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In this paper, we address the completeness problem of certain classes of Riemannian metrics on vector bundles. We first establish a general result on the completeness of the total space of a vector bundle when the projection is a horizontally conformal submersion with a bound condition on the dilation function, and in particular when it is a Riemannian submersion. This allows us to give completeness results for spherically symmetric metrics on vector bundle manifolds and eventually for the class of Cheeger-Gromoll and generalized Cheeger-Gromoll metrics on vector bundle manifolds. Moreover, we study the completeness of a subclass of g-natural metrics on tangent bundles and we extend the results to the case of unit tangent sphere bundles. Our proofs are mainly based on techniques of metric topology and on the Hopf-Rinow theorem.

We consider a family of semiclassically scaled second-order elliptic differential operators on high tensor powers of a Hermitian line bundle (possibly, twisted by an auxiliary Hermitian vector bundle of arbitrary rank) on a Riemannian manifold of bounded geometry. We establish an off-diagonal Gaussian upper bound for the associated heat kernel. The proof is based on some tools from the theory of operator semigroups in a Hilbert space, results on Sobolev spaces adapted to the current setting, and weighted estimates with appropriate exponential weights.

Classical isomorphism theorems due to Dynkin, Eisenbaum, Le Jan, and Sznitman establish equalities between the correlation functions or distributions of occupation times of random paths or ensembles of paths and Markovian fields, such as the discrete Gaussian free field. We extend these results to the case of real, complex, or quaternionic vector bundles of arbitrary rank over graphs endowed with a connection, by providing distributional identities between functionals of the Gaussian free vector field and holonomies of random paths. As an application, we give a formula for computing moments of a large class of random, in general non-Gaussian, fields in terms of holonomies of random paths with respect to an annealed random gauge field, in the spirit of Symanzik's foundational work on the subject., Comment: 51 pages, 10 figures. This version contains a new introduction, an additional Section (6.8) detailing an important example (the case of trace-positive holonomies), and a treatment of the quaternionic case. The introductory material on continuous time random walks on multigraphs in Section 1 was also simplified

We introduce a notion of $$\xi $$ ξ -stability on the affine grassmannian $${\fancyscript{X}}$$ X for the classical groups, this is the local version of the $$\xi $$ ξ -stability on the moduli space of Higgs bundles on a curve introduced by Chaudouard and Laumon. We prove that the quotient $${\fancyscript{X}}^{\xi }/T$$ X ξ / T of the stable part $${\fancyscript{X}}^{\xi }$$ X ξ by the maximal torus $$T$$ T exists as an ind- $$k$$ k -scheme, and we introduce a reduction process analogous to the Harder-Narasimhan reduction for vector bundles over an algebraic curve. For the group $${\mathrm {SL}}_{d}$$ SL d , we calculate the Poincaré series of the quotient $${\fancyscript{X}}^{\xi }/T$$ X ξ / T .

We propose a definition of persistent Stiefel-Whitney classes of vector bundle filtrations. It relies on seeing vector bundles as subsets of some Euclidean spaces. The usual \v{C}ech filtration of such a subset can be endowed with a vector bundle structure, that we call a \v{C}ech bundle filtration. We show that this construction is stable and consistent. When the dataset is a finite sample of a line bundle, we implement an effective algorithm to compute its persistent Stiefel-Whitney classes. In order to use simplicial approximation techniques in practice, we develop a notion of weak simplicial approximation. As a theoretical example, we give an in-depth study of the normal bundle of the circle, which reduces to understanding the persistent cohomology of the torus knot (1,2). We illustrate our method on several datasets inspired by image analysis., Comment: To appear in Journal of Applied and Computational Topology

It is well-known that principal bundles and associated bundles underlie the geometric structure of classical gauge field theories. In this paper, we explore the reformulation of gauge theories in terms of Lie algebroids and their associated bundles. This turns out to be a simple but elegant change, mathematically involving a quotient that removes spurious structure. The payoff is that the entire geometric structure involves only vector bundles over space-time, and we emphasize that familiar concepts such as BRST are built into the geometry, rather than appearing as adjunct structure. Thus the formulation of gauge theories in terms of Lie algebroids provides a fully off-shell account of the BRST complex. We expect that this formulation will have appealing impacts on the geometric understanding of quantization and anomalies, as well as entanglement in gauge theories. The formalism covers all gauge theories, and we discuss Yang-Mills theories with matter as well as gravitational theories explicitly., SCOPUS: ar.j, info:eu-repo/semantics/published

We study the property of \emph{continuous Castelnuovo-Mumford regularity}, for semihomogeneous vector bundles over a given Abelian variety, which was formulated in \cite{Kuronya:Mustopa:2020} by K\"{u}ronya and Mustopa. Our main result gives a novel description thereof. It is expressed in terms of certain normalized polynomial functions that are obtained via the Wedderburn decomposition of the Abelian variety's endomorphism algebra. This result builds on earlier work of Mumford and Kempf and applies the form of the Riemann-Roch Theorem that we established in \cite{Grieve:R-R:abVars}. In a complementary direction, we explain how these topics pertain to the \emph{Index} and \emph{Generic Vanishing Theory} conditions for simple semihomogeneous vector bundles. In doing so, we refine results from \cite{Gulbrandsen:2008}, \cite{Grieve-cup-prod-ab-var} and \cite{Mum:Quad:Eqns}., Comment: Accepted by Ann. Univ. Paedagog. Crac. Stud. Math

The strongly coupled heterotic M-theory vacuum for both the observable and hidden sectors of the $B-L$ MSSM theory is reviewed, including a discussion of the "bundle" constraints that both the observable sector $SU(4)$ vector bundle and the a hidden sector bundle induced from a line bundle must satisfy. Gaugino condensation is then introduced within this context, and the hidden sector bundles that exhibit gaugino condensation are presented. The condensation scale is computed, singling out one line bundle whose associated condensation scale is low enough to be compatible with the energy scales available at the LHC. The corresponding region of K\"ahler moduli space where all bundle constraints are satisfied is presented. The generic form of the moduli dependent $F$-terms due to a gaugino superpotential - which spontaneously break $N=1$ supersymmetry in this sector - is presented and then given explicitly for the unique line bundle associated with the low condensation scale. The moduli dependent coefficients for each of the gaugino and scalar field soft supersymmetry breaking terms are computed leading to a low-energy effective Lagrangian for the observable sector matter fields. We then show that at a large number of points in K\"ahler moduli space that satisfy all "bundle" constraints, these coefficients are initial conditions for the renormalization group equations which, at low energy, lead to completely realistic physics satisfying all phenomenological constraints. Finally, we show that a substantial number of these initial points also satisfy a final constraint arising from the quadratic Higgs-Higgs conjugate soft supersymmetry breaking term., Comment: v2: typos corrected; 58 pages, 7 figures

We study the Heckman-Opdam hypergeometric functions associated to a root system of type $BC$ and a multiplicity function which is allowed to assume some non-positive values (a standard multiplicity function). For such functions, we obtain positivity properties and sharp estimates which imply a characterization of the bounded hypergeometric functions. As an application, our results extend known properties of Harish-Chandra's spherical functions on Riemannian symmetric spaces of the non-compact type $G/K$ to spherical functions over homogeneous vector bundles on $G/K$ which are associated to certain small $K-$types., Comment: The present paper subsumes and extends the results of the preprint arXiv:1705.00277. With respect to arXiv:2008.00337: Corrected Proposition 3.5. Some computations added in Appendix B

We define non-ordinary instanton bundles on Fano threefolds $X$ extending the notion of (ordinary) instanton bundles. We determine a lower bound for the quantum number of a non-ordinary instanton bundle, i.e. the degree of its second Chern class, showing the existence of such bundles for each admissible value of the quantum number when $i_X\ge 2$ or $i_X=1$, $\mathrm{Pic}(X)$ is cyclic and $X$ is ordinary. In these cases we deal with the component inside the moduli spaces of simple bundles containing the vector bundles we construct and we study their restriction to lines. Finally we give a monadic description of non-ordinary instanton bundles on $\mathbb{P}^3$ and the smooth quadric studying their loci of jumping lines, when of the expected codimension., 34 pages. Minor changes. The final version will appear in The Asian Journal of Mathematics

Let $E$ be a vector bundle of rank $n$ on $\mathbb{P}^1$. Fix a positive integer $d$. Let $\mathcal{Q}(E,d)$ denote the Quot scheme of torsion quotients of $E$ of degree $d$ and let $Gr(E,d)$ denote the Grassmann bundle that parametrizes the $d$-dimensional quotients of the fibers of $E$. We compute Seshadri constants of ample line bundles on $\mathcal{Q}(E,d)$ and $Gr(E,d)$., 18 pages; minor changes; final version; to appear in Forum Mathematicum

In this work we present the foundations of generalized scalar-tensor theories arising from vector bundle constructions, and we study the kinematic, dynamical and cosmological consequences. In particular, over a pseudo-Riemannian space-time base manifold, we define a fiber structure with two scalar fields. The resulting space is a 6-dimensional vector bundle endowed with a non-linear connection. We provide the form of the geodesics and the Raychaudhuri and general field equations, both in Palatini and metrical method. When applied at a cosmological framework, this novel geometrical structure induces extra terms in the modified Friedmann equations, leading to the appearance of an effective dark energy sector, as well as of an interaction of the dark mater sector with the metric. We show that we can obtain the standard thermal history of the universe, with the sequence of matter and dark-energy epochs, and furthermore the effective dark-energy equation-of-state parameter can lie in the quintessence or phantom regimes, or exhibit the phantom-divide crossing., 17 pages, 2 figures, version to appear in Phys.Rev.D

We observe, utilize dualities in differential equations and differential inequalities, dualities between comparison theorems in differential equations, and obtain dualities in "swapping" comparison theorems in differential equations. These dualities generate comparison theorems on differential equations of mixed types I and II and lead to comparison theorems in Riemannian geometry with analytic, geometric, P.D.E.'s and physical applications. In particular, we prove Hessian comparison theorems and Laplacian comparison theorem under varied radial Ricci curvature or radial curvature assumptions, generalizing and extending the work of Han-Li-Ren-Wei, and Wei. We also extend the notion of function or differential form growth to bundle-valued differential form growth of various types and discuss their interrelationship. These provide tools in extending the notion, integrability and decomposition of generalized harmonic forms to those of bundle-valued generalized harmonic forms, introducing Condition W for bundle-valued differential forms, and proving duality theorem and unity theorem, generalizing the work of Andreotti and Vesentini, and Wei. We then apply Hessian and Laplacian comparison theorems to obtain comparison theorems in mean curvature, generalized sharp Caffarelli-Kohn-Nirenberg type inequalities, embedding theorem for weighted Sobolev spaces, geometric differential-integral inequalities, generalized sharp Hardy type inequalities on Riemannian manifolds, monotonicity formulas and vanishing theorems for differential forms of degree $k$ with values in vector bundles, such as $F$-Yang Mills fields (when $F$ is the identity map, they are Yang-Mills fields), generalized Yang-Mills-Born-Infeld fields on manifolds, Liouville type theorems for $F$-harmonic maps, and Dirichlet problems on starlike domains for vector bundle valued differential $1$-forms and $F$-harminic maps, etc., 61 pages, to appear in Science China Mathematics

We study the moduli space of Yang--Mills connections on bundles over a conformally compact manifold $\overline{M}$. We prove that, for every Yang--Mills connection $A$ that satisfies an appropriate nondegeneracy condition, and for every small deformation $\gamma$ of $A_{|\partial\overline{M}}$, there is a Yang--Mills connection in the interior that extends $A_{|\partial\overline{M}}+\gamma$. As a corollary, we confirm an expectation of Witten mentioned in his foundational paper about holography [arXiv:hep-th/9802150]., Comment: Replaced with fully revised version. The preliminary sections containing preliminary material have been reduced, and now contain only the background tools needed to prove the main theorem; the main theorem (Theorem 63) and its proof are essentially unchanged; Theorem 65 has been removed for the sake of brevity; Theorem 75 has been removed while I try and fix a gap in the proof

We study a problem of isometric compact 2-step nilmanifolds ${M}/\Gamma $ using some information on their geodesic flows, where $M$ is a simply connected 2-step nilpotent Lie group with a left invariant metric and $\Gamma $ is a cocompact discrete subgroup of isometries of $M$. Among various works concerning this problem, we consider the algebraic aspect of it. In fact, isometry groups of simply connected Riemannian manifolds can be characterized in a purely algebraic way, namely by normalizers. So, suitable factorization of normalizers and expression of a vector bundle as an associated fiber bundle to a principal bundle, lead us to a general framework, namely groupoids. In this way, drawing upon advanced ingredients of Lie groupoids, normal subgroupoid systems and other notions, not only an answer in some sense to our rigidity problem has been given, but also the dependence between normalizers, automorphisms and especially almost inner automorphisms, has been clarified.

A new symmetry of $(1,0)$ supersymmetric non-linear $\sigma$-models in two dimensions with Fermi and mass sectors is introduced. It is a generalisation of the so-called special holonomy $W$-symmetry of Howe and Papadopoulos associated with structure group reductions of the target space $\mathcal{M}$. Our symmetry allows in particular non-trivial flux and instanton-like connections on vector bundles over $\mathcal{M}$. We also investigate potential anomalies and show that cohomologically non-trivial terms in the quantum effective action are invariant under a corrected version of our symmetry. Consistency with heterotic supergravity at first order in $\alpha'$ is manifest and discussed., Comment: 26 pages. Added missing sentence above (4.21)

We introduce a formal $\hbar$-differential operator $\Delta$ that generates higher Koszul brackets on the algebra of (pseudo)differential forms on a $P_{\infty}$-manifold. Such an operator was first mentioned by Khudaverdian and Voronov in \texttt{arXiv:1808.10049}. (This operator is an analogue of the Koszul--Brylinski boundary operator $\partial_P$ which defines Poisson homology for an ordinary Poisson structure.) Here we introduce $\Delta=\Delta_P$ by a different method and establish its properties. We show that this BV type operator generating higher Koszul brackets can be included in a one-parameter family of BV type formal $\hbar$-differential operators, which can be understood as a quantization of the cotangent $L_{\infty}$-bialgebroid. We obtain symmetric description on both $\Pi TM$ and $\Pi T^*M$. For the purpose of the above, we develop in detail a theory of formal $\hbar$-differential operators and also of operators acting on densities on dual vector bundles. In particular, we have a statement about operators that can be seen as a quantization of the Mackenzie--Xu canonical diffeomorphism. Another interesting feature is that we are able to introduce a grading, not a filtration, on our algebras of operators. When operators act on objects on vector bundles, we obtain a bi-grading.

Complex structure moduli of a Calabi-Yau threefold in $N=1$ supersymmetric heterotic compactifications can be stabilized by holomorphic vector bundles. The stabilized moduli are determined by a computation of Atiyah class. In this paper, we study how this mechanism work in the context of elliptically fibered Calabi-Yau manifolds where complex structure moduli space contains two kinds of moduli, ones from base and ones from fibration. With spectral cover bundles, we find three types of situations when holomorphicity of bundles is determined by algebraic cycles supported on special choice of complex structure, which allows us to stabilize both of these two moduli. We present concrete examples for each type and develop practical tools to analyze the stabilized moduli. Finally, by checking the holomorphicity of the four-flux and/or local Higgs bundle data in F-theory, we briefly study the dual complex structure moduli stabilization scenarios., 27 pages

Let $Q\to M$ be a principal $G$-bundle, and $B_0$ a connection on $Q$. We introduce an infinitesimal homogeneity condition for sections in an associated vector bundle $Q\times_GV$ with respect to $B_0$, and, inspired by the well known Ambrose-Singer theorem, we prove the existence of a connection which satisfies a system of parallelism conditions. We explain how this general theorem can be used to prove the known Ambrose-Singer type theorems by an appropriate choice of the initial system of data.We also obtain new applications, which cannot be obtained using the known formalisms, e.g. a classification theorem for locally homogeneous spinors. Finally we introduce natural local homogeneity and local symmetry conditions for triples $(g,P\stackrel{p}{\to} M,A)$ consisting of a Riemannian metric on $M$, a principal bundle on $M$, and a connection on $P$. Our main results concern locally homogeneous and locally symmetric triples, and they can be viewed as bundle versions of the Ambrose-Singer and Cartan theorem., Comment: 32 pages

The goal of this paper is the study of simple rank 2 parabolic vector bundles over a 2 2 -punctured elliptic curve C C . We show that the moduli space of these bundles is a non-separated gluing of two charts isomorphic to P 1 × P 1 \mathbb {P}^1 \times \mathbb {P}^1 . We also showcase a special curve Γ \Gamma isomorphic to C C embedded in this space, and this way we prove a Torelli theorem. This moduli space is related to the moduli space of semistable parabolic bundles over P 1 \mathbb {P}^1 via a modular map which turns out to be the 2:1 cover ramified in Γ \Gamma . We recover the geometry of del Pezzo surfaces of degree 4 and we reconstruct all their automorphisms via elementary transformations of parabolic vector bundles.

In the semiclassical limit$$\hbar \rightarrow 0$$ħ→0, we analyze a class of self-adjoint Schrödinger operators$$H_\hbar = \hbar ^2 L + \hbar W + V\cdot {\mathrm {id}}_{\mathscr {E}}$$Hħ=ħ2L+ħW+V·idEacting on sections of a vector bundle$${\mathscr {E}}$$Eover an oriented Riemannian manifoldMwhereLis a Laplace type operator,Wis an endomorphism field and the potential energyVhas non-degenerate minima at a finite number of points$$m^1,\ldots m^r \in M$$m1,…mr∈M, called potential wells. Using quasimodes of WKB-type near$$m^j$$mjfor eigenfunctions associated with the low lying eigenvalues of$$H_\hbar $$Hħ, we analyze the tunneling effect, i.e. the splitting between low lying eigenvalues, which e.g. arises in certain symmetric configurations. Technically, we treat the coupling between different potential wells by an interaction matrix and we consider the case of a single minimal geodesic (with respect to the associated Agmon metric) connecting two potential wells and the case of a submanifold of minimal geodesics of dimension$$\ell + 1$$ℓ+1. This dimension$$\ell $$ℓdetermines the polynomial prefactor for exponentially small eigenvalue splitting.

We investigate the symplectic geometric and differential geometric aspects of the moduli space of connections on a compact Riemann surface $X$. Fix a theta characteristic $K^{1/2}_X$ on $X$; it defines a theta divisor on the moduli space ${\mathcal M}$ of stable vector bundles on $X$ of rank $r$ degree zero. Given a vector bundle $E \in {\mathcal M}$ lying outside the theta divisor, we construct a natural holomorphic connection on $E$ that depends holomorphically on $E$. Using this holomorphic connection, we construct a canonical holomorphic isomorphism between the following two: \begin{enumerate} \item the moduli space $\mathcal C$ of pairs $(E, D)$, where $E\in {\mathcal M}$ and $D$ is a holomorphic connection on $E$, and \item the space ${\rm Conn}(\Theta)$ given by the sheaf of holomorphic connections on the line bundle on $\mathcal M$ associated to the theta divisor. \end{enumerate} The above isomorphism between $\mathcal C$ and ${\rm Conn}(\Theta)$ is symplectic structure preserving, and it moves holomorphically as $X$ runs over a holomorphic family of Riemann surfaces., Comment: Final version

We discuss various aspects of the positive kernel method of quantization of the one-parameter groups $$\tau _t \in \text{ Aut }(P,\vartheta )$$ τ t ∈ Aut ( P , ϑ ) of automorphisms of a G-principal bundle $$P(G,\pi ,M)$$ P ( G , π , M ) with a fixed connection form $$\vartheta $$ ϑ on its total space P. We show that the generator $${\hat{F}}$$ F ^ of the unitary flow $$U_t = e^{it {\hat{F}}}$$ U t = e i t F ^ being the quantization of $$\tau _t $$ τ t is realized by a generalized Kirillov–Kostant–Souriau operator whose domain consists of sections of some vector bundle over M, which are defined by a suitable positive kernel. This method of quantization applied to the case when $$G=\hbox {GL}(N,{\mathbb {C}})$$ G = GL ( N , C ) and M is a non-compact Riemann surface leads to quantization of the arbitrary holomorphic flow $$\tau _t^{\mathrm{hol}} \in \text{ Aut }(P,\vartheta )$$ τ t hol ∈ Aut ( P , ϑ ) . For the above case, we present the integral decompositions of the positive kernels on $$P\times P$$ P × P invariant with respect to the flows $$\tau _t^{\mathrm{hol}}$$ τ t hol in terms of the spectral measure of $${\hat{F}}$$ F ^ . These decompositions generalize the ones given by Bochner’s Theorem for the positive kernels on $${\mathbb {C}} \times {\mathbb {C}}$$ C × C invariant with respect to the one-parameter groups of translations of complex plane.

We prove the Kobayashi-Hitchin correspondence and the approximate Kobayashi-Hitchin correspondence for twisted holomorphic vector bundles on compact K\"ahler manifolds. More precisely, if $X$ is a compact manifold and $g$ is a Gauduchon metric on $X$, a twisted holomorphic vector bundle on $X$ is $g-$polystable if and only if it is $g-$Hermite-Einstein, and if $X$ is a compact K\"ahler manifold and $g$ is a K\"ahler metric on $X$, then a twisted holomorphic vector bundle on $X$ is $g-$semistable if and only if it is approximate $g-$Hermite-Einstein., Comment: 119 pages

Fix $n\geq 5$ general points $p_1, \dots , p_n\in{\mathbb{P}}^1$ and a weight vector ${\mathcal{A}} = (a_{1}, \dots , a_{n})$ of real numbers $0 \leq a_{i} \leq 1$. Consider the moduli space $\mathcal{M}_{{\mathcal{A}}}$ parametrizing rank two parabolic vector bundles with trivial determinant on $\big ({\mathbb{P}}^1, p_1,\dots , p_n\big )$ that are semistable with respect to ${\mathcal{A}}$. Under some conditions on the weights, we determine and give a modular interpretation for the automorphism group of the moduli space $\mathcal{M}_{{\mathcal{A}}}$. It is isomorphic to $\left (\frac{\mathbb{Z}}{2\mathbb{Z}}\right )^{k}$ for some $k\in \{0,\dots , n-1\}$ and is generated by admissible elementary transformations of parabolic vector bundles. The largest of these automorphism groups, with $k=n-1$, occurs for the central weight ${\mathcal{A}}_{F}= \left (\frac{1}{2},\dots ,\frac{1}{2}\right )$. The corresponding moduli space ${\mathcal M}_{{\mathcal{A}}_F}$ is a Fano variety of dimension $n-3$, which is smooth if $n$ is odd, and has isolated singularities if $n$ is even.

As a natural extension of the theory of uniform vector bundles on Fano manifolds, we consider uniform principal bundles, and study them by means of the associated flag bundles, as their natural projective geometric realizations. In this paper, we develop the necessary background and prove some theorems that are flag bundle counterparts of some of the central results in the theory of uniform vector bundles.

A DFT algebroid is a special case of the metric (or Vaisman) algebroid, shown to be relevant in understanding the symmetries of double field theory. In particular, a DFT algebroid is a structure defined on a vector bundle over doubled spacetime equipped with the C-bracket of double field theory. In this paper we give the definition of a DFT algebroid as a curved $L_\infty$-algebra and show how implementation of the strong constraint of double field theory can be formulated as an $L_\infty$-algebra morphism. Our results provide a useful step towards coordinate invariant descriptions of double field theory and the construction of the corresponding sigma-model.

In this paper we study the relationship between three compactifications of the moduli space of Hermitian-Yang-Mills connections on a fixed Hermitian vector bundle over a projective algebraic manifold of arbitrary dimension. Via the Donaldson-Uhlenbeck-Yau theorem, this space is analytically isomorphic to the moduli space of stable holomorphic vector bundles, and as such it admits an algebraic compactification by Gieseker-Maruyama semistable torsion-free sheaves. A recent construction due to the first and third authors gives another compactification as a moduli space of slope semistable sheaves. In the present article, following fundamental work of Tian generalising the analysis of Uhlenbeck and Donaldson in complex dimension two, we define a gauge theoretic compactification by adding certain ideal connections at the boundary. Extending work of Jun Li in the case of bundles on algebraic surfaces, we exhibit comparison maps from the sheaf theoretic compactifications and prove their continuity. The continuity, together with a delicate analysis of the fibres of the map from the moduli space of slope semistable sheaves allows us to endow the gauge theoretic compactification with the structure of a complex analytic space., Comment: minor changes to the exposition based on referee's comments; final version to appear in Geometry & Topology; 95 pages

The problem of computing quantum mechanical propagators can be recast as a computation of a Wilson line operator for parallel transport by a flat connection acting on a vector bundle of wave functions. In this picture, the base manifold is an odd-dimensional symplectic geometry, or quite generically a contact manifold that can be viewed as a ``phase-spacetime,'' while the fibers are Hilbert spaces. This approach enjoys a ``quantum Darboux theorem'' that parallels the Darboux theorem on contact manifolds which turns local classical dynamics into straight lines. We detail how the quantum Darboux theorem works for anharmonic quantum potentials. In particular, we develop a novel diagrammatic approach for computing the asymptotics of a gauge transformation that locally makes complicated quantum dynamics trivial.