---
title: Haar Non-Unitary Conformal Field Theory
url: https://www.emergentmind.com/topics/haar-non-unitary-conformal-field-theory
type: topic
---

# Haar Non-Unitary Conformal Field Theory

Haar Non-Unitary Conformal Field Theory (CFT) denotes a new class of two-dimensional non-unitary rational conformal field theories (RCFTs) whose modular data replicate the generalized Haagerup-Izumi modular data. These theories arise naturally via a bulk-boundary correspondence with three-dimensional non-unitary Haagerup topological field theories (TQFTs) obtained by twisting 3D $\mathcal{N}=4$ rank-zero superconformal field theories (SCFTs), termed S-fold SCFTs. The explicit construction of such non-unitary RCFTs employs half-index computations to produce explicit Nahm-type characters, which, upon modular completion using the theory of Bantay-Gannon, yield the full set of admissible boundary conformal characters and establish a distinctive non-unitary bulk-boundary correspondence [2310.14877].

## 1. Modular Data and Category Structure

For each integer $k \geq 3$, the associated non-unitary Haagerup TQFT, denoted $\mathrm{TFT}_-[\mathcal{S}_k]$, exhibits $2k+2$ simple objects labeled $\alpha=0,1,\ldots,2k+1$. The modular data—specifically the $S$ and $T$ matrices—are given in closed form. The $S$ matrix, a $(2k+2)\times(2k+2)$ block matrix, incorporates coefficients:
- $a_0=1/\sqrt{8(k-2)}+1/\sqrt{8(k+2)}$
- $a_1=1/\sqrt{2(k-2)}$
- $a_2=1/\sqrt{2(k+2)}$
- $a_3=1/\sqrt{8(k-2)}-1/\sqrt{8(k+2)}$

The upper-left $4\times4$ block is structured as $((-)^k a_0, a_0, a_3, (-)^k a_3;\ldots)$, while lower blocks for $3\leq i,j\leq k-1$ follow $S_{2+i,2+j}=2a_1 \cos(ij \pi/(k-2))$. For the final $k+1$ labels, $S_{i,j}=2a_2 \cos(ij\pi/(k+2))$. The $T$ matrix is diagonal with entries $T_{\alpha\alpha} = \exp[2\pi i (h_\alpha - c/24)]$, where
- Conformal spins $h_\alpha = (k+2)/4,0,0,(k+2)/4,\left\{A^2/[4(k-2)]\right\}_{A=1\ldots k-3}, \left\{B^2/[4(k+2)]\right\}_{B=1\ldots k+1}-(k+2)/4$
- Central charge $c \equiv -(6k+11)\bmod 24$.

The fusion rules, achieved via the Verlinde formula, are non-negative integers and coincide precisely with those of the Haagerup–Izumi fusion category, including $\mathbb{Z}_2$ symmetry generated by simple objects $0,1$ and $A$-type fusion structure in objects related to $SU(2)_{k-2}$, confirming realization as a non-unitary modular tensor category (MTC) [2310.14877].

## 2. Nahm Characters and $q$‑Expansions

Four principal “Nahm” conformal characters (including the vacuum) are constructed using fermionic sum representations via half-index computations. For $m\equiv k-2$ and $r\equiv m+1$, let $K$ denote the relevant rank-$r$ matrix and $Q_1=2$, $Q_{a>1}=1$. The fermionic sums run over ${\mathcal M}=(m_1,\ldots,m_r)\in\mathbb{N}^r$ or $m_1\in\mathbb{N}+1/2, m_{2..r}\in\mathbb{N}$ for twisted sectors.

The explicit expressions for the characters are:
\begin{align*}
\chi_0(q) &= q^{\Delta_0} \sum_{{\mathcal M}\in\mathbb{N}^r} \frac{q^{\frac12 {\mathcal M}^T K {\mathcal M} + (r-1)m_1 + \sum_{a=1}^r (a-1)m_a}}{(q)_{2m_1} \prod_{a=2}^r (q)_{m_a}} \\
\chi_1(q) &= q^{\Delta_1} \sum_{{\mathcal M}\in(\mathbb{N}+1/2)\times\mathbb{N}^{r-1}} \frac{q^{\frac12 {\mathcal M}^T K {\mathcal M} + (r-1)m_1 +\ldots -\frac34(r-1)}}{(q)_{2m_1} \prod_{a>1} (q)_{m_a}} \\
\chi_2(q) &= q^{\Delta_2} \sum_{{\mathcal M}\in\mathbb{N}^r} \frac{q^{\frac12 {\mathcal M}^T K {\mathcal M}}}{(q)_{2m_1} \prod_{a>1} (q)_{m_a}} \\
\chi_3(q) &= q^{\Delta_3} \sum_{{\mathcal M}\in(\mathbb{N}+1/2)\times\mathbb{N}^{r-1}} \frac{q^{\frac12 {\mathcal M}^T K {\mathcal M} - \frac14(r-1)}}{(q)_{2m_1} \prod_{a>1} (q)_{m_a}}
\end{align*}
with exponent shifts
- $\Delta_0 = -(c/24) = (6k+11)/24$
- $\Delta_1 = -1/24$
- $\Delta_2 = -1/24$
- $\Delta_3 = (6k+11)/24\pmod 1$

Their $q$-expansions commence:
- $\chi_0 = q^{\Delta_0}(1 + q^2 + 2q^3 + 3q^4 + \ldots)$
- $\chi_1 = q^{\Delta_1}(1 + 2q + 4q^2 + 6q^3+ \ldots)$
- $\chi_2 = q^{\Delta_2}(1 + q + 3q^2 + 4q^3+ \ldots)$
- $\chi_3 = q^{\Delta_3}(1 + q + 2q^2 + 3q^3 + \ldots)$

## 3. Half-Indices from 3D $\mathcal{N}=2$ Abelian Dual SCFTs

These Nahm characters arise from the half-index computations in three-dimensional $\mathcal{N}=2$ abelian dual SCFTs $\mathcal{S}_k$, which feature $U(1)^r$ gauge factors and chiral fields with charges $Q_a$. With Dirichlet boundary conditions acting on both chirals and vectors (“$D_c$”), the half-index on $D^2\times S^1$ is:
\[
\mathcal{I}_{D_c}(q, \eta, \nu)
= \frac{1}{(q)_{\infty}^r} \sum_{{\mathcal M}\in\mathbb{Z}^r} q^{\frac12 {\mathcal M}^T K {\mathcal M}} \left[(-q^{1/2})^{\nu-1}\eta\right]^{(r-1)m_1 + \sum_{a=1}^r (a-1)m_a} \prod_{a=1}^r (q^{-Q_a m_a + 1};q)_\infty.
\]
In the limit $\eta\to 1$, $\nu\to -1$, this expression reproduces $\chi_0$. Insertion of supersymmetric loops with shifted magnetic charges $m_1\rightarrow m_1+1/2$, or corresponding fugacity modifications, gives rise to the other half-indices $\mathcal{I}^1 = \chi_1, \mathcal{I}^2 = \chi_2, \mathcal{I}^3 = \chi_3$, up to overall $q^\Delta$ factors. The discrete nature of the flux lattices makes contour integration trivial. Thus, the Nahm expression for the vacuum character and its companions arises directly from 3D gauge theory computations [2310.14877].

## 4. Modular Completion and Partition Function Construction

To achieve the full set of $2k+2$ boundary conformal characters transforming under $SL(2,\mathbb{Z})$, two modular completion methodologies apply:

1. **Modular Linear Differential Equation (MLDE):** A unique monic differential equation of order $2k+2$ with minimal Wronskian index is solved, for which $\chi_0\dots\chi_3$ are explicit solutions. The remaining $2k-2$ characters result as further Frobenius-series solutions, and constraints of integrality, positivity, and compatiblity with $S$-matrix transformations fix this solution uniquely for $k\leq 11$.

2. **Bantay–Gannon Riemann–Hilbert Approach:** Direct Riemann–Hilbert completion constructs the characteristic matrix $\Xi(\tau) = q^\Lambda(1+\chi q+\ldots)$, where $\Lambda = \mathrm{diag}(\Delta_\alpha)$ satisfies specific trace and algebraic relations. The recursion relation $\Xi[n]$ (eq. 3.18) and the fixing of leading “principal parts” to known Nahm characters determine the complete set of admissible conformal characters. This method has been carried out explicitly for $3\leq k\leq 11$ [2310.14877].

| Completion Method                | Key Step                                     | Unique Up To        |
|----------------------------------|-----------------------------------------------|---------------------|
| MLDE                            | Solve specific modular ODE                    | $k \leq 11$         |
| Bantay–Gannon                   | Riemann–Hilbert completion using $\Xi(\tau)$  | $3 \leq k \leq 11$  |

## 5. Central Charge, Dimensions, and Fusion Rules

- **Central Charge:** $c \equiv -(6k+11) \bmod 24$. The effective central charge $c_\text{eff} = -24\min_\alpha \Delta_\alpha = 1$ for all $k$.
- **Conformal Dimensions:** $h_\alpha = c/24 + \Delta_\alpha$, with explicit $\Delta_\alpha$ values determined as above.
- **Fusion Rules:** Non-negative integer structure constants $N_{\alpha\beta}^\gamma$ from the Verlinde formula:
  \[
  N_{\alpha\beta}^{\gamma} = \sum_\delta \frac{S_{\alpha\delta} S_{\beta\delta} S^*_{\gamma\delta}}{S_{0\delta}}
  \]
  coincide with the generalized Haagerup–Izumi category, explicitly confirming $\mathbb{Z}_2$ actions and subalgebraic structure matching $SU(2)_{k-2}$ fusion for particular simple objects [2310.14877].

## 6. Non-Unitary Bulk–Boundary Correspondence

A three-dimensional $\mathcal{N}=4$ rank-zero S-fold SCFT $\mathcal{S}_k$ (as in Eq. 1.2 of [2310.14877]) allows both $SU(2)_C$ and $SU(2)_H$ topological twists. The topologically twisted partition function on $\Sigma_g\times S^1$ (indexed by spin structure $\nu=\pm1$) yields TQFT data via Bethe vacua:
- $Z_{\text{TFT}}(S^2 \times S^1) \leftrightarrow \sum_\alpha S_{0\alpha}^2$
- $Z_{\text{TFT}}(L(p,1)) \leftrightarrow \sum_\alpha S_{0\alpha}^2 T_{\alpha\alpha}^p$

The half-indices with holomorphic (“A-model”) boundary condition realize four explicit Nahm characters. Completion to the full RCFT boundary spectrum ensures the same modular data. This non-unitary bulk–boundary correspondence thus demonstrates: the 3D twisted SCFT furnishes a non-unitary modular tensor category in the bulk, whose chiral boundary spectrum is a 2D non-unitary Haagerup RCFT $\mathcal{R}_k$ with identical modular data [2310.14877].

## 7. Context and Implications

The Haar non-unitary CFTs provide explicit realizations of non-unitary RCFTs with generalized Haagerup modular data and establish precise links between 3D non-unitary TQFTs and 2D non-unitary RCFTs via bulk-boundary correspondence. The construction leverages advances in half-index technology, modular completions via MLDE and Riemann–Hilbert methods, and deep categorical relations rooted in Haagerup–Izumi fusion frameworks. A plausible implication is the potential for broader classes of non-unitary RCFTs with non-trivial fusion and modular properties in low-dimensional topology and quantum field theory [2310.14877].

Source: https://www.emergentmind.com/topics/haar-non-unitary-conformal-field-theory