Domenico Fiorenza
Vertex algebras avant Borcherds

Contents

Idea

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})

with

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.

Regularization

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:

:A:=A:A:=A

: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:

𝒜=0.\langle\mathcal{A}\cdots\rangle=0.

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

hence

(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,

where

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

Then

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.

OPEs

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.

Similarly,

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}.

The TT OPE

Set

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')

Then

¯ 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.

Set

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

Then

[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