Tensionless NS-NS Point in AdS3/CFT2

Updated 18 November 2025

The tensionless NS-NS point is defined by minimal NS-NS flux (k=1), leading to a free worldsheet regime that accurately matches symmetric orbifold CFT spectra.
It features a drastic truncation of the bulk spectrum, with localization of partition functions on holomorphic maps and the emergence of infinite higher-spin symmetry.
This regime underpins the AdS3/CFT2 duality, offering a controlled setting to study string/black-hole transitions, integrable deformations, and nonperturbative dynamics.

The tensionless NS-NS point is a distinguished regime in string theory on $\mathrm{AdS}_3 \times \mathrm{S}^3 \times \mathbb{T}^4$ and related backgrounds, defined by having the minimal allowed unit ( $k=1$ ) of worldsheet NS-NS flux. This point exhibits an exact matching between the bulk string spectrum and the symmetric product orbifold CFT of the boundary, enables explicit computations of partition functions and correlators, and manifests enhanced higher-spin symmetry. Its key features arise from the collapse of the world-sheet theory to a free regime, localization mechanisms in path integrals, and the disappearance of long-string states. The tensionless NS-NS point plays a central role in providing a controlled, nonperturbative realization of AdS $_3$ /CFT $_2$ duality and offers a paradigm for studying tensionless strings and higher-spin physics.

1. Definition and Worldsheet Formulation

The tensionless NS-NS point is characterized by the worldsheet WZW level $k=1$ , corresponding to a regime where the string length $\ell_s = \sqrt{\alpha'}$ equals the AdS $_3$ radius $R$ ; equivalently, $k = R^2/\alpha' = 1$ (Giribet, 2020, Gaberdiel et al., 2023). The worldsheet action for AdS $_3$ in the NS-NS sector is that of an SL(2, $\mathbb{R}$ ) $_k$ WZW model, with the compact sectors implemented as SU(2) $_k$ WZW and free bosons/fermions for $\mathbb{T}^4$ .

At $k=1$ , the theory admits a hybrid formulation on the supergroup $\mathrm{PSU}(1,1|2)_1$ , supplemented by the $T^4$ free sector and ghost systems. The action (in Poincaré patch and Wakimoto variables $(\Phi, \gamma, \bar{\gamma}, \beta, \bar{\beta})$ ) collapses to

$S_{\text{AdS}_3} = \frac{1}{4\pi} \int d^2 z \left( \partial\Phi \, \bar{\partial} \Phi + e^{2\Phi} \partial\gamma \, \bar{\partial} \bar{\gamma} \right)$

or, equivalently, a first-order form with an interaction term $- \beta \bar{\beta} e^{-2\Phi}$ that decouples at the boundary (Eberhardt, 2020, Sriprachyakul, 6 May 2024). The total worldsheet central charge balances to zero, rendering the $\mathrm{PSU}(1,1|2)_1$ model "topological."

2. Spectrum, Localization, and Higher-Spin Enhancement

The $k=1$ point truncates the worldsheet spectrum drastically. The only allowed physical excitations stem from short BPS representations and their spectrally-flowed images; long-string (continuous) representations vanish (Gaberdiel et al., 2023, Giribet, 2020). The spectrum is built from $4+4$ transverse free modes on $T^4$ , together with spectrally-flowed short representations of the supergroup. The mass-shell condition in the $w$ -flowed sector is

$L_0 = \tilde{L}_0 + w (\tilde{K}_0^3 - \tilde{J}_0^3) = 0$

yielding conformal weights and R-charges

$h = J_0^3 = \frac{N^\text{ws}}{w} + \frac{w+1}{2} + \sum_i \delta_i, \qquad q = K_0^3 = \frac{w+1}{2} + \sum_i \delta_i$

with $N^\text{ws}$ the oscillator level and $\delta_i$ the $SU(2)$ charges. All states saturate the small $\mathcal{N}=4$ BPS bound $h \geq q$ . This spectrum precisely matches the twisted sectors of Sym $^N(T^4)$ , with worldsheet spectral flow $w$ corresponding to the cycle length in the symmetric orbifold (Gaberdiel et al., 2023).

The theory at $k=1$ exhibits an infinite-dimensional higher-spin symmetry, with the bulk spectrum reorganized as massless higher-spin fields. This is reflected in the boundary by an enhanced chiral algebra, specifically the small $\mathcal{W}_\infty$ with $\mathcal{N}=4$ supersymmetry (Gaberdiel et al., 11 Nov 2025, Eberhardt, 2020).

3. Partition Functions, Localization, and Geometry Independence

A defining feature at the tensionless NS-NS point is that worldsheet partition functions localize on holomorphic covering maps of the conformal boundary. For example, the one-loop worldsheet partition function on thermal $\mathrm{AdS}_3$ takes the form

$Z_\text{psu}(\theta, \zeta; \tau) = \frac{1}{2} \sum_{r, w \in \mathbb{Z}} \delta^2(\theta - w \tau - r) |q|^{w^2} \left[ \frac{\vartheta_1((\theta+\zeta)/2) \vartheta_1((\theta-\zeta)/2)}{\eta^4} \right]^2$

At $k=1$ , the full string partition function depends only on the boundary geometry (the conformal torus), not on the details of the bulk metric. This phenomenon is manifest for backgrounds such as BTZ black holes, conical defects, or Euclidean wormhole geometries: all locally AdS $_3$ backgrounds with the same boundary torus yield identical partition functions (Eberhardt, 2020). The integration over worldsheet moduli is localized by delta functions to those configurations corresponding to holomorphic maps from the worldsheet to the boundary.

The grand-canonical partition function is found to match exactly the Sym $^N(T^4)$ CFT partition function,

$\mathfrak{Z}_\mathrm{string}(t, z, \sigma) = \exp \left[ \sum_{L=1}^{\infty} p^L L \cdot T_L Z^{T^4}(z; t) \right]$

where $T_L$ is the Hecke operator and $p = e^{2\pi i \sigma}$ counts the string "number."

This localization implies the geometric independence of observables—giving evidence for the equivalence of the tensionless string across distinct bulk backgrounds provided the conformal boundary is fixed (Eberhardt, 2020).

4. AdS $_3$ /CFT $_2$ Correspondence and Exact Duality

At $k=1$ , there is an exact holographic duality between string theory on $\mathrm{AdS}_3 \times \mathrm{S}^3 \times \mathbb{T}^4$ and the large- $N$ symmetric orbifold CFT,

$\mathrm{Sym}^N(T^4) \equiv (T^4)^N / S_N$

with central charge $c=6N$ in the large- $N$ limit (Gaberdiel et al., 2023, Gaberdiel et al., 2023, Gaberdiel et al., 11 Nov 2025). The worldsheet spectrum and correlators—specifically sphere-level correlators—precisely coincide with those computed in the symmetric orbifold CFT via branched covering techniques (Lunin–Mathur mechanism).

Vertex operators in the worldsheet theory with spectral flow $w$ correspond to twist sector operators in the boundary CFT. The covering map localization ensures that worldsheet $n$ -point functions reproduce symmetric orbifold correlators in both untwisted and twisted sectors, confirming the AdS $_3$ /CFT $_2$ dictionary at this point (Gaberdiel et al., 11 Nov 2025).

This matching holds also for orbifolds of AdS $_3$ backgrounds, where the dual CFT subsector consists of low-lying excitations atop reference states in twisted sectors (e.g., all cycles of fixed length $k$ ) (Gaberdiel et al., 11 Nov 2025, Gaberdiel et al., 2023). The mapping between worldsheet spectral flow and twisted cycle length remains exact, and the reference twisted state governs the background in which excitations propagate.

5. Black Hole/String Transitions and Factorization Properties

The tensionless NS-NS point provides an explicit microscopic realization of the black-hole/string transition. At low temperature, partition functions are dominated by the vacuum sector and trivial covers; at high temperature, the maximal winding sector dominates, corresponding to a single long string asymptotically winding around thermal $\mathrm{AdS}_3$ —interpreted as the BTZ black hole (Eberhardt, 2020). This effect realizes the Hawking–Page transition as a string/black-hole phase change.

For geometries with disconnected boundaries, the tensionless string partition function factorizes trivially. Both connected wormhole and disconnected handlebody contributions yield the same covering-map terms, so there is no need for ensemble averaging and no loss of factorization in the boundary theory. The mechanism resolves the factorization problem, as the string theory does not distinguish geometries by their connectedness, but only by the structure of their boundaries (Eberhardt, 2020).

The construction and properties of the tensionless NS-NS point extend to orbifolded backgrounds such as $(\mathrm{AdS}_3 \times \mathrm{S}^3)/\mathbb{Z}_k \times T^4$ (Gaberdiel et al., 11 Nov 2025, Gaberdiel et al., 2023), where the worldsheet WZW model is modified by the orbifold action. The exact correspondence persists: the worldsheet spectrum reproduces the single-cycle twisted sector excitations of a reference ground state in the dual symmetric orbifold CFT.

In the case of AdS $_3\times S^3\times S^3\times S^1$ with two units of NS-NS flux through each $S^3$ , a tensionless regime arises with $k^+ = k^- = 2$ (Gaberdiel et al., 25 Nov 2024). The dual CFT is a symmetric orbifold of two bosons and eight free fermions, and both NS-R and hybrid formalisms manifest the collapse of non-trivial $SU(2)$ sectors to null states. The resulting spectrum fills the cycle-twisted sectors, and all correlation functions reduce to those computed in the symmetric orbifold CFT.

7. Deformations, Integrability, and Bound State Structure

Deformations away from the tensionless point, such as switching on R-R flux, break the free regime and centrally extend the superalgebra $\mathrm{psu}(1,1|2)^2$ acting on the symmetric orbifold CFT (Gaberdiel et al., 2023). In the large- $w$ twisted sector, physical excitations—magnons—acquire nontrivial dispersion and interact via an integrable $S$ -matrix satisfying the Yang–Baxter equation. The dispersion relation is found to be

$\epsilon(p) = \sqrt{(1-p)^2 + 4g^2 \sin^2(\pi p)}$

with corrections arising in orders of the perturbation parameter $g$ . Multi-magnon bound states are governed by fusion conditions in rapidity space, and their spectrum organizes into higher Brillouin zones of the infinite spin chain, with dispersion

$\epsilon_Q(p) = \sqrt{(Q-p)^2 + 4g^2 \sin^2(\pi p)}$

thus encoding the precise spectral structure of deformed tensionless strings (Gaberdiel et al., 2023).

In summary, the tensionless NS-NS point ( $k=1$ ) in $\mathrm{AdS}_3$ backgrounds marks a regime of free worldsheet dynamics, exact matching to symmetric orbifold CFTs, and the emergence of infinite higher-spin symmetry. Its hallmark features—collapse of the bulk spectrum, geometric independence of boundary observables, localization of partition functions, and nontrivial integrable deformations—make it a foundational setting for exploring tensionless strings, holography, and stringy phases of quantum gravity.