# Ground State Representations of Some Non-Rational Conformal Nets

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Möbius Covariant Net on Circle

**Möbius covariant net**$(\mathcal{A},U,\mathsf{\Omega})$ is a triple of the map $\mathcal{A}:\mathcal{I}\ni I\mapsto \mathcal{A}\left(I\right)\subset \mathcal{B}\left(\mathcal{H}\right)$, where $\mathcal{A}\left(I\right)$ is a von Neumann algebra on a Hilbert space $\mathcal{H}$ and $\mathcal{B}\left(\mathcal{H}\right)$ is the set of all bounded operators on $\mathcal{H}$, U a strongly continuous representation of the group $\mathrm{PSL}(2,\mathbb{R})$ on $\mathcal{H}$ and a unit vector $\mathsf{\Omega}\in \mathcal{H}$ satisfying the following conditions:

- (MN1)
**Isotony**: If ${I}_{1}\subset {I}_{2}$, then $\mathcal{A}\left({I}_{1}\right)\subset \mathcal{A}\left({I}_{2}\right)$.- (MN2)
**Locality**: If ${I}_{1}\cap {I}_{2}=\varnothing $, then $\mathcal{A}\left({I}_{1}\right)$ and $\mathcal{A}\left({I}_{2}\right)$ commute.- (MN3)
**Möbius covariance**: for $\gamma \in \mathrm{PSL}(2,\mathbb{R})$, $\mathrm{Ad}\phantom{\rule{0.166667em}{0ex}}U\left(\gamma \right)\left(\mathcal{A}\right(I\left)\right)=\mathcal{A}\left(\gamma I\right)$.- (MN4)
**Positive energy**: the generator ${L}_{0}$ of rotations $U\left(\rho \left(t\right)\right)={e}^{it{L}_{0}}$ is positive.- (MN5)
**Vacuum**: $\mathsf{\Omega}$ is the unique (up to a phase) invariant vector for $U\left(\gamma \right),\gamma \in \mathrm{PSL}(2,\mathbb{R})$ and $\overline{\mathcal{A}\left(I\right)\mathsf{\Omega}}=\mathcal{H}$ (the**Reeh-Schlieder property**).

- (MN6)
**Haag duality**: $\mathcal{A}\left({I}^{\prime}\right)=\mathcal{A}{\left(I\right)}^{\prime}$, where ${I}^{\prime}$ is the interior of the complement of I.- (MN7)
**Additivity**: If $I=\bigcup {I}_{j}$, then $\mathcal{A}\left(I\right)={\bigvee}_{j}\mathcal{A}\left({I}_{j}\right)$.

**Strong additivity**: If ${I}_{1}$ and ${I}_{2}$ are the intervals obtained from I by removing one point, then $\mathcal{A}\left({I}_{1}\right)\vee \mathcal{A}\left({I}_{2}\right)=\mathcal{A}\left(I\right)$.**Conformal covariance**: U extends to a projective unitary representation of the group ${\mathrm{Diff}}_{+}\left({S}^{1}\right)$ of orientation preserving diffeomorphisms and $\mathrm{Ad}\phantom{\rule{0.166667em}{0ex}}U\left(\gamma \right)\mathcal{A}\left(I\right)=\mathcal{A}\left(\gamma I\right)$, and $\mathrm{Ad}\phantom{\rule{0.166667em}{0ex}}U\left(\gamma \right)\left(x\right)=x$ if $I\cap \mathrm{supp}\phantom{\rule{0.166667em}{0ex}}\gamma =\varnothing $.

**conformal net**.

#### 2.2. Ground State Representations

**state $\phi $**on ${\mathcal{A}|}_{\mathbb{R}}$ is by definition a state on the ${C}^{\ast}$-algebra ${\overline{{\bigcup}_{I\u22d0\mathbb{R}}\mathcal{A}\left(I\right)}}^{\parallel \xb7\parallel}$. Let $\phi $ be a state on ${\mathcal{A}|}_{\mathbb{R}}$ and let ${\pi}_{\phi}$ be the GNS representation with respect to $\phi $. If ${\pi}_{\phi}$ extends to $\mathcal{A}\left({\mathbb{R}}_{\pm}\right)$ in the $\sigma $-weak topology, then we say that ${\pi}_{\phi}$ is

**solitonic**. If there is a representation ${U}^{\phi}$ of the translation group $\mathbb{R}$ such that $\mathrm{Ad}\phantom{\rule{0.166667em}{0ex}}{U}^{\phi}\left(\tau \left(t\right)\right)\left({\pi}_{\phi}\left(x\right)\right)={\pi}_{\phi}\left(\mathrm{Ad}\phantom{\rule{0.166667em}{0ex}}U\left(\tau \left(t\right)\right)\left(x\right)\right)$, then we say that ${\pi}_{\phi}$ is

**translation-covariant**. Furthermore, if the generator of ${U}^{\phi}$ can be taken positive, then we say ${\pi}_{\phi}$ is a

**positive-energy representation**.

**ground state representation**, and the state $\phi $ is called the

**ground state**. By the Reeh-Schlieder argument (see e.g., [15] (Theorem 1.3.2), [16] (Theorem 3.2.1)), the GNS vector ${\mathsf{\Omega}}_{\phi}$ is cyclic for ${\pi}_{\phi}\left(\mathcal{A}\left(I\right)\right)$ for any $I\u22d0\mathbb{R}$.

**Theorem**

**1**

- Isomorphism classes of strongly additive Möbius covariant nets
- Isomorphism classes of Borchers triples $(\mathcal{M},U,\mathsf{\Omega})$, ($\mathcal{M}$ is a von Neumann algebra, U is a representation of $\mathbb{R}$ with positive generator, Ω is cyclic and separating for $\mathcal{M}$ and $U\left(t\right)\mathsf{\Omega}=\mathsf{\Omega}$ such that $\mathrm{Ad}\phantom{\rule{0.166667em}{0ex}}U\left(t\right)\left(\mathcal{M}\right)\subset \mathcal{M}$ for $t\ge 0$) with the property that Ω is cyclic for $\mathrm{Ad}\phantom{\rule{0.166667em}{0ex}}U\left(t\right){\left(\mathcal{M}\right)}^{\prime}\cap \mathcal{M}$

**automorphism**$\alpha $ of the net ${\mathcal{A}|}_{\mathbb{R}}$ is a family ${\left\{{\alpha}_{I}\right\}}_{I\u22d0\mathbb{R}}$ of automorphisms of $\left\{\mathcal{A}\right(I\left)\right\}$ such that ${\alpha}_{\tilde{I}}{|}_{\mathcal{A}\left(I\right)}={\alpha}_{I}$ for $I\subset \tilde{I}$. $\alpha $ extends naturally to ${\overline{{\bigcup}_{I\u22d0\mathbb{R}}\mathcal{A}\left(I\right)}}^{\parallel \xb7\parallel}$.

**Lemma**

**1.**

**Proof.**

**Proposition**

**1.**

## 3. Main Examples

#### 3.1. The $\mathrm{U}\left(1\right)$-Current Net

**$\mathrm{U}\left(1\right)$-current net**. It is known that it satisfies strong additivity [11] (Equation (4.21) and below), and conformal covariance [22] (Section 5.3).

**Lemma**

**2.**

**Proof.**

#### 3.2. The Virasoro Nets

**Virasoro subnet**. By Haag duality, it holds that ${\mathcal{B}}_{1}\left(I\right)\subset {\mathcal{A}}_{\mathrm{U}\left(1\right)}\left(I\right)$. Furthermore, by the Reeh-Schlieder argument, ${\mathcal{H}}_{{\mathrm{Vir}}_{1}}:=\overline{{\mathcal{B}}_{1}\left(I\right)\mathsf{\Omega}}$ does not depend on I. The restriction of ${\mathcal{B}}_{1}\left(I\right)$ and U to ${\mathcal{H}}_{{\mathrm{Vir}}_{1}}$ together with $\mathsf{\Omega}\in {\mathcal{H}}_{{\mathrm{Vir}}_{1}}$ is called the

**Virasoro net with $c=1$**and we denote it by $({\mathrm{Vir}}_{1},{U}_{1},{\mathsf{\Omega}}_{1})$.

## 4. Ground States

#### 4.1. On the $\mathrm{U}\left(1\right)$-Current Net

**Lemma**

**3.**

**Proof.**

**Theorem**

**2.**

- The states $\{\omega \circ {\alpha}_{q}\}$ are mutually different ground states of ${\mathcal{A}}_{\mathrm{U}\left(1\right)}{|}_{\mathbb{R}}$, and they are connected with each other by dilations. Dilations are not implemented unitarily.
- The GNS vector ${\mathsf{\Omega}}_{q}$ of $\omega \circ {\alpha}_{q}$ is in the domain of ${J}_{q}\left(f\right)$, where ${\alpha}_{q}\left(W\left(f\right)\right)={e}^{i{J}_{q}\left(f\right)}$ and ${J}_{q}\left(f\right)=J\left(f\right)+q{\int}_{\mathbb{R}}f\left(t\right)dt$. In particular, it holds that $\omega \circ {\alpha}_{q}\left(J\left(f\right)\right)=q{\int}_{\mathbb{R}}f\left(t\right)dt$
- The GNS representations $\left\{{\alpha}_{q}\right\}$ do not extend to $\mathcal{A}\left({\mathbb{R}}_{\pm}\right)$ in the weak operator topology, hence are not solitonic.
- The dual nets $\left\{{\widehat{\alpha}}_{q}\left({\mathcal{A}}_{\mathrm{U}\left(1\right)}\left(I\right)\right)\right\}$ are equal to the original net ${\mathcal{A}}_{\mathrm{U}\left(1\right)}$.

**Proof.**

#### The $\mathrm{U}\left(1\right)$-Current as a Multiplier Representation of the Loop Group $SL{S}^{1}$

#### 4.2. On Virasoro Nets

**Theorem**

**3.**

- The states $\{\omega \circ {\alpha}_{q}\}$ are mutually different ground states of ${\mathrm{Vir}}_{c}{|}_{\mathbb{R}}$ for different $\frac{{q}^{2}}{2}$, and they are connected with each other by dilations. Dilations are not implemented unitarily in ${\rho}_{\kappa ,q}$.
- The GNS vector ${\mathsf{\Omega}}_{q}$ of $\omega \circ {\alpha}_{q}$ is in the domain of ${T}_{q}^{\kappa}\left(f\right)$, where ${\alpha}_{q}\left({e}^{is{T}^{\kappa}\left(f\right)}\right)={e}^{i{T}_{q}^{\kappa}\left(f\right)}$ and ${T}_{q}^{\kappa}\left(f\right)=T\left(f\right)+qJ\left(f\right)+\kappa J\left({f}^{\prime}\right)+\frac{{q}^{2}}{2}{\int}_{\mathbb{R}}f\left(t\right)dt$ on ${\mathcal{H}}^{\infty}$.
- If $c>1$, the dual nets $\left\{{\widehat{\rho}}_{q}\left({\mathrm{Vir}}_{c}\left(I\right)\right)\right\}$ are not unitarily equivalent to any of ${\mathrm{Vir}}_{{c}^{\prime}}$, ${c}^{\prime}>1$.

**Proof.**

## 5. Outlook

**Classification**. As we have seen, the ground states we constructed in this paper are characterized by a number q (for the $\mathrm{U}\left(1\right)$-current net) or $\frac{{q}^{2}}{2}$ (for the Virasoro nets). This is a structure very similar to the invariant $\psi $ of [10] (Lemma 5.3). If one studies corresponding Lie algebras, it should be possible to classify ground state representations as in [10] (Theorem 5.6).

**Dual Nets**. The most fundamental properties of the dual nets $\left\{{\widehat{\rho}}_{\kappa ,q}\left({\mathrm{Vir}}_{c}\left(I\right)\right)\right\}$ remain open. Among them is conformal covariance. Conformal covariance implies the split property [34], and even the split property is unknown to hold in these dual nets. The split property may fail in the dual net in two-dimensional Haag-Kastler net [20] (Section 4.2), therefore, it may be worthwhile to try to (dis)prove the split property in these nets.

**More Positive-Energy Representations**. Ground states consist only a particular class of positive-energy representations. It was shown in [35,36] that there is a huge class of locally normal positive-energy representations of the free massless fermion field in $(3+1)$-dimensions. A similar construction should be possible in one dimension. Furthermore, important conformal nets, including some loop group nets, can be realized as subnets of (the tensor product of ) the free fermion field nets (see [37] (Examples 4.13–16)). Among them, there might be counterexamples to [12] (Conjecture 34).

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The functions ${g}_{n}(-{e}^{i\theta})$ (the thick line) and $g(-{e}^{i\theta})$ (the thin line above), restricted to $0\le \theta \le 2\pi $.

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Tanimoto, Y.
Ground State Representations of Some Non-Rational Conformal Nets. *Symmetry* **2018**, *10*, 415.
https://doi.org/10.3390/sym10090415

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Tanimoto Y.
Ground State Representations of Some Non-Rational Conformal Nets. *Symmetry*. 2018; 10(9):415.
https://doi.org/10.3390/sym10090415

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Tanimoto, Yoh.
2018. "Ground State Representations of Some Non-Rational Conformal Nets" *Symmetry* 10, no. 9: 415.
https://doi.org/10.3390/sym10090415