Papers

This Master's thesis develops and applies the multisymplectic field-theory framework to (pure) Yang-Mills theory. Building from the geometric foundations of jet bundles and the Poincaré-Cartan forms, the work constructs the Lagrangian multisymplectic formalism for first-order field theories, recovering the usual Euler-Lagrange field equations and extending the Poisson brackets of symplectic mechanics to a full graded Lie algebra. It then establishes the Legendre correspondence with the multisymplectic Hamiltonian formalism (in which Hamilton equations are retrieved) including the case of singular Lagrangians (where a constraint algorithm is reproduced). Breaking Lorentz covariance via a Cauchy surface slicing, the thesis then relates the covariant multisymplectic picture to an instantaneous presymplectic (infinite-dimensional) formulation in which the instantaneous De Donder equations and presymplectic brackets are obtained in a compatible way. A comprehensive treatment of symmetries then leads to multisymplectic and instantaneous momentum maps generalizations and to several versions of Noether's theorems. Specializing the formalism to principal-bundle gauge theory, the thesis identifies the multisymplectic configuration space with the bundle of connections, gauge symmetries with symmetries in the multisymplectic language and builds the Yang-Mills Lagrangian. Lastly, among many computations, the standard tensorial Yang-Mills equations and the Bianchi identity in local trivializations are retrieved, Yang-Mills conserved currents and charges are obtained and the graded algebra is derived for a special class of Hamiltonian forms.

We extend Mark Kac's kinetic framework from single-species Euclidean spaces to a two-species, non-linear interaction space on the open cone $(0, \infty)^2$ governed by the Matsumoto–Yor rational collision involution. We identify a non-linear conserved energy $E(x,y) = x + y + 1/x$ and an invariant conic metric measure $d\mu = \frac{dx \, dy}{x(x+y)}$. By establishing a two-species $H$-theorem, we prove that entropy production vanishes if and only if post-collision velocities are statistically independent ($I(U;V)=0$). In contrast to the transform-domain proof of Letac and Wesołowski~\cite{letac2000}, which relies on joint Laplace transforms of general measures, our approach yields a direct physical derivation in state space via the transformation law for logarithmic gradients and separation of variables. This uncouples the functional-differential system to uniquely characterize the Generalized Inverse Gaussian (GIG) and Gamma distributions as the non-equilibrium thermodynamic equilibria.

We study the inverse resonance problem for the one-dimensional Schrödinger operator on the half-line with a compactly supported complex-valued potential and a complex Robin boundary condition \[ u'(0)-a u(0)=0,\qquad a\in\mathbb C. \] The proof follows the transformation-operator, entire-function, and Fourier analytic strategy developed by Marletta, Shterenberg and Weikard for the Dirichlet problem \cite{MSW}. The Dirichlet Jost function $\psi(z,0)$ is replaced by the Robin characteristic function \[ F_a(z;q)=\psi_x(z,0;q)-a\psi(z,0;q). \] We establish the corresponding large-energy estimates, Hadamard factorization, finite-data stability, and uniqueness results. The principal new feature is that $F_a(z;q)=iz-a+O(1)$ rather than $1+O(z^{-1})$. In the transformation-operator argument the relevant boundary quantity is $B_x(0,t)-aB(0,t)$, together with the scalar term $B(0,0)$. For fixed $a\in\mathbb C$, and uniformly for $a$ in bounded subsets of $\mathbb C$, the resulting inverse resonance estimates have the same qualitative form as those proved in \cite{MSW} for the Dirichlet problem. In particular, for a fixed Robin parameter $a$, the complete set of Robin eigenvalues and resonances, counted with multiplicity, determines the potential uniquely. The text is produced with the help of AI ChatGPT for Academy.

Let $ \operatorname{MaxCol}(n,d) $ denote the maximal number of nonsimultaneous elastic collisions among $n$ identical hard balls moving freely in $\mathbb{R}^d$. We give a planar construction with exponential lower bound with respect to $n$: $ \operatorname{MaxCol}(n,2)\geq 2^{\lfloor n/2\rfloor}, \quad n\geq3 $.

This article presents an introductory study of $p$-differential forms in spinor variables defined on the Minkowskian (or Lorentzian) manifold. Starting from the algebraic and geometric foundations of spinor spaces and Clifford algebras, we establish the correspondence between Minkowski spacetime and Hermitian matrices in spinor coordinates. Within this framework, we construct differential 1-forms and 2-forms expressed in spinor notation, and formulate the Dirac operator in terms of exterior calculus. Furthermore, we extend this approach to the case of non-monogenic functions, following the method proposed in~\cite{Jaramillo2024}. The results illustrate the consistency of the spinorial exterior calculus and its potential applications in the study of relativistic field equations and geometric formulations of quantum theory.

We show how to perform exact diagonalization of SU($N$)-symmetric Bose-Hubbard models separately in each irreducible representation of SU($N$), by exploiting the Howe duality between U($L$) and U($N$). Three structural consequences follow for any lattice and any Hamiltonian built from the U($L$) generators: (i) the spectrum in a given orbital sector is independent of $N$ and of the particle statistics, (ii) the canonical partition functions for bosons and fermions obey the exact identity $Z_{\mathrm{b}}(-N)=(-1)^{N_\mathrm{b}}Z_{\mathrm{f}}(N)$, (iii) with unfrustrated hopping the groundstate is always a color ferromagnet, so that the color-resolved method we introduce is useful to study excited sectors, frustrated lattices and finite temperatures. As applications, we first derive the finite-temperature phase diagram of attractive SU($N$) bosons on few-site rings at large $N$ for arbitrary filling. We then show the emergence of singlet-cluster states from frustrated hopping, the counterpart of fermionic ferromagnetism, with the example of the sawtooth and kagome lattice.

We develop a homotopical framework for understanding the space of quantum spin systems based on topological $n$-nets, which assign associative unital algebras to local regions in an $n$-dimensional space in a manner that captures the locality structure of the underlying lattice theories. The net-theoretic perspective admits a natural categorification, in which the category of topological $n$-nets carries the structure of a higher Morita category. Moreover, the nets of greatest physical interest form a Karoubi-complete subcategory that shares many of the characteristic structural features of the category of finite TQFTs. Our main result identifies the group of locality-preserving symmetries of (3+1)-dimensional quantum spin systems (i.e. quantum cellular automata modulo stabilization, finite-depth quantum circuits, and translations) with the full Witt group of nondegenerate braided fusion categories, thereby answering a question posed by Haah.

We determine a universal sector of the spectrum of light operators in four-dimensional $\mathcal{N}=1$ class $\mathcal{S}$ superconformal field theories arising from M5-branes wrapped on a compact hyperbolic Riemann surface $\Sigma_2$. We work at large $N$, analysing Kaluza–Klein fluctuations about the dual Bah–Beem–Bobev–Wecht anti-de Sitter backgrounds for general values of the twist parameter $z$. To obtain these spectra, we specialise recently derived mass operators to modes that are constant on $\Sigma_2$ by imposing a suitable $z$-dependent generalised $G$-structure. We then organise the resulting eigenstates into long and protected $\mathcal{N}=1$ supermultiplets with definite flavour quantum numbers. As $z$ varies, rich patterns emerge in the spectrum, including both isolated multiplets and infinite towers of multiplets, which we characterise analytically.

The $q$-deformed Weil-Petersson volumes of Do and Norbury are the top-degree parts of Okuyama's discrete volumes, which are defined by topological recursion on the spectral curve of the double-scaled SYK (DSSYK) model. We show that the same q-deformed volumes can be obtained from a double-scaled matrix model whose disk density is found by zooming in on the soft edges of the DSSYK density. Applying the same procedure to the hard edge of $\mathcal{N}=1$ DSSYK, we obtain candidate $q$-deformed $\mathcal{N}=1$ super Weil-Petersson volumes. They are polynomials of fixed degree, and they reduce to the $\mathcal{N}=1$ super Weil-Petersson volumes in a scaled $q\to1$ limit. These candidate volumes differ from the $\mathcal{N}=1$ volumes of Do and Norbury in (arXiv:2510.12431) up to what we call "irrelevant" contributions which exactly vanish in the scaled $q\to 1$ limit. Together, these results demonstrate how q-deformed volumes can be naturally associated to the spectral edges of various DSSYK models and provides an intuitive way to construct them in supersymmetric contexts.

We develop a unified description of convolution on the sphere by lifting to the rotation group. A number of definitions of spherical convolution have been proposed, differing in whether the kernel is axisymmetric or directional, whether the output is on the sphere or the rotation group, whether the signals carry spin, and in their computational cost. No unified treatment exists. Regarding the sphere as the homogeneous space $S^2 \cong \text{SO(3)}/\text{SO(2)}$, we lift a spin $s$ field to the rotation group, convolve on $\text{SO(3)}$, and project back to the sphere through the adjoint of the lift for a potentially different spin $s'$. We recover every rotation-equivariant convolution considered previously through constraints on the kernel, spin choices and inclusion or omission of the lift and projection. For maps between fields of fixed input and output spin, we prove that directional kernels, output on the sphere and equivariance cannot be achieved simultaneously. We make explicit that spin fields are sections of associated line bundles, linking the physics and bundle literatures, and express the spin rotation phase as the character of the twist of the section. We present harmonic-space algorithms for the exact and efficient computation of lifted spherical convolutions of band-limited fields by appealing to sampling theorems on the sphere and rotation group. All operators are implemented in the open-source s2conv code, and numerical experiments confirm the reductions and the equivariance of the operators to machine precision. Finally, we illustrate the operators on fields measured on the Earth and on the celestial sky.

We study a continuum model of frustrated magnetism that supports particle-like field configurations with nontrivial Hopf index. The energy functional combines competing second and first order exchange interactions with a stabilizing external magnetic field and gives rise to a variational problem for maps from three-dimensional space into the two-sphere. We prove the existence of energy minimizers in infinitely many nontrivial homotopy classes. We further establish the existence of a unit-charge Hopfion in the regime of strong external fields. Finally, we analyze the corresponding large-field limit, identify the natural concentration scale, and prove strong convergence of suitably rescaled minimizers to a minimizer of a limiting variational problem.

We construct a set of conserved charges which are deformations of quantum KdV charges. They are associated with the modular transformation of a GGE with the spin-$\sigma$ KdV charge inserted with chemical potential $\alpha_\sigma$. The charges are formal power series in the chemical potential, with coefficients given by local integrals of motion. We define local recursion relations and equivalent flow equations, prove commutativity to all orders, and establish uniqueness of the currents modulo total derivatives for a fixed generating series. We show that the eigenvalues of these deformed charges can be obtained from the thermodynamic Bethe Ansatz in two models. In the $c=1/2$ free fermion theory, the deformed charges reduce to explicit linear combinations of undeformed KdV charges, and their ground-state eigenvalues agree with the TBA expansion to all orders. In the Lee-Yang model, we find numerical agreement through cubic order in the spin-$5$ deformation for the first few charges evaluated on low-lying states.

Standing waves $e^{i\omega t}u(x)$ of nonlinear dispersive equations, such as the nonlinear Schrödinger and Klein–Gordon equations, have a profile $u$ that solves a nonlinear elliptic ODE of the form $\Delta u+f(u)=0$. We prove the existence of radial bound states with a prescribed number of nodal surfaces for this equation, on a large class of spherically symmetric manifolds and nonlinearities $f$. Under mild assumptions, we prove uniqueness of such bound states for the model nonlinearity $f(u)=|u|^{p-1}u-u$. Both existence and uniqueness results apply to some manifolds with a singularity at the origin.

We establish that bond Hamiltonian densities, up to local difference terms whose sum vanishes under periodic boundary conditions, lying in the commutant of bond operators, $w_j\pm w_{j+1}$, is a necessary and sufficient condition for a one-dimensional nearest-neighbour Hamiltonian to conserve a uniform or staggered sum of the one-site operators $w_j$. Starting from these local commutants, we show that every such Hamiltonian and its conserved charge can be obtained from monodromy matrices constructed using regular and non-regular members of scalene Yang-Baxter triples. The results hold on periodic chains of length $L\geq3$, with even length required in the staggered case. The results do not require translation invariance, and hence the interaction on each bond can be chosen independently from the full corresponding commutant. Our representation-independent construction applies in arbitrary finite local dimension and exhausts the Hamiltonians with either prescribed additive symmetry, without requiring ordinary Yang-Baxter integrability. We illustrate the construction with spin-$\tfrac12$ and spin-1 systems and discuss its relation to the double-commutant theorem. Classifying the one-site generators $w$ and their associated bond commutants provides a route to classifying such Hamiltonians with additive conserved charges.

We consider the normal form of a 1:1 detuned resonance that can be related, via a double averaging process, to the three body problem of celestial mechanics. In view of the symmetry in the gravitational coupling between the planets, the obtained normal form exhibits inversion symmetry. We provide an analytical prediction of the possible bifurcations in terms of the detunings and the other physical parameters of the system. We show that the bifurcation scenario is rich. It includes bifurcation sequences where the unstable orbit appearing after a saddle-node bifurcation can have a reversed pitchfork bifurcation, where it turns into stability with the birth of two unstable orbits, which eventually reconnect with one of the two quasi-normal modes. Furthermore, another scenario shows a standard pitchfork in which one of the quasi-normal modes becomes unstable, giving rise to a pair of stable periodic orbits. By introducing coupling and asymmetry parameters, we provide an effective way to discern all the possible sequences of bifurcations.

We construct a linear map from the space of spin polynomials in the critical Ising model on a lattice to the state space of the $M(4,3)$ minimal model of conformal field theory. The kernel of the map consists precisely of null fields, i.e., spin polynomials that vanish in all correlations. For a polynomial $\Op$ which is not a null field, we define its scaling dimension $\Delta_{\Op}\in\Z_{\geq0}\cup(\Z_{\geq0}+\frac{1}{8}),$ to be the lowest $\Loper{0}+\Loperbar{0}$ eigenvalue in the decomposition of its image into homogeneous components. We show that as the mesh size $\delta$ goes to zero, the rescaled field $\delta^{-\Delta_{\Op}}\Op$ has a non-trivial scaling limit, when inserted in correlations with other such fields. Moreover, all these limits can be expressed in terms of correlations of primary fields of $M(4,3)$ and the Virasoro action in the continuum. The highest weight vectors of $M(4,3)$, the joint eigenvectors of $\Loper{0}$ and $\Loperbar{0}$ with the same eigenvalues $0,1/16,1/2$ for both, are the images of the constant $1$, the spin at a single lattice vertex, and the energy at a single lattice edge, respectively. For the proof, we construct a map from a larger lattice field space, accommodating also fermion and disorder insertions, to a direct sum of two fermionic Fock spaces. This direct sum contains contains the state spaces of both the $M(4,3)$ and the free fermion CFT. The main technical ingredient of the proof is a new version of the exact fermionic current action on the space of lattice local fields due to C. Hongler, K.K. and F. Viklund, combined with some new advances in discrete complex analysis. Our results are valid on arbitrary isoradial graphs.

We establish a Thouless–Anderson–Palmer (TAP) variational formula for the free energy of spherical $2$-spin models whose interaction matrices have a compactly supported limiting spectral distribution. The formula holds at every inverse temperature and, in particular, extends the TAP representation previously obtained for the spherical Sherrington–Kirkpatrick model. Our main contribution is a new microcanonical proof designed to handle marginality. We partition the sphere according to the smallest collection of top eigendirections carrying a prescribed fraction of the norm. On every element of this partition, the remaining degrees of freedom are effectively at high temperature and their contribution can be computed through spherical-integral asymptotics. The resulting auxiliary variational problem separates the macroscopic low-temperature contribution from the high-temperature fluctuation term; its maximization recovers the usual TAP functional and Plefka condition. This decomposition also shows why neighborhoods of the TAP maximizers alone do not capture the Gibbs mass: all eigendirections associated with large eigenvalues contribute on the same exponential scale, a manifestation of the marginality of spherical $2$-spin states.

We introduce a discrete spectral curve from $\mathscr{N}=2$ double-scaled SYK whose topological recursion defines quasi-polynomial volumes in integer boundary lengths. The curve has two edge theories: a local $\mathscr{N}=2$ theory at the lower edge and a bosonic theory at the upper edge. Focusing on the local $\mathscr{N}=2$ theory, we compute its finite-$q$ volumes and independently check the first few using the string equation. In the $q\rightarrow 1$ continuum limit, these recover the super Weil-Petersson volumes of Turiaci and Witten. Finally, we give an all-genus intersection-theoretic interpretation of these local volumes on the moduli space of stable marked curves, in terms of a deformed spin class with a gap-dependent $\kappa$-class dressing.

In an earlier study of the Laplacian eigenvalues with strongly attractive Robin boundary conditions in polygons it was found that if each corner is either concave or non-resonant, then the asymptotic behavior of the individual eigenvalues is determined by an effective one-dimensional operator on the boundary with Dirichlet boundary conditions at the corners. The non-resonance condition was formulated in terms of an eigenvalue bound on truncated sectors, and it was only shown to be satisfied by the right and obtuse opening angles. In the present work, we determine the full range of non-resonant angles: it turns out that the non-resonance condition fails only for the opening angles $\frac{\pi}{2n+1}$ with $n=1,2,3,\dots$, and these critical values are equivalently characterized in terms of the jumps of the eigenvalue counting function and the existence of bounded generalized eigenfunctions. This resolves the numerically supported conjecture stated by Dauge and Lafranche.

We study the de Rham cohomology and Deligne splitting for a class of Eisenstein-Kronecker forms by introducing a modular weight grading. As an application, we compute a particular cohomology class that gives rise to generating series of Gromov-Witten invariants of elliptic curves.

In one dimension, Anderson localization by a disordered potential sets in at any strength, whereas localization by a quasiperiodic potential in the Aubry-André model occurs through a transition at a critical strength. Disordered photonic time crystals exhibit a temporal counterpart of this Anderson localization, in which light, instead of decaying in space, stops propagating and grows in time. Whether a counterpart of the Aubry-André transition exists in photonic time quasicrystals remains an open question. Here we show that it does, and that the transition occurs at mobility edges in momentum rather than in energy. These edges arise even under impulsive modulation, where the field obeys the self-dual almost-Mathieu equation exactly, and for general analytic modulations Avila's global theory reveals the condition under which they form. The Aubry-André transition thus extends from spatial lattices to the time domain, where it sorts light by wave number into components that propagate and components that stop and grow.

On the Riemannian Schwarzschild–Tangherlini metric of dimension $n\geq 4$, we show that the destabilizing TT tensor of Ozuch lies in the image of the “TT-tensor machine”. By continuity, it can then be approximated, in suitable weighted Sobolev spaces, by smooth compactly supported TT tensors, on which the stability quadratic form is still negative. This proves instability in the usual sense for noncompact manifolds. To our knowledge, no exact smooth compactly supported destabilizing TT tensor was known before, even in 4D.

We describe a family of two-dimensional metric deformations that preserve integrability. The seeds are (classically) conformal integrable field theories and the deformations are directly related to different classes of auxiliary field sigma models. We introduce a master action connecting deformations by Lie-algebra-valued auxiliary fields, the so-called $\nu$-frame, to deformations by two Lorentz-charged scalars. These auxiliary scalar fields can be identified as Beltrami differentials of an auxiliary two-dimensional metric, providing a direct connection with the metric approach to $T\bar{T}$-like deformations. For Weyl-invariant seed matter theories, the auxiliary metric sector is described by a reduced gravitational potential. We show that the resulting framework preserves classical Lax integrability whenever the seed matter theory admits a Lax representation of its equations of motion on a generic curved two-dimensional spacetime. Finally, within the Lorentz-invariant family and the zweibein deformations of conformal matter theories considered here, we identify $T\bar{T}$ as the only deformation that can be interpreted as a field-dependent coordinate transformation.

We prove the Berezinskii-Kosterlitz-Thouless (BKT) phase transition in the large-distance behaviour of the two-point correlation function of the $\mathsf{Sine}_\beta$ point process for all $\beta>0$, confirming the first term of an asymptotic expansion predicted by Forrester.

We study the recovery of a three-dimensional Riemannian manifold from the areas of minimal surfaces. We show that areas of minimal surfaces determine the metric on manifolds foliated by minimal surfaces, up to boundary-fixing isometry. This resolves the conjecture on boundary rigidity for least-area data in an exterior formulation, even without topological restrictions on the minimal surfaces. We also prove global rigidity for Einstein metrics on compact three-manifolds with boundary, assuming $\(\pi_1(M,\partial M)=0\)$ and a non-trapping condition. The Einstein result includes the

We discuss General Relativity and some of its modifications as effective field theories of the breaking of the affine general linear group using the coset approach and the Maurer-Cartan form. While the ultimate goal is to break the symmetry down to the Lorentz subgroup, we argue that there are three interesting breaking cascades in which the affine general linear group is first broken to a larger subgroup that contains the Lorentz group. In all cases, the breaking of translations always generates Goldstone modes that can be interpreted as metric degrees of freedom. We argue that the effective field theories that emerge as a result of the different breaking patterns are all dynamically equivalent at the lowest order of the derivative expansion, e.g., using Einstein-Hilbert-like actions, but can differ beyond the leading order.

We prove that for every choice of five positive masses the planar five-body problem has finitely many central configurations up to similarity. This settles Smale's sixth problem for five bodies in the plane. Albouy and Kaloshin proved it for masses outside a closed algebraic subset of codimension two. We follow their method. Infinitely many central configurations would give an algebraic curve of complex solutions. At the places at infinity of the curve the valuations of the mutual distances must balance. We determine the possible valuation vectors at all positive masses, together with the mass conditions under which each can occur. We treat limits in which four bodies form a cluster of zero potential energy by expanding the potential, which is constant along the curve, at a place. This excludes, at every positive mass, one of the two diagrams in which such clusters occur, and part of the other. The same expansion excludes one further diagram at every positive mass. What remains is an exact analysis, by computer algebra, of the sets of mass conditions that can hold together at positive masses. Exact rational certificates show that the balance fails under each of them, except on a few one-parameter families of masses, which we treat separately. On the rays through $(1,4,4,4,4)$ and $(1,1,4,4,4)$ the certificates do not apply. On the first ray they cannot apply: there the complex equations have a continuum of solutions, as Albouy and Kaloshin observed. On these rays we count the real central configurations: there are 306 and 294 up to orientation-preserving similarity. The certificates, the programs that check them and a proof of the theorem in the Lean proof assistant are available.

We study quantum-assisted balanced rendezvous on the $d$-dimensional hypercube in a one-step two-agent quantum-walk framework, where the two agents move the same Hamming distance and meet at a common midpoint. The key structure is a projective $t$-sphere resolution, which governs exact finite-dimensional quantum attainment. Its scalar analogue recovers the corresponding classical characterization. We characterize finite-dimensional quantum attainment of the no-signalling benchmark, construct quantum strategies from projective sphere resolutions, and establish a rigidity theorem for optimal finite-dimensional strategies. A marked difference appears between nearest-neighbor and higher-cardinality updates: quantum advantage can occur in the former, whereas in the latter the geometry of the relevant Hamming spheres rules out exact finite-dimensional attainment of the no-signalling value. Together, these results connect quantum-walk dynamics, Hamming-sphere geometry, and quantum rendezvous within the quantum Bernoulli noises (QBN) framework.

Practical ship design requires localized refinement and performance evaluation. This paper presents a reward-directed diffusion framework for localized hull optimization. Parametric masking during dataset generation restricts geometric changes to selected variables, including the bulbous bow and forebody, while preserving full-hull continuity. Surrogate-assisted sequential Monte Carlo sampling targets reduced predicted calm-water resistance, with stability retained through training data and displacement matched during screening. An automated viscous free-surface CFD pipeline, validated against measurements for two benchmark container ships, evaluates a generated variant and GenHull E, a hull refined by a naval architect within the same representation. At matched KRISO Container Ship (KCS) displacement, the generated variant has comparable total resistance at $F_n=0.195$ but 45-50% higher resistance at $F_n=0.250$ and $0.260$, missed by empirical screening. GenHull E achieves 6-8% lower total resistance up to $F_n=0.240$ but higher resistance above $F_n=0.250$ as opposing friction and pressure trends reverse the gains. Aggregating resistance across this crossover can conceal tradeoffs that emerge when friction and pressure components are evaluated separately. The assessment reveals an inability to represent a faired bulbous bow and a reward lacking viscous pressure and stability terms. These findings show how high-fidelity assessment informs improved representations and objectives for physically realizable generative designs.

We construct a geometric theory on the brane space $S^2M=C^\infty(S^2,M)$ associated to a gerbe with connection on $M$. Normalized logarithmic holonomy around the latitude circles defines an average holonomy line bundle on the brane space, whose connection and averaged flux give an equivariantly flat superconnection. We prove that the completed periodic exotic twisted cohomology defined by this superconnection localizes to the twisted cohomology of $M$. This brane space localization happens in two stages: circle localization lands on the free path space of $M$, and contracting the paths, after a correction of the twisting form, brings us back to $M$. Parallel transport along the meridians lets us average gerbe module connections, after which the circular iterated-integral construction applies. The resulting twisted brane Bismut-Chern forms restrict to the twisted Chern forms on $M$; their variation is described by twisted brane Bismut-Chern-Simons forms. For a T-dual pair, transporting the classical Hori morphisms through localization yields graded genus-0 T-duality on brane cohomology. The composite of each morphism with its dual is minus the Euler operator in the level variable, so these morphisms are isomorphisms at every nonzero level. At level one these morphisms are compatible with the brane character and twisted $K$-theory T-duality.