String Geometry Theory

• String Geometry Theory is a framework where the entire configuration space of strings forms an infinite-dimensional manifold that embeds standard perturbative backgrounds.
• It unifies geometric, topological, and quantum aspects by recovering the full genus expansion through a universal path integral and distinctive renormalization features.
• The theory introduces novel dualities and extends to generalized geometries such as area metric and non-Riemannian phases, with vacuum selection achieved via a non-perturbative potential.

String geometry theory is a non-perturbative, background-independent framework in which the entire configuration space of strings—including all genera, embedding data, and super/Riemann surface moduli—is promoted to an infinite-dimensional “string manifold.” In this approach, not only the matter content (string excitations) but also the space-time itself emerges as a collective geometric property of string configurations. Standard perturbative string backgrounds, including supergravity solutions, are embedded as special field configurations on this manifold, and the fluctuations around these backgrounds reproduce the full perturbative expansion, capturing all worldsheet topologies in a universal path integral. The theory is characterized by new geometric structures, a distinctive renormalization structure, and a principle of vacuum selection via a non-perturbative potential.

1. Fundamental Structure of String Geometry Theory

The foundational data of string geometry theory consists of the string manifold $\mathcal{M}$, whose points are equivalence classes $[\bar{\Sigma},X(\bar\tau),\bar\tau]$ for bosonic or $$[\bar{\mathbf{E}}_T,\mathbf{X}_T(\bar\tau),\bar\tau]$$ for superstrings, where:

• $\bar{\Sigma}$ is a (super-)Riemann surface with chosen global time $\bar\tau$;
• $X(\bar\tau)$ or $\mathbf{X}_T(\bar\tau)$ are embeddings into target space;
• $\bar{\mathbf{E}}_T$ is the supergeometry datum for superstrings.

Dynamical fields on this infinite-dimensional manifold include the string manifold metric $G_{IJ}$, a two-form $B_{IJ}$, a dilaton $\Phi$, and (for superstrings) a tower of RR and gauge fields. The string geometry action is a direct analog of the Einstein-Hilbert action, e.g.,

$$S = \int \mathcal{D}\bar\tau\,\mathcal{D}\bar{h}\,\mathcal{D}X\, \sqrt{G} e^{-2\Phi} \left( - R + 4 \nabla_I\Phi\nabla^I\Phi + \tfrac{1}{2}|H|^2 + \dots \right),$$

where $R$ is the scalar curvature on $\mathcal{M}$ and $|H|^2$ involves contractions with $G_{IJ}$ (Sato, 12 Jul 2024, Sato, 4 Nov 2025).

2. Embedding of Perturbative String Backgrounds

All conventional string backgrounds—including the metric $G_{\mu\nu}(x)$, $B$-field $B_{\mu\nu}(x)$, dilaton $\Phi(x)$, and, in the superstring case, RR and gauge backgrounds—can be embedded as stationary field configurations on $\mathcal{M}$. For bosonic and heterotic strings, this takes the form: $$\bar{G}_{dd} = e^{2\phi(X)}, \quad \bar{G}_{(\mu\bar\sigma)(\nu\bar\sigma')} = \frac{\bar{e}^3(\bar\sigma)}{\sqrt{\bar{h}(\bar\sigma)}} G_{\mu\nu}(X(\bar\sigma))\delta_{\bar\sigma\bar\sigma'}\,,$$ with similar expressions for $\bar B$ and the dilaton (Nagasaki et al., 2023, Nagasaki et al., 5 Nov 2025). The embedding includes all ten-dimensional supergravity sectors (type IIA, IIB, heterotic, type I), and, under the embedding, the equations of motion for the string geometry theory reduce in the $\alpha'\to 0$ limit to the usual supergravity background equations.

3. Path Integral and Recovery of Perturbative String Theory

Fluctuations around any such classical background solution ($\bar{G}_{IJ}$, $\bar{B}_{IJ}$, $\bar{\Phi}$, ...) generate, to quadratic order, an effective action where only a particular scalar mode ($\psi_{dd}$, corresponding to the physical string fluctuation) propagates. The associated two-point function, after suitable gauge fixing and field redefinitions, admits a Schwinger representation: $$\Delta_F = \int_0^\infty dT\, \langle \psi_{dd}| e^{-T\hat{H}} |\psi_{dd}\rangle,$$ where $\hat{H}$ is the "string geometry Hamiltonian" (Sato et al., 2022, Sato et al., 2022, Nagasaki et al., 5 Nov 2025). Integrating out the auxiliary variables and ghosts, and gauge-fixing the one-dimensional reparametrization invariance, this procedure precisely produces the Polyakov (or Green-Schwarz for superstrings) path integral including all worldsheet genera, with target-space fields $G_{\mu\nu}$, $B_{\mu\nu}$, and $\Phi$ appearing as background couplings.

This derivation holds universally for all perturbative backgrounds, and, by construction, the usual moduli space integrals over Riemann (or super-Riemann) surfaces emerge from the integration over the infinite-dimensional metric on the string manifold (Sato, 2017, Sato, 4 Nov 2025).

4. Non-Perturbative Effective Potential and Vacuum Selection

A defining feature of string geometry theory is the existence of a non-perturbative effective potential $V[G,B,\Phi,...]$ for the background fields, obtained by evaluating the classical (zeroth order) part of the string geometry action on a given background: $$V_{\rm eff}[G,B,\Phi] = - S_{\rm str-geom}^{(0)}\big|_{\bar{G}=G,\,\bar{B}=B,\,\bar{\Phi}=\Phi}\;.$$ Stationary backgrounds (satisfying the on-shell and Weyl invariance constraints) serve as candidate vacua. The global minimum of $V_{\rm eff}$ is postulated to be the dynamically selected "string vacuum" (Sato, 12 Jul 2024, Nagasaki et al., 2023, Nagasaki et al., 5 Nov 2025). Perturbative contributions to the potential can be solved explicitly up to quadratic order via Green's function techniques, and non-local terms incorporate the full genus expansion.

New non-perturbative effects arise: semiclassical instanton amplitudes correspond to tunneling between semi-stable vacua, contributing corrections of order $e^{-1/g_s^2}$, and are responsible for driving generic configurations toward the true vacuum (Sato, 4 Nov 2025). No further loop corrections in the string geometry parameter $\beta$ exist, due to a non-renormalization theorem dictated by the supergeometry structure (Sato, 4 Nov 2025). In particular, the genus (the string loop expansion) is already encoded in the background structure, and there is no notion of separate "string loops" in the nonperturbative regime.

5. Symmetries and Dualities: T-Symmetry and Background Independence

"T-symmetry" is a discrete $\mathbb{Z}_2$ symmetry of the dimensionally reduced string geometry theory which generalizes ordinary T-duality. Spatial T-symmetry in compact directions exchanges IIA and IIB perturbative vacua and all corresponding RR/gauge field content, reproducing Buscher's duality; temporal T-symmetry acts in the "string geometry time" direction and exchanges Kaluza-Klein and winding modes in a novel way not visible in perturbative string theory (Sato et al., 2023, Sato, 4 Nov 2025).

The full path integral is manifestly background independent: all classical string backgrounds are field configurations on $\mathcal{M}$, and the selection of a physical vacuum is controlled by extremizing the non-perturbative potential. The theory automatically includes all standard modular dualities of string theory and unifies all perturbative backgrounds in a single non-perturbative framework.

6. Generalizations: Area Metric, Non-Riemannian, and Topological Phases

String geometry theory incorporates, as special limits or deformations:

• Area Metric Theories—Geometric variants in which the basic structure is an area metric $h_{abcd}$, not reducible to a line element. These allow for generalizations of connections, curvature, and Einstein equations, and admit vacuum solutions with and without an underlying metric (Ho et al., 2015).
• Non-Riemannian Geometries—O$(D,D)$-covariant generalized metrics $H_{MN}$ describe situations where no ordinary spacetime metric exists. Specific backgrounds correspond to nonrelativistic string geometries or purely chiral/anti-chiral vacua, with restricted (and finite) physical spectra (Park et al., 2020).
• Topological String Geometry—In a topological phase, the data and functional integral over $\mathcal{M}$ produce the full genus expansion of the perturbative topological string, and non-perturbative "instanton" corrections of order $e^{-1/g_s}$ naturally arise from the new structure of the string manifold (Sato et al., 2019).

7. Analytical and Numerical Methods for Vacuum Search

Practical determination of the true string vacuum proceeds via analytical truncations (e.g. restricting to Calabi–Yau or special holonomy metrics, series expansions in flux, or large-volume moduli) and/or numerical schemes such as Regge calculus discretization and optimization approaches (including gradient descent, simulated annealing, and Monte Carlo sampling) to minimize $V_{\rm eff}[G,B,\Phi]$ (Sato, 12 Jul 2024, Nagasaki et al., 2023). This search is essential for predicting low-energy spectrum, moduli stabilization, and cosmological parameters from first principles.

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String geometry theory thus provides a genuinely non-perturbative framework for string theory, uniting the geometrical, topological, and quantum aspects of string backgrounds in a single master formalism. All perturbative string vacua are embedded as field configurations, the full genus expansion of string amplitudes is recovered from tree-level correlators on the string manifold, and vacuum selection is reduced to a minimization principle for a universal effective potential. The structure and properties of this formulation suggest new avenues for the paper of dualities, emergent geometry, and the quantum structure of spacetime (Sato, 4 Nov 2025, Nagasaki et al., 5 Nov 2025, Nagasaki et al., 2023, Sato, 12 Jul 2024, Sato, 2017).