# Tagged Questions

189 views

### Can eta invariant be written in terms of topological data?

The eta invariant was introduced by Atiyah, Patodi, and Singer. It roughly measures the asymmetry of the spectrum of a self-adjoint elliptic operator with respect to the origin. In ...
67 views

### Short examples that are/are not quantum-ergodic

Are there any considerably short examples of manifolds that are/aren't quantum ergodic, or quantum unique ergodic? Note that a (compact) Riemannian manifold is said to be quantum ergodic if ...
140 views

420 views

### How to characterize Dirac's gamma matrices in differential geometry?

I want to understand what is the interpretation of Dirac gamma matrices in differential geometry. Basically, I am considering the Dirac matrices as 3-indexed tensors, which means a tensor with 1 ...
169 views

### quantization for integral coadjoint orbits

I am looking for a referrence for proof of following statement. Let $G$ be a semi-simple Lie group and $\mathfrak{g}$ be its Lie algebra and $\alpha_k$ be simple roots of $\mathfrak{g}$ and $\mu_0$ ...
215 views

### Curvature as infinitesimal holonomy

Let $P \to M$ be a principal $G$-bundle, assume as much regularity as you want (compact $G$ or compact base manifold, ect). Via parallel transport, a connection $A$ on $P$ gives rise to the holonomy ...
105 views

### When the leaves of a distribution are compact

Let $(M,\omega)$, be a symplectic manifolds. A subbundle $P\subset TM^{\mathbf{C}}$ of the complexified tangent bundle is called a complex polarization if $P$ is Lagrangian P involutive ...
137 views

### Metalinear frame bundle on sphere or $\mathbb{C}P^n$

Let $M$ be a smooth manifold. A complex metalinear frame bundle $\tilde F(P)\to M$ of a rank $n$ complex vector bundle $P\to M$ is a principal $ML(n,\mathbb{C})$-bundle together with a covering map ...
153 views

### An example for the case tht if the leaves of polarization be non-compact then the polarized sections are not square integrable

Let $(M,\omega)$, be a symplectic manifolds. A subbundle $P\subset TM^{\mathbf{C}}$ of the complexified tangent bundle is called a complex polarization if $P$ is Lagrangian P involutive ...
268 views

### The space of holomorphic sections are finite dimensional?

I start my question with a definition and some motivation. Let $M$ be a symplectic manifold. A subbundle $P\subset TM^{\mathbf{C}}$ of the complexified tangent bundle is called a complex ...
77 views

776 views

### How does hyperbolicity of space time affect our lives?

My main research has been in hyperbolic geometry and geometric group theory. I always thought that the only real "application" of my work was that the universe is a 3-manifold. But recently I found ...
113 views

### universal connection for SU(3)

In 1961, Narasimhan and Ramanan (Am J Math 83 563) showed that one could represent an arbitrary SU(3) connection as $i t^c_{a b} A^c_\mu(x) = e^*_a \cdot e_{b, \mu}(x)$ in which the $t^a$'s are ...
129 views

### Self Dual 2-Forms on Complexified Minkowski Space

I'm trying to get my head around integrability in twistor theory, but am struggling with interpreting the concept of self-duality on complexified spaces. One can complexify Minkowski space to ...
364 views

### Wedge Product of Lie Algebra Valued One-Form

I've been reading about the formal structure of gauge theories and am a little confused by the notation. Could someone clarify this for me? Suppose that $A$ is a Lie algebra valued 1-form ...
153 views

Definition: A scalar k-th order differential equation on a smooth manifold $M$ , is $F(x,v,\frac{\partial {^\left | \sigma \right |}v}{\partial x^\sigma })=0$ for $\left | \sigma \right |\leqslant ... 1answer 365 views ### necessary and sufficient condition for existence of$SU(3)$-structure on 6-manifolds Is there any necessary and sufficient condition for existence of$SU(3)$-structure on 6-manifolds$M$? 3answers 467 views ### Space of sections of a fibre bundle with non-compact base space Let$\pi: E \rightarrow M$be a fiber bundle over the manifold M and denote by$\Gamma(E)$the space of smooth sections of$E$. For compact$M$it is well known (Hamilton 1982, Part II Corollary ... 1answer 1k views ### The Dedekind Eta Function in Physics This interesting little fellow popped up in some operator algebra (Witt / Virasoro Lie algebra) I was exploring, so I've been curious about where else the Dedekind$\eta$-function makes a cameo ... 2answers 388 views ### Can a sphere be a phase space? Put in other words, given an even-dimensional sphere$S^{2k}$: is there a manifold$M$such that$T^* M$is diffeomorphic to$S^{2k}$? 2answers 312 views ### Is there a lattice model of E8 manifold? Background I'm using physics terminology because I'm not sure what the right mathematical terminology is, perhaps a simplicial complex? I'm interested, for various physics reasons, in four manifolds ... 1answer 598 views ### Functional/variational derivative and the Leibniz rule I am currently trying to understand the BV-formalism, which makes heavy use of the functional derivative. Let us consider the functional derivative, as defined in for example its Wikipedia article. ... 4answers 589 views ### Applications of Hodge-de Rham Laplacian on p-forms ($p\neq 0,n$) in physics or engineering The Hodge-de Rham Laplacian$L=(d+d^*)^2$, where$d\$ is the boundary operator of the de Rham complex, is well-known in the math community. Recently, I tried very hard to search for examples of its use ...
It is well-known that the equation for stationary solutions of the Einstein-Hilbert functional (without matter and cosmological constant, which is irrelevant here): $$S = \int_M R \mu_g,$$ is given by ...