## 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 $$mathbb {R} ^ 4$$we know that his boulevard is steerable. So $$U partial$$ is a closed orientable $$3$$-manifold, but which one is it?