Lambda G Conjecture
In algebraic geometry, the \lambda_g-conjecture gives a particularly simple formula for certain integrals on the Deligne–Mumford compactification \overline_ of the moduli space of curves with marked points. It was first found as a consequence of the Virasoro conjecture by . Later, it was proven by using virtual localization in Gromov–Witten theory. It is named after the factor of \lambda_g, the ''g''th Chern class of the Hodge bundle, appearing in its integrand. The other factor is a monomial in the \psi_i, the first Chern classes of the ''n'' cotangent line bundles, as in Witten's conjecture. Let a_1, \ldots, a_n be positive integers such that: :a_1 + \cdots + a_n = 2g-3+n. Then the \lambda_g-formula can be stated as follows: : \int_ \psi_1^ \cdots \psi_n^\lambda_g = \binom \int_ \psi_1^\lambda_g. The \lambda_g-formula in combination withge :\int_ \psi_1^\lambda_g = \frac \frac, where the ''B''2''g'' are Bernoulli number In mathematics, the Bernoulli numbers are ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Moduli Space Of Curves
In algebraic geometry, a moduli space of (algebraic) curves is a geometric space (typically a scheme or an algebraic stack) whose points represent isomorphism classes of algebraic curves. It is thus a special case of a moduli space. Depending on the restrictions applied to the classes of algebraic curves considered, the corresponding moduli problem and the moduli space is different. One also distinguishes between fine and coarse moduli spaces for the same moduli problem. The most basic problem is that of moduli of smooth complete curves of a fixed genus. Over the field of complex numbers these correspond precisely to compact Riemann surfaces of the given genus, for which Bernhard Riemann proved the first results about moduli spaces, in particular their dimensions ("number of parameters on which the complex structure depends"). Moduli stacks of stable curves The moduli stack \mathcal_ classifies families of smooth projective curves, together with their isomorphisms. ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Virasoro Conjecture
In algebraic geometry, the Virasoro conjecture states that a certain generating function encoding Gromov–Witten invariants of a smooth projective variety is fixed by an action of half of the Virasoro algebra. The Virasoro conjecture is named after theoretical physicist Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ... Miguel Ángel Virasoro. proposed the Virasoro conjecture as a generalization of Witten's conjecture. gave a survey of the Virasoro conjecture. References * * Algebraic geometry Conjectures Unsolved problems in geometry {{algebraic-geometry-stub ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Chern Class
In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since found applications in physics, Calabi–Yau manifolds, string theory, Chern–Simons theory, knot theory, Gromov–Witten invariants, topological quantum field theory, the Chern theorem etc. Chern classes were introduced by . Geometric approach Basic idea and motivation Chern classes are characteristic classes. They are topological invariants associated with vector bundles on a smooth manifold. The question of whether two ostensibly different vector bundles are the same can be quite hard to answer. The Chern classes provide a simple test: if the Chern classes of a pair of vector bundles do not agree, then the vector bundles are different. The converse, however, is not true. In topology, differential geometry, and algebraic geometry, it is often important to count ho ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Hodge Bundle
In mathematics, the Hodge bundle, named after W. V. D. Hodge, appears in the study of families of curves, where it provides an invariant in the moduli theory of algebraic curves. Furthermore, it has applications to the theory of modular forms on reductive algebraic groups and string theory. Definition Let \mathcal_g be the moduli space of algebraic curves of genus ''g'' curves over some scheme. The Hodge bundle \Lambda_g is a vector bundleHere, "vector bundle" in the sense of quasi-coherent sheaf on an algebraic stack on \mathcal_g whose fiber at a point ''C'' in \mathcal_g is the space of holomorphic differentials on the curve ''C''. To define the Hodge bundle, let \pi\colon \mathcal_g\rightarrow\mathcal_g be the universal algebraic curve of genus ''g'' and let \omega_g be its relative dualizing sheaf. The Hodge bundle is the pushforward of this sheaf, i.e., :\Lambda_g=\pi_*\omega_g. See also *ELSV formula In mathematics, the ELSV formula, named after its four authors , ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Witten's Conjecture
In algebraic geometry, the Witten conjecture is a conjecture about intersection numbers of stable classes on the moduli space of curves, introduced by Edward Witten in the paper , and generalized in . Witten's original conjecture was proved by Maxim Kontsevich in the paper . Witten's motivation for the conjecture was that two different models of 2-dimensional quantum gravity should have the same partition function. The partition function for one of these models can be described in terms of intersection numbers on the moduli stack of algebraic curves, and the partition function for the other is the logarithm of the τ-function of the KdV hierarchy. Identifying these partition functions gives Witten's conjecture that a certain generating function formed from intersection numbers should satisfy the differential equations of the KdV hierarchy. Statement Suppose that ''M''''g'',''n'' is the moduli stack of compact Riemann surfaces of genus ''g'' with ''n'' distinct marked points ''x ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Bernoulli Number
In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, in Faulhaber's formula for the sum of ''m''-th powers of the first ''n'' positive integers, in the Euler–Maclaurin formula, and in expressions for certain values of the Riemann zeta function. The values of the first 20 Bernoulli numbers are given in the adjacent table. Two conventions are used in the literature, denoted here by B^_n and B^_n; they differ only for , where B^_1=-1/2 and B^_1=+1/2. For every odd , . For every even , is negative if is divisible by 4 and positive otherwise. The Bernoulli numbers are special values of the Bernoulli polynomials B_n(x), with B^_n=B_n(0) and B^+_n=B_n(1). The Bernoulli numbers were discovered around the same time by the Swiss mathematician Jacob Bernoulli, after whom they are named, and i ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Algebraic Curves
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane curve can be completed in a projective algebraic plane curve by homogenizing its defining polynomial. Conversely, a projective algebraic plane curve of homogeneous equation can be restricted to the affine algebraic plane curve of equation . These two operations are each inverse to the other; therefore, the phrase algebraic plane curve is often used without specifying explicitly whether it is the affine or the projective case that is considered. More generally, an algebraic curve is an algebraic variety of dimension one. Equivalently, an algebraic curve is an algebraic variety that is birationally equivalent to an algebraic plane curve. If the curve is contained in an affine space or a projective space, one can take a projection for s ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |