Chapter 3 of 3

§Amplitudes with local coordinates

We use the Majorana VOSA of the preceding chapter and the Wick theorem Proposition 2.4.

§3.1 Local coordinates and the amplitude

Choose distinct finite punctures $z_i$ on the sphere. A local coordinate at $z_i$ is an invertible germ

u=f_i(x),\qquad x=z-z_i,\qquad f_i(0)=0,\quad f_i'(0)\ne0.

Thus $f_i$ maps the global displacement $x$ to the local coordinate $u$ $u$ . For boundary insertions in the upper half-plane, take $z_i\in\mathbb R$ and boundary-compatible local coordinates. All inserted fields below remain at $z_i$ $z_{i}$ .

For any origin-fixing germ $f$ $f$ , write

f(x)=\alpha\exp\!\left(\sum_{m\geq1}A_mx^{m+1}\partial_x\right)x, \qquad W(f)=\exp\!\left(-\sum_{m\geq1}A_mL_m\right)\alpha^{-L_0}.

The coefficients $A_m$ are determined recursively. Choose a logarithm of $\alpha=f'(0)$ and use compatible lifts under composition. Then

W(f)W(g)=W(g\circ f),\qquad W(f^{-1})=W(f)^{-1}.

On each state of $F$ $F$ , the positive Virasoro modes lower weight, so the exponential is a finite sum. This is the coordinate-change convention of Meng–Lan–Gu, Appendix D.1.

Definition 3.1 (Amplitude with local coordinates)

For states $a_i\in F$ $F$ , define the normalized chiral vacuum amplitude by

\widehat{\mathcal A}_f(a_1,\ldots,a_n) =\left\langle Y(W(f_1)a_1,z_1)\cdots Y(W(f_n)a_n,z_n) \right\rangle.

The expectation is the vacuum expectation of Proposition 2.4, read as a rational function in the ordered insertion points. In particular, $\widehat{\mathcal A}_f(\mathbf1,\ldots,\mathbf1)=1$ $1$ .

This is the vacuum-sector specialization of the decorated-sphere amplitude in Appendix D.1, with the outgoing vacuum at infinity. For the primary state $\psi$ $ψ$ ,

Y(W(f_i)\psi,z_i)=f_i'(0)^{-1/2}\psi(z_i).

For descendants, the positive Virasoro modes in $W(f_i)$ also contribute. The following identities compute these transformed states.

§3.2 Coordinate covariance and descendant insertions

Put

g_i=f_i^{-1},\qquad \varphi_i(u)=z_i+g_i(u),\qquad \lambda_i=g_i'(0),\qquad s_i(u)^2=g_i'(u).

Here $\varphi_i$ is the insertion map from the local disk to the sphere; it is inverse to the local coordinate function $z\mapsto f_i(z-z_i)$ $z_{i})$ . The square-root branch is fixed by the chosen lift of $f_i$ $f_{i}$ .

Proposition 3.2 (Coordinate covariance of the fermion)

For an invertible germ $g(u)=\lambda u+O(u^2)$ $O (u^{2})$ ,

W(g)^{-1}\psi(u)W(g) =\sqrt{g'(u)}\,\psi(g(u)),

where $\sqrt{g'(0)}=\lambda^{1/2}$ with the chosen logarithm.

Proof.

By Exercise 1.5 and the primarity of $\psi$ $ψ$ ,

[L_m,\psi(u)] =\left(u^{m+1}\partial_u+\frac{m+1}{2}u^m\right)\psi(u).

Write $g(u)=\lambda\exp(v(u)\partial_u)u$ $λ exp (v (u) \partial_{u}) u$ , where $v(u)=\sum_{m\geq1}A_mu^{m+1}$ $\sum_{m \geq 1} A_{m} u^{m + 1}$ , and set $A=\sum_{m\geq1}A_mL_m$ $\sum_{m \geq 1} A_{m} L_{m}$ . Then

[A,\psi(u)]=\left(v(u)\partial_u+\frac12v'(u)\right)\psi(u).

Let $g_t(u)=\exp(t v(u)\partial_u)u$ $exp (t v (u) \partial_{u}) u$ . The field $F_t(u)=e^{tA}\psi(u)e^{-tA}$ satisfies

\partial_tF_t(u)=\left(v(u)\partial_u+\frac12v'(u)\right)F_t(u), \qquad F_0(u)=\psi(u).

The chain rule shows that $\sqrt{g_t'(u)}\,\psi(g_t(u))$ satisfies the same equation and initial condition, with square root $1$ at $t=0$ $0$ . Uniqueness of formal power-series solutions gives

e^A\psi(u)e^{-A}=\sqrt{g_1'(u)}\,\psi(g_1(u)).

Similarly, the $L_0$ commutator gives

\lambda^{L_0}\psi(u)\lambda^{-L_0} =\lambda^{1/2}\psi(\lambda u).

Since $W(g)^{-1}=\lambda^{L_0}e^A$ and $g=\lambda g_1$ $λ g_{1}$ , combining these two identities proves the proposition.

Corollary 3.3 (Generating transformed descendants)

The fields inserted in Definition 3.1 satisfy

\Psi_i(u):=Y(W(f_i)e^{uT}\psi,z_i) =s_i(u)\psi(\varphi_i(u)).

Its coefficients give the transformed descendants at the fixed puncture $z_i$ $z_{i}$ :

[u^m]\Psi_i(u) =\frac1{m!}Y(W(f_i)T^m\psi,z_i),\qquad m\geq0.

Proof.

Since $W(f_i)=W(g_i)^{-1}$ and $W(g_i)\mathbf1=\mathbf1$ $1$ , apply Proposition 3.2 to the vacuum and use creation:

W(f_i)e^{uT}\psi =s_i(u)e^{g_i(u)T}\psi.

Applying $Y(\,\cdot\,,z_i)$ coefficientwise, translation gives

Y(W(f_i)e^{uT}\psi,z_i) =s_i(u)Y(\psi,z_i+g_i(u)),

which is the asserted identity.

For example,

\begin{aligned} [u^0]\Psi_i(u) &=Y(W(f_i)\psi,z_i)=\lambda_i^{1/2}\psi(z_i),\\ [u^1]\Psi_i(u) &=Y(W(f_i)T\psi,z_i)\\ &=\lambda_i^{3/2}\partial\psi(z_i) +\frac{g_i''(0)}{2\lambda_i^{1/2}}\psi(z_i). \end{aligned}

Thus $\Psi_i$ is a way to compute the fields in the amplitude definition. Extracting coefficients at $u=0$ keeps their insertion point at $z_i$ $z_{i}$ .

§3.3 Regularized contractions and Pfaffian amplitudes

For states containing several fermions, use a separate local variable for each constituent and normal order in those local variables. The local vacuum contraction is $1/(u-v)$ $v)$ . After the coordinate change, the contraction of the generating fields is $s_i(u)s_j(v)/(\varphi_i(u)-\varphi_j(v))$ $φ_{j} (v))$ .

Definition 3.4 (Regularized contraction kernel)

Define

K_{ij}(u,v)= \begin{cases} \displaystyle\frac{s_i(u)s_j(v)}{\varphi_i(u)-\varphi_j(v)},&i\ne j,\\[3mm] \displaystyle\frac{s_i(u)s_i(v)}{\varphi_i(u)-\varphi_i(v)}-\frac1{u-v},&i=j. \end{cases}

These are Taylor series at $(0,0)$ and satisfy $K_{ij}(u,v)=-K_{ji}(v,u)$ $- K_{j i} (v, u)$ . For positive half-integers $r,s$ $r, s$ , set

M_{(i,r),(j,s)} =[u^{r-\frac12}v^{s-\frac12}]K_{ij}(u,v).

Thus $M^{\mathsf T}=-M$ $- M$ . The brackets extract Taylor coefficients, including the factorial normalization of derivatives.

The subtraction removes the local vacuum contraction. For $\varphi_i(u)=z_i+u$ $u$ , the entire diagonal block vanishes, recovering the usual rule that contractions occur only between different normal-ordered factors. For a nonlinear coordinate, its regular remainder may be nonzero. These diagonal blocks encode the finite change of normal ordering.

Proposition 3.5 (Pfaffian amplitudes)

Let

a_i=\psi_{-r_{i,1}}\cdots\psi_{-r_{i,k_i}}\mathbf1, \qquad r_{i,1}>\cdots>r_{i,k_i}>0.

Concatenate the occupied labels $(i,r_{i,\alpha})$ in puncture order, preserving the displayed order within each state, to obtain an ordered list $I$ $I$ . Then

\widehat{\mathcal A}_f(a_1,\ldots,a_n) =\operatorname{Pf}M[I,I].

For an odd number of occupied modes the amplitude is zero; for the empty list it is $1$ $1$ . The formula extends multilinearly to arbitrary states in $F$ $F$ .

Proof.

Write each state as the coefficient of $\prod_\alpha u_\alpha^{r_{i,\alpha}-1/2}$ in $:\psi(u_1)\cdots\psi(u_{k_i}):\mathbf1$ $1$ , with normal ordering in the local variables. Apply $W(f_i)$ to this state before applying $Y(\,\cdot\,,z_i)$ $Y (\cdot, z_{i})$ , as required by Definition 3.1. The conjugation identity in Proposition 3.2 transforms each constituent into $\Psi_i$ $Ψ_{i}$ ; the local scalar subtractions are unchanged. Wick’s theorem therefore gives pairwise contractions $s_i(u)s_j(v)/(\varphi_i(u)-\varphi_j(v))$ $φ_{j} (v))$ . Within a group, local normal ordering subtracts $1/(u-v)$ $v)$ , leaving exactly $K_{ii}$ $K_{ii}$ .

In each pairing, every variable occurs in exactly one kernel. Extracting the coefficient prescribed by each occupied mode therefore gives a product of entries of $M$ $M$ . The signed sum over pairings is its Pfaffian. In particular, the coefficient convention agrees with

Y(\psi_{-m-\frac12}\mathbf1,z) =\frac1{m!}\partial_z^m\psi(z).

§3.4 Finite-mode calculation

For a fixed finite set of modes, compute the required Taylor coefficients of $K_{ij}$ once and store $M$ $M$ . Each occupation-basis amplitude then requires only a selected submatrix. A Pfaffian of order $N$ can be evaluated by pivoted antisymmetric elimination in $O(N^3)$ arithmetic operations and $O(N^2)$ storage, without enumerating pairings; see Wimmer. The regularization of $K_{ii}$ must be performed in Taylor series before numerical evaluation.

If $a_i$ has weight $h_i$ $h_{i}$ , fermionic exclusion gives

h_i\geq\frac{k_i^2}{2},\qquad N=\sum_i k_i\leq\sum_i\lfloor\sqrt{2h_i}\rfloor.

For general states, apply the formula to their occupation-basis expansions. The cost of producing every entry of a tensor with $D$ states at each puncture still includes its $D^n$ output entries. All amplitudes here belong to the Majorana vacuum block; an overall geometric vacuum factor and amplitudes involving spin fields require separate data.

Exercise 3.6 (Coordinate changes and Gaussian amplitudes)

Regularity. Put

$H_i(u,v)=\frac{\varphi_i(u)-\varphi_i(v)}{u-v}.$

Show that $H_i$ is an invertible Taylor series and that $s_i(u)s_i(v)-H_i(u,v)$ is divisible by $(u-v)^2$ $v)^{2}$ . Deduce the regularity and antisymmetry of $K_{ii}$ $K_{ii}$ , and use

$K_{ii}(u,v)=\frac{s_i(u)s_i(v)-H_i(u,v)}{(u-v)H_i(u,v)}$

to compute its Taylor coefficients without subtracting poles numerically.
The Schwarzian term. Define

$Sf=\frac{f'''}{f'}-\frac32\left(\frac{f''}{f'}\right)^2.$

Prove

$K_{ii}(u,v)=\frac{S\varphi_i(0)}{12}(u-v) +\text{terms of total degree at least }2.$

Using $\omega=\tfrac12\psi_{-\frac32}\psi_{-\frac12}\mathbf1$ $\frac{1}{2} ψ_{- \frac{3}{2}} ψ_{- \frac{1}{2}} 1$ , deduce

$\widehat{\mathcal A}_f(\mathbf1,\ldots,\omega,\ldots,\mathbf1) =\frac{S\varphi_i(0)}{24}.$

More generally, for three punctures, set $z_{ij}=z_i-z_j$ and verify

$\widehat{\mathcal A}_f(\omega,\psi,\psi) =s_2(0)s_3(0)\left( \frac{\lambda_1^2z_{23}}{2z_{12}^2z_{13}^2} +\frac{S\varphi_1(0)}{24z_{23}}\right).$

Identify the pairings responsible for the two terms.
Size of the Pfaffian. Show that the least possible weight of a state with $k$ occupied fermion modes is $\tfrac12+\tfrac32+\cdots+(k-\tfrac12)=k^2/2$ $k^{2} /2$ . Deduce the bound on $N$ above.
Generating all amplitudes. For a finite ordered set of mode labels and Grassmann variables $\theta_a$ $θ_{a}$ , prove

$\exp\!\left(\frac12\sum_{a,b}\theta_aM_{ab}\theta_b\right) =\sum_{I\text{ even}}\operatorname{Pf}M[I,I]\,\theta_I,$

where $I$ runs through subsets in the inherited order and $\theta_I$ is the corresponding ordered product. Explain how $M$ determines all occupation-basis amplitudes without storing the full tensor.

§3.5 Provenance and license

Author: Chenqi Meng. This independent note is released under Creative Commons Attribution 4.0 International.

The definitions, basic properties, and Majorana example were developed in the VOSA chapter of Introduction to Vertex Operator Algebras, published under CC0. Material previously released under CC0 remains available under those terms. This edition includes the additional treatment of amplitudes with local coordinates.

License · CC BY 4.0 · Peerpedia collaboration terms