5
votes
1answer
192 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 ...
5
votes
0answers
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 ...
4
votes
2answers
140 views

Lagrangians with the same extremal curves

It is well know that (in the sense having the same image not parametrization) the extremal curves of the energy functional on a Riemanian manifold $(M,g)$: $E[\gamma(t)] = \int ...
1
vote
1answer
231 views

A question about complex polarization

Let $M$ be a symplectic manifold, Then a subbundle $P\subset TM^{\mathbf{C}}$ of the complexified tangent bundle is called a complex polarization if $P$ is Lagrangian P involutive dim$P\cap\bar P ...
4
votes
5answers
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 ...
1
vote
0answers
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 $\mu_0$ ...
5
votes
0answers
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 ...
0
votes
0answers
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 ...
0
votes
0answers
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 ...
0
votes
1answer
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 ...
1
vote
1answer
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 ...
2
votes
1answer
77 views

An example to show that when $P$ is a complex polarization the subbundle $P+ \bar P$ is not necessarly involutive

I start my question with some motivation. A subbundle $P\subset TM^{\mathbf{C}}$ of the complexified tangent bundle is called a complex polarization if $P$ is Lagrangian P involutive dim$P\cap\bar ...
-1
votes
1answer
137 views
0
votes
1answer
204 views

On the Geroch's argument

During the study of Geroch's argument to prove positive mass theorem, I faced a problem explained below: Suppose $(M,g_{\mu \nu})$ is a four dimensional Lorentzian Manifold and $\Sigma$ is a ...
1
vote
2answers
478 views

Uniqueness on square root of complex Line Bundle

Let $L$ be a line bundle over a compex manifold $X$, a square-root of $L$ is a line bundle $M$ such that $M^{\otimes2}=L$. My question is when the square-root of Line Bundle is unique?
4
votes
1answer
327 views

Quantization of symplectic vector space and choice of lagrangian subspaces

My question is related to Geometric Quantization. I don't undrestand the philosophy of following assertion If $(V,\omega)$ be a symplectic vector space then the quantizations of $V$ ...
3
votes
0answers
197 views

Looking for a necessary and sufficient condition for the polarization $\mathbb{P}$ being positive

My question is about positivity of polarization in Geometric quantization theory. Let $\mathbb{P}$, be a complex polarization on symplectic manifold $(M,\Omega)$. For every $m\in M$, we can define a ...
2
votes
1answer
320 views

looking for an identity for higher jet bundle $J^kM$?

We know this fact that the first jet Bundle $J^1M$ is diffeomorphic with $T^*M×\mathbb{R}$.i.e, ($J^1M=T^*M×\mathbb{R}$) Is there something like this identity for higher jet bundle $J^kM$? I editted ...
4
votes
0answers
195 views

Local version of a slice (for a Lie group action)

Let $\Upsilon: G \times M \to M$ be a smooth action of a Lie group $G$ on a manifold $M$. Isenberg and Marsden (1982) define a slice at $m \in M$ as a submanifold $S \subseteq M$ containing $m$ such ...
1
vote
1answer
313 views

pre-quantization of Jet bundle

We know that the notion of Jet bundle $J^kM×\mathbb{R}$, is generalization of cotangent bundle. What is the prequantization of $J^kM×\mathbb{R}$?
2
votes
0answers
186 views

a question about geometric quantization background

I don't understand why for geometric description of a regular system, we take always the classical phase space as a symplectic manifold?
3
votes
1answer
151 views

Does fixing the reparameterization invariance of the string action correspond to some kind of orbifolding?

Does fixing the reparameterization invariance of the string action, for example by choosing the light-cone gauge $$ X^{+} = \beta\alpha' p^{+}\tau $$ $$ p^{+} = \frac{2\pi}{\beta} P^{\tau +} $$ ...
16
votes
1answer
573 views

Combinatorial spin structures

I would like to know how to define spin structures combinatorially, for an oriented smooth manifold equipped with a triangulation. In the case of a 2d manifold, spin structures correspond to ...
0
votes
1answer
288 views

A reference about Dolbeault cohomology

I am looking for a reference about Dolbeault cohomology when the line bundle is not supposed to be positive.
3
votes
2answers
384 views

Van Vleck-Morette Determinant

There seems to be something curious about the so-called Van-Vleck-Morette determinant, as I cannot find any source that properly defines it in terms of expressions previously defined in that source ...
1
vote
1answer
132 views

pre-symplectic and foliation and its trajectories

Let $(M,\omega)$, be pre-symplectic, then can we say, we have a foliation of $M$, with tangent spaces $ker\omega$.What can we say about its trajectories. ?
8
votes
2answers
405 views

Hodge decomposition in Minkowski space

This question is motivated by the physical description of magnetic monopoles. I will give the motivation, but you can also jump to the last section. Let us recall Maxwell’s equations: Given a ...
2
votes
1answer
138 views

Extending the variational bicomplex to Hamiltion or Hamiltion-Jacobi formalism

The variational bicomplex seams to provide a modern formulation of the variational problem in terms of modern differential geometry. In particular the bigraded complex of differential forms ...
3
votes
1answer
195 views

What are the invariant definitions of spinorial quantities from mathematical physics?

When physicists write expressions involving spinors $\psi \in S \otimes V$, where $S=S_+ \oplus S_-$ is a complex spinor representation of a spin group $Spin(2d)$ and $V$ is a complex representation ...
3
votes
2answers
462 views

Energy functional

During my study on Ricci Flow I faced some functional known as energy functional. For example Einstein-Hilbert functional is called an energy functional, also in Perelman's works ...
1
vote
0answers
109 views

Felder Kazhdan classical master equation

Has there been any follow up by anyone to Giovanni Felder, David Kazhdan, The classical master equation (arXiv:1212.1631)? Other than on nlab, I haven't found any citations.
7
votes
1answer
312 views

How unique is a conformal compactification?

I'm trying to understand the term "conformal compactification" which is often used in physics. I reckon that most places take this to mean a (sometimes specific) compact conformal completion. That is, ...
3
votes
1answer
337 views

conservation law and generalized Symplectic Monge-Ampere equation arising from 3-variables

If we have a Jacobi PDE system with conservation law $\theta \in \Omega^1(M)$ such that $d \theta$ is non-degenerate 2-form , then we know this fact that it can be written as symplectic 2D ...
6
votes
1answer
484 views

Darboux like theorem for non-degenerate 3-forms in 6-manifolds

we know Darboux theorem for higher-symplectic geometry is not correct in general, but is there any Darboux like theorem for non-degenerate 3-forms in 6-manifolds?
2
votes
1answer
236 views

symmetry of generationg function of PDE

We know that for finding the solutions of PDE equations, one of methods is "reduction of PDE", . For nonlinear equation $v_t=(v^{-4/3}v_x)_x+\lambda v$ how can we compute the generators of Lie ...
1
vote
0answers
278 views

a question about contact manifolds

Let $\mathfrak{g}=< X_{f_1},X_{f_2},...,X_{f_n}> $ be a lie algebra of contact symmetries , $\mathfrak{g}\subset Sym(E_\omega )$ , such that the orbit spact $ B_\mathfrak{g}=E_\mathfrak{g}/ ...
17
votes
2answers
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 ...
2
votes
0answers
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 ...
1
vote
0answers
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 ...
1
vote
1answer
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 ...
1
vote
0answers
153 views

multivalued solution of a equation

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 ...
7
votes
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 $M$?
8
votes
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 ...
12
votes
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 ...
1
vote
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 $S^{2k}$?
1
vote
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 ...
3
votes
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. ...
6
votes
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 ...
4
votes
1answer
557 views

Geometric derivation of the Einstein’s field equation from the Hilbert action.

It is well-known that the equation for stationary solutions of the Einstein-Hilbert functional is given by the Einstein field equation (for a statement, see previous question). The standard derivation ...
7
votes
1answer
458 views

Coordinate-free derivation of the Einstein's field equation from the Hilbert action.

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 ...