dg.differential geometry – Instanton numbers for diverse gauge bundles on diverse manifolds — their relations to characteristic classes

It is standard (?) that the $SU(N)$ gauge theory has the instanton number $n$ quantized as $n in mathbb{Z}$
$$
n = { 1 over 8pi^2} int_{mathcal{M}_{4}} text{tr} left(F wedge Fright) = {1 over 64 pi^2} int_{mathcal{M}_{4}} d^4 x , epsilon^{munurholambda} F_{munu}^alpha F_{rholambda}^alpha
in mathbb{Z}
$$

Here $F$ is the curvature 2-form $F=d A + A wedge A$ and $A$ is the gauge 1-connection of the gauge bundle of gauge group $G$. Also $F=F^alpha T^alpha$ where $alpha$ is the Lie algebra indices, with repeated indices summed over.

We know that:

  • If the gauge group is $G=SU(2)$, the instanton number $n in mathbb{Z}.$ I think this can be understood as
    $$
    n = c_2(V_{SU(2)}) in mathbb{Z}?
    $$
  • If the gauge group is $G=SO(3)$ on the spin 4-manifold $mathcal{M}_{4}$, then $n in frac{1}{2}mathbb{Z}$. (I use the notation to mean that $n in frac{1}{2}mathbb{Z}$ as $n$ takes the half integer values.)
    $$
    n = p_1(V_{SO(3)})/4 in frac{1}{2}mathbb{Z}?
    $$
  • If the gauge group is $G=SO(3)$ on the non-spin 4-manifold $mathcal{M}_{4}$, then $n in mathbb{Z}/4$.
    $$
    n = p_1(V_{SO(3)})/4 in frac{1}{4}mathbb{Z}?
    $$

What are the general statements we can make for other general $G$ and other manifolds?

Questions:

  1. If the gauge group is $G=U(1)$, the instanton number $n in 2 mathbb{Z}.$ True or False? We can express this $n$ as the first Chern class $c_1$ square of associated vector bundle $V_{U(1)}$ as
    $$
    n = c_1(V_{U(1)})^2 in 2 mathbb{Z}?
    $$

    Is this correct?

  2. If the gauge group is $G=SU(N)$, the instanton number $n in mathbb{Z}.$ True or False? We can express this $n$ as the second Chern class $c_2$ of associated vector bundle $V_{SU(N)}$ as
    $$
    n = c_2(V_{SU(N)}) in mathbb{Z}?
    $$

  3. If the gauge group is $G=PSU(N)$ on the spin 4-manifold $mathcal{M}_{4}$, the instanton number can be $1/N$ fractional of $mathbb{Z}$ values:
    $$n in frac{1}{N} mathbb{Z},$$
    True or False? What is the precise mathematical invariant to characterize this $n in frac{1}{N} mathbb{Z}$? Is that Pontryagin class $p_1$ when $G=PSU(N)$ is real $mathbb{R}$? Or some $c_2(V_{PSU(N)})$ when $PSU(N)$ is complex $mathbb{C}$?

  4. If the gauge group is $G=PSU(N)$ on the non-spin 4-manifold $mathcal{M}_{4}$, the instanton number can be $1/N^2$ fractional of $mathbb{Z}$ values: $$n in frac{1}{N^2} mathbb{Z},$$
    True or False? What is the precise mathematical invariant to characterize this $n in frac{1}{N^2} mathbb{Z}$? Is that Pontryagin class $p_1$ when $G=PSU(N)$ is real $mathbb{R}$? Or some $c_2(V_{PSU(N)})$ when $PSU(N)$ is complex $mathbb{C}$?

  5. If the gauge group is $G=U(N)$ on the spin or non-spin 4-manifold $mathcal{M}_{4}$, the instanton numbers carry both the $U(1)$ and $PSU(N)$ part with constraints. So there are two instanton numbers
    $n_{U(1)}$ and $n_{PSU(N)}$.
    What can be their constraints?
    $$n_{U(1)} in ?$$
    $$n_{PSU(N)} in ?$$

differential topology – Rank of mapping betweeen two manifolds

Let $M$ be a $7$-dimentional manifold with nonempty boundary $partial M$ and let $f:Mrightarrow N$ be a smooth mapping where $N$ is $5$-dimentional. Assume tham for some $pinpartial M$ we have $rank(f|_{partial M},p)=3$. I need to prove that $$3leq rank(f,p)leq4$$
I think I have the left side figured out. We know that $partial M$ is $6$-dimentional manifold. The matrix of derivatives has $6$ columns and $5$ rows. We know that rank of that matrix is $3$. On the other hand when we consider a whole manifold $M$, then we have a matrix with $7$ columns and $5$ rows. Rank of such matrix cannot be less than $3$ because it has the matrix for $partial M$ in it.

Since $N$ is $5$-dimetional we have $rank(f,p)leq 5$ but I don’t know how to prove the right inequality.

riemannian geometry – Principal eigenvalue of non self-adjoint elliptic operators on closed manifolds

Consider the elliptic operator $Lu = – Delta u + langle nabla u , X rangle + c , u $ acting on functions on a closed Riemannian manifold $M$. Here $Delta$ denotes the Laplace-Beltrami operator, $X$ is an arbitrary smooth vector field, and $c geq 0$ is a smooth function on $M$ which does not vanish identically. Does $L$ have a so-called `principal eigenvalue’ $lambda_1 > 0$, whose corresponding (unique up to scaling) eigenfunction does not change sign?

A similar statement holds for smooth domains in $mathbb{R}^n$, as shown for instance in Evans’ PDE book, chapter 6. Moreover, in this paper it is sated that this fact is equivalent to the operator satisfying a maximum principle (which is indeed the case for the above $L$).

dg.differential geometry – Equality of the derivative of the exponential map on Kähler manifolds

Let $ M $ be a Kähler collector, $ omega $ its Kähler form and $ J $ the complex structure. Also, let $ V $ to be a smooth vector field (or even analytical, I'm not sure it matters) on $ M $, $ p in M ​​$ and $ c: (- epsilon, epsilon) rightarrow M $ and integral curve of $ V $ such as $ c (0) = p $. Here we assume that $ epsilon $ is small enough. Moreover, $ g = omega ( cdot, J cdot) $ to be the associated Riemannian metric. Consider the card $ F (u, tau) = exp_ {c ( tau)} ((u_ {0} + u) JV (c ( tau))) $, or $ u, u_ {0} $ are small enough and $ exp $ is relative to the riemannian metric $ g $

Equity:
$$ frac {d} {d tau} vert _ { tau = 0} F (0, tau) = J (F (0,0)) frac {d} {du} vert_ {u = 0} F (u, 0) $$
hold on?

Greetings,
Fabien

manifolds – Is a pentagon a surface?

My question arises because a pentagon is homeomorphic to a closed disc. The latter is a surface with limits.

However, a pentagon has vertices, so it does not appear to be a double.

If you consider a pentagon without edges, it is a collector at 2, but is not compact, so it is not a surface.

So … any polygon is a surface?

cordially

ag.algebraic geometry – Do Neron-Severi groups of smooth projective unirational manifolds contain a twist $ ell $?

Let $ X $ to be a smooth projective unirational variety on an algebraically closed characteristic field $ p> 0 $, and $ ell neq p $ a first. My question: the Neron-Severi group of $ X $ contain (not null) $ ell $-torsion? This seems to be closely related to the presence of torsion in the etal cohomology group. $ H ^ 2_ {and} (X, mathbb {Z} _l) $.

ag.algebraic geometry – Rham algebraic cohomology for singular manifolds

I would like to know to what extent Rham's naive algebraic cohomology is a "bad" theory of cohomology. Yes $ X $ is smooth then there is a comparison theorem with a singular cohomology. Yes $ X $ is singular, so Hartshorne integrates $ X $ in a smooth manifold generalizes the definition and proves a similar comparison theorem singular cohomology, Poincaré duality and other things (http://www.numdam.org/article/PMIHES_1975__45__5_0.pdf).

In this thread (Rham's naive cohomology fails for singular varieties), we learn that De Rham's naive cohomology is certainly not the same as singular cohomology and does not satisfy Poincaré's duality .

Despite these negative results, my question is as follows.

Yes $ X $ is singular, can we still prove that his naive Rham cohomology is finely generated? Can we prove that it is zero above its dimension?

All known results or references are welcome.

dg.differential geometry – Theorem13.30 Einstein Manifolds (Besse). Dirac operator index, Chern character calculation of a torsion beam of symmetric product

I am having trouble understanding some calculation lines from the text of the Einstein de Besse collectors (see image below).

We twist the beam of spins (on Einstein 4-manifold) $ Sigma $ with an auxiliary beam $ S ^ 3 Sigma ^ – $. We train the operator Dirac $ mathscr {D} ^ D $ formed by twisting the Levi-Civita connection on $ Sigma $ with a copy $ D $ acting on $ S ^ 3 Sigma ^ – $ (and compose with the Clifford action acting trivially on $ S ^ 3 Sigma $). Besse evaluates the index of this operator using the APS theorem as an integral on the Chern and Pontrjagin classes.

I understand that the $ (1- frac {1} {24} p_1) $ terms comes from the $ widehat {A} $-the genre of the variety, and further the signature theorem, $ tau = frac13 int_M p_1 (M) $. However, I do not understand the evaluation of the Chern character of the twisted beam as $ (4-10c_2) $ and subsequent evaluation in terms of Euler characteristic and signature.

Any idea would be greatly appreciated.

Calculation of the index

dg.differential geometry – are the harmonic functions on hyperbolic manifolds with finite volume constant?

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manifolds – What is the limit of a tubular neighborhood of the projective plane embedded in $ mathbb {R} ^ 4 $?

There are different ways to integrate the projective plane into $ mathbb {R} ^ $ 4 very well, see for example Wikipedia. Suppose now that I take such an integrated projective plan $ P subset mathbb {R} ^ $ 4 and repair a tubular neighborhood $ U $ of $ P $. Now $ overline {U} $ is a $ 4 $-manifold with an integrated border in $ mathbb {R} ^ $ 4we know that his boulevard is steerable. So $ U partial is a closed orientable $ 3 $-manifold, but which one is it?