arXiv subjects

Vishnu Jejjala

Publications and source records attributed to Vishnu Jejjala.

At least 19 recordsLinked to original sources

arXiv2026

A Tale of Two Compact Bosons

Neural network field theory (NN-FT) defines a field theory by a network architecture together with a probability density on its latent variables. For compact theories the local Gaussian sector is only part of the story: one must also sum over discrete topological sectors. We review a mixed continuous/discrete latent-variable construction and apply it to two compact bosons. For the Berezinskii--Kosterlitz--Thouless transition, a random Fourier feature spin-wave sampler supplemented by an explicit Coulomb gas vortex sector reproduces the Gaussian critical line below $T_c$, vortex proliferation above $T_c$, the essential singularity of the correlation length, and the Nelson--Kosterlitz jump. For the bosonic string, oscillator modes augmented by momentum--winding labels reproduce circle T-duality, Buscher transformations on constant toroidal backgrounds, self-dual current algebra enhancement, and a toy T-fold. The common lesson is that the same local neural sampler, paired with different discrete topological data, yields physically distinct compact theories. These proceedings are based on arXiv:2604.02313.

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arXiv2026

Balanced Metrics Know About SYZ

Numerical Ricci-flat metrics on Calabi-Yau manifolds are becoming increasingly accurate. However, they often lack the interpretability required to extract theoretical insights. In this paper, we introduce a novel variant of Donaldson's algorithm based on the Moore-Penrose pseudo-inverse that operates on the global sections of the ambient space rather than the manifold itself. This approach allows us to use the canonical monomial basis to compute interpretable balanced metrics even at large degrees $k$. Applying our ambient algorithm to multiple families, including the Dwork family and complete intersection Calabi-Yau manifolds, we discover that the metric parameters obey novel power laws near the Large Complex Structure Limit (LCSL). We connect these to the Gromov-Hausdorff metric collapse predicted by the SYZ conjecture.

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arXiv2026

What to do with a Ricci-flat Calabi--Yau metric?

Numerical approximations to Ricci-flat Calabi--Yau metrics make it possible to move beyond the topological and holomorphic data that have traditionally dominated explicit string compactifications. This article explains what new physics and mathematics become accessible once the metric, and eventually the associated Hermitian Yang--Mills bundle data, can be computed. In heterotic compactifications, such data are needed to determine matter K\"ahler metrics, canonically normalized Yukawa couplings, Kaluza--Klein spectra, threshold effects, soft terms, and other non-holomorphic ingredients of the four-dimensional effective action. More broadly, numerical Calabi--Yau geometry provides quantitative input for moduli stabilization, $\alpha'$-corrected backgrounds, de~Sitter model building, axion physics, swampland distance tests, and compactifications in which the internal geometry varies over spacetime. Geometric data permit a computational approach to long-standing mathematical questions involving special Lagrangian submanifolds, SYZ fibrations, mirror symmetry, calibrated geometry, metric degeneration, restrictions of Ricci-flat metrics to fibers, and the search for analytic or semi-analytic structures. We present these directions as a roadmap for future work.

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arXiv2026

Fortuity and Complexity in a Simple Quark Model

We observe and elaborate on a structural similarity between the categorization of monotone and fortuitous BPS operators in supersymmetric theories and gauge invariant quark operators in $SU(N_c)$ QCD. Our designation of fortuity does not rely on supersymmetry and instead uses the BRST cohomology. We argue that within this designation, baryon states are fortuitous while meson states are monotone. We illustrate that in the Veneziano limit of large number of flavors and colors, this designation displays features resembling the fortuitous vs. monotone categorization of BPS operators, e.g., an exponential vs. polynomial dichotomy in the counting of operators. We explore these ideas explicitly in a toy qubit model of quarks. We further investigate the stabilizer R\'enyi entropy of meson and baryon states as a proxy for the complexity of classical simulation for these states. We show that all mesons display power law complexity and present evidence that typical baryons display super-exponential complexity in the Veneziano limit.

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arXiv2026

Anomalies in Neural Network Field Theory

Neural network field theory (NN-FT) formulates field theory in terms of a network architecture and a density on its parameters. We derive Schwinger--Dyson equations and Ward identities in NN-FT and utilize them to study anomalies. The equations depend on a conserved parameter space current that characterizes symmetries and how they break. It is relevant even in non-local NN-FTs, but can recover local currents in the case of a local Lagrangian by an appropriate fiber-wise average. In machine learning, this formalism is applied to feedforward networks and the attention mechanism. In physics, we use this machinery to study $U(1)$ symmetry for a complex scalar, the scale anomaly in $4d$ massless $\phi^4$ theory, the Weyl anomaly for the bosonic string (including a new computation of the critical dimension), and examples involving discrete topological data, such as winding numbers and T-duality. Since the results are obtained in network parameter space rather than the standard field space, they represent a new way to understand symmetries in quantum field theories.

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arXiv2026

Topological Effects in Neural Network Field Theory

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.

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arXiv2026

Machine learning modularity

Based on a transformer based sequence-to-sequence architecture combined with a dynamic batching algorithm, this work introduces a machine learning framework for automatically simplifying complex expressions involving multiple elliptic Gamma functions, including the $q$-$\theta$ function and the elliptic Gamma function. The model learns to apply algebraic identities, particularly the SL$(2,\mathbb{Z})$ and SL$(3,\mathbb{Z})$ modular transformations, to reduce heavily scrambled expressions to their canonical forms. Experimental results show that the model achieves over 99\% accuracy on in-distribution tests and maintains robust performance (exceeding 90\% accuracy) under significant extrapolation, such as with deeper scrambling depths. This demonstrates that the model has internalized the underlying algebraic rules of modular transformations rather than merely memorizing training patterns. Our work presents the first successful application of machine learning to perform symbolic simplification using modular identities, offering a new automated tool for computations with special functions in quantum field theory and the string theory.

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arXiv2025

On isomorphisms between quiver Yangians

Quiver Yangians are infinite-dimensional algebras capturing the BPS structure of a large class of supersymmetric models. Quiver theories related by Seiberg duality are expected to have isomorphic quiver Yangians, and this isomorphism has previously been shown for quivers corresponding to generalised conifold geometries. In this work, we present an explicit isomorphism for the two Seiberg dual phases of the F0 quiver theory, which falls outside of the above class. Some aspects of our construction are similar to the known cases, while others appear to be specific to the F0 quiver. In particular, the map involves square roots of operators bilinear in the fermionic fields of the mode being dualised.

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arXiv2025

Vacuum Geometry of the Standard Model

Vacuum structure of a quantum field theory is a crucial property. In theories with extended symmetries, such as supersymmetric gauge theories, the vacuum is typically a continuous manifold, called the vacuum moduli space, parametrized by the expectation values of scalar fields. Starting from the R-parity preserving superpotential at renormalizable order, we use Gr\"obner bases to determine the explicit structure, as an algebraic variety, of the vacuum geometry of the minimal supersymmetric extension of the Standard Model. Gr\"obner bases have doubly exponential computational complexity (for this case, $7^{2^{1023}}$ operations); we exploit symmetry and multigrading to render the computation tractable. This geometry has three irreducible components of complex dimensions $1$, $15$, and $29$, each being a so-called rational variety. The defining equations of the components express the solutions to F-terms and D-terms in terms of the gauge invariant operators and are interpreted in terms of classical geometric constructions.

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arXiv2025

The Vacuum Moduli Space of the Minimal Supersymmetric Standard Model

A starting point in the study of the minimal supersymmetric Standard Model (MSSM) is the vacuum moduli space, which is a highly complicated algebraic variety: it is the image of an affine variety $X \subset \mathbb{C}^{49}$ under a symplectic quotient map $\phi$ to $\mathbb{C}^{973}$. Previous work computed the vacuum moduli space of the electroweak sector; geometrically this corresponds to studying a restriction of $\phi$: $\mathbb{C}^{13} \stackrel{\phi^{\texttt{res}}}{\longrightarrow} \mathbb{C}^{22}$. We analyze the geometry of the full vacuum moduli space for superpotentials $W_{\rm minimal}$ (without neutrinos) and $W_{\rm MSSM}$ (with neutrinos) in $\mathbb{C}^{973}$. In both cases, we prove that $X$ consists of three irreducible components $X_1$, $X_2$, and $X_3$, and determine the images $M_i$ of the $X_i$ under $\phi$. For $W_{\rm minimal}$ we show they have, respectively, dimensions $1$, $15$, and $29$, and prove that each of the $M_i$ is a rational variety, while for $W_{\rm MSSM}$ we show that $M_3$ is the only component. Restricting the $M_i$ to the electroweak sector, we recover known results. We describe the components of the vacuum moduli space geometrically in terms of incidence varieties to a product of Segre varieties.

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arXiv2025

Machine learning automorphic forms for black holes

Modular, Jacobi, and mock-modular forms serve as generating functions for BPS black hole degeneracies. By training feed-forward neural networks on Fourier coefficients of automorphic forms derived from the Dedekind eta function, Eisenstein series, and Jacobi theta functions, we demonstrate that machine learning techniques can accurately predict modular weights from truncated expansions. Our results reveal strong performance for negative weight modular and quasi-modular forms, particularly those arising in exact black hole counting formulae, with lower accuracy for positive weights and more complicated combinations of Jacobi theta functions. This study establishes a proof of concept for using machine learning to identify how data is organized in terms of modular symmetries in gravitational systems and suggests a pathway toward automated detection and verification of symmetries in quantum gravity.

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arXiv2025

Dynamical Dark Energy, Dual Spacetime, and DESI

In this paper we discuss possible consequences of a manifestly non-commutative and $T$-duality covariant formulation of string theory on dark energy, when the correspondence between short distance (UV) and long distance (IR) physics is taken into account. We demonstrate that the dark energy is dynamical, \textit{i.e.}, time-dependent, and we compute the allowed values of $w_0$ and $w_a$, given by $w(a) = w_0+w_a(1-a)$, which compare favorably to the most recent observations by DESI. From this point of view, the latest results from DESI might point to a fundamentally new understanding of quantum spacetime in the context of quantum gravity.

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arXiv2025

Colored Jones Polynomials and the Volume Conjecture

Using the vertex model approach for braid representations, we compute polynomials for spin-1 placed on hyperbolic knots up to 15 crossings. These polynomials are referred to as 3-colored Jones polynomials or adjoint Jones polynomials. Training a subset of the data using a fully connected feedforward neural network, we predict the volume of the knot complement of hyperbolic knots from the adjoint Jones polynomial or its evaluations with 99.34% accuracy. A function of the adjoint Jones polynomial evaluated at the phase $q=e^{ 8 \pi i / 15 }$ predicts the volume with nearly the same accuracy as the neural network. From an analysis of 2-colored and 3-colored Jones polynomials, we conjecture the best phase for $n$-colored Jones polynomials, and use this hypothesis to motivate an improved statement of the volume conjecture. This is tested for knots for which closed form expressions for the $n$-colored Jones polynomial are known, and we show improved convergence to the volume.

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arXiv2024

Metaheuristic Generation of Brane Tilings

The combinatorics of dimer models on brane tilings describe a large class of four-dimensional $\mathcal{N}=1$ gauge theories that afford quiver descriptions and have toric moduli spaces. We introduce a combinatorial optimization method leveraging simulated annealing to explicitly construct geometrically consistent brane tilings, providing a proof of concept for efficient generation of gauge theories using metaheuristic techniques. The implementation of this idea recovers known examples and allows us to derive a new brane tiling with $26$ quantum fields, illustrating the potential of metaheuristic techniques as a valuable addition to the toolbox for constructing and analyzing gauge theories from brane tilings.

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arXiv2024

cymyc -- Calabi-Yau Metrics, Yukawas, and Curvature

We introduce \texttt{cymyc}, a high-performance Python library for numerical investigation of the geometry of a large class of string compactification manifolds and their associated moduli spaces. We develop a well-defined geometric ansatz to numerically model tensor fields of arbitrary degree on a large class of Calabi-Yau manifolds. \texttt{cymyc} includes a machine learning component which incorporates this ansatz to model tensor fields of interest on these spaces by finding an approximate solution to the system of partial differential equations they should satisfy.

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arXiv2024

Precision String Phenomenology

Calabi--Yau compactifications of the $E_8\times E_8$ heterotic string provide a promising route to recovering the four-dimensional particle physics described by the Standard Model. While the topology of the Calabi--Yau space determines the overall matter content in the low-energy effective field theory, further details of the compactification geometry are needed to calculate the normalized physical couplings and masses of elementary particles. In this work, we present numerical computations of physical Yukawa couplings in a number of heterotic models in the standard embedding and demonstrate the existence of natural hierarchies, a coveted feature in string model building.

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arXiv2024

Generating Triangulations and Fibrations with Reinforcement Learning

We apply reinforcement learning (RL) to generate fine regular star triangulations of reflexive polytopes, that give rise to smooth Calabi-Yau (CY) hypersurfaces. We demonstrate that, by simple modifications to the data encoding and reward function, one can search for CYs that satisfy a set of desirable string compactification conditions. For instance, we show that our RL algorithm can generate triangulations together with holomorphic vector bundles that satisfy anomaly cancellation and poly-stability conditions in heterotic compactification. Furthermore, we show that our algorithm can be used to search for reflexive subpolytopes together with compatible triangulations that define fibration structures of the CYs.

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arXiv2024

Physical Yukawa Couplings in Heterotic String Compactifications

One of the challenges of heterotic compactification on a Calabi-Yau threefold is to determine the physical $(\mathbf{27})^3$ Yukawa couplings of the resulting four-dimensional $\mathcal{N}=1$ theory. In general, the calculation necessitates knowledge of the Ricci-flat metric. However, in the standard embedding, which references the tangent bundle, we can compute normalized Yukawa couplings from the Weil-Petersson metric on the moduli space of complex structure deformations of the Calabi-Yau manifold. In various examples (the Fermat quintic, the intersection of two cubics in $\mathbb{P}^5$, and the Tian-Yau manifold), we calculate the normalized Yukawa couplings for $(2,1)$-forms using the Weil-Petersson metric obtained from the Kodaira-Spencer map. In cases where $h^{1,1}=1$, this is compared to a complementary calculation based on performing period integrals. A third expression for the normalized Yukawa couplings is obtained from a machine learned approximate Ricci-flat metric making use of explicit harmonic representatives. The excellent agreement between the different approaches opens the door to precision string phenomenology.

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