Topological Effects in Neural Network Field Theory

Christian Ferko; James Halverson; Vishnu Jejjala; Brandon Robinson

hep-thApr 2, 2026

Topological Effects in Neural Network Field Theory

Authors: Christian Ferko, James Halverson, Vishnu Jejjala, Brandon Robinson

Organizations: The NSF Institute for Artificial Intelligence and Fundamental Interactions (IAIFI) and Department of Physics, Northeastern University, Boston, MA 02115, USA · Mandelstam Institute for Theoretical Physics, School of Physics, and NITheCS, University of the Witwatersrand, Johannesburg, WITS 2050, South Africa · Institute of Physics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, Netherlands

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.

Figures & tables

Figure 1: Spin-wave correlator

C(r)

below

T_{c}

(log-log). Solid lines: data. Dashed lines: predicted power law

r^{-b^{2}}

$b$	$\eta_{\text{exp}}=b^{2}$	$\eta_{\text{fit}}$	$R^{2}$
0.050	0.00250	0.00252	0.99196
0.089	0.00791	0.00796	0.99196
0.158	0.02500	0.02517	0.99196
0.281	0.07906	0.07963	0.99198
0.500	0.25000	0.25204	0.99204

Table 1: Below-

T_{c}

power law exponents.

\eta_{\text{fit}}

agrees with

b^{2}

to better than

1\%

throughout, confirming ( 44 ).

Figure 2: Spin-wave correlator above

T_{c}

(log-log). The correlator remains power law because the RFF field is Gaussian by construction: the spin-wave sector alone has no disordered phase.

Figure 3: Correlation length

\xi

extracted from the full two-point function via ( 48 ).

R^{2}

values of the per-point fit ( 48 ) are annotated. Dashed curve: BKT essential singularity fit ( 49 ) (

R^{2}=0.98

Figure 4: Vortex density

\rho_{v}

vs.

b

. The vertical dashed line marks

b_{c}=1/2

. Below

T_{c}

the vortex density is consistent with zero; above

T_{c}

it rises steeply as vortices unbind.

Figure 5: Vortex angle field $\theta_{v}(x)\;\text{mod}\;2\pi$ for nine values of $b$ . Red (blue) markers denote vortices (antivortices). At small $b$ the field is featureless; as $b$ increases past $b_{c}=1/2$ , vortex–antivortex pairs appear and eventually form a dense plasma.

Figure 6: Vortex–antivortex pair correlation

g_{+-}(r)

. At large

T

b\to\infty

and

g_{+-}=1

for all

r

, since the unbinding makes it an uncorrelated Coulomb gas (dashed line). At lower temperatures, we expect excess (deficit) probability at low (large)

r

, which is reflected in the indigo (

b=0.60

) and blue (

b=0.75

) curves, due to the logarithmic potential in

P_{\text{vort}}

. The noisy statistics is due to a paucity of vortices.

Figure 7: Renormalized stiffness

K_{R}

vs.

b

. Gray dashed: bare coupling

K_{0}=1/(2\pi b^{2})

. Blue dotted: universal value

2/\pi

. Red dashed:

b_{c}=1/2

. Below

b_{c}

K_{R}\approx K_{0}

; above

b_{c}

, vortex screening drives

K_{R}

sharply downward.

Figure 8: Helicity modulus

\Upsilon_{R}=2\pi b^{2}K_{R}

vs. temperature

T=2\pi b^{2}

. Blue dashed: Nelson–Kosterlitz line

\Upsilon_{R}=2T/\pi

. Red dashed:

T_{c}=\pi/2

. The data cross the NK line near

T_{c}

as predicted, and

\Upsilon_{R}

collapses toward zero in the disordered phase.

Observable	$b<b_{c}$ (ordered)	$b=b_{c}$ (critical)	$b>b_{c}$ (disordered)
$\eta_{\text{sw}}$	$b^{2}<1/4$	$1/4$	$b^{2}>1/4$
$G_{2}(r)$	$\sim r^{-b^{2}}$	$\sim r^{-1/4}$	$\sim r^{-b^{2}}e^{-r/\xi}$
$\xi$	$\infty$	$\infty$	finite, $\sim e^{c/\sqrt{b^{2}-b_{c}^{2}}}$
$\rho_{v}$	$\approx 0$	onset	$>0$ , increasing
$K_{R}$	$\approx K_{0}=1/(2\pi b^{2})$	$2/\pi$	$\ll 1/(2\pi b^{2})$
$\Upsilon_{R}$	above NK line	on NK line	below NK line

Table 2: Observable signatures across the BKT transition in NN-FT.

Test	NN-FT data	Exact statement	Numerical signal
Circle duality	Oscillator sampler + discrete $(n,w)$ labels + zero-mode	$R\leftrightarrow\alpha^{\prime}/R$ , $n\leftrightarrow w$ , $p_{L}$ fixed, $p_{R}\to-p_{R}$	Lattice weights, monodromies, and zero-mode slopes agree under duality

Buscher on $T^{2}$	Vector-valued oscillator covariance + compact lattice with $G,B$ dependence	Buscher map on $(G,B)$ , compact charges, and the dilaton invariant	Transformed weights and marginals match; oscillator covariance tracks $G^{-1}$ ; $e^{-2\Phi}\sqrt{\det G}$ is preserved

Vertex operators	Chiral finite-mode sampler with explicit $X_{L},X_{R}$ and zero-mode averaging	Neutrality rules and dual matching of momentum/winding correlators	Non-neutral channels are suppressed; dual two-point functions and fitted dimensions agree

Table 3: Summary of the T-duality tests realized in the NN-FT mixed ensemble.

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