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We present a concise explicit expression for the heat trace coefficients of spheres. Our formulas yield certain combinatorial identities which are proved following ideas of D. Zeilberger. In particular, these identities allow to recover in a surprising way some known formulas for the heat trace asymptotics. Our approach is based on a method for computation of heat invariants developed in math.DG/9905073.
According to a conjecture of E. Witten proved by M. Kontsevich, a certain generating function for intersection indices on the Deligne -- Mumford moduli spaces of Riemann surfaces coincides with a certain tau-function of the KdV hierarchy. The generating function is naturally generalized under the name the {\em total descendent potential} in the theory of Gromov -- Witten invariants of symplectic manifolds. The papers arXiv: math.AG/0108100 and arXive: math.DG/0108160 contain two equivalent constructions, motivated by some results in Gromov -- Witten theory, which associate a total descendent potential to any semisimple Frobenius structure. In this paper, we prove that in the case of K.Saito's Frobenius structure on the miniversal deformation of the $A_{n-1}$-singularity, the total descendent potential is a tau-function of the $n$KdV hierarchy. We derive this result from a more general construction for solutions of the $n$KdV hierarchy from $n-1$ solutions of the KdV hierarchy.
The cyclic (co)homology of Hopf algebras is defined by Connes and Moscovici [math.DG/9806109] and later extended by Khalkhali et.al [math.KT/0306288] to admit stable anti-Yetter-Drinfeld coefficient module/comodules. In this paper we will show that one can further extend the cyclic homology of Hopf algebras with coefficients non-trivially. The new homology, called the bialgebra cyclic homology, admits stable coefficient module/comodules, dropping the anti-Yetter-Drinfeld condition. This fact allows the new homology to use bialgebras, not just Hopf algebras. We will also give computations for bialgebra cyclic homology of the Hopf algebra of foliations of codimension $n$ and the quantum deformation of an arbitrary semi-simple Lie algebra with several stable but non-anti-Yetter-Drinfeld coefficients.
When ${\frak g}$ is a complex semisimple Lie algebra, we study the variety ${\mathcal L}$ of subalgebras of ${\frak g}\oplus{\frak g}$ that are maximally isotropic with respect to $K_1 - K_2$, where $K_i$ is the Killing form on the ith factor. We show the irreducible components of ${\mathcal L}$ are smooth, classify them in terms of the generalized Belavin-Drinfeld triples introduced by Schiffmann, and relate them to orbits of the adjoint group $G\times G$. Building on ideas of Yakimov, we give a new proof of Karolinsky's classification of the diagonal $G$-orbits in ${\mathcal L}$. Our proof enables us to compute of the normalizer in ${\frak g}$ of a subalgebra in ${\mathcal L}$ under the diagonal action. As a consequence, we recover the classification of Belavin-Drinfeld triples. By results of math.DG/9909005, ${\mathcal L}$ is a Poisson variety and we determine the rank of the symplectic leaf at each point of ${\mathcal L}$ in terms of combinatorial data and relate the symplectic leaves to intersections of orbits of subgroups of $G\times G$. As a consequence, an intrinsically defined Poisson structure on each conjugacy class on $G$ has an open symplectic leaf and we determine the rank at each point of the conjugacy class.
This is a short version of math.DG/0505537. For an acyclic representation of the fundamental group of a compact oriented odd-dimensional manifold, which is close enough to a unitary representation, we define a refinement of the Ray-Singer torsion associated to this representation. This new invariant can be viewed as an analytic counterpart of the refined combinatorial torsion introduced by Turaev. The refined analytic torsion is a holomorphic function of the representation of the fundamental group. When the representation is unitary, the absolute value of the refined analytic torsion is equal to the Ray-Singer torsion, while its phase is determined by the eta-invariant. The fact that the Ray-Singer torsion and the eta-invariant can be combined into one holomorphic function allows to use methods of complex analysis to study both invariants.
We introduce generalized calibrations that take into account the gauge field on the D-brane so that calibrated submanifolds minimize the Dirac-Born-Infeld energy. We establish the calibration bound and show that the calibration form is closed in a supersymmetric background with non-vanishing NS-NS 3-form H and dilaton. We show that the calibration conditions are equivalent to the existence of unbroken supersymmetry on the D-brane. We study the problem of supersymmetric D-branes in the presence of non-vanishing H also from the world-sheet approach and find exactly the same conditions. Finally, we show that our notion of generalized calibrations is equivalent to the calibrations introduced in the context of generalized Calabi-Yau geometry in math.DG/0401221.
An embedded curve in a symplectic surface $Σ\subset X$ defines a smooth deformation space $\mathcal{B}$ of nearby embedded curves. A key idea of Kontsevich and Soibelman arXiv:1701.09137 [math.AG], is to equip the symplectic surface $X$ with a foliation in order to study the deformation space $\mathcal{B}$. The foliation, together with a vector space $V_Σ$ of meromorphic differentials on $Σ$, endows an embedded curve $Σ$ with the structure of the initial data of topological recursion, which defines a collection of symmetric tensors on $V_Σ$. Kontsevich and Soibelman define an Airy structure on $V_Σ$ to be a formal quadratic Lagrangian $\mathcal{L}\subset T^*(V_Σ^*)$ which leads to an alternative construction of the tensors of topological recursion. In this paper we produce a formal series $θ$ on $\mathcal{B}$ of meromorphic differentials on $Σ$ which takes it values in $\mathcal{L}$, and use this to produce the Donagi-Markman cubic from a natural cubic tensor on $V_Σ$, giving a generalisation of a result of Baraglia and Huang, arXiv:1707.04975 [math.DG].
Using double 2+2 and 3+1 nonholonomic fibrations on Lorentz manifolds, we extend the concept of W-entropy for gravitational fields in the general relativity, GR, theory. Such F- and W-functionals were introduced in the Ricci flow theory of three dimensional, 3-d, Riemannian metrics by G. Perelman, arXiv: math.DG/0211159. Nonrelativistic 3-d Ricci flows are characterized by associated statistical thermodynamical values determined by W--entropy. Generalizations for geometric flows of 4-d pseudo-Riemannian metrics are considered for models with local thermodynamical equilibrium and separation of dissipative and non-dissipative processes in relativistic hydrodynamics. The approach is elaborated in the framework of classical filed theories (relativistic continuum and hydrodynamic models) without an underlying kinetic description which will be elaborated in other works. The 3+1 splitting allows us to provide a general relativistic definition of gravitational entropy in the Lyapunov-Perelman sense. It increases monotonically as structure forms in the Universe. We can formulate a thermodynamic description of exact solutions in GR depending, in general, on all spacetime coordinates. A corresponding 2+2 splitting with nonholonomic deformation of linear connection and frame structures is necessary for generating in very general form various classes of exact solutions of the Einstein and general relativistic geometric flow equations. Finally, we speculate on physical macrostates and microstate interpretations of the W-entropy in GR, geometric flow theories and possible connections to string theory (a second unsolved problem also contained in Perelman's works) in the Polyakov's approach.
For each affine Kac-Moody algebra $X_n^{(r)}$ of rank $\ell$, $r=1,2$, or $3$, and for every choice of a vertex $c_m$, $m=0,\dots,\ell$, of the corresponding Dynkin diagram, by using the matrix-resolvent method we define a gauge-invariant tau-structure for the associated Drinfeld-Sokolov hierarchy and give explicit formulas for generating series of logarithmic derivatives of the tau-function in terms of matrix resolvents, extending the results of [Mosc. Math. J. 21 (2021), 233-270, arXiv:1610.07534] with $r=1$ and $m=0$. For the case $r=1$ and $m=0$, we verify that the above-defined tau-structure agrees with the axioms of Hamiltonian tau-symmetry in the sense of [Adv. Math. 293 (2016), 382-435, arXiv:1409.4616] and [arXiv:math.DG/0108160].
Let $(X,L_{X})$ be an $n$-dimensional polarized manifold. Let $D$ be a smooth hypersurface defined by a holomorphic section of $L_{X}$. In this paper, we study the existence of a complete scalar-flat Kähler metric on $X \setminus D$ on the assumption that $D$ has a constant positive scalar curvature Kähler metric.
We continue to explore, in the context of a toy model, the hypothesis that the interacting universe we see around us could result from single particle (undergraduate) quantum mechanics via a novel spontaneous symmetry breaking (SSB) acting at the level of probability distributions on Hamiltonians (rather than on states as is familiar from both Ginzburg-Landau superconductivity and the Higgs mechanism). In an earlier paper [7] we saw qubit structure emerge spontaneously on $\mathbb{C}^4$ and $\mathbb{C}^8$, and in this work we see $\mathbb{C}^6$ spontaneously decomposing as $\mathbb{C}^2\otimes \mathbb{C}^3$ and very curiously $\mathbb{C}^5$ (and $\mathbb{C}^7$) splitting off one (one or three) directions and then factoring. This evidence provides additional support for the broad hypothesis: Nature will seek out tensor decompositions where none are present. We consider how this finding may form a basis for the origins of interaction and ask if it can be related to established foundational discussions such as string theory.
On the universal bundle of unit spinors we study a natural energy functional whose critical points, if dim M \geq 3, are precisely the pairs (g, ϕ) consisting of a Ricci-flat Riemannian metric g together with a parallel g-spinor ϕ. We investigate the basic properties of this functional and study its negative gradient flow, the so-called spinor flow. In particular, we prove short-time existence and uniqueness for this flow.
In this paper, we prove that for any Kähler metrics $ω_0$ and $χ$ on $M$, there exists $ω_φ=ω_0+\sqrt{-1}\partial\bar\partialφ>0$ satisfying the J-equation $\mathrm{tr}_{ω_φ}χ=c$ if and only if $(M,[ω_0],[χ])$ is uniformly J-stable. As a corollary, we can find many constant scalar curvature Kähler metrics with $c_1<0$. Using the same method, we also prove a similar result for the deformed Hermitian-Yang-Mills equation when the angle is in $(\frac{nπ}{2}-\fracπ{4},\frac{nπ}{2})$.
We explore the nonperturbative aspects of the chiral algebras of N = (0,2) sigma models, which perturbatively are intimately related to the theory of chiral differential operators (CDOs). The grading by charge and scaling dimension is anomalous if the first Chern class of the target space is nonzero. This has some nontrivial consequences for the chiral algebra. As an example, we study the case where the target space is CP^1, and show that worldsheet instantons trivialize the chiral algebra entirely. Consequently, supersymmetry is spontaneously broken in this model. We then turn to a closer look at the supersymmetry breaking from the viewpoint of Morse theory on loop space. We find that instantons interpolate between pairs of perturbative supersymmetric states with different fermionic numbers, hence lifting them out of the supersymmetric spectrum. Our results reveal that a "quantum" deformation of the geometry of the target space leads to a trivialization of the kernels of certain twisted Dirac operators on CP^1.
Let a compact Lie group act isometrically on a non-collapsing sequence of compact Alexandrov spaces with fixed dimension and uniform lower curvature and upper diameter bounds. If the sequence of actions is equicontinuous and converges in the equivariant Gromov--Hausdorff topology, then the limit space is equivariantly homeomorphic to spaces in the tail of the sequence. As a consequence, the class of Riemannian orbifolds of dimension $n$ defined by a lower bound on the sectional curvature and the volume and an upper bound on the diameter has only finitely many members up to orbifold homeomorphism. Furthermore, any class of isospectral Riemannian orbifolds with a lower bound on the sectional curvature is finite up to orbifold homeomorphism.
The critical CR GJMS operator on a strictly pseudoconvex CR manifold is a non-hypoelliptic CR invariant differential operator. We prove that, under the embeddability assumption, it is essentially self-adjoint and has closed range. Moreover, its spectrum is discrete, and the eigenspace corresponding to each non-zero eigenvalue is a finite-dimensional subspace of the space of smooth functions. As an application, we obtain a necessary and sufficient condition for the existence of a contact form with zero CR $Q$-curvature.
We study the fundamental group of an open $n$-manifold $M$ of nonnegative Ricci curvature with additional stability condition on $\widetilde{M}$, the Riemannian universal cover of $M$. We prove that if any tangent cone of $\widetilde{M}$ at infinity is a metric cone, whose cross-section is sufficiently Gromov-Hausdorff close to a prior fixed metric space, then $π_1(M)$ is finitely generated and contains a normal abelian subgroup of finite index; if in addition $\widetilde{M}$ has Euclidean volume growth of constant at least $L$, then we can bound the index of that abelian subgroup in terms of $n$ and $L$. In particular, our result implies that if $\widetilde{M}$ has Euclidean volume growth of constant at least $1-ε(n)$, then $π_1(M)$ is finitely generated and $C(n)$-abelian.
Submission was withdrawn by authors. arXiv:math/0305140