Inverse String Theory KLT Kernel

• Inverse string theory KLT kernel is a universal mathematical structure that decomposes closed string amplitudes into bilinear combinations of open string amplitudes.
• It employs diagrammatic rules, intersection theory, and hypergeometric formulations to capture stringy corrections and bridge quantum field theories like BAS and NLSM.
• Its analytic and combinatorial representations reveal deep links between monodromy, color-kinematics duality, and positive geometries, advancing scattering amplitude understanding.

The inverse string theory KLT kernel is a mathematical structure central to the decomposition of closed string amplitudes into bilinear combinations of open string amplitudes, extended from flat spacetime to curved backgrounds and beyond. At its core, this kernel acts as a universal function whose different evaluations encode the scattering amplitudes for bi-adjoint scalar (BAS) theory, the non-linear sigma model (NLSM), mixed BAS+NLSM theories, as well as string-theoretic, gauge-theory, and gravitational S-matrices. Its properties are deeply tied to color-kinematics duality, intersection theory (specifically, intersection numbers of twisted cycles in moduli space), the combinatorics of polytopes (associahedra), and the analytic and geometric structure of string scattering amplitudes.

1. Definition and Structural Properties

The inverse string theory KLT kernel, denoted typically as $m^{(\alpha')}$ or $S^{(\alpha')^{-1}}$, is the inverse of the momentum kernel $S^{(\alpha')}$ that glues two sets of color-ordered open string amplitudes to produce closed string amplitudes through the KLT double-copy procedure. Explicitly, at $n$ points: $$\mathcal{M}_n^\text{closed} = \sum_{\sigma, \rho \in S_{n-3}} \mathcal{A}_n^\text{open}[\sigma] \; S^{(\alpha')}[\sigma|\rho] \; \mathcal{A}_n^\text{open}[\rho]$$ where $S^{(\alpha')}$ is a matrix of generalized sine and tangent kinematic factors of Mandelstam invariants, and $m^{(\alpha')}$ is its inverse: $m^{(\alpha')} = S^{(\alpha')^{-1}}$ This inversion encodes all stringy corrections (in $\alpha'$) and captures both the trigonometric structure and the combinatorics of factorization channels.

The inverse kernel can be written in terms of:

• Graph-based/diagrammatic rules: Standard Feynman propagators $1/p^2$ are replaced by $1/\sin(\pi \alpha' p^2/2)$ or $1/\tan(\pi \alpha' p^2/2)$, depending on whether the case is off-diagonal or diagonal in the color orderings. For higher-multiplicity, vertex factors follow Catalan number weights (e.g., 1, 1, 2, 5, ... for 3-, 5-, 7-, 9-valent vertices) (Mizera, 2016).
• Intersection numbers: In twisted de Rham theory, $m^{(\alpha')}(\beta|\gamma)$ appears as the intersection number of two twisted cycles labeled by color orderings $\beta$ and $\gamma$ (Mizera, 2017).

2. Physical and Algebraic Universality

A remarkable property, recently established, is the universality of $m^{(\alpha')}$ as the generator of scattering amplitudes for several distinct quantum field theories:

• Bi-adjoint scalar (BAS) amplitudes: $m^{(\alpha')}$ evaluated at the “unshifted” kinematics and in the $\alpha' \to 0$ limit reproduces tree-level BAS amplitudes.
• Non-linear sigma model (NLSM) amplitudes (pions): After an "abelianization" or “$\alpha'$-shift”—comprising rescaling $\alpha' \to \alpha'/2$ and channel-dependent trigonometric shifts—the same $m^{(\alpha')}$ encodes NLSM (pion) amplitudes.
• Mixed amplitudes: More general evaluations interpolate between the BAS and NLSM results, reproducing mixed NLSM+$\phi^3$ amplitudes.

In precise terms (Bartsch et al., 2 May 2025): $$\text{BAS}_{(\alpha')} = \text{NLSM}_{(\alpha')} = (\text{NLSM} + \phi^3)_{(\alpha')} = KLT_{(\alpha')}^{-1}$$ The $\alpha'$-shift, a key concept, involves evaluating $m^{(\alpha')}$ on $\tau_{ij} = \tan(\tfrac{\pi}{2} \alpha' X_{ij})$ instead of $t_{ij} = \tan(\pi \alpha' X_{ij})$, with appropriate sign conventions depending on channel parity.

This equivalence, complemented by the $\delta$-shift of Arkani-Hamed et al., demonstrates the embedding of QFT amplitudes for colored scalars and Goldstone pions in a common stringy framework, and highlights that their differences are purely kinematic.

3. Analytical and Combinatorial Constructions

The inverse KLT kernel admits several constructive formulations:

Representation	Structure	Context
Graph/diagrammatic	Trigonometric propagators, Catalan vertex weights	$\alpha'$-corrected partial amplitudes (Mizera, 2016)
Intersection numbers	Pairing of twisted cycles (associahedra) in moduli space	Twisted de Rham theory (Mizera, 2017)
Hypergeometric/Aomoto-Gelfand	Multiple polylogarithmic integrals for AdS generalizations	AdS KLT, worldsheet methods (Alday et al., 28 Apr 2025)
Algebraic (momentum kernel)	Inversion of $S^{(\alpha')}$ matrix in minimal color basis	BCJ/KLT monodromy, color-kinematic duality

In the intersection theory approach, the moduli space $M_{0,n}$ is blown up to an associahedron $K_{n-1}$, and the intersection numbers reduce to sums over faces and vertices labeled by trivalent diagrams (factorization channels). The monodromy properties of the Koba-Nielsen integrand determine the trigonometric structure (e.g., $1/\tan(\pi s_H)$ and $1/\sin(\pi s_H)$ factors).

4. Specialization and Field Theory Limit

Taking the infinite tension limit ($\alpha' \to 0$), the trigonometric (stringy) structure linearizes and $m^{(\alpha')}$ reduces to the minimal-basis bi-adjoint scalar amplitude matrix $m(\beta|\gamma)$. In this regime, the KLT relations reproduce the double-copy of Yang-Mills theory into gravity, with the field-theory KLT kernel mediating between minimal bases of color orderings. The monodromy (BCJ) relations, encoded in the annihilation property of the kernel, imply that only a $(n-3)!$-dimensional subspace is independent.

Conversely, for finite $\alpha'$ (full string theory), the kernel captures the entire massive tower of intermediate states via its infinite series of simple and higher-order trigonometric poles.

5. Extensions to AdS and Curved Backgrounds

Recent developments have extended the stringy KLT kernel to anti-de Sitter (AdS) space (Alday et al., 28 Apr 2025). Here, the open- and closed-string building blocks generalize the flat-space Euler and complex beta functions to integrals with multiple polylogarithmic insertions and their single-valued counterparts. The AdS KLT kernel is derived via holomorphic/anti-holomorphic factorization and monodromy analysis, yielding a closed trigonometric expression: $$\mathcal{K}(s,t;\mathbf{e}_0,\mathbf{e}_1) = -\frac{1}{2\pi i}\left[1 + \frac{e^{2\pi i s} M_0}{1 - e^{2\pi i s} M_0} + \frac{e^{2\pi i t} M_1}{1 - e^{2\pi i t} M_1}\right]^{-1}$$ where $M_0$, $M_1$ are monodromy matrices depending on the multiple polylogarithm variables. The open string integrals can be mapped to Aomoto-Gelfand hypergeometric functions, and explicit results up to weight four are given.

6. Loop-Level Extensions and KLT Bootstrap

At one loop, the KLT kernel generalizes to a loop-momentum dependent structure arising from the splitting of the torus worldsheet into cylinder (open string) pieces. The one-loop closed string amplitude is then a sum over products of one-loop open string cylinder amplitudes, glued via a loop-dependent KLT kernel (Stieberger, 2022, Stieberger, 2023). In the field theory limit, this produces the one-loop double-copy relation between gauge theory and gravity amplitudes, now controlled by a loop-momentum-dependent generalization of $S^{(\alpha')}$. The KLT bootstrap program identifies the necessary algebraic and locality constraints that any candidate double-copy kernel must satisfy at tree and loop level (Chi et al., 2021).

7. Implications and Theoretical Significance

The inverse string theory KLT kernel is a unifying object that connects seemingly distinct quantum field theories, elucidates the universality of amplitude structures, and links color and kinematic properties. Its evaluation at distinct kinematic points abstracts BAS, NLSM, and mixed amplitudes into a single analytic function; its combinatorics reflect deep geometric properties (associahedra, intersection theory); its analytic structure encodes massive (stringy) spectrum information; and its representation theory relates to monodromy, permutation, and factorization symmetries.

This kernel provides the backbone for the modern understanding of double-copy relations, the universality of soft theorems, the geometry of scattering (positive geometries and polytopes), and ongoing attempts to understand quantum gravity from the perspective of scattering amplitudes.

Key References: (Mizera, 2016, Mizera, 2017, Chi et al., 2021, Stieberger, 2022, Stieberger, 2023, Massidda, 13 Mar 2024, Alday et al., 28 Apr 2025, Bartsch et al., 2 May 2025).