Haar Non-Unitary Conformal Field Theory

• Haar Non-Unitary CFT defines a novel class of 2D non-unitary rational conformal field theories characterized by generalized Haagerup–Izumi modular data and explicit fusion rules.
• Explicit Nahm characters obtained through half-index computations yield detailed q‑expansions and modular completions via MLDE and Bantay–Gannon methods.
• The theory exhibits a non-unitary bulk–boundary correspondence linking 3D twisted SCFTs to 2D RCFTs, offering fresh insights into modular tensor categories.

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 (Gang et al., 2023).

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) (Gang et al., 2023).

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 (Gang et al., 2023).

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 $3\leq k\leq 11$ (Gang et al., 2023).

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 (Gang et al., 2023).

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 (Gang et al., 2023)) 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 (Gang et al., 2023).

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 (Gang et al., 2023).