Topological Effects in Neural Network Field Theory

Abstract: Neural network field theory formulates field theory as a statistical ensemble of fields defined by a network architecture and a density on its parameters. We extend the construction to topological settings via the inclusion of discrete parameters that label the topological quantum number. We recover the Berezinskii--Kosterlitz--Thouless transition, including the spin-wave critical line and the proliferation of vortices at high temperatures. We also verify the T-duality of the bosonic string, showing invariance under the exchange of momentum and winding on $S^1$, the transformation of the sigma model couplings according to the Buscher rules on constant toroidal backgrounds, the enhancement of the current algebra at self-dual radius, and non-geometric T-fold transition functions.

• The paper introduces a neural network field theory framework that encodes topological sectors directly in its network parameters.
• It successfully models the Berezinskii–Kosterlitz–Thouless transition with precise critical exponents and clear vortex dynamics.
• The study extends the framework to string theory, reproducing T-duality and non-geometric T-fold phenomena through a mixed continuous-discrete ensemble.

Topological Effects in Neural Network Field Theory

Foundations of Neural Network Field Theory

Neural Network Field Theory (NN-FT) is a constructive formalism for quantum field theory in which the architecture and parameter density of a neural network define the statistical ensemble of fields. Rather than representing fields as functionals in the path integral, NN-FT leverages ensembles over network parameters, $\theta$, with architectures $\phi_\theta$, such that $\phi_\theta(x)$ serves as the field at point $x$. This approach is not limited to variational ansätze; architecture and parameter densities encode locality, symmetry, operator content, and phase structure directly in parameter space.

Correlation functions are computed as expectations over parameter space:

$$\langle \mathcal{O} \rangle = \int d\theta~P(\theta)~\mathcal{O}[\phi_\theta]$$

For theories with explicit topological sectors (e.g., winding numbers, vortices), the expectation extends to discrete network parameters:

$$\langle \mathcal{O} \rangle = \sum_Q \int d\theta~P(\theta, Q)~\mathcal{O}[\phi_{\theta,Q}]$$

Gaussian process limits are realized for wide architectures, enabling free field theory representations. Deviations from this limit or non-Gaussian parameter densities induce interactions. Symmetries are implemented by design in the architecture and/or $P(\theta)$. This framework is universal under broad conditions (Ferko et al., 20 Jan 2026).

Topological Sectors: Constructive Representation

The paper introduces explicit topological sector labels as discrete neural network parameters, enabling NN-FT to realize compact boson models and associated phenomena. This mixed continuous-discrete ensemble mirrors the standard field theory sum over topological sectors, providing a unified approach to sampling smooth fluctuations and global topological data.

Two Principal Case Studies

• Berezinskii–Kosterlitz–Thouless (BKT) Transition: A paradigmatic finite-temperature phase transition in two-dimensional $U(1)$ systems. NN-FT realizes both the critical spin-wave line and the vortex proliferation mechanism.
• T-Duality in String Theory: Demonstrates that NN-FT can represent worldsheet string backgrounds and reproduce the invariance under exchange of momentum and winding on $S^1$, including Buscher rules, symmetry enhancement, and non-geometric T-folds.

Berezinskii–Kosterlitz–Thouless Transition via NN-FT

Architecture and Sector Construction

• Spin-wave Sector: Random Fourier Feature (RFF) network, $\theta_\text{sw}(x)$, capturing smooth field fluctuations.
• Vortex Sector: Coulomb gas ensemble of vortex-antivortex pairs, sampled with explicit discrete charges.
• Composite Field: $\phi_\theta$0.

The spin-wave correlator exhibits power-law decay $\phi_\theta$1 below the critical coupling $\phi_\theta$2, matching analytic predictions with high numerical precision.

Figure 1: Spin-wave correlator $\phi_\theta$3 below $\phi_\theta$4, confirming power-law scaling with exponent $\phi_\theta$5.

Above $\phi_\theta$6, in the pure spin-wave sector, correlations remain power-law due to the Gaussian nature, but the inclusion of vortex sector gives rise to a mass gap and exponential decay.

Figure 2: Spin-wave correlator above $\phi_\theta$7; RFF sector remains critical, requiring vortex sector for disordering.

Figure 3: Correlation length $\phi_\theta$8 diverges as essential singularity near transition, reflecting BKT physics.

Vortex density $\phi_\theta$9 sharply distinguishes the phases:

Figure 4: Vortex density $\phi_\theta(x)$0 versus $\phi_\theta(x)$1, rising steeply above $\phi_\theta(x)$2 as vortices unbind.

Spatial representations elucidate the proliferation of vortices:

Figure 5: Vortex angle field $\phi_\theta(x)$3 captures vortex-antivortex configurations across $\phi_\theta(x)$4.

Pair correlations confirm tight binding below $\phi_\theta(x)$5 and plasma behavior above:

Figure 6: Vortex-antivortex pair correlation $\phi_\theta(x)$6; Coulomb attraction dominates at low $\phi_\theta(x)$7, transitions to uncorrelated gas.

Spectral estimates of the stiffness $\phi_\theta(x)$8 and helicity modulus $\phi_\theta(x)$9 track the universal jump and vanishing in the disordered phase:

Figure 7: Renormalized stiffness $x$0 closely follows $x$1 below $x$2, drops sharply above due to screening.

Figure 8: Helicity modulus $x$3 collapses at $x$4 and aligns with Nelson-Kosterlitz universal jump.

The NN-FT reproduces:

• Exact critical exponent $x$5.
• BKT-type essential singularity in $x$6.
• Universal jump in stiffness.
• Vortex proliferation marks the phase boundary.

T-Duality in NN-FT

Sigma Model Realization

NN-FT architectures represent worldsheet fields $x$7 as Gaussian processes, supplemented by discrete momentum and winding labels for compact directions. For $x$8 and $x$9 backgrounds, sampled compact sectors match the well-known T-duality exchanges.

• Oscillator Modes: Gaussian process sampler matches local fluctuations.
• Compact Zero-Modes: Discrete $$\langle \mathcal{O} \rangle = \int d\theta~P(\theta)~\mathcal{O}[\phi_\theta]$$0 labels represent momentum and winding.

Buscher rules transform sigma model couplings, including metric, $$\langle \mathcal{O} \rangle = \int d\theta~P(\theta)~\mathcal{O}[\phi_\theta]$$1-field, and dilaton, in line with string-theoretic duality.

Operator Structure and Symmetry Enhancement

Chiral vertex operator correlators reproduce compact boson selection rules, two-point functions, and scaling dimensions under duality transformations. At the self-dual radius, the sampled ensemble organizes operators into $$\langle \mathcal{O} \rangle = \int d\theta~P(\theta)~\mathcal{O}[\phi_\theta]$$2 multiplets, placing charged currents at dimension one and confirming enhanced symmetry.

Non-Geometric T-Folds

Patchwise NN-FT constructions with Buscher gluing realize T-fold backgrounds: local torus patches are glued non-geometrically, demonstrating NN-FT's flexibility for non-geometric global identifications.

Implications and Future Directions

Practical and Theoretical Implications

• Parameter Space Encoding: NN-FT accommodates both local field fluctuations and global topological data, providing a universal constructive framework for quantum field theory.
• Phase Structure: Explicit representation of topological quanta enables the study of transitions and dualities inaccessible to purely Gaussian architectures.
• Stringy Dualities: Exact equivalence under T-duality demonstrates the suitability of NN-FT for string theory and compactification studies.

Future Developments

• Intrinsic Topological Sector Learning: Investigating architectures and priors that can dynamically learn topological sectors.
• Extension to Strong/Weak Coupling Dualities: Generalizing the mixed ensemble framework to encompass genuine strong/weak-coupling dualities and nonperturbative phenomena.
• Topological Phases and Gauge Theory: Applying NN-FT to gauge theories, defect sectors, and systems with higher-form symmetries.
• Hybrid Training Schemes: Incorporating gradient-based and genetic algorithms for continuous and discrete parameter sectors.

Conclusion

The paper rigorously demonstrates that neural network field theory, when extended to mixed continuous/discrete parameter ensembles, can faithfully reproduce both dynamical and kinematical topological effects, including the BKT transition and T-duality phenomena. The explicit representation of topological sectors, rather than relying on emergent behavior from unconstrained priors, provides a constructive and robust framework for studying phases and dualities in quantum field theory and string theory. The NN-FT paradigm opens new avenues for systematic exploration of global and nonperturbative effects in field theoretic models and offers promising applications for machine learning architectures that require non-trivial bundle or superselection sector representations.