto X^\nu+\delta^{\nu}_\mu\epsilon f. Postulating the volume form on MM is such that the vector fields i μ,fi_{\mu,f} are divergence-free, we have (dS|i μ)=0\langle(d S|i_\mu)\cdots\rangle=0 for any μ\mu. One then computes (dS|i μ)=¯X μ(d S|i_\mu)=\overline{\partial}\partial X^\mu, hence

¯X μ(z)=0 \langle \overline{\partial}\partial X^\mu(z)\cdots\rangle=0

by the general argument above. This in particular means that the distribution ¯ z zX μ(z)X ν(w)\langle \overline{\partial}_z\partial_z X^\mu(z)X^\nu(w) is supported at z=wz=w, and indeed one computes ¯ z zX μ(z)X ν(w)=πδ μνδ(zw)\langle \overline{\partial}_z\partial_z X^\mu(z)X^\nu(w)\cdots\rangle=-\pi\langle \delta^{\mu\nu}\delta(z-w)\cdots \rangle. Since δ(zw)=¯ z zlog|zw| 2\delta(z-w)=\overline{\partial}_z\partial_z\log|z-w|^2, this is conveniently rewritten as

¯ z z(X μ(z)X μ(w)+12δ μνlog|zw| 2)=0 \overline{\partial}_z\partial_z\langle (X^\mu(z) X_\mu(w)+\frac{1}{2}\delta^{\mu\nu}\log|z-w|^2)\cdots\rangle=0

In particular, X μ(z)X μ(w)+12δ μνlog|zw| 2\langle X^\mu(z)X_\mu(w)+\frac{1}{2}\delta^{\mu\nu}\log|z-w|^2\rangle is an harmonic function and so we have the regularization

:X μ(z)X ν(w):=X μ(z)X ν(w)+12δ μνlog|zw| 2 :X^\mu(z)X^\nu(w):=X^\mu(z) X^\nu(w)+\frac{1}{2}\delta^{\mu\nu}\log|z-w|^2

In other words,

φ X μX ν=12δ μνlog|zw| 2. \varphi_{X^\mu X^\nu}=-\frac{1}{2}\delta^{\mu\nu}\log|z-w|^2.


φ zX μ zX ν=12δ μν1(zw) 2. \varphi_{\partial_z X^\mu\partial_z X^\nu}=-\frac{1}{2}\delta^{\mu\nu}\frac{1}{(z-w)^2}.



T(z)=R zX μ zX μ(z)=lim zz: zX μ(z) zX μ(z) T(z)=R_{\partial_z X^\mu\partial_z X_\mu}(z)=\lim_{z'\to z}:\partial_z X^\mu(z)\partial_{z'}X_\mu(z')


¯ zT(z)=(R ¯ z zX μ zX μ+R X μ¯ z zX μ)(z)=0, \overline{\partial}_zT(z)=(R_{\overline\partial_z \partial_z X^\mu\partial_z X_\mu}+R_{\partial X^\mu\overline{\partial}_z\partial_z X_\mu})(z)=0,

that is, T(z)T(z) is holomorphic! We have

T(z)T(w)=lim (z,w)(z,w): zX μ(z) zX μ(z):: wX ν(w) wX ν(w):=D/2(zw) 4+21(zw) 2: zX μ(z) wX μ(w):+ T(z)T(w)=\lim_{(z',w')\to (z,w)}:\partial_zX^\mu(z)\partial_{z'}X_\mu(z'): :\partial_wX^\nu(w)\partial_{w'}X_\nu(w'): = \frac{D/2}{(z-w)^4}+ 2\frac{1}{(z-w)^2}:\partial_zX^\mu(z)\partial_w X_\mu(w):+\cdots
T(z)T(w)=D/2(zw) 4+2(zw) 2T(w)+1(zw) wT(w)+ \phantom{T(z)T(w)} =\frac{D/2}{(z-w)^4}+\frac{2}{(z-w)^2}T(w)+\frac{1}{(z-w)}\partial_w T(w)+\cdots

Therefore we have found the TTT T OPE

T(z)T(w)D/2(zw) 4+2(zw)2T(w)+1(zw) wT(w)+ T(z)T(w)\sim \frac{D/2}{(z-w)^4}+\frac{2}{(z-w)2}T(w)+\frac{1}{(z-w)}\partial_w T(w)\phantom{+\cdots}

The Virasoro algebra

Fix w=0w=0. Holomorphic vector fields on 0\mathbb{C}\setminus0 act as symmetries of the action. The charge associated with the vector field z n+1zz^{n+1}\frac{\partial}{\partial z} is

j n(z)dz=z n+1T(z)dz. j_n(z)dz=z^{n+1}T(z)dz.


L n=Res z0j n(z)dz L_n=Res_{z\to 0}\langle j_n(z)\cdots\rangle dz


[L m,L n]=Res z0(Res wzj m(w)j n(z)dw)dz=Res z0(Res wzz n+1w m+1T(w)T(z)dw)dz [L_m,L_n]=Res_{z\to 0}\left(Res_{w\to z}\langle j_m(w)j_n(z)\cdots\rangle dw\right)dz=Res_{z\to 0}\left( Res_{w\to z} z^{n+1}w^{m+1}\langle T(w)T(z)\cdots \rangle dw\right)dz

Look at the expression Res wzz n+1w m+1T(w)T(z)dw=z n+1Res wz(z+(wz)) m+1T(w)T(z)dwRes_{w\to z} z^{n+1}w^{m+1}\langle T(w)T(z)\cdots \rangle dw=z^{n+1}Res_{w\to z}\bigl(z+(w-z)\bigr)^{m+1}\langle T(w)T(z)\cdots\rangle dw and use the TTT T OPE to find

Res wz(z+(wz)) m+1T(w)T(z)dw=Res wz(z+(wz)) m+1(2(wz) 2T(z)+1(wz) zT(z)+D/2(wz) 4) Res_{w\to z}\bigl(z+(w-z)\bigr)^{m+1}\langle T(w)T(z)\cdots\rangle dw= Res_{w\to z}\bigl(z+(w-z)\bigr)^{m+1}( \frac{2}{(w-z)^2}\langle T(z)\cdots\rangle+ \frac{1}{(w-z)} \langle \partial_z T(z)\cdots\rangle+\frac{D/2}{(w-z)^4}\langle\cdots\rangle)
Res wz(z+(wz)) m+1T(w)T(z)dw=D2(m+13)z m2+2(m+11)z mT(z)+z m+1 zT(z) \phantom{Res_{w\to z}\bigl(z+(w-z)\bigr)^{m+1}\langle T(w)T(z)\cdots\rangle dw}= \frac{D}{2}\binom{m+1}{3}z^{m-2} \langle\cdots\rangle+2\binom{m+1}{1}z^m\langle T(z)\cdots\rangle + z^{m+1}\langle \partial_z T(z)\cdots\rangle

Therefore we find

[L m,L n]=Res z0(Dm 3m12z n+m1+2(m+1)z m+n+1T(z)+z n+m+2 zT(z))dz [L_m,L_n]=Res_{z\to 0} \biggl(D \frac{m^3-m}{12}z^{n+m-1}+2(m+1)z^{m+n+1}\langle T(z)\cdots\rangle+ z^{n+m+2}\partial_z \langle T(z)\cdots\rangle\biggr)dz
[L m,L n]=Dm 3m12δ m,n2(m+1)Res z0j n+m(z)dz+Res z0( zz n+m+2)T(z)dz \phantom{[L_m,L_n]}= D\frac{m^3-m}{12}\delta_{m,-n}-2(m+1)Res_{z\to 0}j_{n+m}(z)dz+Res_{z\to 0}(\partial_z z^{n+m+2})\langle{T}(z)\cdots\rangle dz
[L m,L n]=Dm 3m12δ m,n+(mn)Res z0j m+n(z)dz \phantom{[L_m,L_n]}= D\frac{m^3-m}{12}\delta_{m,-n}+(m-n)Res_{z\to 0}j_{m+n}(z)dz
[L m,L n]=(mn)L m+n+Dm 3m12δ m,n. \phantom{[L_m,L_n]}=(m-n)L_{m+n}+D\frac{m^3-m}{12}\delta_{m,-n}.
Revised on May 19, 2010 at 13:43:39 by Domenico Fiorenza