Genus of a multiplicative sequence

Last updated
A cobordism (W; M, N). Cobordism.svg
A cobordism (W; M, N).

In mathematics, a genus of a multiplicative sequence is a ring homomorphism from the ring of smooth compact manifolds up to the equivalence of bounding a smooth manifold with boundary (i.e., up to suitable cobordism) to another ring, usually the rational numbers, having the property that they are constructed from a sequence of polynomials in characteristic classes that arise as coefficients in formal power series with good multiplicative properties.

Contents

Definition

A genus assigns a number to each manifold X such that

  1. (where is the disjoint union);
  2. ;
  3. if X is the boundary of a manifold with boundary.

The manifolds and manifolds with boundary may be required to have additional structure; for example, they might be oriented, spin, stably complex, and so on (see list of cobordism theories for many more examples). The value is in some ring, often the ring of rational numbers, though it can be other rings such as or the ring of modular forms.

The conditions on can be rephrased as saying that is a ring homomorphism from the cobordism ring of manifolds (with additional structure) to another ring.

Example: If is the signature of the oriented manifold X, then is a genus from oriented manifolds to the ring of integers.

The genus associated to a formal power series

A sequence of polynomials in variables is called multiplicative if

implies that

If is a formal power series in z with constant term 1, we can define a multiplicative sequence

by

,

where is the kth elementary symmetric function of the indeterminates . (The variables will often in practice be Pontryagin classes.)

The genus of compact, connected, smooth, oriented manifolds corresponding to Q is given by

where the are the Pontryagin classes of X. The power series Q is called the characteristic power series of the genus . A theorem of René Thom, which states that the rationals tensored with the cobordism ring is a polynomial algebra in generators of degree 4k for positive integers k, implies that this gives a bijection between formal power series Q with rational coefficients and leading coefficient 1, and genera from oriented manifolds to the rational numbers.

L genus

The L genus is the genus of the formal power series

where the numbers are the Bernoulli numbers. The first few values are:

(for further L-polynomials see [1] or OEIS:  A237111 ). Now let M be a closed smooth oriented manifold of dimension 4n with Pontrjagin classes . Friedrich Hirzebruch showed that the L genus of M in dimension 4n evaluated on the fundamental class of , denoted , is equal to , the signature of M (i.e., the signature of the intersection form on the 2nth cohomology group of M):

.

This is now known as the Hirzebruch signature theorem (or sometimes the Hirzebruch index theorem).

The fact that is always integral for a smooth manifold was used by John Milnor to give an example of an 8-dimensional PL manifold with no smooth structure. Pontryagin numbers can also be defined for PL manifolds, and Milnor showed that his PL manifold had a non-integral value of , and so was not smoothable.

Application on K3 surfaces

Since projective K3 surfaces are smooth complex manifolds of dimension two, their only non-trivial Pontryagin class is in . It can be computed as -48 using the tangent sequence and comparisons with complex chern classes. Since , we have its signature. This can be used to compute its intersection form as a unimodular lattice since it has , and using the classification of unimodular lattices. [2]

Todd genus

The Todd genus is the genus of the formal power series

with as before, Bernoulli numbers. The first few values are

The Todd genus has the particular property that it assigns the value 1 to all complex projective spaces (i.e. ), and this suffices to show that the Todd genus agrees with the arithmetic genus for algebraic varieties as the arithmetic genus is also 1 for complex projective spaces. This observation is a consequence of the Hirzebruch–Riemann–Roch theorem, and in fact is one of the key developments that led to the formulation of that theorem.

 genus

The  genus is the genus associated to the characteristic power series

(There is also an  genus which is less commonly used, associated to the characteristic series .) The first few values are

The  genus of a spin manifold is an integer, and an even integer if the dimension is 4 mod 8 (which in dimension 4 implies Rochlin's theorem) – for general manifolds, the  genus is not always an integer. This was proven by Hirzebruch and Armand Borel; this result both motivated and was later explained by the Atiyah–Singer index theorem, which showed that the  genus of a spin manifold is equal to the index of its Dirac operator.

By combining this index result with a Weitzenbock formula for the Dirac Laplacian, André Lichnerowicz deduced that if a compact spin manifold admits a metric with positive scalar curvature, its  genus must vanish. This only gives an obstruction to positive scalar curvature when the dimension is a multiple of 4, but Nigel Hitchin later discovered an analogous -valued obstruction in dimensions 1 or 2 mod 8. These results are essentially sharp. Indeed, Mikhail Gromov, H. Blaine Lawson, and Stephan Stolz later proved that the  genus and Hitchin's -valued analog are the only obstructions to the existence of positive-scalar-curvature metrics on simply-connected spin manifolds of dimension greater than or equal to 5.

Elliptic genus

A genus is called an elliptic genus if the power series satisfies the condition

for constants and . (As usual, Q is the characteristic power series of the genus.)

One explicit expression for f(z) is

where

and sn is the Jacobi elliptic function.

Examples:

The first few values of such genera are:

Example (Elliptic genus for quaternionic projective plane) :

Example (Elliptic genus for octonionic projective plane (Cayley plane)):

Witten genus

The Witten genus is the genus associated to the characteristic power series

where σL is the Weierstrass sigma function for the lattice L, and G is a multiple of an Eisenstein series.

The Witten genus of a 4k dimensional compact oriented smooth spin manifold with vanishing first Pontryagin class is a modular form of weight 2k, with integral Fourier coefficients.

See also

Notes

  1. McTague, Carl (2014) "Computing Hirzebruch L-Polynomials".
  2. Huybrechts, Daniel. "14.1 Existence, uniqueness, and embeddings of lattices". Lectures on K3 Surfaces (PDF). p. 285.

Related Research Articles

Hydrogen atom Atom of the element hydrogen

A hydrogen atom is an atom of the chemical element hydrogen. The electrically neutral atom contains a single positively charged proton and a single negatively charged electron bound to the nucleus by the Coulomb force. Atomic hydrogen constitutes about 75% of the baryonic mass of the universe.

In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space under the operation of composition. By definition, a rotation about the origin is a transformation that preserves the origin, Euclidean distance, and orientation. Every non-trivial rotation is determined by its axis of rotation and its angle of rotation. Composing two rotations results in another rotation; every rotation has a unique inverse rotation; and the identity map satisfies the definition of a rotation. Owing to the above properties, the set of all rotations is a group under composition. Rotations are not commutative, making it a nonabelian group. Moreover, the rotation group has a natural structure as a manifold for which the group operations are smoothly differentiable; so it is in fact a Lie group. It is compact and has dimension 3.

In continuum mechanics, the infinitesimal strain theory is a mathematical approach to the description of the deformation of a solid body in which the displacements of the material particles are assumed to be much smaller than any relevant dimension of the body; so that its geometry and the constitutive properties of the material at each point of space can be assumed to be unchanged by the deformation.

Spherical harmonics

In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields.

In number theory, the study of Diophantine approximation deals with the approximation of real numbers by rational numbers. It is named after Diophantus of Alexandria.

In the calculus of variations, a field of mathematical analysis, the functional derivative relates a change in a functional to a change in a function on which the functional depends.

In mathematics, the Hodge star operator or Hodge star is a linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. Applying the operator to an element of the algebra produces the Hodge dual of the element. This map was introduced by W. V. D. Hodge.

In mathematics, the Carlson symmetric forms of elliptic integrals are a small canonical set of elliptic integrals to which all others may be reduced. They are a modern alternative to the Legendre forms. The Legendre forms may be expressed in terms of the Carlson forms and vice versa.

In theoretical physics, the Batalin–Vilkovisky (BV) formalism was developed as a method for determining the ghost structure for Lagrangian gauge theories, such as gravity and supergravity, whose corresponding Hamiltonian formulation has constraints not related to a Lie algebra. The BV formalism, based on an action that contains both fields and "antifields", can be thought of as a vast generalization of the original BRST formalism for pure Yang–Mills theory to an arbitrary Lagrangian gauge theory. Other names for the Batalin–Vilkovisky formalism are field-antifield formalism, Lagrangian BRST formalism, or BV–BRST formalism. It should not be confused with the Batalin–Fradkin–Vilkovisky (BFV) formalism, which is the Hamiltonian counterpart.

In quantum mechanics, the canonical commutation relation is the fundamental relation between canonical conjugate quantities. For example,

In general relativity, the Gibbons–Hawking–York boundary term is a term that needs to be added to the Einstein–Hilbert action when the underlying spacetime manifold has a boundary.

Voigt effect

The Voigt effect is a magneto-optical phenomenon which rotates and elliptizes linearly polarised light sent into an optically active medium. Unlike many other magneto-optical effects such as the Kerr or Faraday effect which are linearly proportional to the magnetization, the Voigt effect is proportional to the square of the magnetization and can be seen experimentally at normal incidence. There are several denominations for this effect in the literature: the Cotton–Mouton effect, the Voigt effect, and magnetic-linear birefringence. This last denomination is closer in the physical sense, where the Voigt effect is a magnetic birefringence of the material with an index of refraction parallel and perpendicular ) to the magnetization vector or to the applied magnetic field.

An a priori probability is a probability that is derived purely by deductive reasoning. One way of deriving a priori probabilities is the principle of indifference, which has the character of saying that, if there are N mutually exclusive and collectively exhaustive events and if they are equally likely, then the probability of a given event occurring is 1/N. Similarly the probability of one of a given collection of K events is K / N.

In 1927, a year after the publication of the Schrödinger equation, Hartree formulated what are now known as the Hartree equations for atoms, using the concept of self-consistency that Lindsay had introduced in his study of many electron systems in the context of Bohr theory. Hartree assumed that the nucleus together with the electrons formed a spherically symmetric field. The charge distribution of each electron was the solution of the Schrödinger equation for an electron in a potential , derived from the field. Self-consistency required that the final field, computed from the solutions was self-consistent with the initial field and he called his method the self-consistent field method.

Multipole radiation is a theoretical framework for the description of electromagnetic or gravitational radiation from time-dependent distributions of distant sources. These tools are applied to physical phenomena which occur at a variety of length scales - from gravitational waves due to galaxy collisions to gamma radiation resulting from nuclear decay. Multipole radiation is analyzed using similar multipole expansion techniques that describe fields from static sources, however there are important differences in the details of the analysis because multipole radiation fields behave quite differently from static fields. This article is primarily concerned with electromagnetic multipole radiation, although the treatment of gravitational waves is similar.

Lagrangian field theory is a formalism in classical field theory. It is the field-theoretic analogue of Lagrangian mechanics. Lagrangian mechanics is used to analyze the motion of a system of discrete particles each with a finite number of degrees of freedom. Lagrangian field theory applies to continua and fields, which have an infinite number of degrees of freedom.

The Pomeranchuk instability is an instability in the shape of the Fermi surface of a material with interacting fermions, causing Landau’s Fermi liquid theory to break down. It occurs when a Landau parameter in Fermi liquid theory has a sufficiently negative value, causing deformations of the Fermi surface to be energetically favourable. It is named after the Soviet physicist Isaak Pomeranchuk.

In two-dimensional conformal field theory, Virasoro conformal blocks are special functions that serve as building blocks of correlation functions. On a given punctured Riemann surface, Virasoro conformal blocks form a particular basis of the space of solutions of the conformal Ward identites. Zero-point blocks on the torus are characters of representations of the Virasoro algebra; four-point blocks on the sphere reduce to hypergeometric functions in special cases, but are in general much more complicated. In two dimensions as in other dimensions, conformal blocks play an essential role in the conformal bootstrap approach to conformal field theory.

The two-dimensional critical Ising model is the critical limit of the Ising model in two dimensions. It is a two-dimensional conformal field theory whose symmetry algebra is the Virasoro algebra with the central charge . Correlation functions of the spin and energy operators are described by the minimal model. While the minimal model has been exactly solved, the solution does not cover other observables such as connectivities of clusters.

References