Little string theory

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Little string theory

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In theoretical physics, little string theory is a non-gravitational non-local theory in six spacetime dimensions that can be obtained as an effective theory of NS5-branes in the limit in which gravity decouples. Little string theories exhibit T-duality, much like the full string theory.

References

[edit]

Aharony, Ofer (2000). "A brief review of "little string theories"". Classical and Quantum Gravity. 17 (5): 929–938. arXiv:hep-th/9911147. Bibcode:2000CQGra..17..929A. doi:10.1088/0264-9381/17/5/302. S2CID 14143964.

David Kutasov (2001). Introduction to Little String Theory (PDF). Spring School on Superstrings and Related Matters. Archived from the original on 2024-02-21. Retrieved 2024-02-21.{{cite conference}}: CS1 maint: bot: original URL status unknown (link)

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Little string theory (LST) is a non-gravitational, non-local quantum theory in six spacetime dimensions that describes string-like objects with finite tension, emerging as an effective description in the decoupling limit of NS5-brane systems in type II string theory.^[1]^[2] First conceptualized in the late 1990s, LST arises from taking the string coupling

g_s

to zero while keeping the string scale

M_s

finite in configurations involving multiple parallel NS5-branes, resulting in a theory with no gravity but exhibiting stringy behaviors such as a Hagedorn temperature and T-duality invariance.^[1]^[2] It serves as an intermediate framework between local quantum field theories and full string theory, providing insights into non-local physics without the complications of relativistic gravity.^[1]^[2] LST is particularly notable for its supersymmetric variants, including those with

\mathcal{N} = (2,0)

supersymmetry in the type IIA limit and

\mathcal{N} = (1,1)

in type IIB, both preserving 16 supercharges and derived from the worldvolume dynamics of k NS5-branes for

k > 1

.^[1] In the low-energy regime below

M_s

, (2,0) LST reduces to the six-dimensional

\mathcal{N} = (2,0)

superconformal field theory associated with k M5-branes in M-theory, while (1,1) LST flows to a

U(k)

gauge theory with coupling

g_{YM}^2 = 1/M_s^2

, which becomes strongly coupled at scales around

M_s / \sqrt{k}

Ms/k

for large k.^[1] The theory lacks a local Lagrangian description for interactions, reflecting its non-local nature, and its moduli space—corresponding to the positions of the NS5-branes—is protected from quantum corrections, taking the form $\mathbb{R}^{4k} / S_k$ for (1,1) LST and $(\mathbb{R}^4 \times S^1)^k / S_k$ for (2,0) LST with the circle radius tied to $M_s$ .^[1] A defining feature of LST is its spectrum of BPS states, including fundamental "little strings" with tension $T = M_s^2$ , which are bound states of ordinary strings and NS5-branes, alongside additional massless particles like gluons in the (1,1) case at the origin of moduli space.^[1] The theory exhibits T-duality invariance even in the absence of gravity, such as the equivalence between (2,0) and (1,1) LST upon compactification on a circle, underscoring its non-local structure as it lacks a conventional energy-momentum tensor after toroidal reduction.^[1]^[2] At finite temperature or energy density, LST displays a Hagedorn-like phase with a limiting temperature $[T_H](/page/Hagedorn_temperature) \approx M_s / \sqrt{6k}$

, above which the density of states grows exponentially, preventing local descriptions at distances shorter than $T_H^{-1}$ and highlighting its stringy, non-perturbative dynamics.^[1]^[2] LST's significance extends to its applications in understanding broader string theory phenomena, including holographic dualities with linear dilaton backgrounds and compactifications that yield lower-dimensional theories like four-dimensional $\mathcal{N} = 4$ super Yang-Mills.^[1] It also provides a non-perturbative definition via discrete light-cone quantization (DLCQ), making it a valuable toy model for exploring non-locality, supersymmetry enhancement, and the limits of field theory validity in higher dimensions.^[1] These properties position LST as a key tool in high-energy physics for bridging quantum field theories and gravitational string theories without introducing relativistic issues.^[2]

Introduction

Definition and Overview

Little string theory (LST) is a six-dimensional quantum theory describing extended string-like objects that possess tension but are decoupled from gravity, thereby exhibiting inherently stringy behavior without the inclusion of general relativity. This framework captures the dynamics of strings with finite tension in a non-gravitational setting, where interactions are governed by a finite tension scale, leading to a regime of non-local physics that transcends traditional quantum field theory while avoiding the full complexities of gravitational string theory.^[1] Positioned as an intermediate theoretical construct, LST bridges the gap between quantum field theories (QFTs), which describe point-like particles with local interactions and relativistic invariance, and full string theories that incorporate gravity through closed string modes and spacetime curvature. In contrast to QFTs, where particles are zero-dimensional and locality holds at short distances, LST features one-dimensional extended objects whose interactions exhibit stringy non-locality, such as exponential growth in the density of states at the Hagedorn temperature—a hallmark property reflecting its intermediate status. Meanwhile, unlike conventional string theory, LST decouples bulk gravitational effects, allowing focused study of pure string dynamics without relativistic complications or black hole formation. This "halfway" nature makes LST a valuable model for exploring non-perturbative string phenomena in a controlled, gravity-free environment. LST is realized theoretically as a specific decoupling limit within type IIA or type IIB string theory involving NS5-branes, where the string coupling constant

g_s

is taken to zero while keeping the string length scale

l_s

fixed and finite. In this limit, the dynamics on the branes become independent of the bulk string theory, yielding a six-dimensional theory of little strings confined to the brane worldvolume, with no propagation into higher dimensions. This setup effectively isolates the stringy sector, providing a simplified arena akin to the hydrogen atom model in atomic physics, which facilitates the study of non-local interactions and extended object behaviors without the confounding influences of gravity or point-particle approximations.^[1]

Key Characteristics

Little string theory (LST) is fundamentally defined in six non-compact dimensions, where the theory emerges as a UV-complete framework decoupled from bulk gravity, allowing for the study of stringy dynamics without the complications of relativistic spacetime curvature at low energies. This six-dimensional structure arises naturally from the worldvolume theory of NS5-branes in type IIA string theory, with the non-compact nature ensuring that the theory describes an infinite volume spacetime without periodic boundary conditions that would otherwise introduce compactification artifacts. While compactifications to lower dimensions are possible, the core non-compact 6D formulation captures the essential features of LST as a standalone theory.^[1]^[2]^[3] A defining characteristic of LST is the absence of gravity within the theory itself, distinguishing it from full string theory where gravitons propagate freely in the bulk. In LST, gravity is confined to a higher-dimensional bulk outside the 6D worldvolume, achieved through a specific decoupling limit where the string coupling and length scales are tuned such that closed string modes associated with gravity decouple from the open string sector describing the little strings. This results in a non-gravitational theory that avoids the low-energy effective description by general relativity, yet retains the full non-perturbative structure of string interactions, making LST a valuable model for exploring quantum gravity alternatives without the infrared divergences of gravity.^[1]^[2]^[4] LST features extended objects known as "little strings," which are 1-dimensional extended objects confined to the 6D spacetime, possessing a fixed tension that sets the intrinsic scale of the theory. These little strings replace point-like particles, leading to inherently non-perturbative effects even at weak coupling, as interactions involve the exchange of these extended objects rather than local vertex operators. The fixed tension ensures that the spectrum includes a Hagedorn-like behavior, indicating stringy thermodynamics without gravitational backreaction.^[1]^[2]^[5] The physics of LST is non-local, with interactions occurring at the string scale rather than through point-like exchanges, which manifests in the absence of a local field theory description and the presence of a fundamental length scale that prevents ultraviolet divergences in a conventional sense. This non-locality is evident in the theory's invariance under T-duality, such as upon compactification on a circle, allowing for a rich structure of dualities that connect different realizations of LST without altering the underlying extended object dynamics. In the infrared limit, the (2,0) LST flows to the six-dimensional \mathcal{N} = (2,0) superconformal field theory, providing a bridge to local quantum field theories.^[1]^[2]^[4]

Historical Development

Origins in String Theory Research

The conceptual roots of little string theory (LST) trace back to key developments in superstring theory during the 1990s, particularly advances in understanding brane dynamics and decoupling limits. Building on this, the 1990s saw pivotal advances in D-brane dynamics that motivated the decoupling limits central to LST. Introduced by Joseph Polchinski in 1995, D-branes are extended objects on which open strings end, preserving supersymmetry and enabling the description of gauge theories as low-energy limits of string theory on brane worldvolumes. These developments, part of the second superstring revolution, emphasized T-duality and brane interactions, revealing how gauge dynamics could persist in limits where the closed string sector, including gravity, decouples. Such configurations, particularly involving NS5-branes related to D-branes via dualities like S-duality, inspired explorations of intermediate scales where stringy effects dominate without relativistic gravity, positioning LST as an extension of these brane-world models.^[6] Key pre-LST papers from 1995-1996 further shaped these ideas by examining decoupled gauge theories in higher dimensions in the context of brane systems. For instance, works on the worldvolume theories of multiple D-branes highlighted non-perturbative effects and UV completions free from gravitational singularities, motivating gravity-free stringy frameworks to resolve ultraviolet issues in six-dimensional field theories. These studies, influenced by dualities and matrix models, underscored the need for non-local models that capture Hagedorn-like spectra while avoiding full string theory's gravitational complications, thus paving the way for LST as a theoretical bridge. Theoretical motivations stemmed from the quest to understand strongly coupled gauge theories and their stringy origins, where decoupling gravity allowed focused study of intrinsic string scales and T-duality invariance.

Key Milestones and Discoveries

The foundational discovery of little string theory (LST) occurred in 1997, when Nathan Seiberg and collaborators analyzed the decoupling limit of NS5-brane systems in type II string theory, isolating a non-gravitational regime of string-like excitations in six dimensions.^[7] The term "little string theory" was introduced in 1998 by A. Losev, G. Moore, and S. L. Shatashvili to characterize this intermediate framework between quantum field theories and full string theory.^[8] This work built on the worldvolume dynamics of coincident NS5-branes, revealing a theory with a characteristic string tension but without gravity, marking a key breakthrough in understanding non-local physics in higher dimensions.^[9] Between 1998 and 2000, significant developments advanced the understanding of LST's symmetries and dualities. In 2015, Amit Giveon, David Kutasov, and Nathan Seiberg examined T-duality properties in LST, demonstrating its invariance under T-duality transformations and linking it to heterotic-type II dualities, which provided evidence for LST as a distinct stringy phase.^[10] Earlier works by Nathan Seiberg and Edward Witten on six-dimensional fixed points laid the groundwork, connecting LST to superconformal field theories (SCFTs) through analyses of brane configurations and non-perturbative effects, solidifying LST's role in unifying gauge theories with string-like behaviors.^[11] These efforts, including explorations of double scaling limits, established LST's theoretical consistency and duality structure.^[12] Post-2010 advancements focused on holographic dualities and classifications beyond traditional AdS spaces. In 2015, Lakshya Bhardwaj used F-theory to classify supersymmetric LSTs, providing a geometric framework that encompassed previously missing theories and enabled systematic constructions.^[13] Further progress in 2017–2018 explored holography beyond AdS, with studies of string theory backgrounds interpolating between AdS_3 in the infrared and linear dilaton spacetimes in the ultraviolet, offering insights into LST's dual gravity descriptions without asymptotic AdS confinement.^[14] Works during 2015–2020 extended these ideas, investigating holographic realizations of LST that incorporate non-AdS geometries and testing consistency with six-dimensional SCFT limits.^[14] Theoretical validations of LST have included consistency checks with six-dimensional SCFTs, particularly through anomaly cancellation and symmetry analyses. In 2020, studies of 2-group symmetries in 6D LSTs under T-duality provided rigorous consistency checks, confirming that dual pairs of LSTs exhibit matching symmetry structures, including discrete gauging effects.^[15] Additional milestones involved geometric classifications, such as those using F-theory, which verified the existence of predicted LSTs by matching tensor branch dimensions and flavor symmetries with SCFT expectations, though full lattice simulations remain challenging due to the non-local nature of LST.^[13] These validations have reinforced LST's viability as a UV completion for 6D gauge theories.^[15]

Theoretical Framework

Dimensional Structure and Setup

Little string theory (LST) is formulated in a six-dimensional spacetime with Lorentzian signature (5,1), specifically on the flat space

\mathbb{R}^{5,1}

, which corresponds to the worldvolume directions of a stack of NS5-branes in type II string theory.^[16] This setup can be extended by compactifying some spatial directions on tori or circles, enabling the study of T-duality invariances that map the theory between different compactification radii while preserving the underlying structure.^[1] The theory emerges as a non-gravitational, non-local framework where the dynamics are governed by an intrinsic string scale

M_s = 1/l_s

, with

l_s

the fundamental string length, distinguishing it from relativistic quantum field theories or full ten-dimensional string theory.^[16] The core construction of LST involves a specific decoupling limit applied to configurations of

N

coincident NS5-branes in type II string theory. In this limit,

[g_s](/page/Coupling_constant) \to 0

with the string scale

M_s = 1/l_s

held fixed, isolating the worldvolume dynamics from bulk closed string modes.^[16] For heterotic string variants of LST, analogous limits apply to NS5-brane configurations in the Spin(32)/

\mathbb{Z}_2

[E_8 \times E_8](/page/Heterotic_string_theory)

heterotic strings, yielding theories with distinct gauge structures.^[17] LST exhibits rich symmetry structures inherited from the underlying string theories. In type IIB constructions, the theory possesses an SL(2,

\mathbb{Z}

) duality symmetry, analogous to that of type IIB superstring theory, which acts on the axio-dilaton and related fields, ensuring invariance under transformations of the coupling.^[18] These symmetries are manifest upon compactification and play a crucial role in relating different realizations of the theory. Quantization of LST proceeds via a perturbative expansion in the string coupling

g_s

, leveraging worldsheet descriptions in the linear dilaton background of the NS5-brane system. In the weak-coupling regime (large dilaton value), physical observables are computed using worldsheet correlation functions of vertex operators in the conformal field theory on the string worldsheet, incorporating fields for the six-dimensional coordinates, the linear dilaton

\phi

, and the transverse SU(2) WZW model at level

N-2

.^[16] The stress tensor includes a term for the dilaton,

T_\phi = -\frac{1}{2} (\partial \phi)^2 - Q^2 \partial^2 \phi

with

Q = \sqrt{2 N}

Q=2N

, ensuring conformal invariance, while the perturbative series organizes interactions beyond the free theory.^[16] This worldsheet approach provides a controlled expansion for scattering amplitudes and BPS states, consistent with the decoupled nature of the theory.

String-like Objects and Absence of Gravity

In little string theory (LST), the fundamental degrees of freedom are extended, infinite strings that possess a finite tension, distinguishing them from point-like particles in quantum field theories. These strings exhibit a discrete mass spectrum where the mass of the nth excited state scales linearly as

M_n \sim \frac{n}{\ell_s}

, with

\ell_s

denoting the characteristic string length scale, leading to an exponential Hagedorn growth in the density of states at high energies.^[2]^[5] A defining feature of LST is the complete decoupling of gravity, achieved in the decoupling limit (g_s → 0) where the closed string modes responsible for gravitational interactions become massive and are effectively removed from the low-energy spectrum. Specifically, the graviton acquires a mass on the order of the inverse string length,

m_g \sim \frac{1}{\ell_s}

, gapping out the massless graviton multiplet and ensuring that the theory remains non-gravitational without emergent spacetime curvature effects.^[19]^[20] This decoupling arises naturally in the near-horizon geometry of NS5-brane configurations, where bulk modes, including those mediating gravity, are suppressed as the string coupling tends to zero while maintaining a finite string scale.^[5] While LST incorporates both open and closed strings, the theory emphasizes the closed string sector, where the absence of gravitons is most pronounced, as open strings typically end on the underlying NS5-branes and contribute to gauge-like dynamics in the effective description. In the closed string sector, the spectrum consists of massive string excitations without the zero-mode graviton, preserving the stringy non-locality while avoiding relativistic gravitational propagation.^[21]^[22] The role of open strings, confined to the brane worldvolume, supports additional perturbative interactions but does not introduce gravity, aligning with the overall gravity-decoupled framework of LST.^[21] Non-perturbative effects in LST are dominated by instanton contributions from these strings, which generate corrections to the effective action and partition function through worldsheet instantons wrapping non-trivial cycles in the geometry. These instantons provide essential non-perturbative insights, such as modifications to the prepotential in the underlying gauge theories, and are captured by the instanton partition function that encodes multi-instanton sectors.^[4]^[23] In particular, string instantons in the NS5-brane background contribute to the exact solvability of certain observables, highlighting the interplay between perturbative string dynamics and non-perturbative brane effects.^[4]

Physical Properties

Hagedorn Temperature and Thermodynamics

In Little String Theory (LST), the Hagedorn temperature represents a fundamental upper limit on the temperature of the system, beyond which the thermodynamic description breaks down due to the proliferation of string-like excitations. This temperature is defined as

T_H \sim 1/(2\pi \sqrt{k} l_s)

TH∼1/(2πk

ls), where $l_s$ is the characteristic string length scale and $k$ is the number of NS5-branes, arising from the exponential growth in the density of states $\rho(m) \sim \exp(m / T_H)$ for massive string modes with mass $m$ .^[24] This behavior mirrors the Hagedorn phenomenon in critical string theory but occurs in a non-gravitational context, where the density of states reflects the underlying six-dimensional string-like degrees of freedom without gravitational backreaction. The thermodynamics of LST is captured by the partition function $Z(\beta)$ , with $\beta = 1/T$ the inverse temperature, which exhibits a divergence as $T$ approaches $T_H$ from below. This divergence stems from the summation over an infinite number of string excitations, leading to a free energy $F = -T \log Z$ that becomes singular at the Hagedorn point, indicating an instability in the perturbative vacuum and a phase transition, potentially akin to deconfinement in gauge theories. Unlike in quantum field theories (QFTs), where high-temperature phases often involve deconfinement transitions, LST's behavior near $T_H$ involves a Hagedorn/deconfinement-like phase transition, as suggested by analyses of the entropy and energy relations.^[25] A key distinction in LST thermodynamics compared to full string theory lies in the absence of black hole contributions to the entropy. In conventional string theory, the high-temperature entropy includes both string excitation terms and black hole microstates, leading to a smoother thermodynamic behavior; in contrast, LST's purely stringy entropy, dominated by the exponential density of states up to $T_H$ , lacks this gravitational component, resulting in a sharper Hagedorn transition without the extended phase space of black holes. This purity highlights LST as a controlled model for studying stringy thermodynamics in isolation.

Non-local Behavior and T-duality

Little string theory (LST) manifests non-locality through interactions that occur at the string scale, characterized by correlations extending over distances comparable to the string length

l_s = \sqrt{\alpha'}

ls=α′

, rather than being confined to points as in local quantum field theories.^[26] This non-locality is evident in the theory's structure, where fundamental degrees of freedom behave like strings with tension $T = 1/(2\pi \alpha')$ , leading to extended object dynamics without the emergence of gravity.^[26] A defining symmetry of LST is T-duality, which acts on compactified directions within the theory's six-dimensional spacetime, transforming the compactification radius $R$ to $l_s^2 / R$ while preserving the overall physics.^[26] This transformation interchanges type IIA and IIB versions of LST for a single circle inversion but serves as an internal symmetry when an even number of radii are inverted, ensuring the invariance of the LST spectrum across dual descriptions.^[26] The implementation of T-duality in LST follows the Buscher rules, which prescribe transformations for the background metric $g_{\mu\nu}$ , the Kalb-Ramond field $B_{\mu\nu}$ , and the dilaton $\phi$ along the dualized direction; for instance, the metric components transform as $g_{ij} \to g_{ij} - \frac{g_{i r} g_{j r}}{g_{rr}}$ and $g_{r i} \to \frac{g_{r i}}{g_{rr}}$ , where $r$ denotes the compact direction, maintaining consistency without gravitational effects.^[27]^[26] The implications of T-duality in LST are profound, as it mixes winding modes—where strings wrap around compact dimensions—and momentum modes—corresponding to quantized momenta along those directions—without introducing gravitational interactions, thereby highlighting the theory's purely stringy nature.^[26] This mixing underscores LST's role as a bridge between local field theories and full string theory, preserving duality symmetries in a gravity-free regime.^[26] Recent developments in the 2020s have extended T-duality frameworks in LST to non-compact settings, exploring dualities that operate beyond toroidal compactifications and incorporating twisted or flavor-deforming variants, as seen in analyses of heterotic LSTs and their symmetry structures.^[28]

Applications

UV-Completion for 6D Gauge Theories

Six-dimensional quantum field theories (QFTs), particularly gauge theories with supersymmetry, face significant ultraviolet (UV) challenges due to their non-renormalizability and the requirement for anomaly cancellation. In 6D, pure Yang-Mills theories with gauge groups like SU(N) exhibit perturbative non-renormalizability beyond one loop, leading to divergences that cannot be absorbed into a finite set of counterterms, necessitating a UV completion beyond standard QFT.^[29] Additionally, gravitational and mixed anomalies in these theories demand the inclusion of tensor multiplets for cancellation, as seen in N=(1,0) supersymmetric SU(N) gauge theories where the anomaly polynomial must be factored appropriately.^[30] Little string theory (LST) provides a natural UV completion for these 6D gauge theories by introducing string-like degrees of freedom that act as a stringy regulator at the little string scale, resolving the non-renormalizability while maintaining the low-energy gauge dynamics. In this framework, the gauge theory emerges in the weak-coupling regime of the LST, with the UV behavior governed by non-local string interactions decoupled from gravity.^[3] This completion is often engineered using NS5-brane configurations in string theory.^[31] Consistency between the low-energy gauge theory and its LST completion is verified through matching of beta functions, where the one-loop beta function of the 6D SYM aligns with the perturbative expansion in the LST tensor branch, and through the identification of operator spectra, such as chiral operators whose dimensions and multiplicities match across the duality.^[29] These checks confirm that the LST provides a faithful UV embedding without introducing extraneous degrees of freedom.^[31]

Testing Ground for Holographic Principle

Little string theory (LST) has been proposed as a holographic dual to certain seven-dimensional gravity theories formulated on backgrounds involving NS5-branes, providing a framework to test holographic principles in non-standard geometries. In this duality, the six-dimensional LST is conjectured to correspond to a seven-dimensional theory with a linear dilaton background, where the entropy of states in the LST matches that computed from the gravitational side through thermodynamic relations derived from the holographic description. This matching is evident in the high-energy regime, where the entropy-energy relation

S = \beta_H E + \alpha \log E

emerges from the holographic dual, aligning with the Hagedorn behavior characteristic of LST. Such proposals extend holography beyond the typical Anti-de Sitter (AdS) setups, leveraging the NS5-brane geometry to realize a duality without a global AdS factor. Unlike the AdS/CFT correspondence, which relies on conformal invariance and asymptotic AdS boundaries, LST offers a testbed for holography in asymptotically flat spaces, avoiding pitfalls such as the need for a large

N

limit in field theories or the confinement issues in non-conformal cases. The flat-space nature of LST holography allows exploration of non-local physics in a regime where gravity is decoupled, providing insights into holographic dualities that do not require curved confining geometries. This setup is particularly advantageous for studying T-duality invariance in holographic contexts, as the dual gravity theory incorporates stringy effects that preserve such symmetries without introducing relativistic complications. By residing at spacelike infinity in the dual description, LST holography naturally accommodates asymptotic flatness, offering a controlled environment to probe gravitational phenomena in non-AdS regimes. Key results from holographic calculations in LST include computations of two-point correlation functions using supergravity approximations in the dual backgrounds, which reveal non-local behaviors consistent with the string-like excitations of LST. For instance, in the context of critical non-Abelian vortices, holographic methods have been applied to evaluate vertex operator correlators, demonstrating agreement between normalizable and non-normalizable modes in most channels. More recent advancements, particularly from 2018 to 2022, have focused on LST black holes, such as analyses of small black holes in string theory limits where LST emerges, and studies of quantum chaos and information scrambling via holographic duals, showing exponential growth in out-of-time-order correlators indicative of chaotic dynamics. These calculations, including those for BTZ-like black holes arising from fivebrane dynamics in LST, have provided explicit tests of holographic entanglement and thermal properties beyond AdS. Despite these advances, non-AdS holography in LST faces challenges, including potential violations of unitarity in the dual gravity descriptions due to the absence of a stabilizing AdS boundary, which can lead to instabilities in the spectrum. Additionally, information paradoxes arise in the context of black hole evaporation analogs within LST, where the non-local nature complicates the recovery of information from gravitational collapse, echoing broader issues in flat-space holography without the unitary safeguards of AdS/CFT. These challenges highlight the need for careful regularization in the linear dilaton backgrounds to ensure consistent dualities.

Connection to 6D (2,0) Theory

Little string theory (LST) emerges as a UV completion for the enigmatic 6D (2,0) superconformal field theory (SCFT), providing a stringy framework that resolves longstanding puzzles about its non-local and non-gravitational nature. In the low-energy limit below the little string scale

1/l_s

, with

l_s

fixed, LST flows to the 6D (2,0) SCFT, which describes the dynamics of N coincident M5-branes in M-theory or NS5-branes in type IIA string theory. This limit decouples the stringy excitations, leaving a conformal theory with maximal (2,0) supersymmetry, characterized by a self-dual 2-form field, five scalars, and symplectic Majorana-Weyl fermions transforming under the SO(5) R-symmetry. On the tensor branch of the moduli space, the SCFT deforms by giving vacuum expectation values to the scalars, breaking the gauge symmetry (e.g., SU(N) for A_{N-1} type) to its Cartan subalgebra U(1)^{N-1} and separating the branes, which corresponds to a point in the Coulomb branch parameterized by the deformation parameter

\mu

in the Calabi-Yau singularity.^[31] Self-dual strings play a central role in bridging LST and the 6D (2,0) SCFT, serving as BPS solitons that source the self-dual 2-form field

H_+

satisfying

*dH = J

, where J is the string current. In LST, these strings originate from fundamental strings or M2-branes stretched between NS5-branes, acquiring a finite tension of order

1/l_s

and exhibiting non-local dynamics without gravity. For instance, in type IIA LST, fractional self-dual strings arise from M2-branes wrapping an M-theory circle connecting NS5-branes, with their worldsheet described by a (4,4) supersymmetric sigma model whose elliptic genus encodes the spectrum via Young tableaux contributions. This stringy origin in LST provides a holographic or worldsheet description of the tensionless self-dual strings in the (2,0) SCFT, where they become massless excitations on the tensor branch, facilitating computations of correlation functions and partition functions that are otherwise inaccessible in the SCFT alone.^[31] Anomaly matching between LST and the 6D (2,0) SCFT ensures consistency in their UV completion, particularly through the central charges that count degrees of freedom. The (2,0) SCFT for A_{N-1} type has a cubic central charge scaling as

c \sim N^3

, reflecting its large-N limit behavior, while LST matches this via its Green-Schwarz mechanism for anomaly cancellation, where the anomaly polynomial involves tensor multiplets coupling to gauge and gravitational fields. Specifically, the 2-group structure constants in LST, such as

k = \sum_I (C^{-1})_{II} q_I q_I

(with C the intersection matrix and q_I the little string charges), reproduce the SCFT anomalies after integrating out massive modes, resolving the UV puzzle of the (2,0) theory by embedding it in a controllable stringy regime with Hagedorn growth in the spectrum. This matching extends to the gravitational anomaly coefficient, confirming unitarity and supersymmetry preservation across the flow.^[32]^[31] Different realizations of LST highlight variants in their connection to the 6D (2,0) SCFT. In type IIB LST, arising from NS5-branes with (1,1) supersymmetry, the low-energy limit yields a 6D (1,1) Yang-Mills theory rather than the full (2,0) SCFT, but T-duality to type IIA LST maps it to the (2,0) sector with ADE gauge groups, exchanging momentum and winding modes while preserving the self-dual string spectrum. Heterotic LSTs connect to the (2,0) SCFT through compactifications involving M5-branes, with the low-energy limit flowing to an SCFT consistent with (2,0) anomaly data. These variants underscore T-duality chains that unify the theories, with heterotic LSTs providing non-Lagrangian examples lacking perturbative descriptions yet consistent with (2,0) anomaly data.^[31]

Role of NS5-Branes

Little string theory (LST) emerges from the dynamics of NS5-branes in type II string theory, where these branes serve as the foundational objects for engineering the theory. In type IIA string theory, a stack of N coincident NS5-branes extends along the worldvolume directions x^0 to x^5 and is pointlike in the transverse directions x^6 to x^9, preserving (2,0) supersymmetry with 16 supercharges and breaking the Lorentz symmetry from SO(9,1) to SO(5,1) × SO(4).^[26] Similarly, in type IIB string theory, the NS5-branes preserve (1,1) supersymmetry, with the near-horizon geometry featuring a linear dilaton background coupled to an SU(2) Wess-Zumino-Witten (WZW) model.^[26]^[33] This configuration creates a strongly coupled regime near the branes, independent of the asymptotic string coupling, setting the stage for non-gravitational string-like physics.^[33] The decoupling limit that yields LST involves taking the string coupling g_s → 0 while keeping the string scale m_s = 1/√α' fixed, which suppresses bulk emission processes proportional to g_s without requiring a low-energy limit like α' → 0.^[26] In this near-horizon regime of the NS5-branes, the worldvolume theory becomes independent of gravity and bulk dynamics, resulting in a six-dimensional LST that remains interacting for N > 1.^[26]^[34] For type IIB, the low-energy limit on the brane worldvolume is a U(N) gauge theory with (1,1) supersymmetry and fixed gauge coupling, while in type IIA, it flows to a (2,0) superconformal fixed point.^[26] This process highlights LST as a UV-complete theory of strings without relativistic locality or gravity.^[33] In LST, fundamental strings in the type II background transform into solitonic "little strings" bound to the NS5-branes.^[26] These little strings have a tension T = M_s^2 (in units where 2π α' = 1/ M_s^2), fixed in the decoupling limit and of the same order as the fundamental string tension.^[1] In the type IIB case, such strings correspond to instanton solutions in the low-energy U(N) gauge theory, with tension matching 1/g_{YM}^2, while in type IIA, tensionless strings arise from D2-branes stretched between coincident branes, indicating strong interactions.^[26] For multiple NS5-branes, interactions manifest through a rich moduli space of vacua, parameterized by the transverse positions of the branes, such as \mathbb{R}^{4N}/S_N for type IIB and (\mathbb{R}^4 \times S^1)^N / S_N for type IIA.^[26] This space, described by complex N \times N matrices for the scalar fields, allows for weakly coupled regimes away from coincidence points, resolvable via double scaling limits like g_s \to 0 and r_0 m_s \to 0 (with r_0 the separation radius), leading to geometries like the N=2 Liouville model or cigar.^[26]^[35] Multi-brane effects thus reveal bound states and continua in the spectrum, influencing the overall dynamics of LST.^[35]

Implications and Challenges

Challenging String Theory Axioms

Little string theory (LST) serves as a prominent counterexample to the longstanding axiom in string theory that identifies the framework primarily as a theory of quantum gravity, demonstrating that stringy physics can manifest in a purely non-gravitational context.^[1] Traditional string theory incorporates gravity through the closed string sector, which includes the massless graviton as a fundamental excitation; however, LST emerges in the limit where the string coupling

g_s \to 0

, decoupling gravitational interactions while preserving essential string-like features such as a Hagedorn spectrum and T-duality invariance.^[2] This construction, realized through configurations like stacks of NS5-branes, yields a six-dimensional theory with no massless spin-2 particle, thus challenging the universality of gravity as an inherent component of string dynamics.^[5] The independence of LST's stringy behavior from the gravitational sector underscores how extended objects can engender non-local physics without relying on closed strings. In LST, non-locality arises intrinsically from the string scale

M_s

, leading to a breakdown of locality at distances shorter than the inverse Hagedorn temperature, where correlation functions fail to be well-defined for small separations.^[1] Unlike conventional string theory, where gravity mediates long-range interactions, LST exhibits T-duality symmetries that mix momentum and winding modes in compact directions, implying a fundamental non-local structure decoupled from any gravitational background.^[36] This separation highlights that the non-local, extended nature of strings can produce rich dynamics—such as an exponential density of states—independent of the closed string graviton, thereby questioning assumptions about the necessity of gravity for capturing string-theoretic phenomena.^[1] Philosophically, LST impacts unification paradigms by illustrating a regime of string physics that bridges quantum field theories and full string theory without invoking gravity, prompting reevaluations of how string theory achieves consistency across dimensions. In the context of the string landscape, LST provides a controlled example of non-perturbative stringy effects in non-gravitational settings, influencing discussions on the moduli space and effective field theory completions. Regarding swampland conjectures, which aim to delineate the subspace of effective theories embeddable in string theory, LST is discussed as a non-gravitational theory consistent with certain duality constraints derived in string theory frameworks.^[37] This suggests that swampland criteria may require consideration of non-gravitational string limits, broadening the landscape beyond gravity-dominated scenarios. Post-2015 debates have increasingly highlighted LST's role in the string landscape, particularly through F-theory realizations and dualities that embed LST into broader explorations. For instance, studies of little strings in F-theory have proposed dualities among these theories.^[38] These developments underscore LST's utility in testing landscape conjectures, revealing tensions in scale separation and holography that question the completeness of gravity-centric paradigms.^[37]

Open Questions and Future Directions

One major unresolved issue in little string theory (LST) is the lack of a fully non-perturbative definition that captures its dynamics without relying on holographic or perturbative approximations. While discrete light-cone quantization (DLCQ) offers an explicit non-perturbative framework for LSTs, practical computations remain challenging due to the complexity of the underlying conformal field theories, even in the large N limit.^[1] Similarly, in the context of heterotic/type II duality, LST lacks a direct non-perturbative formulation, with holographic descriptions providing insights into observables but leaving the fundamental dynamics near NS5-brane singularities unresolved.^[39] Recent discussions in fuzzball and microstate geometries highlight the need for non-perturbative approaches to describe strong-coupling regimes where world-sheet perturbation theory breaks down, particularly for five-brane collisions that resolve singularities.^[40] The construction of an exact S-matrix for LST also remains an open problem, as observables in LST are off-shell correlation functions rather than on-shell scattering amplitudes typical of critical string theory. Holographic duals, such as those in linear dilaton backgrounds, allow computation of certain correlation functions corresponding to scattering in specific directions, but extracting a full S-matrix requires identifying normalizable modes and amputated correlators, which is limited to particular regimes.^[39] In topological LST variants, techniques like LSZ reduction have been proposed to derive S-matrix elements from two-point functions, yet extending this to higher-point functions or verifying consistency across dualities poses significant challenges.^[39] Unitary holography in LST presents another key unresolved issue, particularly in understanding how LST contributes to the holographic description of non-gravitational, non-local theories while preserving unitarity. The duality to string theory on near-horizon NS5-brane geometries (e.g., CHS backgrounds) provides a framework for low-energy correlators, but refining the holographic dictionary for normalizable operators and addressing strong-coupling singularities at the geometry's tip remains incomplete.^[39] In the context of black hole microstates, LST's role in bulk dynamics suggests potential bridges to unitary boundary CFT states, but establishing a precise, unitary holographic correspondence for these systems, especially with infrared conformal breaking, is an ongoing problem.^[40] Gaps in knowledge include the full structure of the T-duality group for LST, particularly upon toroidal compactification, where an O(d,d,Z) symmetry emerges, indicating non-locality, but the evolution of this group in lower-dimensional or less supersymmetric compactifications is not fully understood, as non-local theories behave independently upon compactification.^[1] Additionally, the thresholds for gravity emergence in LST are unclear, given its inherently non-gravitational nature in the decoupling limit where the Planck scale diverges; however, embeddings in string theory backgrounds suggest gravity may arise at specific scales in microstate geometries resolving singularities, requiring further exploration of transitions from stringy to effective gravitational descriptions.^[40] Future directions involve deeper analysis of LST compactifications to classify lower-dimensional versions and probe their high-energy behavior, where indications from holography suggest weak coupling at asymptotically high energies, potentially revealing asymptotically free-like properties.^[1] In moduli space configurations, such as those with broken gauge symmetry, holographic descriptions may become weakly coupled, enabling reliable perturbation theory for computations and shedding light on LST structure.^[1] Interdisciplinary links to quantum information arise in the context of LST's role in black hole microstates, where entanglement entropy and information recovery in fuzzball proposals could tie to quantum error-correcting codes, though adapting holographic codes to LST's non-local framework remains a promising but undeveloped avenue.^[40]

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