Papers

These notes are aimed at providing a relatively concise and self-contained introduction to anomalies in quantum field theory. A first part reviews the necessary notions of quantum field theory, in particular its functional integral formulation which we use throughout. Then we discuss in some detail how and why anomalies arise as non-invariance of the functional integral measure or through the absence of a regulator preserving all symmetries, and how they manifest themselves as current non-conservation in anomalous loop diagrams and anomalous Ward identities, or equivalently as a non-invariance of the effective action. Specifically, we deal with the so-called abelian anomaly under chiral transformations, and the so-called chiral anomaly which is a violation of non-abelian gauge invariance in the presence of chiral fermions. We perform a detailed triangle Feynman diagram computation for both cases. Then we move on to provide a brief overview of some important properties of anomalies (locality, finiteness, one-loop exactness), and how they are characterized in general by the Wess-Zumino consistency condition. We show how this condition can be reformulated in terms of BRST cohomology and how it can be solved, for arbitrary (even) space-time dimensions, by the descent equations. Finally, we briefly discuss the general relation between anomalies in arbitrary even dimensions 2n and index theorems in 2n+2 dimensions, generalizing also to gravitational anomalies. We conclude with the exhibition of the highly non-trivial anomaly cancellation in ten-dimensional IIB supergravity.

In genuinely rotating holographic backgrounds, Wilson-loop observables are usually constructed for a quark–antiquark pair static with respect to the rotating plasma. We investigate how this choice affects the heavy-quark sector by comparing two descriptions of the same rotating geometry: a corotating frame, in which the pair is static with respect to the plasma, and a static frame, in which it is static with respect to the static observer. We consider an equal-rotation Myers–Perry black hole in asymptotically global AdS and study Wilson loops oriented longitudinally and transversely to the rotation. We analyze the effective string tension, physical quark–antiquark separation, and heavy-quark potential at fixed thermodynamic temperature measured in the corresponding frame. We find a strong frame dependence of the rotational response. In the corotating frame, the longitudinal and transverse configurations respond qualitatively differently: rotation can either decrease or increase the separation and effective string tension, depending on orientation and, for the latter, on the bulk position and parameter range. The heavy-quark potential increases with angular velocity in both orientations. In the static frame, by contrast, the effective tension, physical separation, and maximal separation decrease with angular velocity in both orientations, while the heavy-quark potential increases. Thus, the anisotropic response is substantially stronger in the corotating frame, both qualitatively and quantitatively. In both frames, rotational modifications generally become less pronounced at higher temperature. These results show that the frame in which the quark pair is static is an essential part of the physical specification of Wilson-loop observables in rotating holographic systems.

A quantum field-theoretical description of neutrino oscillations in matter in adiabatic approximation is developed. Estimates of the oscillation coherence lengths have been made for several solar neutrino production and detection processes. It has been shown that, within the framework of the developed approach, the results of solar neutrino detection experiments can be correctly described in terms of neutrino mass eigenstates without taking oscillations into account.

Key properties of a large sparse operator can often be determined by only a small, spectrally localized subset of its eigenmodes. In lattice QCD with Wilson fermions, the sign of the fermion determinant is determined by the parity of the number of eigenvalues lying on the negative real axis of the non-Hermitian Wilson-Dirac operator. Determination of this signature can be challenging, particularly in parameter regimes where near-zero modes can lead to exceptional configurations. We present a contour-integral-based spectral projection method to count the negative-real eigenvalues of the non-Hermitian Wilson-Dirac operator $D$ that determine the sign of its determinant. The spectral projection is supplemented by an adapted deflation-accelerated solver for shifted linear systems and singular-value exclusions to reliably set spectral bounds and contour boundaries. We demonstrate the robustness of the method through systematic convergence tests, including variations of the contour size, and show its ability to resolve eigenvalues close to the negative real axis. Beyond fermion determinant sign evaluation, the approach provides a systematic means of isolating physically relevant eigenvalues embedded in dense complex spectra, with potential applications to non-Hermitian quantum systems, stability analyses, and large-scale eigenvalue problems.

We construct a holographic open quantum system by coupling two CFTs through a double trace deformation and tracing out the bath degrees of freedom within the Schwinger-Keldysh formalism. This procedure yields an effective description of the nonunitary reduced dynamics of the remaining sector, whose linear response is encoded in its retarded Green's function. Working perturbatively in the double trace coupling, we derive closed form expressions for the quasinormal modes of several coupled geometries involving pure AdS and rotating or nonrotating BTZ black holes, considering both possible system/bath assignments. For the nonrotating configurations previously investigated in arXiv:2509.14312, our analytic expressions agree with the numerical results, while our treatment extends the analysis to configurations involving rotating BTZ black holes. Finally, moving beyond perturbation theory, we derive a relation between dissipation and transmission in the general AdS-BTZ setup.

We study the conservative and dissipative forces on a Newtonian binary interacting with a nonrelativistic ultralight scalar medium. We derive integral expressions for the instantaneous force and the orbit-averaged energy flux generated by wake perturbations of a general background field. For a homogeneous coherent background, the conservative force suffers from a well-known infrared divergence. We trace this divergence to the failure of perturbation theory about a constant scalar state and identify the gravitational Bohr scale at which the homogeneous approximation breaks down. In the long-wavelength regime, the leading dissipation into the medium is quadrupolar, and we obtain an explicit expression for the leading energy flux from an eccentric binary. In the short-wavelength regime, we identify a local ”hard” region in momentum space that produces a universal Coulomb logarithm, while the nonlogarithmic contribution remains orbit dependent. We then promote the background to a stochastic ensemble with a general velocity distribution. Three independent length scales, together with a derived geometric-mean scale, produce eight distinct scale hierarchies, for which we characterize the ensemble-mean response. Finally, we derive the two-point correlation function of the orbit-averaged energy flux and evaluate it in ”coherent-response” regimes. Wave interference produces order-unity density fluctuations, so the flux in an individual realization can differ substantially from its ensemble mean.

In this paper, we study color kinematics duality and related double-copy and BCJ structures in the cosmological Grassmannian framework. The simplicity of these correlators in this framework makes it possible to test the double copy construction beyond flat space. We show that the four-point gluon correlator admits a representation whose kinematic numerators satisfy an algebraically exact Jacobi identity, $n_s$ + $n_t$ + $n_u$ = 0. Replacing color factors with kinematic numerators gives a graviton correlator that reproduces the correct factorization residues and flat-space limit, which matches with the BCFW result up to terms regular in all exchange channels. In addition, we construct a differential bi-adjoint scalar matrix whose null vector gives rise to a differential BCJ relation among color-ordered gluon correlators. We further explore an s-copy construction for general integer-spin correlators and show that for odd spin the corresponding numerators satisfy a Jacobi identity. Finally, we develop a differential representation for scalar four-point correlators with gluon exchange. These results establish algebraic and differential aspects of graviton and gluon correlators in curved spacetime.

The cancellation of renormalon ambiguities in the operator product expansion is usually formulated as a matching between perturbative Borel ambiguities and prescription dependence in non-perturbative condensates. We give this cancellation a contour realization in the two-dimensional Gross–Neveu model at large $N$. The exact order-$1/N$ fermion self-energy in the massive vacuum is a finite real integral; ambiguities arise only after its large-momentum behavior is decomposed into products of perturbative Wilson coefficients and the matrix elements of the associated operators. A reduced integral isolates the leading renormalon and reveals a two-dimensional contour geometry in which the perturbative series is generated at the boundary corner where the exchanged momentum reaches the external hard scale, while its Stokes discontinuity is controlled by a distinct boundary critical point at zero exchanged momentum. After renormalizing the ultraviolet divergences of the condensate scale integral, we derive its compensating lateral ambiguity without fixing it by matching to the perturbative sector, distinguish the renormalon saddle from the apparent IR Landau pole, and obtain the reduced contour variables from the large-$N$ auxiliary-field path integral. The renormalon saddle is a boundary critical point of this reduced mode-space integral. The perturbative Wilson coefficient and the renormalized condensate thus define complementary relative cycles of the same reduced integrand whose complex tails cancel, reconstructing the original real integration cycle.

When spatial sections of spacetime are topologically nontrivial, the symmetry arguments usually used to choose the Bunch-Davies vacuum are no longer guaranteed to apply. We study this problem for a scalar field on de Sitter spacetime with toroidal spatial sections. The Hadamard conditions, which ensure the same short-distance structure as in flat space, are local and naturally insensitive to the compactification. By contrast, we find that the infrared sector is affected by the toroidal identifications that break global de Sitter invariance, thereby removing the usual argument for discarding zero-mode contributions. We treat the remaining zero-mode freedom as an effective topology-induced source and analyze its effects on an inflationary de Sitter expansion within a semiclassical setting. We show that, for a massless scalar, the zero-mode sector can contain a contribution to the stress-energy tensor with equation of state $P=-\rho$, behaving as a cosmological constant. Thus, the source driving an early-Universe de Sitter phase need not be determined solely by local terms in the action, but can also depend on the infrared structure of the quantum state. Nontrivial topology, as a concrete realization of de Sitter symmetry violation, turns the otherwise-unfixed infrared freedom in vacuum selection to a contribution directly relevant to the semiclassical expansion. This contribution could potentially be applicable to the current accelerated expansion of the Universe.

We study non-stabilizerness (magic) in the ground state of $(2+1)$-dimensional $\mathrm{SU}(2)$ Hamiltonian lattice gauge theory with matter, formulated in the dressed-site basis in the hardcore-gluon truncation and restricted to the zero baryon-number sector. Using matrix product states, we compute three facets of magic: the second-order stabilizer Rényi entropy (SRE) $M_2$, its non-local component $M_2^{\rm NL}$, and a lower bound in terms of the anti-flatness $F$ of the entanglement spectrum. We also prove a stronger form of the sandwich relation: $-\log_2(1-4F)\le M_2^{\rm NL}\le M_2$; the lower bound rests on a stronger inequality that we obtain for arbitrary Schmidt bases and rank, thus resolving the open problem of finding the maximal lower bound. We emphasize a structural distinction that makes the non-local quantities the physically preferred diagnostics: whereas the full SRE depends on the (non-unique) encoding of the gauge-invariant local Hilbert space into qubits, both the non-local magic and the anti-flatness are invariant under site-local re-encodings and are therefore intrinsic to the state and bipartition. By varying the gauge coupling on lattices up to $6\times 6$ with bond dimension up to $128$, we find that the non-local magic furnishes a sharper and more bond-dimension-friendly probe of the gauge-matter delocalization crossover compared to the full SRE or the gauge-invariant entanglement entropy, retaining a clear signal at bond dimensions well below those needed to converge the ground state itself.

The ${L}_\Lambda w_{1+\infty}$ algebra is the cosmological-constant deformation of the celestial $Lw_{1+\infty}$ symmetry algebra of gravity, providing an infinite-dimensional extension of the three-dimensional conformal algebra at the AdS$_4$ boundary. We construct the Hamiltonian charges associated with these symmetries from boundary ambitwistor-space data, using LeBrun's Heaven on Earth correspondence, and translate them to spacetime via the Penrose transform. We relate the resulting expressions to light-ray operators in the boundary CFT$_3$, including ANEC operators, their conformal descendants, and their commutators.

We derive thermal Polyakov bootstrap equations for CFT two-point functions on $S^1_\beta\times\mathbb R^{d-1}$. Combining the thermal OPE with a dispersion relation and the method of images gives a representation in blocks that satisfy Kubo-Martin-Schwinger (KMS) covariance. Their local expansions contain additional thermal blocks at classical double-twist dimensions. Matching the prescribed physical OPE requires cancellation of these generated contributions, yielding explicit linear equations for thermal OPE coefficients, including any compensating terms required by the reconstruction. We formulate the equations for bosonic scalar correlators and antiperiodic scalar structures extracted from fermion correlators, and find consistent solutions in analytically tractable thermal CFTs. The coefficient matrix, which we call the Polyakov matrix, connects the large-spin expansion to KMS consistency and reduces to minus the identity on the classical double-twist spectrum. This structure suggests a numerical approach to the three-dimensional Ising CFT that combines known low-lying vacuum data with a self-consistent large-spin tail.

We find the shape of the modes containing most of the entanglement between two disjoint spherical regions in the vacuum of a free massless real scalar field in 1+3 Minkowski spacetime. Using the Klco-Beck-Savage method we approximate the field and momentum profile of the vacuum most entangled modes (MEMs) between two spheres. We define a non-perturbative mode-swap entanglement harvesting protocol where two spacelike separated probes extract the maximal amount of entanglement between two regions. Our results are a first step in the direction of understanding the local entanglement structure of quantum fields, pinpointing the exact degrees of freedom that encode accessible quantum correlations in QFT.

Given any four-point scattering amplitude, the partial wave expansion resolves its residues into contributions from individual spin-$s$ internal particles, where each coefficient in the expansion must be non-negative due to unitarity. In this paper, we introduce a partial wave expansion tailored to "half-ladder" Feynman diagrams at generic multiplicity $n$ and spacetime dimension $d$, with scalars on the external lines and massive bosons on the internal lines. Derived by enforcing that the half-ladder residue is an eigenvector of each internal particle's angular momentum, each basis element is written as a sum over products of special functions related to the Gegenbauer polynomials, and we give a set of simple graphical rules for explicitly building any of them at generic $n$ and $d$. To put the formalism into action, we apply our technology to deformed "string theory" amplitudes, defined a priori in any dimension and with arbitrary Regge intercept $\alpha_0$. By constructing multi-particle positivity tests for the half-ladder and "twisted" half-ladder diagrams, we show that the only amplitudes consistent with unitarity are those belonging to open bosonic string theory ($\alpha_0 = -1$) below the critical dimension $d \leq 26$. In particular, we confirm the expectation that $Z$-theory ($\alpha_0 = 0$) fails to define a unitary string theory at finite $\alpha'$. Finally, for the open bosonic string, we show how knowledge of the partial wave expansion allows us to resolve the contributions to the half-ladder diagram from individual degenerate states at a given level and spin in a purely on-shell analysis. As an ancillary file, we include a Python script combwaves.py which implements the partial wave expansion given a generic (twisted) half-ladder residue.

We study $O(4)$ symmetric Euclidean wormholes in four-dimensional gravity with a massless scalar linearly coupled to the Gauss-Bonnet invariant. At nonzero coupling, the Einstein wormhole becomes a regular complex saddle that satisfies the Kontsevich-Segal-Witten criterion until its branch ends at a finite coupling. In asymptotically flat space, the Gauss-Bonnet term modifies the scalar displacement and the on-shell action for finite wormholes, but the size dependent parts of both corrections vanish in the large throat limit. The original imaginary distance threshold therefore remains unchanged. A negative cosmological constant already lowers the Einstein large throat imaginary distance relative to its flat space value. When the Gauss-Bonnet coupling is also present, the scalar is no longer a modulus as it develops a logarithmic running and its trajectory has infinite length. Subtracting the logarithmic asymptotic behavior leaves a finite regulated displacement. Its large throat value is shifted upward from the Einstein AdS result by the Gauss-Bonnet interaction and determines the leading large charge behavior of the renormalized fixed charge action.

We investigate topological defects produced by a chiral phase transition in a fermionic ultraviolet completion of dynamical Chern-Simons (dCS) gravity. The model contains a complex scalar field with a chiral Yukawa coupling to fermions. Below the chiral symmetry breaking scale, integrating out the gapped bulk modes produces a gravitational Chern-Simons coupling which, we argue, modifies cosmic string solutions from the phase transition. The resulting corrected string metric is supported by a generalization of the Callan-Harvey fermionic zero mode solution, in which the null current generated by these zero modes provides a mechanism for parity violation. In the weak-coupling and weak-gravity regime, we derive the leading order-reduced dCS correction to this exterior metric. We also discuss the effective action description for generic string configurations in dCS gravity. The gravitational Pontryagin term provides the anomaly inflow required to cancel the gravitational anomaly of the chiral worldsheet theory.

Entropy has emerged as a key concept at the intersection of gravity, quantum theory, and cosmology, revealing deep connections between geometry, information, and thermodynamics. The realization that black holes are thermodynamic systems, characterized by an entropy proportional to their horizon area, has provided evidence that spacetime itself may possess a microscopic structure. Since then, entropy has acquired a central role across quantum field theory, holography, quantum gravity, and cosmology. In this review, we present a unified account of entropy in gravitational and cosmological contexts. We first survey entropy in gravitational systems, focusing on black-hole thermodynamics, entropy bounds, the holographic principle, entanglement entropy, and the area law, together with the main corrections and generalizations arising from quantum field theory, generalized uncertainty principles, modified statistics, and extended theories of gravity. We then discuss three conceptually distinct ways in which entropy is employed in cosmology. The first is through the spacetime-thermodynamics conjecture, in which gravitational dynamics emerge from thermodynamic relations applied to cosmological horizons, and generalized entropy forms consequently lead to modified cosmological evolution. The second is the holographic approach, in which entropy bounds constrain the vacuum energy of the Universe, giving rise to the holographic dark energy scenario and its various extensions. The third is entropic gravity, where gravity is interpreted as an emergent phenomenon of entropic origin, leading to interesting cosmological phenomenology. By reviewing the theoretical foundations and cosmological implications of these entropic frameworks, we aim to clarify the role of entropy, discuss open issues and outline directions toward a deeper thermodynamic and informational understanding of gravity and cosmology.

The IIB system is a system of PDEs for an SU(3)–structure and a positive function on a 6-manifold. Its solutions describe conformally balanced pluriclosed Hemitian metrics on complex 3-folds with holomorphically trivial canonical line bundle. In particular, solutions of the IIB system are steady solitons for the generalized Ricci-flow and Bismut–Hermitian–Einstein metrics. In this paper we study cohomogeneity one solutions of the IIB system. We construct 1-parameter families (up to scaling symmetries) of complete non-compact non-Kähler solutions on the smoothing of the conifold with controlled geometry at infinity. The generic member of the family is asymptotically conical with tangent cone at infinity the Calabi–Yau cone metric on the conifold. As a limit of the family we recover an explicit solutions known in the physics literature as the Chamseddine–Volkov/Maldacena–Nuñez solution, which has an exotic asymptotic geometry. We show that there are no complete cohomogeneity one solutions on crepant resolutions of the conifold (or its quotient), but we also establish the existence of an analogous 1-parameter family of forward complete solutions defined on exterior domains of the conifold with prescribed incomplete behaviour along the interior boundary and similar asymptotic behaviour at infinity. As a byproduct of these existence results, we produce infinitely many complete non-compact non-Kähler Bismut–Hermitian–Einstein metrics in complex dimension 3 with full holonomy of the Bismut connection, in contrast to the holonomy reduction forced upon compact examples. We also find infinitely many such complete non-compact full-holonomy non-Kähler examples (with at least two ends) in complex dimension 2 by revisiting a construction due to Callan–Harvey–Strominger in the physics literature.

This is an English translation of a book originally published in Japanese by Saiensu-sha in 2015. The book is an introduction to the 3d-3d correspondence, the relation between 3d $\mathcal{N}=2$ supersymmetric gauge theories and the geometry of 3-manifolds, which arises from the compactification of the 6d $\mathcal{N}=(2,0)$ theory. The presentation is bottom-up rather than top-down: instead of starting from the 6d theory, we begin by asking what a quantum field theory is, and let the geometry emerge on its own. Along the way we discuss renormalization and low-energy effective theories, gauging as an operation which glues field theories together, the resulting $Sp(2n, \mathbb{Z})$-action on 3d theories, 3d $\mathcal{N}=2$ supersymmetric theories and their dualities, the squashed three-sphere partition function and supersymmetric localization, duality domain walls of 4d $\mathcal{N}=2$ theories, quantum Teichmuller theory, complex Chern-Simons theory and the quantization of the moduli space of flat $SL(2, \mathbb{C})$-connections on 3-manifolds, with quantum dilogarithm functions as a recurring thread. The emphasis throughout is on geometric and algebraic structures which are invisible in a single field theory, and appear only in the theory space of quantum field theories. Appendices summarize supersymmetry in various dimensions, classical and quantum dilogarithm functions, and cluster algebras. Exercises with difficulty ratings are included in each chapter.

Gravitational-wave (GW) observations offer a promising probe of new physics associated with a strong first-order electroweak phase transition. Null results from direct searches and constraints from precision measurements, together with the comparatively weak constraints on Higgs self-interactions, motivate scenarios in which non-decoupling effects in the Higgs potential coexist with additional decoupling new physics. Previous Fisher matrix studies within the Standard Model effective field theory (SMEFT) have explored the sensitivity of future GW observations to new physics, focusing on transitions driven by the dimension-six operator $(H^\dagger H)^3$. However, the validity of the truncated EFT expansion can become questionable in the parameter regions relevant to such transitions. We instead consider transitions driven by non-decoupling effects described by the nearly aligned Higgs effective field theory (naHEFT). We investigate whether GW observations can probe small corrections to the Higgs effective potential induced by decoupling effects in interactions other than Higgs self-interactions. As a representative benchmark for additional decoupling effects, we consider the dimension-six SMEFT operator $(H^\dagger H)(\overline{q}_3 t_R\widetilde{H})$ with Wilson coefficient $C_{uH}$. Although we focus on this benchmark, the same framework can be applied to other dimension-six SMEFT operators. We perform two-parameter Fisher matrix analyses for DECIGO and BBO to quantify their expected sensitivity to $C_{uH}$ and the naHEFT mass scale $\Lambda$. For the benchmark configurations considered, the non-decoupling dynamics generates a detectable GW signal. We find that these small corrections can produce potentially measurable changes in the GW spectrum, even after accounting for the correlation between $C_{uH}$ and $\Lambda$.

We derive an analytic black-hole family in three-dimensional scalar–tensor Einstein–Gauss–Bonnet gravity with positive coupling. A single implicit equation determines the static circular metric and a scalar linear in time. We show that these solutions exhaust regular nonextremal exteriors with the stated AdS boundary conditions and a fixed nonzero coefficient of time in the scalar. The coupled metric and scalar perturbations have one propagating degree of freedom. On one branch, its kinetic coefficient is positive, its equation is hyperbolic throughout the exterior, and its bulk spatial energy is positive for perturbations of compact support. Scalar signals can cross the metric horizon outward, so exterior evolution needs information from the interior. For linear perturbations with the original metric and scalar boundary values fixed, nonzero compact initial master displacements with zero velocity can evolve only until their first contact with the AdS boundary. We derive this restriction from the original metric and scalar equations.

We compare the superconformal index of three-dimensional ADHM theory in fixed monopole sectors with the index of the BMN matrix model around fuzzy-sphere vacua. The monopole charges determine the sizes of the fuzzy-sphere blocks, and their multiplicities determine how many identical blocks occur. At a fixed number of fundamental hypermultiplets, we increase all positive monopole charges by the same amount while keeping their differences and multiplicities fixed. After dividing the ADHM index by the contribution of the bare monopole, it agrees order by order with the BMN index in the corresponding limit of large blocks. Fundamental excitations move to arbitrarily high orders in the index expansion, while the upper angular-momentum cutoffs of the BMN harmonics diverge, leaving the same vector and adjoint contribution.

We prove the Kazhdan-Lusztig correspondence for the Lie superalgebra $\mathfrak{osp}_{1|2n}$ at irrational levels, that is, we show the category $\mathrm{KL}_k^{\rm ev}(\mathfrak{osp}_{1|2n})$ of finite-length even ordinary modules for the affine vertex operator superalgebra of $\mathfrak{osp}_{1|2n}$ at level $k \in \mathbb{C} \setminus \mathbb{Q}$ is braided tensor equivalent to the category of finite-dimensional even weight modules for the quantum group of $\mathfrak{osp}_{1|2n}$ at parameter $q = e^{\pi i/(2k+2n+1)}$. We also prove that ${\rm KL}_k^{\rm ev}(\mathfrak{osp}_{1|2n})$ is braided tensor equivalent to the category ${\rm KL}_\ell^{\rm ns}(\mathfrak{so}_{2n+1})$ of finite-length ordinary modules with non-spinorial top level for the affine vertex operator algebra of $\mathfrak{so}_{2n+1}$ at level $\ell$ such that $ \frac{1}{\ell+ 2n-1} = \frac{1}{2k+2n+1} + 1 \ \ ({\rm mod}\ 2\mathbb Z).$ Consequently, by gluing vertex operator (super)algebras via tensor categories, we construct a few new families of simple conformal vertex (super)algebras, including the mixed kernel VOAs that were the missing ingredient for proving certain Feigin-Frenkel type dualities in previous work of the first-named author with Linshaw, Nakatsuka, and Sato.

In a recent paper arxiv.org/abs/2609.21031, we have introduced the concept of a relational background field as a new approach to the renormalization group (RG) for discrete quantum gravity models, focusing on random matrix models. The proposed method offers the dual advantage of preserving the model's gauge symmetry (typically $U(N)$ or $O(N)$), whose breaking in conventional approaches has remained an open problem, and providing a more satisfactory treatment of the continuum or infrared (IR) limit. In this limit, the theory is described by a non-local field theory in dimension $D = 2k + 3$, where $k$ denotes the number of cuts in the effective spectrum as dictated by random matrix theory. This pedagogical paper reviews this construction for real random vector models in zero dimensions, with $O(N)$ as the symmetry group. Following the general strategy of background-field-type formulations, we show, using the BBP (Baik-Ben Arrous-P'ech'e) phase transition, that a partial Hubbard–Stratonovich decomposition of the quartic interaction, involving a matrix-valued intermediate field, allows a preferred notion of scale to emerge. This scale, in turn, defines an RG flow, whose perturbative structure is analyzed. Our calculation demonstrates the existence of an asymptotic Wilson–Fisher-type fixed point, whose sole relevant critical exponent is in qualitative agreement with the double-scaling critical exponent of these models.

Nature, at the shortest distances accessible to contemporary colliders, is described by the Standard Model of particle physics with remarkable precision. Since the discovery of the Higgs boson in 2012, progress has shifted from finding new resonances to precision studies of established theory. Here, quantum chromodynamics (QCD), describing the interactions of quarks and gluons, plays a central role. Collider observables involving hadrons are made theoretically accessible by factorization, which separates universal long-distance hadron dynamics encoded in parton distributions from process-specific short-distance coefficients calculable in perturbation theory. This thesis develops analytic methods for perturbative QCD with applications to collider phenomenology and proton structure. A major part is devoted to angular integrals, key building blocks of phase-space integrals. Methods originally developed for Feynman integrals, including integration-by-parts identities, expansion by regions, differential equations, and dimensional shifts, are adapted to angular integrals and yield several new results. The analytic structure of one-loop integrals is also studied, with spurious branch cuts removed using single-valued polylogarithms, leading to a compact representation of coefficient functions for semi-inclusive deep-inelastic scattering. A systematic expansion in higher powers of transverse momentum is developed and applied to the Drell-Yan process, refining the understanding of its factorization structure. Finally, a new semi-analytical ansatz for the evolution of parton distributions is explored. Together, these methods improve analytic control over radiative corrections, kinematic dependence, and scale evolution in processes relevant to the Electron-Ion Collider and other collider experiments.

Supercooled phase transitions can generate observable gravitational waves, but their bubble dynamics may also imprint large spatial inhomogeneities in the baryon asymmetry. For phase transitions below the TeV scale, ordinary baryon diffusion can be too slow to erase such inhomogeneities before Big Bang Nucleosynthesis, potentially leading to stringent constraints from the observed light-element abundances. We show that this problem can be naturally avoided in darkogenesis scenarios, where the phase transition first generates an asymmetry in a dark-sector particle that is subsequently transferred to visible baryons. The dark-sector particle can travel over much larger distances than ordinary baryons before decaying, erasing the inhomogeneity before the baryon asymmetry is established. We derive the conditions for efficient erasure in both the diffusive and collisionless transport regimes and translate them into constraints on the dark-sector interactions. For a representative dark phase transition with a reheating temperature of $\mathcal{O}(1) \,\mathrm{GeV}$, $\beta/H \simeq 10$, and a dark-sector particle lifetime of $0.1\,\mathrm{s}$, we find that the inhomogeneity is efficiently erased for a viable range of the interactions. Importantly, baryogenesis before the phase transition is challenged both by dilution from strong supercooling and by the subsequent imprinting of spatial inhomogeneities. This points toward darkogenesis associated with the phase transition as a consistent framework, in which dark-sector transport erases the inhomogeneity before it is transferred to visible baryons.

We study the topological non-Abelian string and its classical stability in models with two Higgs fields to achieve the symmetry breaking of ${\rm SU}(N) \otimes {\rm U}(1)_X\to {\rm SU}(N-1) \otimes {\rm U}(1)_{X^\prime}$. The topological origin is due to the global $\widetilde {\rm U}(1)$ symmetry and its breaking in the scalar potential. The most generic winding configurations of arbitrary integers of $(n_1\,, n_2)$ are considered. The stability of the topological string is analyzed based on the time-dependent perturbations to the string background and numerical solutions to the coupled Helmholtz equations of the perturbed Fourier modes. We present the constraints on the stable regions for the $(n_1\,, 0)$ configuration (or equivalently the $(0\,, n_2)$ configuration). For the general winding configurations of $(n_1\neq 0\,, n_2\neq 0)$, we point out the stable regions only exist in the semilocal limit of the large mixing angle of $\vartheta_N\to \frac{\pi}{2}$.

We study the quantized six-vertex (Q6V) model on arbitrary admissible diagrams at $q=\varepsilon$, where $\varepsilon$ is a primitive $N$th root of unity with $N$ odd. For the commuting layer transfer matrices $\mathbb{T}(y)$ with mixed boundary conditions, we establish the quantum Frobenius property \[ \mathbb{T}(y)\mathbb{T}(\varepsilon y)\cdots \mathbb{T}(\varepsilon^{N-1}y) = \mathscr{P}_N(y^N)\mathbb{I}, \] where $\mathscr{P}_N$ is a scalar Laurent polynomial obtained directly from the monomial expansion of $\mathbb{T}(y)$. Our proof exploits the underlying three-dimensional integrability, in particular the tetrahedron equations and their consequences. The quantum Frobenius relation essentially determines the joint eigenvalues of the associated Hamiltonians up to multiplicities and shows that they take a free-parafermion form, thereby extending such structures from previously known one-dimensional examples to a broad class of genuinely two-dimensional quantized six-vertex models. The construction also admits independent local gauge choices and recovers the long-range-interacting $\tau_{\; 2}$ model, which includes the original free-parafermion model as a special case. We also apply the same framework to the relativistic quantum Toda chain at odd roots of unity and obtain the free-parafermion spectra for the relevant Q6V transfer matrices with fixed boundary states.

Universal information extracted from a critical quantum state depends on how the state is probed. Spatial entanglement, participation statistics, and stabilizer Rényi entropy correspond to different replica geometries and need not isolate the same boundary data. We show that the complete distribution of Pauli magnitudes in critical Ising chains carries an exact geometric fingerprint of boundary termination, distinguishing lattice realizations that flow to the same infrared boundary fixed point. This structure is organized by discrete Selberg ensembles associated with the classical root systems $A$, $B$, $C$, and $D$: three open chains sharing the free–free Ising boundary condition retain distinct $B_L$, $C_L$, and $D_L$ trigonometric wall geometries. The standard symmetry-breaking representatives instead produce rectangular, pinned, and character-inserted $C$ ensembles. Their four boundary moments contain exactly two independent multiplicative combinations invariant under local endpoint normalizations. One of these reduces exactly at finite rank to a moment of the fundamental symplectic character and provides a lattice target for a defect–measurement-boundary junction amplitude. A conditional Gaussian-character law predicts its values throughout the unlocked phase, while exact-arithmetic finite-rank results at the $\alpha=4$ transition depart sharply from the Gaussian continuation. These results show that the full Pauli spectrum resolves boundary information invisible to the infrared boundary fixed point alone and provide concrete targets for replicated boundary conformal field theory.

We revisit the classic 1959 paper of L. D. Landau on possible singularities of the integral associated to a Feynman graph. Focusing on the real setup, we relate it to convex geometry by representing a collection $p$ of incoming momenta as the weighted normals of a convex polytope $Q$ via the Minkowski problem. Then, a regular polyhedral subdivision $\mathcal{P}$ of $Q$ exhibits $p$ as a Landau singularity labelled by the dual graph of $\mathcal{P}$ with masses being the areas of the faces. Positivity of the Landau/Feynman/Schwinger multipliers is interpreted as strict convexity of a PL-function. This gives an interesting class of “polyhedral” Landau singularities. In the planar case going back to the original 1959 paper, Landau graphs can also be identified with plane webs of Gaiotto-Moore-Witten that provide a language dual to that of regular polygonal subdivisions.