Papers

The $F$-symbols of an anyon theory encode the associativity of fusion and constitute some of the theory's most fundamental and, simultaneously, subtle data. Much of the subtlety lies in the gauge-dependence of the $F$-symbols. Despite their generic complexity, many braided anyon theories admit gauges in which all $F$-symbols are real, a phenomenon for which no general organising principle has been known. We identify a physical mechanism underlying this reality. For a unitary ribbon fusion category admitting an appropriate braided charge-conjugation symmetry, we show that the complex-conjugated $F$-symbols are related to the original ones by a gauge transformation. Finding a real gauge is thereby reduced to a condition on these transformations. When the charge-conjugation symmetry is suitably "flat,” or equivalently when the associated "twisted” Frobenius-Schur (or "generalized” Kawanaka-Matsuyama) data respects a grading, these local transformations can be trivialised and a real gauge exists. This framework unifies a broad range of previously disparate examples, including families of Chern-Simons theories whose $F$-symbols are difficult to directly compute. Finally, we exhibit a unitary ribbon category that realizes a novel obstruction to the existence of a real gauge and therefore has inherently complex $F$-symbols.

We study the one-sided time evolution of thermofield double (TFD) states in random matrix theory, where the Hamiltonian is taken to be a $D\times D$ random matrix drawn from a unitarily invariant emsemble of Hermitian matrices. We argue that the Krylov basis with the maximally entangled state taken as the initial vector gives a semiclassical Hilbert space description of these TFD states in random matrix theory at any $O(1)$ temperature and time in the $D\to \infty$ limit, very analogous to the “chord Hilbert space” construction in the double-scaled SYK (DSSYK) model. We study this semiclassical description in detail for ensembles where the spectral density in the $D\to \infty$ limit is even, compactly supported on an interval and has square root edges. With a few more conditions on the analytic structure of the spectral density, we observe that the semiclassical Hamiltonian has the same asymptotic behavior at large Krylov depth as that of DSSYK, with the corresponding parameter $\mathfrak{q}=e^{-\lambda}$ being related to the location of the nearest zero of the spectral density away from the spectral cut. Furthermore, in a large class of models corresponding to ultraviolet deformations of the DSSYK spectral density, i.e., where the spectrum in the UV is modified while leaving the near-edge behavior unchanged, we show that the semiclassical effective Hamiltonian in the Krylov basis reduces to the Liouville Hamiltonian of JT gravity in a low-energy, continuum limit. This suggests that our semiclassical Hilbert space should be interpreted as the bulk Hilbert space of a dual gravity description.

This comment gives dimension-independent analytic proofs of the local inequalities arising in the odd- and even-dimensional constructions of higher-dimensional partition charge functions. The prior works established these inequalities by exhaustive computation in low dimensions and tested them numerically in selected higher dimensions.

In the Eulerian walker model (also known as rotor walk), each site of the square lattice begins with an arrow pointing to one of its four neighbors. A walker that starts at the origin repeatedly turns the arrow at its current site clockwise by $90^\circ$ and steps in the new direction. Priezzhev, Dhar, Dhar, and Krishnamurthy (1996) introduced this as a model of self-organized criticality and conjectured that, for independent uniform initial directions, the region explored in the first $t$ steps has radius of order $t^{1/3}$. We establish this conjecture and further show that the walker visits every lattice site infinitely often, and that the region it has visited by time $t$, rescaled by $t^{1/3}$, converges to a convex body.

We study a one-dimensional scalar model for an effective-mass quasiparticle field coupled to a continuum of two-level atoms with at most one excitation present. We prove a general theorem on the existence of bound states under sign-definite spatially localized perturbations of constant atomic density and analyze the bound states for several exactly solvable examples.

Conventional full counting statistics assigns commuting phases to integrated stochastic currents and therefore resolves net transport but not, in general, the temporal ordering of events associated with different cycles. We introduce a non-Abelian counting construction in which crossings of two fundamental cycles rotate an auxiliary spin about different axes. The resulting ordered trajectory statistic has an exact finite-dimensional evolution equation for its first moment. The two rotation angles form a counting torus, and whenever the mean spin is nonzero its normalized direction defines a map $\mathbb T^2\to\mathbb S^2$, equivalently a complex eigenline bundle with a Chern number. Our main analytical result is a Chern–parity correspondence. Reflection symmetry equips this eigenline with a real structure and expresses $C_T\bmod2$ through first Stiefel–Whitney classes on the circles fixed by reflection. When the transverse polarization has no additional zeros along the reflection-fixed circles, these classes reduce to ordinary current-parity statistics at the four high-symmetry counting points. For a five-state nonequilibrium figure-eight network, varying only the observation time produces four polarization-gap closings and the reentrant sequence $C_T=0\to-1\to0\to-1\to0$. Every transition occurs at $(\pi,\pi)$ and coincides with a sign reversal of $\mathbb E[(-1)^{Q_1+Q_2}]$, while the full integer Chern number is obtained independently from the two-dimensional spin texture. Finite observation time can therefore organize a fixed stochastic process into distinct topological sectors of its ordered path ensemble.

We prove a three-term asymptotic formula for the eigenvalues of the fractional Laplacian on the bounded interval $(-1,1)$. This improves the eigenvalue asymptotics of Kulczycki–Kwaśnicki–Małecki–Stós and Kwaśnicki, and confirms the conjectural $O_\alpha(n^{-2})$ remainder suggested by the numerical simulations of Kaleta–Kwaśnicki–Małecki. Moreover, we prove that the normalized eigenfunctions are bounded uniformly in the eigenvalue index $n$ and the fractional order $\alpha$. This settles the conjecture proposed by Kwaśnicki through numerical experiments. Furthermore, we prove that the $n$-th eigenfunction has exactly $n-1$ zeros in the interval $(-1,1)$ and every zero is simple, and hence there are exactly $n$ nodal domains. A key ingredient in the proof is an explicit representation of the eigenfunction.

We consider the Sherrington–Kirkpatrick model with inverse temperature $\beta>0$ and deterministic external field $h>0$. Let $q$ be the replica-symmetric fixed point: $q=\mathbb E{\rm tanh}^2 (h+\beta\sqrt q\,Z)$, where $Z$ is a standard normal. We prove that, uniformly on compact subsets of the strict de Almeida–Thouless region: $\beta^2\mathbb E{\rm sech}^4(h+\beta\sqrt{q} Z) <1,$ the overlap satisfies the concentration: $$ \mathbb E\langle (R_{12}-q)^2\rangle=O(N^{-1}). $$ As a consequence, we obtain an $O(N^{-1})$ replica-symmetric free-energy correction and identify the finite-volume replicon susceptibility. Our proof is self-contained and does not use the identification of the limiting free energy with the Parisi variational formula. Moreover, we prove the central limit theorem for the overlap in this region. The present paper provides an alternative proof of the replica-symmetric free energy formula in the de Almeida–Thouless region, which was recently established by Lopatto [arXiv:2604.11921]. An advantage of our approach is that it establishes an explicit quantitative bound and yields the replica-symmetric free energy formula as a consequence. The main result supersedes the corresponding result in our recent preprint arXiv:2607.23427, extending the replica-symmetric bounds to the strict de Almeida-Thouless region. However, we keep the previous preprint, since its argument is different and substantially simpler than the one given here.

I prove that a collapsing charged scalar field with scalar charge $\mathfrak{e}$ that forms an exactly extremal Reissner–Nordström black hole à la Kehle–Unger with radius $r_+$ must satisfy $|\mathfrak{e}|r_+ > \frac{1}{3}$. In contrast, I also show that no such bound exists for subextremal collapse: a scalar field with an arbitrary $\mathfrak{e} \ne 0$ can form a subextremal black hole with an arbitrary (subextremal) charge-to-mass ratio. Complementing the extremal bound, I prove that a Schwarzschild black hole can become extremal if $|\mathfrak{e}|r_+ > \sqrt{\frac{3 + \sqrt{33}}{3}}$. The fact that $|\mathfrak{e}|r_+$ can be taken to be of order unity could be considered evidence that the third law of black hole mechanics in vacuum is false.

We give a rigorous realization of the dilute instanton picture for a semiclassical Schrödinger operator with a symmetric double-well potential on $\mathbb{R}^n$. Using a localized Feynman–Kac representation, we decompose the heat-kernel trace according to the number of passages made by a Brownian bridge between shrinking neighborhoods of the two wells. We identify the weight of one passage with a hopping coefficient $\rho_\lambda$, $\displaystyle \rho_\lambda = \int_{\partial\Omega} \left( \nabla\overline{\varphi_{\lambda,0}^{\Omega}}\, \varphi_{\lambda,0}^{-\Omega} - \overline{\varphi_{\lambda,0}^{\Omega}}\, \nabla\varphi_{\lambda,0}^{-\Omega} \right)\cdot\nu. $ On the exponentially long time scale $\beta=N/\lvert\rho_\lambda\rvert$, the number of passages converges, for every fixed $N>0$, to a Poisson random variable of mean $N$. We identify $-\frac{1}{\lambda}\log\lvert\rho_\lambda\rvert\to S(d,-d)$ and obtain $\displaystyle E_1(\lambda)-E_0(\lambda) = 2\lvert\rho_\lambda\rvert\left(1+o(1)\right). $ Thus the familiar instanton expansion of the double-well eigenvalue splitting emerges directly from a factorization of the heat-kernel trace.

We study perturbative anomaly inflow for effective theories supported on orbifold fixed loci of the internal geometry. Donnelly's equivariant APS theorem gives a fixed-point density, whose degree-$(d+2)$ component is identified as the anomaly polynomial of the $d$-dimensional effective theory. We apply the construction to six-dimensional A-type orbi-instanton theories engineered by $N$ M5-branes probing a transverse $\mathbb{C}^2/\mathbb{Z}_k$ singularity at an end-of-the-world M9-brane. For a flat $E_8$ connection specified by $\rho:\mathbb{Z}_k\to E_8$, we compute the non-identity ALE fixed-point class on the M9 wall. Its degree-eight component agrees with the complete Kac-label-dependent remainder conjectured in \cite{MOTZ} by Mekareeya-Ohmori-Tachikawa-Zafrir, including the $SU(2)_R$ and tangent-bundle curvatures. Together with the known M5/Hořava–Witten and ALE/Hořava–Witten terms, this reproduces the unflavored tensor-branch anomaly polynomial. This local equality further suggests a complete M-theory corner-inflow interpretation. The same prescription also be extended to include flavor symmetries as the centralizer of $E_8$.

We prove the black hole uniqueness conjecture in the axially symmetric, stationary vacuum setting: there is no regular, asymptotically flat equilibrium configuration with more than one horizon component. More precisely, we establish that in every such multi-horizon configuration, the logarithmic angle defect along every bounded axis rod is negative, which implies that the net interaction force across each such segment is attractive. The proof is based on a refined asymptotic analysis of the associated singular harmonic maps, partially drawn from the companion work [16], together with a global maximum principle bound for the Weyl conformal factor arising from a novel scalar curvature related differential inequality.

This book is a self-contained, tutorial introduction to topology, differential geometry, and general relativity for students and researchers in physics. Its distinguishing feature is a Mathematica-based approach that makes abstract constructions concrete through explicit, runnable notebooks and worked examples. A key contribution is a large collection of original Mathematica notebooks, written by the authors themselves, that let readers run, verify, and extend every demonstration. The first part covers point-set topology, topological spaces, continuity of maps, homotopy, and the fundamental group with applications that highlight the role of topology in modern physics. The second and largest part develops differential geometry from the ground up: manifolds, tangent and dual spaces, vector fields, pullbacks and pushforwards, Lie brackets and Lie algebras, local flows, and the Lie derivative. It then treats tensors, differential forms, the exterior derivative, volume forms, the metric tensor, and Hodge duality, emphasizing coordinate-free formulations and their computational realization. These tools are applied to Maxwell's equations in the language of forms and the generalized Stokes theorem, while Lie groups, fiber bundles, connections, and curvature bridge geometry and gauge-theoretic physics. The final part uses this framework to present general relativity, computing curvature tensors and field equations for standard spacetimes with dedicated Mathematica packages. Throughout, the book balances mathematical rigor with hands-on computation, enabling readers both to understand the theory and to reproduce every result themselves.

The sewing operation is the fundamental algebraic structure of the (infinite-dimensional) Segal moduli spaces as the conformal cobordism category. We prove a universal property of local $\mathbb{R}_+$-extensions of the sewing operation, characterizing them by one real parameter: their central charge. Our result is an analogue of “Mumford's theorem,” that appeared in Segal's monograph [Seg88, Seg04], classifying complex extensions, also known as one-dimensional modular functors. Algebraically, the real extensions are characterized by real-valued cocycles on pairs of surfaces under sewing. We identify the disk-disk cocycle as the universal Liouville action, also known as loop Loewner energy. The real determinant line bundle encodes the trace anomaly of euclidean two-dimensional conformal field theories (CFT) and the restriction function of Schramm-Loewner evolution (SLE) loop measures. It serves as the guiding example in our setup and proof strategy, providing the universal $\mathbb{R}_+$-extension. Our results thus give a mathematical explanation for the appearance of the central charge in CFT and SLE, and prove the conjectured uniqueness of Brownian loop measures as the canonical restriction function for SLE.

We present a variational–adjoint interpretation of quantum mechanics in which the interaction between forward and adjoint wavefunctions defines a general sensitivity kernel of the system. This Fréchet interaction density quantifies how perturbations in the wavefunction (or system parameters) influence a chosen observable. The familiar Born probability density appears as a special case when the adjoint wavefunction is chosen as the complex conjugate of the forward wavefunction, for which the interaction density becomes real and non-negative. Within this framework, probability is a particular positive-definite form of sensitivity described by the Fréchet interaction density. In general, the variational–adjoint interpretation also produces Fréchet sensitivity kernels associated with other quantum observables, including momentum, energy, and spin. This suggests that the Born probability density belongs to a broader class of Fréchet sensitivity kernels associated with quantum observables. The proposed interpretation also provides a connection with time-symmetric interpretations of quantum mechanics and possible future applications in quantum control and quantum metrology.

We study the class of transversal field models with pair interactions, including the quantum Curie-Weiss and the transversal field Ising model, as well as quantum spin glasses based on the Sherrington-Kirkpatrick (SK) and Edwards-Anderson models. Our main result is a general correlation inequality based on a Gaussian convolution estimate, which is derived from an extension of the Ding-Song-Sun inequality and an application of Brascamp-Lieb convexity techniques. The results apply for sufficiently strong transversal fields and, for quantum spin glasses, yield a high-field regime without replica order. Combined with the previously established low-field replica-symmetry-breaking regime, this gives distinct low- and high-field overlap phases. As a corollary of the logarithmic Sobolev inequalities for the underlying functional integrals derived here, we show the concentration of the self- and replica-overlap, and determine the high-field asymptotics of the pressure in the quantum SK model.

We study the disc counting statistics of the real Ginibre ensemble, whose spectrum consists of real and non-real eigenvalues, the latter occurring in complex-conjugate pairs. As the matrix dimension increases, we derive the asymptotic behaviour of the joint cumulants of the numbers of real and non-real eigenvalues contained in a centred disc in the bulk, edge, and exterior regimes of the circular law. As consequences, we establish a joint central limit theorem and obtain the conjectured asymptotics for the number variance. Furthermore, our results reveal a universality phenomenon: although the fluctuations of the real and non-real eigenvalues separately retain dependence on the symmetry class, their combined fluctuations exhibit the same limiting behaviour as in the previously established complex and symplectic Ginibre ensembles.

In this paper, we study the renormalization of a three-dimensional quartic model on a compact connected Riemannian manifold with a boundary. We show that, in addition to the standard shift of the mass parameter, it is necessary to take into account boundary contributions involving singular coefficients. This indicates the need to extend the classical action within the framework of standard renormalization theory. The requirements of consistency regarding the gluing of manifolds lead to renormalization of an operator on the boundary.

The notation $\psi(x,t)$ is among the first pieces of quantum mechanics that students learn. It is also among the easiest to over-interpret. Because $x$ and $t$ occur as arguments of the same function, students may ask whether they have the same mathematical status. They may also ask whether $\psi(t)$ should require a generalized bra $\bra{t}$ in the same way that $\psi(x)=\braket{x}{\psi}$ is often written. A related question is whether the absence of a universal time operator follows simply from Pauli's argument. These questions mix several structures that are usually introduced in different parts of the curriculum. We present a unified pedagogical treatment built around two maps hidden in $\psi(x,t)$. Time evolution selects a state along a trajectory in Hilbert space. A spectral representation then maps that state to amplitudes labelled by outcomes of a chosen observable. We formulate the position representation without generalized eigenkets. We recover Dirac's $\ket{x}$ notation as a controlled continuum shorthand and use a finite-grid limit to show where delta normalization enters. We distinguish background coordinates, translation parameters, observables, spectral labels, and physical records. We also clarify the Stone-theorem analogy, compare prescribed-time position measurements with arrival-time measurements, state what the strong form of Pauli's argument excludes, and exhibit an exactly solvable boundary case in which a canonical self-adjoint time observable exists. Spin, circuit-QED, and optical-clock examples provide experimentally grounded checks. The aim is not a new interpretation of time. It is a reusable teaching framework for separating mathematical role from notation.

The Berry phase arises naturally from the adiabatic dynamics of a pure quantum state, whereas the Uhlmann phase is usually formulated kinematically for a prescribed path of mixed states. For a qubit subject to a uniformly gapped conical drive, we obtain the Uhlmann connection and holonomy in closed form for the equilibrium Gibbs cycle. We then solve the corresponding Lindblad equation exactly in the rotating frame and show that its steady state forms a nonequilibrium limit cycle that lags the instantaneous Gibbs cycle. As the driving becomes slow, the Uhlmann phase approaches the equilibrium Uhlmann phase, which cooling then reduces to the Berry phase: in the joint adiabatic and low-temperature limit, the dynamical Uhlmann phase therefore converges to the Berry phase. This grounds the Uhlmann–Berry correspondence in the physical dynamics of an open system. We also characterize the finite-driving corrections, together with the discontinuous $\pi$ jump that the Uhlmann phase undergoes on the equatorial cycle as the temperature is varied. Finally, we examine an isolated transversal gap closing, where this zero-temperature correspondence can break down: the pure-state path becomes open, its closure is ambiguous, and gap-opening regularizations close it along a geodesic set by the direction of the bias field, splitting the Berry phase into a one-parameter family of values. The Gibbs path instead closes smoothly through the maximally mixed state, so the Uhlmann holonomy requires no regularization and remains unique and continuous at every finite temperature. In the low-temperature limit the Uhlmann phase selects a single member of the Berry family—the geodesic closure in the osculating plane, fixed by the velocity and acceleration of the driving field. The Uhlmann–Berry correspondence of the gapped regime thus survives the gap closing as a geodesic selection rule.

We provide a geometric foundation for the kinetic theory of photons. Although the induced metric $\hat h$ on the light cone bundle $\Gamma_0^+$ is degenerate, which causes the corresponding volume element to vanish, we can still use a method similar to the Hodge dual to construct a volume element $\eta_{\Gamma_0^+}$ for the light cone bundle. Based on this geometric structure $(\Gamma^+_0,\eta_{\Gamma_0^+},\hat h)$, we establish the fully covariant Boltzmann equation for photons. More importantly, the volume element and the induced metric are linked in a nontrivial way, which allows the physical distributions to be defined consistently. This yields the corresponding hydrodynamic quantities and their divergences, which take the same form as in the case of massive particles.

We study the one-dimensional quasilinear Majda–McLaughlin–Tabak (MMT) equation on a large torus $[0,L]$: \begin{align*} i \partial_t u +2\pi|\nabla|^\sigma u +\lambda^{2}|\nabla|^\beta\left[ \left||\nabla|^\beta u\right|^{2} |\nabla|^\beta u\right]=0. \end{align*} Our focus is on the well-posedness of its dynamics and the emergence of kinetic behavior where the domain size $L$ tends to infinity and the nonlinearity $\alpha=\lambda^2L^{-1}$ vanishes. In contrast to semilinear dispersive models, the quasilinear structure leads to unavoidable derivative loss, which prevents the construction of solutions via iteration of the Duhamel formula. Our results exhibit a dichotomy depending on the dispersion exponent $\sigma$. For $\sigma\in(1,2]$, we prove that, with high probability, solutions exist up to time scales $T_0 \sim \alpha^{-\frac54+} \wedge \alpha^{-\frac1{1-\beta}+}$, and that only trivial resonances occur, leading to a degenerate wave kinetic equation. For $\sigma\in(0,1)$, we prove the existence up to time scales $T_0 \sim \alpha^{-1-}$ and show that the second-order statistics are well approximated by the wave kinetic equation. In both cases, the solutions remain smooth while exhibiting smallness in suitable $L^\infty$-based norms despite having large total energy. The proof proceeds in two main steps. First, we establish the propagation of randomness for a suitably truncated equation, which allows us to overcome the derivative loss and recover the kinetic description. Then, we perform deterministic high-order energy estimates and a bootstrap argument to extend the solution up to time $T_0$.

Cooperative emission from three-level ensembles is conventionally diagnosed through the radiated intensity alone, which cannot say which channels carry the photons at each instant. We promote every transition of the cooperative-decay graph into a time-dependent edge flux and show that node-wise conservation of this flux is an exact restatement of the Pauli master equation. Suppose the graph carries a depth function that decreases by one along every edge. Then the total cooperative rate is the descent speed of the mean graph depth, and its energy-weighted form gives the radiated intensity as $I(t)=-\dd\langle E\rangle/\dd t$ for arbitrary level spacings. The same ordering triangularizes the generator. This delivers the relaxation spectrum from the state exit rates, together with a generalized Bateman closed form for every population on an arbitrary finite directed acyclic graph, in which all exit-rate degeneracies are absorbed into a polynomial-times-exponential recursion.

A null mode of the bordered Faddeev–Jackiw matrix is not, in general, a gauge generator. We determine exactly what the residual kernel encodes and identify the additional condition required for genuine gauge symmetry. Writing $f^{(m)}=\begin{pmatrix}f^{(0)}&B\\-B^{\mathsf T}&0\end{pmatrix}$ and $\Gamma=N^{\mathsf T}B$, we prove $\dim\ker f^{(m)}=\dim\ker\Gamma^{\mathsf T}+\dim\ker\Gamma-\operatorname{rank}(\Pi M\Pi)$, where $M=B^{\mathsf T}f^{(0)+}B$ and $\Pi$ projects onto $\ker\Gamma$. The two kernel contributions have distinct meanings: the first consists of null directions of $f^{(0)}$ tangent to the constraint surface, while the second counts independent first-class combinations. We then show that $\Gamma$ is not the Dirac constraint matrix: it is the primary–secondary block of the reduced constraint matrix, with $M$ providing the secondary–secondary block. For a residual null mode with nonzero multiplier component, we derive the associated two-step chain and prove that it closes to an exact gauge symmetry precisely when the chain contraction vanishes in the effective ring modulo the constraint ideal. A four-variable counterexample shows that the Faddeev–Jackiw stopping condition can hold while the system has no gauge freedom, and that the chain criterion rejects the false generators. Mechanical examples then exhibit the criterion and show that parameter strata can carry distinct gauge and dynamical content that is erased by premature cancellation.

We investigate the behavior of the singular spectrum of self-adjoint operators under families of Hermitian perturbations, even non-compact ones. Under mild assumptions we have the “shift” of the singular spectrum for almost all values of the parameter. Moreover, if we consider trace class perturbations, we observe the “shift” outside a countable set of the values of the parameter. While similar results were known for finite rank perturbations, the extensions to trace class perturbations are far from easy, and the proofs require significant new ideas. In addition, some of our results hold (surprisingly enough!) even for all positive and bounded perturbations. This is a consequence of our generalized Aleksandrov disintegration theorem established in this paper. We use some advanced techniques, involving Sz.-Nagy–Foia\c s theory and the operator ${\bf A}_2$ condition.

The observable of the Talbot effect is the intensity behind the grating: the density of the evolving field, not the field itself. Its fractality was previously known only for step data with rational jumps. We prove the general case. For arbitrary real data of bounded variation with at least one jump, the density of the free Schrödinger evolution on the torus is fractal at almost every time, with graph of upper box dimension exactly three halves. Its critical Sobolev mass diverges logarithmically, at a rate given in closed form by Wiener's jump statistic of the datum and independent of the positions and phases of the jumps. The same rate is halved along time traces and doubled for the Airy flow, so the critical mass probes the dispersion relation. Transported by the rank calculus of derangetropy operators, the theory equips every probability law with a quantum carpet: Gauss-sum revivals on quantile cells at rational times, a universal fractal at almost every other. The spectral growth rate of a single measured intensity profile returns the jump content of the grating, a prediction open to test in optical and matter-wave interferometry.

We introduce a natural class of hyperbolic surfaces called flippered surfaces that generalize crowned hyperbolic surfaces (i.e.: worldsheets for open strings). We develop their Teichmüller and moduli-space theory, construct generalized Weil-Petersson volume forms and Chekhov's action, and prove that the resulting generalized Mirzakhani volumes are finite. We establish three geometric recursion formulae-neck chopping, disk excision, and crown extraction-which express these volumes in terms of those of topologically simpler surfaces. For the fundamental polygonal and annular cases, we derive integral representations involving conical Legendre functions, as well as explicit formulae in terms of elliptic integrals and polylogarithms. We further describe the arithmetic structure of the Taylor coefficients of these volumes, show that suitable specializations are Kontsevich-Zagier periods, and prove identities at the imaginary boundary length $2\pi\sqrt{-1}$ that generalize the Do-Norbury paraphrasing of string and dilaton-type equations.

We consider the $2W\times 2W$ transfer matrices associated to the block Anderson model with GOE potential blocks. We make the classical Lyapunov exponent theory quantitative in two ways. First, we prove a quantitative limit theorem for the top Lyapunov exponent. Second, we prove every gap between Lyapunov exponents is at least $c/W$. As a corollary, this implies the localization length of this $1$d block Anderson model is at most $CW^2$. The proof uses Furstenberg type formulas for the Lyapunov exponents, and Malliavin calculus style arguments in the symplectic group to show the product of sufficiently many transfer matrices has a sufficiently smooth density. The main technical input for the latter is a least singular value estimate for a structured random matrix.

In this paper, we show how to adapt a derivation of Hartree equations due to Knowles and Pickl in the case where the one-particle hamiltonian is an Anderson-Hermite operator in dimension 1 and 2 or a formal Anderson-Hermite operator in dimension 3. By proving energy estimates for solution of the limit equation, uniform in the noise in bounded sets, we achieve a double limit. Namely, we show that we can commute the mean-field limit with a limit in noise.

We consider the open XXZ spin chain with unparallel boundary fields, in the special case in which the Vertex-IRF transform of the K-matrix labelling the boundary field at site 1 is diagonal and invariant under the value of the gauge parameters, the other K-matrix being arbitrary. We discuss here several properties related to this special gauge invariant boundary condition. Such a model can be solved in the Vertex-IRF framework with arbitrary gauge parameters: we can diagonalise the transfer matrix by means of generalised Sklyanin's Separation of Variable (SoV) approach in a family of SoV bases that depend on two arbitrary gauge parameters $\alpha$ and $\beta$, $\beta$ being the so-called dynamical parameter and $\alpha$ an arbitrary shift of the spectral parameter. We show here that such SoV bases depend actually on $\alpha$ and $\beta$ through the difference $\alpha-\beta$ only. Considering the ungauged limit in which $\alpha-\beta$ tends to infinity, we show how to compute all elementary building blocks for the correlation functions of this model. We finally show how to compute overlaps between eigenstates of this model and eigenstates of a spin chain sharing the same boundary condition at site $N$ and in which the gauge invariant boundary condition at site 1 has been changed to a more general one.