Questions tagged [index-theory]
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140
questions
3
votes
0answers
111 views
Discrete spectrum of Dirac operator
It is said that if we take the spacetime manifold to be a sphere $S^d$ of large volume so
that the spectrum of Dirac operator $$i\gamma^\mu D_{\mu}$$ is discrete.
For example at least for $d=4$, this ...
14
votes
1answer
822 views
Atiyah's proof of the moduli space of SD irreducible YM connections
In the paper "Self-duality in Four-dimensional Riemannian Geometry" (1978), Atiyah, Hitchin and Singer present a proof that the space of self-dual irreducible Yang-Mills connections is a ...
4
votes
0answers
90 views
Push forward of Chern character and index theorem
I have some trouble understanding a proposition in Leung's paper "Symplectic Structures on Gauge Theory" published in Commun. Math. Phys. 193, 47 – 67 (1998).
I expose here the setup for my ...
2
votes
1answer
179 views
can the actions of fundamental groups annihilate homology?
Let $X$ be a path-connected manifold (or a CW complex).
Let $\pi_1(X)$ be the fundamental group of $X$.
Let $\pi: \tilde X\longrightarrow X$ be a covering map.
For each $m\geq 0$, let $C_m(\tilde X)$ ...
4
votes
0answers
61 views
Analyticity of the regularized $\eta$-invariant
The APS $\eta$- invariant of an operator $B$ with eigenvalues $\lambda$ is defined as
$$\eta = \sum_\lambda sgn (\lambda)$$
which is a divergent sum and it can be regularized as follows:
$$\eta(s) = \...
0
votes
1answer
82 views
How to define an equivariant Kasparov's KK-theory map?
I'm looking for some references about how to construct an equivariant Kasparov's KK-theory map $$ \psi \ : \ KK^{G_{1}} ( A,B ) \to KK^{G_{2}} ( C,D ) $$, where, $ G_1 $ and $ G_2 $ are two distinct ...
1
vote
0answers
49 views
A map from a $ G_1 $ - equivariant KK-theory of Kasparov, to a $ G_2 $ - equivariant KK-theory of Kasparov
Let $ G $ be a locally compact group.
Let $ H $ and $ K $ be two normal subgroups of $ G $.
In order to construct a map, $$ \psi \ : \ \ F(G/H,G/K) \to F(G/K,G/H) $$
where, $$ F(G/H,G/K) = KK^{G/H} ( ...
2
votes
0answers
88 views
Spectrum of the Witten Laplacian on compact Riemannian manifolds
Below I have given what I am calling as the ${\rm Witten{-}Laplacian}_{s,p}$ on a Riemannian manifold $(M,g)$ for any constant $s >0$ and $p \in C^2(M,g)$
How generally is it true that this ${\rm ...
4
votes
1answer
143 views
Equivalence of families indexes of Fredholm operators
Let $F=F(H,H)$ be the space of bounded Fredholm operators in a Hilbert space $H$ with topology inherited from the norm operator topology, and let $X$ be a compact topological space.
For a continuous ...
7
votes
0answers
202 views
Atiyah-Singer theorem in heat kernels and Dirac operators
I'm reading "Heat kernels and Dirac operators" by Berline, Getzler and Vergne.
I have some trouble to understand a identity on the bottom of page 146 which is essential for the proof of the ...
5
votes
1answer
180 views
McKean-Singer formula in Heat Kernels and Dirac Operators book
I'm reading "Heat Kernels and Dirac Operators" by Berline, Getzler and Vergne. The setting is: $E \to M$ is a $\mathbb{Z}_2$-graded vector bundle on a compact Riemannian manifold $M$
and $D :...
6
votes
0answers
203 views
Spectral flow of Dirac operator twisted by instanton
Suppose $E$ is a $SU(2)$-bundle over a closed three manifold $M$ and $S$ is the spinor bundle over $M$. Also assume $D_{A(t)}:\Gamma(S\otimes_{\mathbb c} E)\to \Gamma(S\otimes_{\mathbb c} E)$ is a ...
1
vote
0answers
48 views
Splitting formulas for spectral flows
I'm asking if there are splitting formulas for equivariant spectral flow and higher spectral flow (of Dai-Zhang) for paths of Dirac operators, concerning gluing together two smooth compact Spinc ...
1
vote
1answer
89 views
Is being a deg 0 vector field equivalent to being locally non-surjective?
Let $X:\mathbb R^n\to\mathbb R^n$ be a $C^1$ (or smooth)-vector field, such that $X(0)=0$ is an isolated zero. So we can talk about the mapping of $0$ for $X$.
For convenience assume $0$ be the only ...
3
votes
0answers
72 views
Dixmier trace, Wodzicki residue and topological index
There are a well-known facts about Dixmier trace and Wodzicki residue. Let $P$ be an elliptic pseudodifferential operator of degree $−n$ on a compact Riemannian manifold $(M,g)$, than its Dixmier ...
1
vote
0answers
47 views
About the orientation of index formula on orbifold
Let $X$ be a closed oriented orbifold with singularity $\Sigma X$. The singularity is defined as
$$ \Sigma X=\{(x)|~x\in X,~G_x\neq1\}, $$
where $G_x$ is the isotropy group.
For $u\in K_v(TX)$, as ...
7
votes
0answers
208 views
Applications of the Atiyah-Patodi-Singer eta-function $\eta(s)$
The eta function of a differential operator was used by Atiyah, Patodi and Singer to derive their famous index theorem, and is given by
$$
\eta(s)=\sum_{\lambda\neq 0}(\mathrm{sign}\lambda)|\lambda|^...
2
votes
0answers
86 views
$\mathbb{Z}_2$-grading by Hodge star operator (for signature theorem)
This question may be a bit low level for MO but I have not received any attention from the SE post.
Consider the algebra of exterior forms $\bigwedge T^*M$ on an even dimensional $n$-manifold $M$. We ...
10
votes
1answer
450 views
Index of Dirac operator and Chern character of symmetric product twisting bundle
I am having trouble understanding a couple of lines of computation from Theorem 13.30 in Besse's Einstein Manifolds text
We are twisting the spinor bundle (on Einstein 4-manifold) $\Sigma$ with an ...
2
votes
0answers
185 views
Twisting holomorphic vector bundles and Euler characteristics
Given a holomorphic vector bundle $\mathcal{V}$ over a compact complex manifold $M$, it seems that even if $\mathcal{V}$ is non-trivial, then it can still have trivial Euler characteristic, that it,
$$...
4
votes
0answers
102 views
Fredholm theory of non elliptic operators
In this question we search for a big list of non elliptic operators whose Fredholm index is finite or whose Fredholm theory is extensively discussed. The main motovation is the conference linked in ...
11
votes
0answers
370 views
Elementary-ish geometric proof of Hirzebruch signature theorem for Riemannian 4-manifolds?
The Hirzebruch signature theorem tells us that for a smooth compact oriented 4-manifold, the signature $\sigma(M)$ is proportional to the first Pontryagin number of $M$:
$$
3\sigma(M)= p_1(M) = k \...
0
votes
0answers
53 views
A uniform upper bound for Fredholm index of Laplace quasi-operators on a compact parallelizable manifold
Assume that $M$ is a compact parallelizable manifold. Is there an upper bound for the absolute value of Fredholm index of all operators in the form $D=\sum_{i=1}^n \partial^2/\partial{X_i^2}$...
2
votes
1answer
118 views
Is there a fixed-point index theorem that treats the fixed points on the boundary?
Suppose $f$ is a continuous function from a unit cube $[0,1]^n$ to itself, then $f$ has at least a fixed point. Further suppose $f$ is smooth, $0$ is a regular value of $f(x)-x$, and the fixed points ...
0
votes
1answer
53 views
An adjoint characterization of (unbounded) Fredholm operators
Let $\mathcal{H}_i$, for $i=1,2$, be Hilbert spaces, and $T:{\frak Dom}(T) \subseteq \mathcal{H}_1 \to \mathcal{H}_2$ a densely-defined closed operator. If the kernel of $T$, and the kernel of its ...
6
votes
1answer
146 views
Coarse index of Dirac operator on $\mathbb{R}$
Let $D=i\frac{d}{dx}$ be the Dirac operator on $\mathbb{R}$, acting on the spinor bundle $\mathbb{R}\times\mathbb{C}$. The bounded operator $F=\frac{D}{\sqrt{D^2+1}}$ has a coarse index
$$\text{Ind}(...
6
votes
0answers
222 views
Can spin structures and Arf invariants be defined in terms of local quantities, like Chern classes and Chern numbers?
I'm interested in if it's possible to represent a spin structure and the Arf invariant associated to it in terms of some sort of local fields.
For example, the first Chern class of a complex line ...
9
votes
2answers
393 views
Matrix of cosecants appearing in equivariant index computations
In a computation of characters of certain representations of finite cyclic groups which appear as equivariant indices of Dirac operators (using the Atiyah-Bott fixed point formula, cf. [1, Theorem 8....
4
votes
1answer
203 views
Producing $K$-homology cycles from $KK$-cycles
For two unital (separable) $C^*$-algebras $A$ and $B$, let $(H,\rho,F)$ be a $KK$-cycle in the sense of Kasparov, or in the sense of Wikipedia :)
I wonder if there us a natural way to "forget" the ...
4
votes
1answer
376 views
An integral of the Hodge-Neumann Laplacian on a Riemannian manifold
Background
Let $M$ be a compact, oriented Riemannian manifold with boundary, with $\text{vol}$ the Riemannian volume form. Let $\nu$ be the outwards unit normal vector field on $\partial M$ and $\nu^\...
1
vote
0answers
40 views
The Morse Index of a $T$- periodic geodesics is a integer number?
It is well known that compact Riemannian manifolds $(M, g)$ with
periodic geodesic flows have ( Besse Book) exceptional spectral properties: the spectrum
of $ \sqrt{ - \Delta}$, the square root of ...
6
votes
1answer
216 views
How to compute the eta invariant of torus
I was wondering how to compute the eta invariant $\eta(T^3)$ of a flat torus $T^3$, with respect to the signature operator.
In general, how can we compute the $\eta(T^3/\Gamma)$ of a finite quotient ...
2
votes
1answer
122 views
Perturbation of the adiabatic limit
Let $(M,g_M)$ be a closed oriented Riemannian manifold that has a fibration structure
$$
Y \rightarrow M \overset{\pi}{\rightarrow} B
$$
where $(Y,g_Y)$ and $(B,g_B)$ are closed Riemannian manifolds ...
2
votes
0answers
86 views
A quantity associated to a compact Riemannian manifold with boundary(The pair of Laplacian)
Let $M$ be a Riemannian manifold with boundary $\partial M$.
Are the following operators, Fredholm operators?Is there a geometric terminology and geometric interpretation for the fredholm index of ...
8
votes
1answer
251 views
Noncommutative Fredholm operators
Let $A$ be a unital $C^*$-algebra and $F:H_A\rightarrow H_A$ a Fredholm operator on the standard Hilbert $A$-module $H_A:=l^2(A)$. Is it true that $\mbox{ker}(F)$ and $\mbox{coker}(F)$ are finitely ...
4
votes
0answers
73 views
Complexifed Gauge action on determinant line bundle and change of metric
Within the GIT setup for Hermitian-Einstein connection, Donaldson exhibited a holomorphic line bundle over $\mathcal{A}$ the space of unitary connections such that its cutvature equals $−2πi\Omega$ ...
3
votes
0answers
75 views
Some questions on defining the analytic index
The questions I have are about the definition of the analytic index of a family of self-adjoint Fredholm operators parameterized by a compact space $B$ (say a closed manifold). Actually, the ...
12
votes
0answers
611 views
Eta-Invariant and Atiyah-Patodi-Singer Index Theorem
In Quantum Field Theory and Jones Polynomial (equation 2.16), Witten used a formula relating the APS eta-invariant to the Chern-Simons action. Witten claimed that it is derived from the Atiyah-Patodi-...
6
votes
0answers
159 views
Variation Formula of APS $\eta$-Invariant and Chern-Simons Theory
In Perturbative Expansion of Chern-Simons Theory with Noncompact Gauge Group, the author proved the variation formula of the APS $\eta$-invariant. I have a few questions about their proof. I will ...
3
votes
0answers
121 views
eta invariant and spectral flow
We know that for a family of first-order self adjoint elliptic (Fredholm) operator $A_t$, for $t\in [0,1]$ we have the formula
$$\eta(A_1)-\eta(A_0)=spfl(A_t)_{t\in[0,1]}+\int^1_0 \omega(s)ds,$$
where ...
9
votes
0answers
185 views
Torsion in Atiyah Singer index formula
In the papers of Atiyah and Singer, they first show their index theorem and then derive some index formulas.
For the Fredholm index living in the integers, they use the fact that on spheres the Chern ...
2
votes
1answer
120 views
Dirac operator on manifold with periodic end
Let $\tilde W$ be a spin closed oriented manifold, $Y$ is a codimension $1$ closed oriented submanifold of $\tilde W$, and denote the $W$ the cobordism from $Y$ to
itself obtained from cutting $\...
5
votes
0answers
133 views
Integrand in equivariant Atiyah-Patodi-Singer index theorem
I am trying to determine the integrand in the equivariant Atiyah-Patodi-Singer index theorem by Donnelli ("Eta invariants for G-spaces") for spin manifolds if one only has isolated fixed points. In "...
15
votes
0answers
306 views
Beyond smoothness-the clear picture about the notion of a differential form
In this paper N.Teleman constructs the signature operator on an arbitrary (closed, oriented) Lipschitz manifold with coefficients in a vector bundle $\xi$. In particular the notion of a differential $...
5
votes
0answers
149 views
About mod 2 Index of Dirac Operators in 3D on Non-Orientable Manifold
I was reading Witten's paper "Fermion Path Integrals and Topological Phases" (https://arxiv.org/abs/1508.04715). He claimed that
Indeed, on
an orientable 3-manifold, the eigenvalues of the Dirac ...
9
votes
1answer
384 views
Functoriality for wrong way maps
In the K-theory formulation of the index theorem one defines the topological index in terms of the so called wrong way maps. Those maps are defined for embeddings of compact manifolds $i:X \to Y$: see ...
1
vote
1answer
65 views
Example of a certain partitioned manifold
I'm looking for an example of a non-compact spin manifold $M$ and a compact subset $K\subseteq M$ such that $\partial K$ is a compact hypersurface in $M$ with $\hat{A}(\partial K)\neq 0$.
(At first I ...
5
votes
2answers
212 views
Elements of graded algebra associated with the algebra of differential operators as smooth sections
Let $M$ be a compact manifold and $E$ a complex vector bundle. We will consider differential operators $P$ acting between $\Gamma^{\infty}(M,E)$. Let $\mathcal{P}$ be the algebra of all differential ...
7
votes
0answers
108 views
Vanishing of K-theoretic index and positive scalar curvature
I'm confused about a seemingly basic point about a classical result on positive scalar curvature and would appreciate it if an expert could help me out.
Let $M^n$ be a closed spin manifold with ...
4
votes
1answer
215 views
Index formula with nonisolated fixed points
Consider a compact Riemannian manifold of even dimension $n$ admitting a $U(1)$ action. If the fixed points of the action are isolated, then Witten [1; eq. 35] gives the character-valued index of the ...
