Our ab initio all-electron fully relativistic Dirac–Fock (DF) and Dirac–Fock–Breit–Gaunt (DFBG) and nonrelativistic (NR) Hartree–Fock (HF) calculations for seaborgium hexacarbonyl Sg(CO)6 predict atomization energies (Ae) of 68.81, 69.28, and 67.69 eV, respectively, at the corresponding optimized octahedral geometry. However, our DF, DFBG, and NR HF calculations for the isomer Sg(OC)6 yield atomization energies of 64.30, 64.77, and 62.62 eV, respectively, at the optimized geometry for this species. The optimized Sg–C and C–O bond distances (in Å) for octahedral Sg(CO)6 using our DF (NR) calculations are 2.15 (2.32) and 1.11 (1.11), respectively. However, the optimized Sg–O and O–C bond distances (in Å) for the isomer octahedral Sg(OC)6 obtained with our DF (NR) calculations are 2.80 (2.73) and 1.10 (1.11), respectively. Our prediction of the greater stability of Sg(CO)6 isomer at both the relativistic (DF and DFBG) and the NR HF levels of theory lends further support to the detection of Sg(CO)6 in the state-of-the-art gas-phase experimental studies of the carbonyl complex of seaborgium reported by Even et al. [Science 345(6203), 1491 (2014)].
Abstract In Ozsvath and Szabo (Holomorphic triangles and invariants for smooth four-manifolds, math. SG/0110169, 2001), we introduced absolute gradings on the three-manifold invariants developed in Ozsvath and Szabo (Holomorphic disks and topological invariants for closed three-manifolds, math.SG/0101206, Ann. of Math. (2001), to appear). Coupled with the surgery long exact sequences, we obtain a number of three- and four-dimensional applications of this absolute grading including strengthenings of the “complexity bounds” derived in Ozsvath and Szabo (Holomorphic disks and three-manifold invariants: properties and applications, math.SG/0105202, Ann. of Math. (2001), to appear), restrictions on knots whose surgeries give rise to lens spaces, and calculations of HF + for a variety of three-manifolds. Moreover, we show how the structure of HF + constrains the exoticness of definite intersection forms for smooth four-manifolds which bound a given three-manifold. In addition to these new applications, the techniques also provide alternate proofs of Donaldson's diagonalizability theorem and the Thom conjecture for CP 2 .
We summarise recent work (arXiv:2203.07405 [math.SG]) on the classical result of Kirillov that any simply-connected homogeneous symplectic space of a connected group $G$ is a hamiltonian $\widehat{G}$-space for a one-dimensional central extension $\widehat{G}$ of $G$, and is thus (by a result of Kostant) a cover of a coadjoint orbit of $\widehat{G}$. We emphasise that existing proofs in the literature assume that $G$ is simply-connected and that this assumption can be removed by application of a theorem of Neeb. We also interpret Neeb's theorem as relating the integrability of one-dimensional central extensions of Lie algebras to the integrability of an associated Chevalley--Eilenberg 2-cocycle.
The Abelian/non-Abelian correspondence for cohomology (Martin in Symplectic quotients by a nonabelian group and by its maximal torus. arXiv:math/0001002 [math.SG], 2000; Ellingsrud–Strømme in On the chow ring of a geometric quotient, 1989) gives a novel description of the cohomology ring of the Grassmannian. We show that the natural generalization of this result to small quantum cohomology applies to Fano quiver flag varieties. Quiver flag varieties are generalisations of type A flag varieties. As a corollary, we see that the Gu–Sharpe mirror to a Fano quiver flag variety computes its quantum cohomology. The second focus of the paper is on applying this description to computations inside the classical and quantum cohomology rings. The rim-hook rule for quantum cohomology of the Grassmannian allows one to reduce quantum calculations to classical calculations in the cohomology of the Grassmannian. We use the Abelian/non-Abelian correspondence to prove a rim-hook removal rule for the cohomology and quantum cohomology (in the Fano case) of quiver flag varieties. This result is new even in the flag case. This gives an effective way of computing products in the (quantum) cohomology ring, reducing computations to that in the cohomology ring of the Grassmannian.
We study the classical and semiclassical time evolutions of subsystems of a Hamiltonian system; this is done using a generalization of Heller’s thawed Gaussian approximation introduced by Littlejohn. The key tool in our study is an extension of Gromov’s “principle of the symplectic camel” obtained in collaboration with Dias, de Gosson, and Prata [arXiv:1911.03763v1 [math.SG] (2019)]. This extension says that the orthogonal projection of a symplectic phase space ball on a phase space with a smaller dimension also contains a symplectic ball with the same radius. In the quantum case, the radii of these symplectic balls are taken equal to ℏ and represent the ellipsoids of minimum uncertainty, which we called “quantum blobs” in previous work.
. This paper is devoted to studying some properties of the Courant algebroids: we explain the so-called ”conducting bundle construction” and use it to attach the Courant algebroid to Dixmier-Douady gerbe (following ideas of P. Severa). We remark that WZNW-Poisson condition of Klimcik and Strobl (math.SG/0104189) is the same as Dirac structure in some particular Courant algebroid. We propose the construction of the Lie algebroid on the loop space starting from the Lie algebroid on the manifold and conjecture that this construction applied to the Dirac structure above should give the Lie algebroid of symmetries in the WZNW-Poisson σ -model, we show that it is indeed true in the particular case of Poisson σ -model.
In [L-Y5] (D(6): arXiv:1003.1178 [math.SG]) we introduced the notion of Azumaya C 1 manifolds with a fundamental module and morphisms therefrom to a complex manifold. In the current sequel, we use this notion to give a prototypical definition of supersymmetric D-branes of A-type (i.e. A-branes) – in an appropriate region of the Wilson’s theory-space of string theory – as special Lagrangian morphisms with a unitary, minimally flat connectionwith-singularity. This merges Donaldson’s picture of special Lagrangian submanifolds (1999) and the Polchinski-Grothendieck Ansatz for D-branes in a Calabi-Yau space (Sec. 2.1). The Higgsing/un-Higgsing and the large- vs. small-brane wrapping of A-branes in string theory can be achieved via deformations of such morphisms (Sec. 2.2 and Sec. 2.3). For the case of Calabi-Yau 3-folds, classical results of Alexander (1920), Hilden (1974) and Montesinos (1976), Thurston (1982), and Hilden-Lozano-Montesinos (1983) on 3-manifolds branchedcovering S 3 implies that any embedded special Lagrangian submanifold with a complex vector bundle with a unitary flat connection on a Calabi-Yau 3-fold is the image of a special Lagrangian morphism from an Azumaya 3-sphere with a fundamental module, with a unitary minimally flat connection. This suggests a genus-like expansion of the path-integral of D3-branes in type IIB string theory compactified on Calabi-Yau 3-folds that resembles the genus expansion of the path-integral of strings (Sec. 2.4.2). Similarly, for the path-integral of D2-branes and M2-branes respectively. In Sec. 3, we use the technical results of Joyce (2002-2003) on desingularizations of special Lagrangian submanifolds with conical singularities to explain how supersymmetric D3-branes thus defined can be driven and re-assemble under a reverse split attractor flow at a point on the wall of marginal stability in Type IIB superstring theory compactified on varying Calabi-Yau 3-folds, studied by Denef (2001). This last section is to be read alongside the works [De3] (arXiv:hep-th/0107152) of Denef and [Joy3: V] (arXiv:math.DG/0303272) of Joyce. To cover the basic type of deformations of morphisms from Azumaya spaces in this note and its sequel, we discuss in Sec. 1 Morse cobordisms of manifolds and their promotion to Morse cobordisms of Azumaya manifolds with a fundamental module, and of morphisms from Azumaya manifolds to complex manifolds. The notion of cone of special Lagrangian cycles in a Calabi-Yau manifold is brought out in Sec. 2.4.1 for further study. A summary of the needed results from Joyce is given in the appendix.
This article analyzes the interplay between symplectic geometry in dimension four and the invariants for smooth four-manifolds constructed using holomorphic triangles introduced in math.SG/0110169. Specifically, we establish a non-vanishing result for the invariants of symplectic four-manifolds, which leads to new proofs of the indecomposability theorem for symplectic four-manifolds and the symplectic Thom conjecture. As a new application, we generalize the indecomposability theorem to splittings of four-manifolds along a certain class of three-manifolds obtained by plumbings of spheres. This leads to restrictions on the topology of Stein fillings of such three-manifolds.
Let $\Delta$ be a Delzant polytope in ${\mathbb R}^n$ and ${\bf b}\in{\mathbb Z}^n$. Let $E$ denote the symplectic fibration over $S^2$ determined by the pair $(\Delta, {\bf b})$. We prove the equivalence between the fact that $(\Delta, {\bf b})$ is a mass linear pair (D. McDuff, S. Tolman, {\em Polytopes with mass linear functions, part I.} {\tt arXiv:0807.0900 [math.SG]}) and the vanishing of a characteristic number of $E$ in the following cases: When $\Delta$ is a $\Delta_{n-1}$ bundle over $\Delta_1$; when $\Delta$ is the polytope associated with the one point blow up of ${\mathbb C}P^n$; and when $\Delta$ is the polytope associated with a Hirzebruch surface.
We show that the L^p norms of random sequences {s_N} of L^2 normalized holomorphic sections of increasing powers of an ample line bundle on a compact Kahler manifold are almost surely bounded for 2<p< infinity, and are almost surely O((log N)^{1/2}) for p= infinity. This estimate also holds for almost-holomorphic sections of positive line bundles on symplectic manifolds (in the sense of math.SG/0212180) and we give almost sure bounds for the C^k norms. Our methods involve asymptotics of Bergman-Szego kernels and the concentration of measure phenomenon.
The paper explores some algebraic constructions arising in the theory of Lefschetz fibrations. Specifically, it covers in a fair amount of detail the algebraic issues outlined in ``Symplectic homology as Hochschild homology'' (math.SG/0609037). We also explain how the theory works when applied to a simple example, namely the Landau-Ginzburg mirror of P^2. Version 2: revised, many technical assumptions dropped, statement of one of the main results improved by using dg quotients. I changed the title accordingly.
In math.SG/0605587, we studied Yang-Mills functional on the space of connections on a principal G_R-bundle over a closed, connected, nonorientable surface, where G_R is any compact connected Lie group. In this sequel, we generalize the discussion in "The Yang-Mills equations over Riemann surfaces" by Atiyah and Bott, and math.SG/0605587. We obtain explicit descriptions (as representation varieties) of Morse strata of Yang-Mills functional on orientable and nonorientable surfaces for non-unitary classical groups SO(n) and Sp(n). It turns out to be quite different from the unitary case. we use Laumon and Rapoport's method in "The Langlands lemma and the Betti numbers of stacks of G-bundles on a curve" to invert the Atiyah-Bott recursion relation, and write down explicit formulas of rational equivariant Poincaré series of the semistable stratum of the space of holomorphic structures on a principal $SO(n,\bC)$-bundle or a principal $Sp(n,\bC)$-bundle.
In this paper we provide a criterion for the quasi-autonomous Hamiltonian path (``Hofer's geodesic'') on arbitrary closed symplectic manifolds $(M,\omega)$ to be length minimizing in its homotopy class in terms of the spectral invariants $\rho(G;1)$ that the author has recently constructed (math.SG/0206092). As an application, we prove that any autonomous Hamiltonian path on arbitrary closed symplectic manifolds is length minimizing in {\it its homotopy class} with fixed ends, when it has no contractible periodic orbits {\it of period one}, has a maximum and a minimum point which are generically under-twisted and all of its critical points are nondegenerate in the Floer theoretic sense. This is a sequel to the papers math.SG/0104243 and math.SG/0206092.