arXiv subjects

Emil Albrychiewicz

Publications and source records attributed to Emil Albrychiewicz.

14 recordsLinked to original sources

arXiv2026

Interpreting the Synchronization Gap: The Hidden Mechanism Inside Diffusion Transformers

Recent theoretical models of diffusion processes, conceptualized as coupled Ornstein-Uhlenbeck systems, predict a hierarchy of interaction timescales, and consequently, the existence of a synchronization gap between modes that commit at different stages of the reverse process. However, because these predictions rely on continuous time and analytically tractable score functions, it remains unclear how this phenomenology manifests in the deep, discrete architectures deployed in practice. In this work, we investigate how the synchronization gap is mechanistically realized within pretrained Diffusion Transformers (DiTs). We construct an explicit architectural realization of replica coupling by embedding two generative trajectories into a joint token sequence, modulated by a symmetric cross attention gate with variable coupling strength g. Through a linearized analysis of the attention difference, we show that the replica interaction decomposes mechanistically. We empirically validate our theoretical framework on a pretrained DiT-XL/2 model by tracking commitment and per layer internal mode energies. Our results reveal that: (1) the synchronization gap is an intrinsic architectural property of DiTs that persists even when external coupling is turned off; (2) as predicted by our spatial routing bounds, the gap completely collapses under strong coupling; (3) the gap is strictly depth localized, emerging sharply only within the final layers of the Transformer; and (4) global, low frequency structures consistently commit before local, high frequency details. Ultimately, our findings provide a mechanistic interpretation of how Diffusion Transformers resolve generative ambiguity, isolating speciation transitions to the terminal layers of the network.

cs.LG↗

arXiv2026

Dynamical Regimes of Multimodal Diffusion Models

Diffusion based generative models have achieved unprecedented fidelity in synthesizing high dimensional data, yet the theoretical mechanisms governing multimodal generation remain poorly understood. Here, we present a theoretical framework for coupled diffusion models, using coupled Ornstein-Uhlenbeck processes as a tractable model. By using the nonequilibrium statistical physics of dynamical phase transitions, we demonstrate that multimodal generation is governed by a spectral hierarchy of interaction timescales rather than simultaneous resolution. A key prediction is the ``synchronization gap'', a temporal window during the reverse generative process where distinct eigenmodes stabilize at different rates, providing a theoretical explanation for common desynchronization artifacts. We derive analytical conditions for speciation and collapse times under both symmetric and anisotropic coupling regimes, establishing strict bounds for coupling strength to avoid unstable symmetry breaking. We show that the coupling strength acts as a spectral filter that enforces a tunable temporal hierarchy on generation. We support these predictions through controlled experiments with diffusion models trained on MNIST datasets and exact score samplers. These results motivate time dependent coupling schedules that target mode specific timescales, offering a potential alternative to ad hoc guidance tuning.

cs.LG↗

arXiv2025

Nil-Equivariant Tropological Sigma Models on Filtered Geometries

We investigate the behavior of tropological (tropical topological) sigma models on higher dimensional target spaces and show that higher dimensional spaces explicitly admit nested Maslov dequantizations which lead to nontrivial anisotropic filtration structures. We provide a classification of all inequivalent tropological sigma models that can be constructed for the case of 4D targets and show that, generically, the corresponding sigma-models are not defined on foliated geometries like in the 2D case but instead are defined on filtered manifolds. We find that the nontrivial filtration structures lead to enhanced global symmetries characterized by noncompact nilpotent Lie algebras given by the 4 dimensional step 3 Engel algebra on the space of fields. We provide a Nilmanifold lattice regularization of the noncompact symmetry group and use this Nilmanifold symmetry to construct a natural equivariant extension of the tropological sigma model. We conjecture that these equivariant tropological sigma models are associated with a new version of GW invariants on filtered manifolds known as \textit{filtered Gromov Witten invariants

hep-th↗

arXiv2025

Notes on the Quantization of Tropological Yang Mills Theory

We continue our investigation into anisotropic topological field theories which arise from a tropical limit of conventional isotropic topological field theories. We analyze both the TBF theory and the tropical analogue of 2D topological Yang-Mills theory (TrYM) through a direct path integral calculation which probes a deformed analytic torsion and also through canonical quantization. The explicit construction of the Hilbert space of TrYM theory demonstrates that the TrYM theory provides an example of a solvable field theory where anisotropy properties and topological invariance can simultaneously hold. We show that the partition function has an asymptotic limit, which verifies that the dimension of the moduli space of tropicalized flat connection on a Riemann surface of genus $g>1$ is precisely given by $(g-1) \operatorname{rank}(\mathfrak{g})$. We show that the interpretation of this result is that the random matrix model associated to the U$(N)$ TrYM is in fact a novel random matrix theory whose integration space is still the same space of hermitian matrices (similar to a GUE) however the Dyson index matches that of a GOE consistent with the usual intuition that tropicalization reduces down complex objects to their real counterparts.

hep-th↗

arXiv2025

Tropical BF Theory and Tropical Limits of TQFTs

We study anisotropic scaling limits of topological field theories using tropical geometry. The resulting topological field theories are characterized by foliated geometries and are invariant under foliation-preserving gauge transformations. We demonstrate the tropicalization for the 2D BF theory and generalize the prescription to topological Yang-Mills and Chern-Simons theories. We call the tropical limit of the BF theory, the \textit{TBF} theory, which is an anisotropic generalization of the BF theory with an additional adjoint-valued field $T$ that enforces a projectability condition onto the leaves of the foliation. The TBF theory localizes onto the moduli space of tropicalized flat connections $\mathcal{M}(\Sigma_g,G)$ on a foliated Riemann surface $\Sigma_g$ of genus $g$. The tropical connections exhibit anisotropic behavior; their holonomy is sensitive only to the leaves of the foliation. We analyze this moduli space two distinct ways, Firstly, they are classified by leaf-wise holonomy whose dimension can be explicitly calculated for the case of tropical projective space $\mathbb{TP}^1$ by the moduli space isomorphism $\mathcal{M}\left(\mathbb{TP} ^1, G\right) \cong \operatorname{Hom}(\mathbb{Z}, G) / G$. The second way is through Kodaira-Spencer theory which gives a twisted cohomology argument to argue that $\operatorname{dim} \mathcal{M}\left(\mathbb{T} P^1, G\right)=\operatorname{rank}(\mathfrak{g})$ and we demonstrate their equivalence for the case of SU$(N)$. We show that we can glue together several $\mathbb{TP}^1$ to obtain $\operatorname{dim} \mathcal{M}\left(\Sigma_g, G\right)=(g-1)\operatorname{rank}(\mathfrak{g})$ for $g \geq 2$ which is precisely $\frac{1}{2}$ of the usual result through an application of a foliated refinement of the Atiyah-Segal axioms. We leave several open questions such as potential connections to JT gravity and anisotropic conformal field theory.

hep-th↗

arXiv2025

Tropical Limits of Dirac Operators

We explore the tropical analog of spinors by representing tropical geometries as foliated Riemann surfaces endowed with degenerate complex structures. We investigate tropical limits of the Laplace-Beltrami operator and explicitly construct its square root, which defines a tropical Dirac operator. We find that the tropical Clifford algebra is classified as a degenerate Clifford algebra with nilpotent generators. The nilpotent generator allows us to work with a new kind of representation that allows for Grassmann odd numbers, effectively supersymmetrizing the tropical spin bundle. We show through Dirac-Bergmann's quantization procedure, that the corresponding tropicalized quantum field theories enjoy a purely fermionic topological symmetry which can be expected to give a new class of path integral localization that we call tropical localization similar to the alternative localization method recently constructed by Choi and Takhtajan. We also discuss how the tropical Dirac operator, when twisted by gauge fields, obeys a tropical version of the Lichnerowicz identity, thereby demonstrating how some elements of Yang-Mills curvature should arise in the tropical limit.

hep-th↗

arXiv2025

A Tropical Look at Coisotropic Branes and Quantization

We continue our investigation of tropical branes by exploring the tropicalization of topological sigma models with boundaries. We show that the tropical limit naturally decomposes conventional A-branes into two distinct classes: tropical Lagrangian branes and tropical coisotropic branes. By carefully analyzing the modified boundary conditions emerging from the tropological sigma models, we construct these tropical branes explicitly and demonstrate their utility in an example where we quantize a symplectic manifold through the use of a tropical version of brane quantization.

hep-th↗

arXiv2025

Analytic Torsion from Chern-Simons theory via the $(2,0)$-theory on Dicyclic Orbifolds of $S^3$

The Witten index of the $(2,0)$-theory compactified on spaces of the form $S^3/\Gamma\times S^2$, with a freely acting group $\Gamma$, and with external string sources implemented via timelike surface operator insertions, is expressed in terms of Ray-Singer torsion of $S^3/\Gamma$ and characters of irreducible representations of $\Gamma$. We compute it explicitly for the Dicyclic groups $\Gamma=\text{Dic}_k$. The torsion and characters are generally irrational numbers, but they nicely combine to an integer index. Alternatively, the Witten index can be computed from Chern-Simons theory on $S^2$, and Ray-Singer torsion on $S^3/\text{Dic}_k$ is thus computable from Chern-Simons theory. The matching of the Witten index calculated by these dual approaches reveals new details about the partition function of the $(2,0)$-theory with surface operators.

hep-th↗

arXiv2024

Tropical Branes

We investigate canonically quantized open string solutions associated to the analytically continued action for the recently proposed tropical limit of topological A-type models, tropological sigma models, with various boundary conditions. These solutions naturally give rise to a non-relativistic counterpart of branes, which we name tropical branes. We provide a preliminary worldsheet description of these tropical branes, laying the groundwork for an upcoming paper that will explore the role of tropical branes in the context of brane quantization.

hep-th↗

arXiv2024

Timelike Kasner singularities and Floquet States in 2+1d AdS/CFT

We consider a model of a holographic 2+1d CFT interacting with an oscillating background gauge field. It is solved by an AdS-Vaidya metric describing Ohmic heating of the boundary field theory. However, we also show that if timelike singularities of Kasner type are permitted then a time independent solution that may be interpreted as a Floquet state of the system can be constructed. In this state the system exhibits either Hall conductivity or kinetic induction, and we numerically evaluate the Kasner exponents for a range of boundary conditions. This model may contribute to the ongoing discussion on the validity and meaning of the Kasner metric in the AdS/CFT correspondence and its application in cosmology.

hep-th↗

arXiv2024

Ground States of Class S Theory on ADE Singularities and dual Chern-Simons theory

In radial quantization, the ground states of a gauge theory on ADE singularities $\mathbb{R}^4/\Gamma$ are characterized by flat connections that are maps from $\Gamma$ to the gauge group. We study Class $\mathcal{S}$ theory of type $\mathfrak{a}_1=\mathfrak{su}(2)$ on a Riemann surface of genus $g>1$, without punctures. The fundamental building block of Class $\mathcal{S}$ theory is the trifundamental Trinion theory - a low energy limit of two M5 branes compactified on the three-punctured Riemann sphere. We show, through the superconformal index, that the supersymmetric Casimir energy of the trifundamental theory imposes a constraint on the set of allowed flat connections, which agrees with the prediction of a duality relating the ground state Hilbert space of Class $\mathcal{S}$ on ADE singularities to the Hilbert space of a certain dual Chern-Simons theory whose gauge group is given by the McKay correspondence. The conjecture is shown to hold for $\Gamma=\mathbb{Z}_k$, agreeing with the previous results of Benini et al. and Alday et al. A non-abelian generalization of this duality is analyzed by considering the example of the dicyclic group $\Gamma=\text{Dic}_2$, corresponding to Chern-Simons gauge group SO$(8)$.

hep-th↗

arXiv2023

Tropological Sigma Models

With the use of mathematical techniques of tropical geometry, it was shown by Mikhalkin some twenty years ago that certain Gromov-Witten invariants associated with topological quantum field theories of pseudoholomorphic maps can be computed by going to the tropical limit of the geometries in question. Here we examine this phenomenon from the physics perspective of topological quantum field theory in the path integral representation, beginning with the case of the topological sigma model before coupling it to topological gravity. We identify the tropicalization of the localization equations, investigate its geometry and symmetries, and study the theory and its observables using the standard cohomological BRST methods. We find that the worldsheet theory exhibits a nonrelativistic structure, similar to theories of the Lifshitz type. Its path-integral formulation does not require a worldsheet complex structure; instead, it is based on a worldsheet foliation structure.

hep-th↗

arXiv2021

MHV amplitudes and BCFW recursion for Yang-Mills theory in the de Sitter static patch

We study the scattering problem in the static patch of de Sitter space, i.e. the problem of field evolution between the past and future horizons of a de Sitter observer. We formulate the problem in terms of off-shell fields in Poincare coordinates. This is especially convenient for conformal theories, where the static patch can be viewed as a flat causal diamond, with one tip at the origin and the other at timelike infinity. As an important example, we consider Yang-Mills theory at tree level. We find that static-patch scattering for Yang-Mills is subject to BCFW-like recursion relations. These can reduce any static-patch amplitude to one with N^{-1}MHV helicity structure, dressed by ordinary Minkowski amplitudes. We derive all the N^{-1}MHV static-patch amplitudes from self-dual Yang-Mills field solutions. Using the recursion relations, we then derive from these an infinite set of MHV amplitudes, with arbitrary number of external legs.

hep-th↗

arXiv2020

Scattering in the static patch of de Sitter space

We study the scattering problem in the static patch of de Sitter space, i.e. the problem of field evolution between the past and future horizons of a de Sitter observer. We calculate the leading-order scattering for a conformally massless scalar with cubic interaction, as both the simplest case and a warmup towards Yang-Mills and gravity. Our strategy is to decompose the static-patch evolution problem into a pair of more symmetric evolution problems in two Poincare patches, sewn together by a spatial inversion. To carry this out explicitly, we end up developing formulas for the momentum-space effect of inversions in flat spacetime. The geometric construction of an electron's 4-momentum and spin vectors from a Dirac spinor turns out to be surprisingly relevant.

hep-th↗