mathematical-physics

Contents Context -Chern-Weil theory Differential cohomology differential cohomology Ingredients Connections on bundles Higher abelian differential cohomology Higher nonabelian differential cohomology Fiber integration Application to gauge theory Contents Idea A Chern-Simons form is a differential form naturally associated to a differential form with values in a Lie algebra : it is the form trivia…

Designing irregular structures often means wrestling with huge systems of equations. Princeton engineers have found a shortcut, using a mathematical bridge between origami and tensegrity to preserve known mechanical properties as a structure shifts into a more complex shape. Tensegrity is a structural principle where a continuous network of tension (cables or strings) and a… The post Princeton re…

While our previous papers on the Axiom of Structural Identity (ASI) and Entropic Dispersion established a robust philosophical and meta-mathematical framework for navigating the set-theoretic multiverse, the strict formalization of these concepts necessitates precise model-theoretic boundaries. The conceptual architecture of the Methodological Principle of Operational Integrity (MPOI) fundamental…

This paper provides a structural analysis of a topological unification framework based on the universal Hopf fibration and the identification of $\mathbb{CP}^{\infty}$ as a physical base space. We establish explicit criteria for what constitutes a physically admissible theory and show that the analyzed construction fails to meet these conditions. The framework replaces dynamical derivation with t…

This paper isolates a common spectral grammar behind three Millennium Problems which, in their classical formulations, appear to belong to different worlds. Each problem is associated with a shadow-symmetric spectral datum: a Hilbert space, an involution exchanging two spectral half-planes, and a fixed self-dual interface. For the Riemann zeta-function the interface is the critical line Re(s) = 1…

We prove that the computational successes of string theory are consequences of two mathematical structures: the four normed division algebras and the axioms of two-dimensional conformal field theory. Eight results are established. (1) The Green-Schwarz critical dimensions D in {3,4,6,10} are classified by normed division algebras via the Hurwitz theorem. (2) The Veneziano amplitude and the celest…

This paper develops a mathematical theory of identity persistence under admissible transformation. It asks what structure is required for a same/not-same judgment across recurrence to be meaningful, non-arbitrary, and non-trivial. The theory defines identity-bearing units, state spaces, admissible transformations, admissible redescriptions, quotient state spaces, continuation relations, invariant…

_Zenodo_. 2026This paper presents the Noological Unification Theorem: a formal proof that all physical laws are the surjective projection of a single minimal algebraic structure, the algebra M₃(ℂ) of 3×3 complex matrices, onto the category of physical laws L_phys. The composition ρ∘Π : M₃(ℂ) → L_phys is shown to be surjective while its inverse is structurally undefined. Surjectivity is establishe…

Let ##A(t)## be an ##m \times m## matrix continuously dependent on ##t \in \mathbb{R},## and let X be the fundamental matrix satisfying $$\dot X = A(t)X, \quad X(0) = E.$$ In the text attached below it is shown that Liouville's formula $$\det X(t) = e^{\int_0^t \mathrm{tr} A(s) ds},$$ is a... Read more

This paper presents an exhaustive historical and mathematical survey of the seventeen Permanent Axioms underlying the first machine-verified Coq formalisation of a global regularity result for the three-dimensional incompressible Navier-Stokes equations on the periodic torus T³. The formalisation establishes subcritical energy estimates for arbitrary smooth initial data in the Sobolev regularity …

On Lie integration of L-infinity algebras to smooth infinity-groups (Lie's third theorem in higher Lie theory): On simplicial principal bundles in descent categories as models for (smooth) principal -bundles: Jesse Wolfson: On Simplicial Principal Bundles in Descent Categories [arXiv:2305.01630] Jesse Wolfson: Descent for -bundles, Advances in Mathematics 288 (2016) 527–575 [doi:10.1016/j.aim.201…

physics, mathematical physics, philosophy of physics theory (physics), model (physics) experiment, measurement, computable physics Axiomatizations Tools Structural phenomena Types of quantum field thories examples The Yang-Mills Mass Gap problem is an open conceptual problem in the quantization of Yang-Mills theory (cf. general spectral gaps of quantum systems), closely related to what in the phe…

Odrzywołek's Exp-Minus-Log operator, eml(x, y) = exp(x) ln(y), is provably a complete basis for the entire class of elementary functions when paired with the constant 1 such that any finite expression built from variables, constants, arithmetic operations, and the exp and ln functions will eventually materialize given enough runtime. This paper strips away even this dependence to study a modified…

This entry provides some broad pointers. For a detailed introduction see geometry of physics – perturbative quantum field theory. algebraic quantum field theory (perturbative, on curved spacetimes, homotopical) quantum mechanical system, quantum probability interacting field quantization Quantum field theory is the general framework for the description of the fundamental processes in physics as u…

On lifting the Witten genus of the heterotic string to topological modular forms: Yuji Tachikawa, Topological modular forms and the absence of a heterotic global anomaly (arXiv:2103.12211) Yuji Tachikawa, Mayuko Yamashita, Topological modular forms and the absence of all heterotic global anomalies, Comm. Math. Phys. 402 (2023) 1585-1620 [arXiv:2108.13542, doi:10.1007/s00220-023-04761-2] and in re…

physics, mathematical physics, philosophy of physics theory (physics), model (physics) experiment, measurement, computable physics Axiomatizations Tools Structural phenomena Types of quantum field thories examples representation, 2-representation, ∞-representation Grothendieck group, lambda-ring, symmetric function, formal group principal bundle, torsor, vector bundle, Atiyah Lie algebroid Eilenb…

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical) quantum mechanical system, quantum probability interacting field quantization In mathematical/theoretical physics, semi-topological or semi-holomorphic D=4 Chern-Simons theory is a variant of ordinary D=3 Chern-Simons theory where, roughly, one of the three coordinate functions is promoted from an ordinary real-value…

physics, mathematical physics, philosophy of physics theory (physics), model (physics) experiment, measurement, computable physics Axiomatizations Tools Structural phenomena Types of quantum field thories examples Models in theoretical physics are often given in the form of differential equations. Informally, integrability is the property of a concrete model which enables one to solve these equat…

On rigorous semi-topological 4d Chern-Simons theory via homotopical AQFT: and its relation to 2d integrable models via -algebras: and generalization to higher dimensions:

physics, mathematical physics, philosophy of physics theory (physics), model (physics) experiment, measurement, computable physics Axiomatizations Tools Structural phenomena Types of quantum field thories examples The special case of super Yang-Mills theory over a spacetime of dimension 4 and with supersymmetry. A speciality of , SYM is that its moduli space of vacua has two “branches” called the…