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1
Angella, Daniele, et Federico Alberto Rossi. « Cohomology of D-complex manifolds ». Differential Geometry and its Applications 30, no 5 (octobre 2012) : 530–47. http://dx.doi.org/10.1016/j.difgeo.2012.07.003.
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ANGELLA, DANIELE, et ADRIANO TOMASSINI. « ON THE COHOMOLOGY OF ALMOST-COMPLEX MANIFOLDS ». International Journal of Mathematics 23, no 02 (février 2012) : 1250019. http://dx.doi.org/10.1142/s0129167x11007604.
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Following [T.-J. Li and W. Zhang, Comparing tamed and compatible symplectic cones and cohomological properties of almost complex manifolds, Comm. Anal. Geom.17(4) (2009) 651–683], we continue to study the link between the cohomology of an almost-complex manifold and its almost-complex structure. In particular, we apply the same argument in [T.-J. Li and W. Zhang, Comparing tamed and compatible symplectic cones and cohomological properties of almost complex manifolds, Comm. Anal. Geom.17(4) (2009) 651–683] and the results obtained by [D. Sullivan, Cycles for the dynamical study of foliated manifolds and complex manifolds, Invent. Math.36(1) (1976) 225–255] to study the cone of semi-Kähler structures on a compact semi-Kähler manifold.
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FANG, FUQUAN, et XIAOCHUN RONG. « FIXED POINT FREE CIRCLE ACTIONS AND FINITENESS THEOREMS ». Communications in Contemporary Mathematics 02, no 01 (février 2000) : 75–86. http://dx.doi.org/10.1142/s0219199700000062.
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We prove a vanishing theorem of certain cohomology classes for an 2n-manifold of finite fundamental group which admits a fixed point free circle action. In particular, it implies that any Tk-action on a compact symplectic manifold of finite fundamental group has a non-empty fixed point set. The vanishing theorem is used to prove two finiteness results in which no lower bound on volume is assumed. (i) The set of symplectic n-manifolds of finite fundamental groups with curvature, λ ≤ sec ≤ Λ, and diameter, diam ; ≤ d, contains only finitely many diffeomorphism types depending only on n, λ, Λ and d. (ii) The set of simply connected n-manifolds (n ≤ 6) with λ ≤ sec ≤ Λ and diam ≤ d contains only finitely many diffeomorphism types depending only on n, λ, Λ and d.
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GILMER, PATRICK M. « REMARKS ON CONGRUENCE OF 3-MANIFOLDS ». Journal of Knot Theory and Its Ramifications 16, no 10 (décembre 2007) : 1357–60. http://dx.doi.org/10.1142/s021821650700583x.
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We give two proofs that the 3-torus is not weakly d-congruent to #3S1 × S2, if d > 2. We study how cohomology ring structure relates to weak congruence. We give an example of three 3-manifolds which are weakly 5-congruent but are not 5-congruent.
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Hai-bao, Duan. « Some Newman-type theorems for maps from Riemannian manifolds into manifolds ». Proceedings of the Royal Society of Edinburgh : Section A Mathematics 111, no 1-2 (1989) : 53–59. http://dx.doi.org/10.1017/s0308210500025002.
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SynopsisLetf: Mm→Nnbe a map from a Riemannianm-manifold(Mm, d)into ann-manifold Nn. The major purpose of this paper is to give a lower bound for the numberby examining the behaviour of the cohomology homomorphisms induced byf. This idea will be used to generalise the classical Newman theorem and present a geometric background for a well-known non-embedding theorem in topology.
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Chen, Tai-Wei, Chung-I. Ho et Jyh-Haur Teh. « Aeppli and Bott–Chern cohomology for bi-generalized Hermitian manifolds and d′d″-lemma ». Journal of Geometry and Physics 93 (juillet 2015) : 40–51. http://dx.doi.org/10.1016/j.geomphys.2015.03.006.
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JOHANSEN, ANDREI. « REALIZATION OF W1+∞ AND VIRASORO ALGEBRAS IN SUPERSYMMETRIC THEORIES ON FOUR MANIFOLDS ». Modern Physics Letters A 09, no 28 (14 septembre 1994) : 2611–22. http://dx.doi.org/10.1142/s0217732394002458.
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We demonstrate that a supersymmetric theory twisted on a Kähler four-manifold M= Σ1×Σ2, where Σ1,2 are 2-D Riemann surfaces, possesses a “left-moving” conformal stress tensor on Σ1 (Σ2) in the BRST cohomology. The central charge of the Virasoro algebra has a purely geometric origin and is proportional to the Euler characteristic of the Σ2 (Σ1) surface. This structure is shown to be invariant under renormalization group. We also give a representation of the algebra W1+∞ in terms of a free chiral supermultiplet.
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SUGIYAMA, KATSUYUKI. « THREE-POINT FUNCTIONS ON THE SPHERE OF CALABI-YAU d-FOLDS ». International Journal of Modern Physics A 11, no 02 (20 janvier 1996) : 229–52. http://dx.doi.org/10.1142/s0217751x96000110.
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Using mirror symmetry in Calabi-Yau manifolds M, we study three-point functions of A(M) model operators on the genus 0 Riemann surface in cases of one-parameter families of d-folds realized as Fermat type hypersurfaces embedded in weighted projective spaces and a two-parameter family of d-folds embedded in a weighted projective space Pd+1 [2,2,2,...,2,2,1,1] (2 (d + 1)). These three-point functions [Formula: see text] are expanded by indeterminates [Formula: see text] associated with a set of Kähler coordinates {tl}, and their expansion coefficients count the number of maps with a definite degree which map each of the three-points 0, 1 and ∞ on the world sheet on some homology cycle of M associated with a cohomology element. From these analyses, we can read the fusion structure of Calabi-Yau A(M) model operators. In our cases they constitute a subring of a total quantum cohomology ring of the A(M) model operators. In fact we switch off all perturbation operators on the topological theories except for marginal ones associated with Kähler forms of M. For that reason, the charge conservation of operators turns out to be a classical one. Furthermore, because their first Chern classes c1 vanish, their topological selection rules do not depend on the degree of maps (in particular, a nilpotent property of operators [Formula: see text] is satisfied). Then these fusion couplings {κl} are represented as some series adding up all degrees of maps.
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Kawamura, Kazuhiro. « Point derivations and cohomologies of Lipschitz algebras ». Proceedings of the Edinburgh Mathematical Society 62, no 4 (18 juillet 2019) : 1173–87. http://dx.doi.org/10.1017/s0013091519000142.
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AbstractFor a compact metric space (K, d), LipK denotes the Banach algebra of all complex-valued Lipschitz functions on (K, d). We show that the continuous Hochschild cohomology Hn(LipK, (LipK)*) and Hn(LipK, ℂe) are both infinite-dimensional vector spaces for each n ≥ 1 if the space K contains a certain infinite sequence which converges to a point e ∈ K. Here (LipK)* is the dual module of LipK and ℂe denotes the complex numbers with a LipK-bimodule structure defined by evaluations of LipK-functions at e. Examples of such metric spaces include all compact Riemannian manifolds, compact geodesic metric spaces and infinite compact subsets of ℝ. In particular, the (small) global homological dimension of LipK is infinite for every such space. Our proof uses the description of point derivations by Sherbert [‘The structure of ideals and point derivations in Banach algebras of Lipschitz functions’, Trans. Amer. Math. Soc.111 (1964), 240–272] and directly constructs non-trivial cocycles with the help of alternating cocycles of Johnson [‘Higher-dimensional weak amenability’, Studia Math.123 (1997), 117–134]. An alternating construction of cocycles on the basis of the idea of Kleshchev [‘Homological dimension of Banach algebras of smooth functions is equal to infinity’, Vest. Math. Mosk. Univ. Ser. 1. Mat. Mech.6 (1988), 57–60] is also discussed.
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JOHANSEN, A. « TWISTING OF N=1 SUSY GAUGE THEORIES AND HETEROTIC TOPOLOGICAL THEORIES ». International Journal of Modern Physics A 10, no 30 (10 décembre 1995) : 4325–57. http://dx.doi.org/10.1142/s0217751x9500200x.
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It is shown that D=4N=1 SUSY Yang-Mills theory with an appropriate supermultiplet of matter can be twisted on a compact Kähler manifold. The conditions for cancellation of anomalies of BRST charge are found. The twisted theory has an appropriate BRST charge. We find a nontrivial set of physical operators defined as classes of the cohomology of this BRST operator. We prove that the physical correlators are independent of the external Kähler metric up to a power of a ratio of two Ray-Singer torsions for the Dolbeault cohomology complex on a Kähler manifold. The correlators of local physical operators turn out to be independent of antiholomorphic coordinates defined with a complex structure on the Kähler manifold. However, a dependence of the correlators on holomorphic coordinates can still remain. For a hyper-Kähler metric the physical correlators turn out to be independent of all coordinates of insertions of local physical operators.
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Crabb, M. C. « On the KOℤ/2-Euler class, II ». Proceedings of the Royal Society of Edinburgh : Section A Mathematics 117, no 1-2 (1991) : 139–54. http://dx.doi.org/10.1017/s0308210500027669.
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SynopsisLet ξ be an oriented n-dimensional real vector bundle over an oriented closed m-manifold X. An r-field on ξ defined outside a finite subset of X has an index in the homotopy group πm−l(Vn,r) of the Stiefel manifold of r-frames in ℝn. The principal theorems of this paper relate the d and e-invariants of an associated ℝ/2-equivariant stable homotopy class, in certain cases, to computable cohomology characteristic numbers. Results of this type were first obtained by Atiyah and Dupont [5].
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Youssef, A. « Контрпример к гипотезе Андреотти - Грауэрта ». Владикавказский математический журнал, no 2 (22 juin 2022) : 14–24. http://dx.doi.org/10.46698/a8931-0543-3696-o.
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In 1962, Andreotti and Grauert showed that every $q$-completecomplex space $X$ is cohomologically $q$-complete, that is for every coherent analytic sheaf${\mathcal{F}}$ on $X$, the cohomology group $H^{p}(X, {\mathcal{F}})$ vanishes if $p\geq q$.Since then the question whether the reciprocal statements of these theorems are true have been subject to extensive studies, where more specific assumptions have been added. Until now it is not known if these two conditions are equivalent. Using test cohomology classes, it was shown however that if $X$ is a Stein manifold and, if $D\subset X$ is an open subset which has $C^{2}$ boundary such that $H^{p}(D, {\mathcal{O}}_{D})=0$ for all $p\geq q$, then $D$ is $q$-complete.The aim of the present article is to give a counterexample tothe conjecture posed in $1962$ by Andreotti and Grauert [1]to show that a~cohomologically $q$-complete space is not necessarily $q$-complete.More precisely, we show that there exist for each $n\geq 3$ open subsets$\Omega\subset\mathbb{C}^{n}$ such that for every ${\mathcal{F}}\in coh(\Omega)$,the cohomology groups $H^{p}(\Omega, {\mathcal{F}})$ vanish for all $p\geq n-1$but $\Omega$ is not $(n-1)$-complete.
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BYTSENKO, A. A. « BRST-INVARIANT DEFORMATIONS OF GEOMETRIC STRUCTURES IN SIGMA MODELS ». International Journal of Modern Physics A 26, no 22 (10 septembre 2011) : 3769–80. http://dx.doi.org/10.1142/s0217751x11054231.
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The closed string correlators can be constructed from the open ones using topological string theories as a model. The space of physical closed string states is isomorphic to the Hochschild cohomology of (A,Q) (operator Q of ghost number one), - this statement has been verified by means of computation of the Hochschild cohomology of the category of D -branes. We study a Lie algebra of formal vector fields Wn with its application to the perturbative deformed holomorphic symplectic structure in the A -model, and a Calabi-Yau manifold with boundaries in the B -model. We show that equivalent classes of deformations are describing by a Hochschild cohomology theory of the DG-algebra, [Formula: see text], [Formula: see text], which is defined to be the cohomology of (-1)nQ+d Hoch . Here [Formula: see text] is the initial non-deformed BRST operator while ∂ deform is the deformed part whose algebra is a Lie algebra of linear vector fields gl n. We assume that if in the theory exists a single D -brane then all the information associated with deformations is encoded in an associative algebra A equipped with a differential [Formula: see text]. In addition equivalence classes of deformations of these data are described by a Hochschild cohomology of (A,Q), an important geometric invariant of the (anti)holomorphic structure on X. We also discuss the identification of the harmonic structure (HT•(X); HΩ•(X)) of affine space X and the group [Formula: see text] (the HKR isomorphism), and bulk-boundary deformation pairing.
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BYTSENKO, A. A. « BRST-INVARIANT DEFORMATIONS OF GEOMETRIC STRUCTURES IN SIGMA MODELS ». International Journal of Modern Physics : Conference Series 03 (janvier 2011) : 75–86. http://dx.doi.org/10.1142/s2010194511001164.
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The closed string correlators can be constructed from the open ones using topological string theories as a model. The space of physical closed string states is isomorphic to the Hochschild cohomology of (A, Q) (operator Q of ghost number one), - this statement has been verified by means of computation of the Hochschild cohomology of the category of D-branes. We study a Lie algebra of formal vector fields Wn with its application to the perturbative deformed holomorphic symplectic structure in the A-model, and a Calabi-Yau manifold with boundaries in the B-model. We show that equivalent classes of deformations are describing by a Hochschild cohomology theory of the DG-algebra [Formula: see text], [Formula: see text], which is defined to be the cohomology of (-1)n Q + d Hoch . Here [Formula: see text] is the initial non-deformed BRST operator while ∂deform is the deformed part whose algebra is a Lie algebra of linear vector fields gl n. We assume that if in the theory exists a single D-brane then all the information associated with deformations is encoded in an associative algebra A equipped with a differential [Formula: see text]. In addition equivalence classes of deformations of these data are described by a Hochschild cohomology of (A, Q), an important geometric invariant of the (anti)holomorphic structure on X. We also discuss the identification of the harmonic structure (HT•(X); HΩ•(X)) of affine space X and the group [Formula: see text] (the HKR isomorphism), and bulk-boundary deformation pairing.
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BYTSENKO, A. A., M. CHAICHIAN, A. TUREANU et F. L. WILLIAMS. « BRST-INVARIANT DEFORMATIONS OF GEOMETRIC STRUCTURES IN TOPOLOGICAL FIELD THEORIES ». International Journal of Modern Physics A 28, no 16 (28 juin 2013) : 1350069. http://dx.doi.org/10.1142/s0217751x13500693.
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We study a Lie algebra of formal vector fields Wn with its application to the perturbative deformed holomorphic symplectic structure in the A-model, and a Calabi–Yau manifold with boundaries in the B-model. A relevant concept in the vertex operator algebra and the BRST cohomology is that of the elliptic genera (the one-loop string partition function). We show that the elliptic genera can be written in terms of spectral functions of the hyperbolic three-geometry (which inherits the cohomology structure of BRST-like operator). We show that equivalence classes of deformations are described by a Hochschild cohomology theory of the DG-algebra [Formula: see text], which is defined to be the cohomology of (-1)n Q + d Hoch . Here, [Formula: see text] is the initial nondeformed BRST operator while ∂ deform is the deformed part whose algebra is a Lie algebra of linear vector fields gl n. We discuss the identification of the harmonic structure (HT•(X);HΩ•(X)) of affine space X and the group [Formula: see text] (the HKR isomorphism), and bulk-boundary deformation pairing.
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RIERA, IGNASI MUNDET I. « LIFTS OF SMOOTH GROUP ACTIONS TO LINE BUNDLES ». Bulletin of the London Mathematical Society 33, no 3 (mai 2001) : 351–61. http://dx.doi.org/10.1017/s0024609301007937.
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Let X be a compact manifold with a smooth action of a compact connected Lie group G. Let L → X be a complex line bundle. Using the Cartan complex for equivariant cohomology, we give a new proof of a theorem of Hattori and Yoshida which says that the action of G lifts to L if and only if the first Chern class c1(L) of L can be lifted to an integral equivariant cohomology class in H2G(X; ℤ), and that the different lifts of the action are classified by the lifts of c1(L) to H2G(X; ℤ). As a corollary of our method of proof, we prove that, if the action is Hamiltonian and ∇ is a connection on L which is unitary for some metric on L, and which has a G-invariant curvature, then there is a lift of the action to a certain power Ld (where d is independent of L) which leaves fixed the induced metric on Ld and the connection ∇[otimes ]d. This generalises to symplectic geometry a well-known result in geometric invariant theory.
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Kashiwara, Masaki, et Pierre Schapira. « Microlocal Euler classes and Hochschild homology ». Journal of the Institute of Mathematics of Jussieu 13, no 3 (18 juillet 2013) : 487–516. http://dx.doi.org/10.1017/s1474748013000169.
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AbstractWe define the notion of a trace kernel on a manifold $M$. Roughly speaking, it is a sheaf on $M\times M$ for which the formalism of Hochschild homology applies. We associate a microlocal Euler class with such a kernel, a cohomology class with values in the relative dualizing complex of the cotangent bundle ${T}^{\ast } M$ over $M$, and we prove that this class is functorial with respect to the composition of kernels.This generalizes, unifies and simplifies various results from (relative) index theorems for constructible sheaves, $\mathscr{D}$-modules and elliptic pairs.
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FARBER, MICHAEL, JEAN-CLAUDE HAUSMANN et DIRK SCHÜTZ. « The Walker conjecture for chains in ℝd ». Mathematical Proceedings of the Cambridge Philosophical Society 151, no 2 (5 mai 2011) : 283–92. http://dx.doi.org/10.1017/s030500411100020x.
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AbstractA chain is a configuration in ℝd of segments of length ℓ1, . . ., ℓn−1 consecutively joined to each other such that the resulting broken line connects two given points at a distance ℓn. For a fixed generic set of length parameters the space of all chains in ℝd is a closed smooth manifold of dimension (n − 2)(d − 1) − 1. In this paper we study cohomology algebras of spaces of chains. We give a complete classification of these spaces (up to equivariant diffeomorphism) in terms of linear inequalities of a special kind which are satisfied by the length parameters ℓ1, . . ., ℓn. This result is analogous to the conjecture of K. Walker which concerns the special case d=2.
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ANCONA, VINCENZO, et BERNARD GAVEAU. « BOCHNER–MARTINELLI FORMULAS ON SINGULAR COMPLEX SPACES ». International Journal of Mathematics 21, no 02 (février 2010) : 225–53. http://dx.doi.org/10.1142/s0129167x10005994.
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Let U be a complex space of complex dimension n ≥ 2, P a point of U, π : Ũ → U a modification such that Ũ is nonsingular and D = π-1 (P) is a divisor with normal crossings. A Bochner–Martinelli form on U\{P} is a [Formula: see text]-closed differential form ω on Ũ\D, of pure type (n,n - 1), logarithmic along D. Such form detects a cohomology class of H2n - 1 (U\{P},ℂ) on the singular space U\{P}. Thanks to a general residue formula we prove that the forms ω give rise to an integral formula of Bochner–Martinelli type for holomorphic functions. If U satisfies the following assumption that {there exists a compact complex space X bimeromorphic to a Kähler manifold, and a closed subspace T ⊂ X, such that X\T = U (an affine, or a quasi-projective variety satisfies the above property), we relate Bochner–Martinelli forms to the mixed Hodge structure carried by H2n-1 (U\{P},ℂ). Most of our results hold for complex spaces which are not Stein.
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Schaftingen, Jean Van. « Estimates by gap potentials of free homotopy decompositions of critical Sobolev maps ». Advances in Nonlinear Analysis 9, no 1 (10 décembre 2019) : 1214–50. http://dx.doi.org/10.1515/anona-2020-0047.
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Abstract A free homotopy decomposition of any continuous map from a compact Riemannian manifold 𝓜 to a compact Riemannian manifold 𝓝 into a finite number maps belonging to a finite set is constructed, in such a way that the number of maps in this free homotopy decomposition and the number of elements of the set to which they belong can be estimated a priori by the critical Sobolev energy of the map in Ws,p(𝓜, 𝓝), with sp = m = dim 𝓜. In particular, when the fundamental group π1(𝓝) acts trivially on the homotopy group πm(𝓝), the number of homotopy classes to which a map can belong can be estimated by its Sobolev energy. The estimates are particular cases of estimates under a boundedness assumption on gap potentials of the form $$\begin{array}{} \displaystyle \iint\limits_{\substack{(x, y) \in \mathcal{M} \times \mathcal{M} \\ d_\mathcal{N} (f (x), f (y)) \ge \varepsilon}} \frac{1}{d_\mathcal{M} (y, x)^{2 m}} \, \mathrm{d} y \, \mathrm{d}x. \end{array}$$ When m ≥ 2, the estimates scale optimally as ε → 0. When m = 1, the total variation of the maps appearing in the decomposition can be controlled by the gap potential. Linear estimates on the Hurewicz homomorphism and the induced cohomology homomorphism are also obtained.
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Bulnes, Francisco. « Motivic Hypercohomology Solutions in Field Theory and Applications in H-States ». Journal of Mathematics Research 13, no 1 (23 janvier 2021) : 31. http://dx.doi.org/10.5539/jmr.v13n1p31.
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Triangulated derived categories are considered which establish a commutative scheme (triangle) for determine or compute a hypercohomology of motives for the obtaining of solutions of the field equations. The determination of this hypercohomology arises of the derived category $\textup{DM}_{\textup {gm}}(k)$, which is of the motivic objects whose image is under $\textup {Spec}(k)$ that is to say, an equivalence of the underlying triangulated tensor categories, compatible with respective functors on $\textup{Sm}_{k}^{\textup{Op}}$. The geometrical motives will be risked with the moduli stack to holomorphic bundles. Likewise, is analysed the special case where complexes $C=\mathbb{Q}(q)$, are obtained when cohomology groups of the isomorphism $H_{\acute{e}t}^{p}(X,F_{\acute{e}t})\cong (X,F_{Nis})$, can be vanished for $p>\textup{dim}(Y)$. We observe also the Beilinson-Soul$\acute{e}$ vanishing conjectures where we have the vanishing $H^{p}(F,\mathbb{Q}(q))=0, \ \ \textup{if} \ \ p\leq0,$ and $q>0$, which confirms the before established. Then survives a hypercohomology $\mathbb{H}^{q}(X,\mathbb{Q})$. Then its objects are in $\textup{Spec(Sm}_{k})$. Likewise, for the complex Riemannian manifold the integrals of this hypercohomology are those whose functors image will be in $\textup{Spec}_{H}\textup{SymT(OP}_{L_{G}}(D))$, which is the variety of opers on the formal disk $D$, or neighborhood of all point in a surface $\Sigma$. Likewise, will be proved that $\mathrm{H}^{\vee}$, has the decomposing in components as hyper-cohomology groups which can be characterized as H- states in Vec$_\mathbb{C}$, for field equations $d \textup{da}=0$, on the general linear group with $k=\mathbb{C}$. A physics re-interpretation of the superposing, to the dual of the spectrum $\mathrm{H}^{\vee}$, whose hypercohomology is a quantized version of the cohomology space $H^{q}(Bun_{G},\mathcal{D}^{s})=\mathbb{H}^{q}_{G[[z]]}(\mathrm{G},(\land^{\bullet}[\Sigma^{0}]\otimes \mathbb{V}_{critical},\partial))$ is the corresponding deformed derived category for densities $\mathrm{h} \in \mathrm{H}$, in quantum field theory.
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Mare, Augustin-Liviu. « A Characterization of the Quantum Cohomology Ring of G/B and Applications ». Canadian Journal of Mathematics 60, no 4 (1 août 2008) : 875–91. http://dx.doi.org/10.4153/cjm-2008-037-8.
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AbstractWe observe that the small quantum product of the generalized flag manifold G/B is a product operation ★ on H*(G/B) ⊗ ℝ[q1, . . . , ql] uniquely determined by the facts that it is a deformation of the cup product on H*(G/B); it is commutative, associative, and graded with respect to deg(qi ) = 4; it satisfies a certain relation (of degree two); and the corresponding Dubrovin connection is flat. Previously, we proved that these properties alone imply the presentation of the ring (H*(G/B)⊗ℝ[q1, . . . , ql], ★) in terms of generators and relations. In this paper we use the above observations to give conceptually new proofs of other fundamental results of the quantum Schubert calculus for G/B: the quantumChevalley formula of D. Peterson (see also Fulton andWoodward) and the “quantization by standard monomials” formula of Fomin, Gelfand, and Postnikov for G = SL(n, ℂ). The main idea of the proofs is the same as in Amarzaya–Guest: from the quantum -module of G/B one can decode all information about the quantum cohomology of this space.
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D’AURIA, RICCARDO, PIETRO FRE’, GUIDO DE MATTEIS et IGOR PESANDO. « SUPERSPACE CONSTRAINTS AND CHERN-SIMONS COHOMOLOGY IN D=4 SUPERSTRING EFFECTIVE THEORIES ». International Journal of Modern Physics A 04, no 14 (20 août 1989) : 3577–613. http://dx.doi.org/10.1142/s0217751x89001412.
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The linear multiplet, composed of the dilaton ϕ, of an antisymmetric gauge field Bμν and of a spinor χ is always present in any superstring induced N=1D=4 supergravity model. We consider its coupling to supergravity using only superspace Bianchi identities and the rheonomy approach. In this way, our results are fully general and independent from the choice of any Lagrangian, a concept which is never mentioned in this paper. We consider two situations corresponding to two different free differential algebras: (1) the case where there are no Chern-Simons terms in the Bμν field strength Hμνρ and (2) the case where such terms are included in Hμνρ. Case (2) is obviously the one chosen by string theory on the ground of anomaly cancellation. In both cases, we must solve the H-Bianchi identity using a solution of the super Poincare’ Bianchi identities as a background. Such a solution, besides the physical fields displays a certain number of auxiliary fields. The most general solution of the super Poincare’ Bianchis we have to consider corresponds to 16⊕16 off-shell multiplet which, by suitable choices can be reduced either to the so-called old minimal or to the new minimal 12⊕12 multiplet. We give the general solution of the H-Bianchi within the 16⊕16 formulation both with and without Chern-Simons terms. This is done through the D=4 analogue of Bonora-Pasti-Tonin theorem of the 10D anomaly free supergravity. By specializing our parameters, we obtain the form of the coupling in the new minimal model retrieving in this case the results of Cecotti, Ferrara and Villasante. In addition we clarify the geometrical meaning of R-symmetry showing that in the absence of Chern-Simons forms, the condition for the embedding of the linear multiplet into the Kaehler manifold [Formula: see text] spanned by the chiral multiplets (existence on [Formula: see text] of a U(1) Killing vector) is the same condition which guarantees the existence of a local Weyl transformation by means of which the 16⊕16 curvatures can be reduced to the new minimal form and the scalar complex scalar auxiliary field S can be set to zero. Finally, we discuss the arbitrariness contained in the solution of the H-Bianchi identities at the level of the (0, 3) superspace sector. We derive the D=4 analogue of the superspace cocycle which is responsible for the Grisaru-Zanon R4-terms in the D=10 case.
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Knutson, Allen, Thomas Lam et David E. Speyer. « Positroid varieties : juggling and geometry ». Compositio Mathematica 149, no 10 (19 août 2013) : 1710–52. http://dx.doi.org/10.1112/s0010437x13007240.
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AbstractWhile the intersection of the Grassmannian Bruhat decompositions for all coordinate flags is an intractable mess, it turns out that the intersection of only the cyclic shifts of one Bruhat decomposition has many of the good properties of the Bruhat and Richardson decompositions. This decomposition coincides with the projection of the Richardson stratification of the flag manifold, studied by Lusztig, Rietsch, Brown–Goodearl–Yakimov and the present authors. However, its cyclic-invariance is hidden in this description. Postnikov gave many cyclic-invariant ways to index the strata, and we give a new one, by a subset of the affine Weyl group we call bounded juggling patterns. We call the strata positroid varieties. Applying results from [A. Knutson, T. Lam and D. Speyer, Projections of Richardson varieties, J. Reine Angew. Math., to appear, arXiv:1008.3939 [math.AG]], we show that positroid varieties are normal, Cohen–Macaulay, have rational singularities, and are defined as schemes by the vanishing of Plücker coordinates. We prove that their associated cohomology classes are represented by affine Stanley functions. This latter fact lets us connect Postnikov’s and Buch–Kresch–Tamvakis’ approaches to quantum Schubert calculus.
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ABBONDANDOLO, ALBERTO, et MATTHIAS SCHWARZ. « ESTIMATES AND COMPUTATIONS IN RABINOWITZ–FLOER HOMOLOGY ». Journal of Topology and Analysis 01, no 04 (décembre 2009) : 307–405. http://dx.doi.org/10.1142/s1793525309000205.
Texte intégralRésumé :
The Rabinowitz–Floer homology of a Liouville domain W is the Floer homology of the Rabinowitz free period Hamiltonian action functional associated to a Hamiltonian whose zero energy level is the boundary of W. This invariant has been introduced by K. Cieliebak and U. Frauenfelder and has already found several applications in symplectic topology and in Hamiltonian dynamics. Together with A. Oancea, the same authors have recently computed the Rabinowitz–Floer homology of the cotangent disk bundle D* M of a closed Riemannian manifold M, by means of an exact sequence relating the Rabinowitz–Floer homology of D* M with its symplectic homology and cohomology. The first aim of this paper is to present a chain level construction of this exact sequence. In fact, we show that this sequence is the long homology sequence induced by a short exact sequence of chain complexes, which involves the Morse chain complex and the Morse differential complex of the energy functional for closed geodesics on M. These chain maps are defined by considering spaces of solutions of the Rabinowitz–Floer equation on half-cylinders, with suitable boundary conditions which couple them with the negative gradient flow of the geodesic energy functional. The second aim is to generalize this construction to the case of a fiberwise uniformly convex compact subset W of T* M whose interior part contains a Lagrangian graph. Equivalently, W is the energy sublevel associated to an arbitrary Tonelli Lagrangian L on TM and to any energy level which is larger than the strict Mañé critical value of L. In this case, the energy functional for closed geodesics is replaced by the free period Lagrangian action functional associated to a suitable calibration of L. An important issue in our analysis is to extend the uniform estimates for the solutions of the Rabinowitz–Floer equation — both on cylinders and on half-cylinders — to Hamiltonians which have quadratic growth in the momenta. These uniform estimates are obtained by the Aleksandrov integral version of the maximum principle. In the case of half-cylinders, they are obtained by an Aleksandrov-type maximum principle with Neumann conditions on part of the boundary.
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McOrist, Jock, et Eirik Eik Svanes. « Heterotic quantum cohomology ». Journal of High Energy Physics 2022, no 11 (15 novembre 2022). http://dx.doi.org/10.1007/jhep11(2022)096.
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Abstract It is believed, but not demonstrated, that the large radius massless spectrum of a heterotic string theory compactified to four-dimensional Minkowski space should obey equations that split into ‘F-terms’ and ‘D-terms’ in ways analogous to that of four-dimensional supersymmetric field theories. This is not easy to do directly as string theory is first quantised. Nonetheless, in this paper we demonstrate this splitting. We construct an operator $$ \overline{\mathcal{D}} $$ D ¯ whose kernel amounts to deformations solving ‘F-term’ type equations. In many previous works in this field, the spin connection is treated as an independent degree of freedom (and so is spurious or fake); here our results apply on the physical moduli space in which these fake degrees of freedom are eliminated. We utilise the moduli space metric, constructed in previous work, to define an adjoint operator $$ \overline{\mathcal{D}} $$ D ¯ †. The kernel of $$ \overline{\mathcal{D}} $$ D ¯ † amounts to ‘D-term’ type equations. Put together, we show there is a $$ \overline{\mathcal{D}} $$ D ¯ -operator in which the massless spectrum are harmonic representatives of $$ \overline{\mathcal{D}} $$ D ¯ . We conjecture that one could better study the moduli space of heterotic theories by studying the corresponding cohomology, a natural counterpart to studying the $$ \overline{\partial} $$ ∂ ¯ -cohomology groups relevant to moduli of Calabi-Yau manifolds.
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CAMERE, CHIARA, ALBERTO CATTANEO et ANDREA CATTANEO. « NON-SYMPLECTIC INVOLUTIONS ON MANIFOLDS OF -TYPE ». Nagoya Mathematical Journal, 27 février 2020, 1–25. http://dx.doi.org/10.1017/nmj.2019.43.
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We study irreducible holomorphic symplectic manifolds deformation equivalent to Hilbert schemes of points on a $K3$ surface and admitting a non-symplectic involution. We classify the possible discriminant quadratic forms of the invariant and coinvariant lattice for the action of the involution on cohomology and explicitly describe the lattices in the cases where the invariant lattice has small rank. We also give a modular description of all $d$ -dimensional families of manifolds of $K3^{[n]}$ -type with a non-symplectic involution for $d\geqslant 19$ and $n\leqslant 5$ and provide examples arising as moduli spaces of twisted sheaves on a $K3$ surface.
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Knutson, Allen, et Kevin Purbhoo. « Product and Puzzle Formulae for $GL_n$ Belkale-Kumar Coefficients ». Electronic Journal of Combinatorics 18, no 1 (31 mars 2011). http://dx.doi.org/10.37236/563.
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The Belkale-Kumar product on $H^*(G/P)$ is a degeneration of the usual cup product on the cohomology ring of a generalized flag manifold. In the case $G=GL_n$, it was used by N. Ressayre to determine the regular faces of the Littlewood-Richardson cone. We show that for $G/P$ a $(d-1)$-step flag manifold, each Belkale-Kumar structure constant is a product of $d\choose 2$ Littlewood-Richardson numbers, for which there are many formulae available, e.g. the puzzles of [Knutson-Tao '03]. This refines previously known factorizations into $d-1$ factors. We define a new family of puzzles to assemble these to give a direct combinatorial formula for Belkale-Kumar structure constants. These "BK-puzzles" are related to extremal honeycombs, as in [Knutson-Tao-Woodward '04]; using this relation we give another proof of Ressayre's result. Finally, we describe the regular faces of the Littlewood-Richardson cone on which the Littlewood-Richardson number is always $1$; they correspond to nonzero Belkale-Kumar coefficients on partial flag manifolds where every subquotient has dimension $1$ or $2$.
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Govc, Dejan, Wacław Marzantowicz et Petar Pavešić. « Estimates of covering type and minimal triangulations based on category weight ». Forum Mathematicum, 31 mai 2022. http://dx.doi.org/10.1515/forum-2021-0216.
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Abstract In a recent publication [D. Govc, W. A. Marzantowicz and P. Pavešić, Estimates of covering type and the number of vertices of minimal triangulations, Discrete Comput. Geom. 63 2020, 1, 31–48], we have introduced a new method, based on the Lusternik–Schnirelmann category and the cohomology ring of a space X, that yields lower bounds for the size of a triangulation of X. In this current paper, we present an important extension that takes into account the fundamental group of X. In fact, if π 1 ( X ) {\pi_{1}(X)} contains elements of finite order, then one can often find cohomology classes of high ‘category weight’, which in turn allow for much stronger estimates of the size of triangulations of X. We develop several weighted estimates and then apply our method to compute explicit lower bounds for the size of triangulations of orbit spaces of cyclic group actions on a variety of spaces including products of spheres, Stiefel manifolds, Lie groups and highly-connected manifolds.
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Ashmore, Anthony, Charles Strickland-Constable, David Tennyson et Daniel Waldram. « Generalising G2 geometry : involutivity, moment maps and moduli ». Journal of High Energy Physics 2021, no 1 (janvier 2021). http://dx.doi.org/10.1007/jhep01(2021)158.
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Abstract We analyse the geometry of generic Minkowski $$ \mathcal{N} $$ N = 1, D = 4 flux compactifications in string theory, the default backgrounds for string model building. In M-theory they are the natural string theoretic extensions of G2 holonomy manifolds. In type II theories, they extend the notion of Calabi-Yau geometry and include the class of flux backgrounds based on generalised complex structures first considered by Graña et al. (GMPT). Using E7(7) × ℝ+ generalised geometry we show that these compactifications are characterised by an SU(7) ⊂ E7(7) structure defining an involutive subbundle of the generalised tangent space, and with a vanishing moment map, corresponding to the action of the diffeomorphism and gauge symmetries of the theory. The Kähler potential on the space of structures defines a natural extension of Hitchin’s G2 functional. Using this framework we are able to count, for the first time, the massless scalar moduli of GMPT solutions in terms of generalised geometry cohomology groups. It also provides an intriguing new perspective on the existence of G2 manifolds, suggesting possible connections to Geometrical Invariant Theory and stability.
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Buring, Ricardo, Dimitri Lipper et Arthemy V. Kiselev. « The hidden symmetry of Kontsevich's graph flows on the spaces of Nambu-determinant Poisson brackets ». Open Communications in Nonlinear Mathematical Physics Volume 2 (2 décembre 2022). http://dx.doi.org/10.46298/ocnmp.8844.
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Kontsevich's graph flows are -- universally for all finite-dimensional affine Poisson manifolds -- infinitesimal symmetries of the spaces of Poisson brackets. We show that the previously known tetrahedral flow and the recently obtained pentagon-wheel flow preserve the class of Nambu-determinant Poisson bi-vectors $P=[\![ \varrho(\boldsymbol{x})\,\partial_x\wedge\partial_y\wedge\partial_z,a]\!]$ on $\mathbb{R}^3\ni\boldsymbol{x}=(x,y,z)$ and $P=[\![ [\![\varrho(\boldsymbol{y})\,\partial_{x^1}\wedge\ldots\wedge\partial_{x^4},a_1]\!],a_2]\!]$ on $\mathbb{R}^4\ni\boldsymbol{y}$, including the general case $\varrho \not\equiv 1$. We detect that the Poisson bracket evolution $\dot{P} = Q_\gamma(P^{\otimes^{\# Vert(\gamma)}})$ is trivial in the second Poisson cohomology, $Q_\gamma = [\![ P, \vec{X}([\varrho],[a]) ]\!]$, for the Nambu-determinant bi-vectors $P(\varrho,[a])$ on $\mathbb{R}^3$. For the global Casimirs $\mathbf{a} = (a_1,\ldots,a_{d-2})$ and inverse density $\varrho$ on $\mathbb{R}^d$, we analyse the combinatorics of their evolution induced by the Kontsevich graph flows, namely $\dot{\varrho} = \dot{\varrho}([\varrho], [\mathbf{a}])$ and $\dot{\mathbf{a}} = \dot{\mathbf{a}}([\varrho],[\mathbf{a}])$ with differential-polynomial right-hand sides. Besides the anticipated collapse of these formulas by using the Civita symbols (three for the tetrahedron $\gamma_3$ and five for the pentagon-wheel graph cocycle $\gamma_5$), as dictated by the behaviour $\varrho(\mathbf{x}') = \varrho(\mathbf{x}) \cdot \det \| \partial \mathbf{x}' / \partial \mathbf{x} \|$ of the inverse density $\varrho$ under reparametrizations $\mathbf{x} \rightleftarrows \mathbf{x}'$, we discover another, so far hidden discrete symmetry in the construction of these evolution equations.
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Onishchik, Arkady. « Non-split supermanifolds associated with the cotangent bundle ». Communications in Mathematics Volume 30 (2022), Issue 3... (21 décembre 2022). http://dx.doi.org/10.46298/cm.9613.
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Here, I study the problem of classification of non-split supermanifolds having as retract the split supermanifold $(M,\Omega)$, where $\Omega$ is the sheaf of holomorphic forms on a given complex manifold $M$ of dimension $> 1$. I propose a general construction associating with any $d$-closed $(1,1)$-form $\omega$ on $M$ a supermanifold with retract $(M,\Omega)$ which is non-split whenever the Dolbeault class of $\omega$ is non-zero. In particular, this gives a non-empty family of non-split supermanifolds for any flag manifold $M\ne \mathbb{CP}^1$. In the case where $M$ is an irreducible compact Hermitian symmetric space, I get a complete classification of non-split supermanifolds with retract $(M,\Omega)$. For each of these supermanifolds, the 0- and 1-cohomology with values in the tangent sheaf are calculated. As an example, I study the $\Pi$-symmetric super-Grassmannians introduced by Yu. Manin.
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Kourliouros, Konstantinos. « Gauss-Manin Connections for Boundary Singularities and Isochore Deformations ». Demonstratio Mathematica 48, no 2 (1 juin 2015). http://dx.doi.org/10.1515/dema-2015-0020.
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AbstractWe study here the relative cohomology and the Gauss-Manin connections associated to an isolated singularity of a function on a manifold with boundary, i.e. with a fixed hyperplane section. We prove several relative analogs of classical theorems obtained mainly by E. Brieskorn and B. Malgrange, concerning the properties of the Gauss-Manin connection as well as its relations with the Picard-Lefschetz monodromy and the asymptotics of integrals of holomorphic forms along the vanishing cycles. Finally, we give an application in isochore deformation theory, i.e. the deformation theory of boundary singularities with respect to a volume form. In particular, we prove the relative analog of J. Vey's isochore Morse lemma, J .-P. Fran~oise's generalisation on the local normal forms of volume forms with respect to the boundary singularity-preserving diffeomorphisms, as well as M. D. Garay's theorem on the isochore version of Mather's versa! unfolding theorem.