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δ-invariants

In Riemannian geometry, δ-invariants or Chen invariants, after Bang-Yen Chen, are a type of intrinsic geometric invariants of a Riemannian manifold.

Definition

Let $M$ be a d-dimensional Riemannian manifold, $K(\pi )$ the sectional curvature associated with a plane section $\pi$ of the tangent space $T_{p}M$ at a point $p\in M$, $e_{1},\ldots ,e_{d}$ an orthonormal basis of $T_{p}M$ and $\tau (L)$ the scalar curvature of of the subspace $L$ of $T_{p}M$, for each unordered k-tuple $(n_{1},\ldots ,n_{k})$ of integers larger than or equal to two, satisfying $d>n_{1}$ and $n_{1}+\ldots +n_{k}<d$ the δ-invariant $\delta (n_{1},\ldots ,n_{k})(p)$ is defined as

$\delta (n_{1},\ldots ,n_{k})(p)=\tau (T_{p}M)-inf\lbrace \tau (L_{1})+\ldots +\tau (L_{k})\rbrace$

with $L_{1},\ldots ,L_{k}$running over all $k$ mutually orthogonal subspaces of the tangent space of $M$ at $p$ such that $dimL_{j}=n_{j}$

Vector Harmonics

Connection with representation theory

Vector spherical harmonics are related to irreducible representations of $\mathrm {SO} (3)$ on tangent vector fields to the sphere just as spherical harmonics are for ordinary functions on the sphere. When considering Euclidean space there is an additional normal (radial) direction which transforms trivially such that a vector field in this direction decomposes essentially as an ordinary function, as seen from $\mathbf {Y} _{lm}$.

Highest weight construction

Consider first the sphere $S^{2}$ rather than the Euclidean space. The (oriented) sphere $S^{2}\cong \mathrm {SO} (3)/\mathrm {SO} (2)$ is a homogeneous space for the Lie group $\mathrm {SO} (3)$ of rotations. This means in particular that there is a group action of rotations on the manifold, i.e. each $g\in \mathrm {SO} (3)$ determines an isomorphism $\phi _{g}:S^{2}\to S^{2}$ such that these isomorphisms form a representation of the group under composition. Similarly, the pushforward of these isomorphisms form a representation on the tangent bundle $TS^{2}$. The infinitesimal action associated to the related Lie algebra on a differentiable manifold is generated by the Lie derivative. If $\rho ({\mathfrak {g}})$ is the representation of an element of the algebra on $TS^{2}$, the action of the Lie derivative will be denoted simply as $L_{g}$. An irreducible representation of the algebra ${\mathfrak {so}}(3)$ on $TS^{2}$ can be constructed by the method of highest weights. Let $J_{+},J_{-}\in {\mathfrak {so}}(3)$ be respectively raising and lowering operators for $J_{z}\in {\mathfrak {so}}(3)$ and let $V_{\mu \nu }$ be a section of $TS^{2}$ such that

$L_{J_{z}}V_{\mu \nu }=i\nu V_{\mu \nu }$,

$(L_{J_{-}}L_{J_{+}}/2+L_{J_{-}}L_{J_{+}}/2+L_{J_{z}}L_{J_{z}})V_{\mu \nu }=-\mu (\mu +1)V_{\mu \nu }$,

$L_{J_{+}}V_{\mu \nu }=0$.

In words, $V_{\mu \nu }$ diagonalizes $J_{z}$ and the Casimir element $J^{2}$, and is of highest weight. Denote also

$L_{J_{-}}V_{\mu \nu }=V_{\mu (\nu -1)}$,

by the definition of the lowering operator $[J_{z},J_{-}]=-iJ_{-}$ this is consistent with

$L_{J_{z}}V_{\mu (\nu -1)}=i(\nu -1)V_{\mu (\nu -1)}$.

Now using $[J_{+},J_{-}]=2iJ_{z}$ one can derive for the highest weight state $\mu =\nu$ and similarly for the lowest weight state $\mu =-\nu$. In terms of the spherical coordinates (θ, φ) the algebra can be realized explicitly as $J_{z}=\partial _{\phi }$, $J_{+}=e^{i\phi }(\partial _{\theta }+i\cot {\theta }\partial _{\phi })$, $J_{-}=e^{-i\phi }(\partial _{\theta }-i\cot {\theta }\partial _{\phi })$ to find for the components of $V_{\mu \nu }=V_{\mu \nu }^{\theta }\partial _{\theta }+V_{\mu \nu }^{\phi }\partial _{\phi }$.

$\partial _{\phi }V_{\mu \nu }^{\theta }=i\nu V_{\mu \nu }^{\theta }$, $\partial _{\phi }V_{\mu \nu }^{\phi }=i\nu V_{\mu \nu }^{\phi }$

$r^{2}\Delta V_{\mu \nu }^{\theta }=-\mu (\mu +1)V_{\mu \nu }^{\theta }+\csc ^{2}{(\theta )}V_{\mu \nu }^{\theta }+2\cot {(\theta )}{\frac {\partial V_{\mu \nu }^{\phi }}{\partial \phi }}$,

$r^{2}\Delta V_{\mu \nu }^{\phi }=-\mu (\mu +1)V_{\mu \nu }^{\phi }+2(V_{\mu \nu }^{\phi }-\cot {(\theta )}{\frac {\partial V_{\mu \nu }^{\phi }}{\partial \theta }})-2\cot {(\theta )}\csc ^{2}{(\theta )}{\frac {\partial V_{\mu \nu }^{\theta }}{\partial \phi }}$.

We can now extend easily extend this discussion to the tangent space of ... or other differentiable manifolds with a natural group action associated to rotations

combining these as recovers the expressions for $\mathbf {Z} _{lm}=\mathbf {Y} _{lm},\mathbf {\Psi } _{lm},\mathbf {\Phi } _{lm}$

Ward identity

One of the big problems that had been plaguing the theory of quantum electrodynamics (QED) since the early 1930s was the appearance of divergences at all but the lowest order in perturbation theory. This had been pointed out, for instance by, Robert Oppenheimer.. Near the end of the 1940s, this difficulty was being resolved following the observation of Hans Bethe that the experimental value of the the Lamb shift could be recovered to excellent agreement by attaching infinities to corrections of mass and charge that were fixed to a finite value by experiments . In this way, the infinities get absorbed in those constants and yield a finite result in good agreement with experiments. This procedure was named renormalization. After fundamental work by Shin'ichirō Tomonaga, Julian Schwinger and Feynman, who were awarded with a Nobel prize in physics in 1965 for this , Freeman Dyson then showed that renormalization could be used to consistently remove divergences to each order in perturbation theory . Starting from the QED Lagrangian in terms of the bare electron–positron field ψ_B , four-potential of the electromagnetic field A_B^μ, electromagnetic field tensor F_B^μν, the coupling constant e_B and the electron-positron mass m_B

${\mathcal {L}}={\bar {\psi }}_{B}\left[i\gamma _{\mu }\left(\partial ^{\mu }+ie_{B}A_{B}^{\mu }\right)-m_{B}\right]\psi _{B}-{\frac {1}{4}}F_{B\mu \nu }F_{B}^{\mu \nu }$

with Dirac matrices γ^μ. The renormalization can be expressed in a way such that the bare Lagrangian terms are multiples of the renormalized one

$\left({\bar {\psi }}m\psi \right)_{B}=Z_{0}{\bar {\psi }}m\psi$

$\left({\bar {\psi }}\partial ^{\mu }\psi \right)_{B}=Z_{1}{\bar {\psi }}\partial ^{\mu }\psi$

$\left({\bar {\psi }}ieA^{\mu }\psi \right)_{B}=Z_{2}{\bar {\psi }}ieA^{\mu }\psi$

$\left(F_{\mu \nu }F^{\mu \nu }\right)_{B}=Z_{3}\,F_{\mu \nu }F^{\mu \nu }.$

Dyson had conjectured that Z₁=Z₂ to all orders in perturbation theory and this is what Ward set out to prove in 1950 . To do so, Ward used an identity the QED interaction vertex with equal electron momenta to the electron propagator. It did not take long before it was pointed out explicitly that this was a consequence of the gauge symmetry of QED .

Airy structures

Introduced by Kontsevich and Soibelman? named after the archetypical example of the Airy spectral curve $x=y^{2}$.

Not to be confused with Airy structure in reference to the appearance of a Airy disk.

Classical Airy structure

A classical Airy structure is a set of Hamiltonians

$l_{i}=y_{i}-{\frac {1}{2}}A_{ijk}x^{j}x^{k}-B_{ij}^{k}x^{j}y_{k}-{\frac {1}{2}}C_{j}^{jk}y_{j}y_{k}$,

which satisfy

$\lbrace l_{i},l_{j}\rbrace =f_{ij}^{k}l_{k}$,

with respect to the Poisson bracket

$\lbrace f,g\rbrace ={\frac {\partial f}{\partial y_{i}}}{\frac {\partial g}{\partial x^{i}}}-{\frac {\partial f}{\partial x^{i}}}{\frac {\partial g}{\partial y_{i}}}$.

Remark we are using the summation convention.