Symbols and conventions

This page is assembled mechanically from the LaTeX conventions appendices of the two papers and the conventions block of the phase-6 brief. Cross-references survive only as their raw labels, and there is no curated symbol table yet: a human pass is still owed. Read the appendices themselves if a symbol here is unclear.

Every constant quoted in this project depends on the choices collected here. This page is a first assembly from the conventions of the two papers and of the Phase 6 brief; it is meant to be read once by a human and corrected, after which the marker line at the top stops the generator from touching it again.

Assembled from paper1/app_conventions.tex, paper2/app_conventions.tex, rigor/phase6_brief.md. Mathematics is reproduced as LaTeX; a renderer that supports it will typeset it, and the sources above are authoritative.

From paper1/app_conventions.tex

Fourier transform and the Hardy projection

$$ \widehat f(k)=\int_{\mathbb R}f(x)e^{-ikx} dx, \qquad f(x)=\frac1{2\pi}\int_{\mathbb R}\widehat f(k)e^{ikx} dk . $$ With this convention the positive-energy subspace for the light-ray coordinate used here is ${k<0}$, so $P_+=\mathbf1_{{k<0}}$ and $P_-=1-P_+$; the distributional kernel of $P_+$ is (eq:hardy). (In (RigorUB) and in the Bogoliubov numerics the opposite labelling $P_+=\mathbf1_{{k>0}}$ is used for the same object after $x\to-x$; the Hilbert--Schmidt norms and determinants quoted are unaffected.)

Modular frame

For the interval $I=(-a,L)$, $$ \xi(x)=\log\frac{x+a}{L-x}, \qquad \beta(x)=\frac{dx}{d\xi}=\frac{(x+a)(L-x)}{a+L}, \qquad \beta_0=\beta(0)=\frac{aL}{a+L}, $$ and the modular modes are (eq:modes), normalized by $\int_{\mathbb R}d\kappa |\psi_\kappa\rangle\langle\psi_\kappa|=1$ on $L^2(I)$, with $Q\psi_\kappa=q_\kappa\psi_\kappa$, $q_\kappa=(1+e^\kappa)^{-1}$; thus $\kappa$ is the one-particle modular Hamiltonian $K=\log\frac{1-Q}Q$, and $\kappa\to+\infty$ labels the nearly empty modes. Modular translation is $\xi\mapsto\xi-2\pi t$, corresponding to the KMS asymmetry $\widehat W(-k)=e^{2\pi k}\widehat W(k)$ of the thermal spectral density at inverse temperature $2\pi$. A symbol $X$ with bounded kernel on $I\times I$ has modular kernel $$ X(\kappa,\kappa') =\frac1{(2\pi)^2}\iint d\xi d\xi' e^{-i\kappa\xi/2\pi} \widetilde X(\xi,\xi') e^{i\kappa'\xi'/2\pi}, \qquad \widetilde X(\xi,\xi')=\sqrt{\beta(x)\beta(y)} X(x,y), $$ and $ \operatorname{Tr} X=\int d\kappa X(\kappa,\kappa)$ when $X$ is trace class.

Cross-ratio variables

Four variables occur, all Möbius invariants of the same configuration: $$ \eta=\frac{b(a+b+c)}{(a+b)(b+c)},\quad z=\frac1\eta-1=\frac{ac}{b(a+b+c)},\quad \eta_{\rm V}=1-\eta=\frac z{1+z},\quad \zeta=\frac{as}{a+L} . $$ $\eta$ is the cross ratio of Note 1, in which the Longo--Xu conditional mutual information is $-\frac r6\log\eta$; $\eta_{\rm V}$ is the cross ratio of (VWZ), the complement of $\eta$; $\zeta$ is the recovery variable, with $1+\zeta$ the cross ratio of $J_s$ inside $I$. For the ordinary Petz map $\zeta=2z$, for the rotated map $\zeta_t=z(1+e^{-2\pi t})$, and for the pure geometric compression $\zeta=z$.

Central charge versus interval length

The letter $c$ denotes the length of the third interval $C$ throughout (sec:setting to sec:petz); where the central charge appears it is written $c_{\rm cft}$, and $c_{\rm cft}=r$ for the net of $r$ complex chiral fermions. In (prop:twobranch) the pair $(c,\bar c)$ denotes the two central charges of a two-dimensional CFT, the interval lengths being written $L_A,L_B,L_C$ there.

From paper2/app_conventions.tex

Fourier transform

$\hat f(p)=\int_{\mathbb R}f(x)e^{-ipx}dx$ on the line; $\hat f_{n}=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(\theta)e^{-in\theta}d\theta$ on the circle. Sobolev norms: $|f|_{\beta}=\sum_{n}|\hat f_{n}|(1+|n|^{\beta})$ (the Carpi--Weiner norm), and $|G|_{3/2}^{2}=\int_{\mathbb R}|p|^{3}|\hat G(p)|^{2}\frac{dp}{2\pi}$ for the homogeneous $\dot H^{3/2}$ (Weil--Petersson) norm on the line.

Inner products, modular objects

$\langle\cdot,\cdot\rangle$ is conjugate linear in the first variable. $S=J\Delta^{1/2}$, $F=S^{*}=J\Delta^{-1/2}$, $J\Omega=\Omega$, $\Delta\Omega=\Omega$, $J \mathcal{M} J= \mathcal{M}'$. In the line chart with $I=(-\infty,L)$ and $u=L-x$, (BW) says that $\Delta^{it}$ is the dilation $u\mapsto e^{2\pi t}u$ and $J$ implements $u\mapsto-u$, i.e. $x\mapsto 2L-x$.

Hardy projections

For the free fermion the one-particle space is $L^{2}(\mathbb R)$ and $\Pi_{\pm}$ are the spectral projections of the momentum $-i\partial_{x}$ onto $\pm(0,\infty)$; we use $\Pi_{+=$ positive momenta}, so that $\Pi_{+}L^{2}$ is the Hardy space of the upper half plane and the vacuum is the Fock state of $\Pi_{+}$. The opposite convention ($\Pi_{+}=$ negative momenta) is also common; it replaces $V=\Pi_{-}W\Pi_{+}$ by $\Pi_{+}W\Pi_{-}$, which has the same Hilbert--Schmidt norm whenever $\Pi_{+}W\Pi_{+}$ has Fredholm index zero (in general the two squared norms differ by that index), so none of the quoted numbers changes in that case. With $\mathfrak L_{f}=f\partial_{x}+\frac12f'$ one has $T(f)= : d\Gamma(-i\mathfrak L_{f}) :$ and $T(f)\Omega\cong\Pi_{-}(-i\mathfrak L_{f})\Pi_{+}$, and $|\Pi_{-}\mathfrak L_{f}\Pi_{+}|_{\mathfrak S_{2}}^{2}=\frac{1}{48\pi^{2}}\int_{0}^{\infty} p^{3}|\hat f(p)|^{2}dp$, i.e. (eq:appnorm) at $c=1$ (checked numerically to $1.5\cdot10^{-11}$).

The constants, and where they come from

From rigor/phase6_brief.md

MODULAR FRAME of I

y = log((x-p_1)/(p_{n+1}-x)), beta_I(x) = dx/dy = (x-p_1)(p_{n+1}-x)/T. The first-order field of a corner (p, kappa) is EXACTLY zeta w(y - y_p) d/dy with w(u) = -4 sinh^2(u/2) 1_{u>0} (Phase 5 F1 Prop 6.2, f3_t0.py) and zeta = kappa beta_I(p). [Derivation: in the (0,inf)-frame X = e^y, -sigma'(X-X_p)^2 d/dX = -sigma' X_p (X/X_p - 2 + X_p/X) d/dy = sigma' X_p w(y-y_p) d/dy, and sigma' X_p = sigma beta_I(p) by the (-1)-density law sigma' = sigma/M'(p) of parabolic parameters under a Moebius change of coordinate M. Theorem A's zeta = a s/(a+L) is the case p = 0, I = (-a, L): beta_I(0) = aL/(a+L), sigma = s/L.]

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