Questions tagged [clifford-algebras]
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Equivalent definition of Spin group in terms of automorphisms
Let $\mathrm{Cl}(\mathbb{R}^n)$ denote the (real) Clifford algebra on $\mathbb{R}^n$ with respect to the Euclidean inner product. Let $\mathrm{Cl}^0({\mathbb{R}^n})$ denote the even part of $\mathrm{...
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Geometric explanation of Fueter-Sce-Qian Theorem and similar situations
In Clifford analysis there is a fundamental theorem due to Fueter and extended by Sce and Qian that says (in modern terminology) that the given a slice regular function $f:\mathbb{R}^{m+1}\to\mathbb{R}...
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Definition of Clifford super-connections
I have some questions concerning the definition of Clifford super-connections in Heat Kernels and Dirac Operators:
Definition 3.39. If $A$ is a super-connection on a Clifford module $E\to M$, we say ...
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Can any Clifford module bundle be extended to a Dirac bundle?
I assume that the question in the title is clear, so let me talk about its relevance:
According to theorem 4.3 in Heat Kernels and Dirac Operators the index theorem
\begin{equation}\tag{1}
\mathrm{ind}...
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"The index is independent of the Dirac operator"
Fix a Clifford module bundle $E$ on a compact Riemannian manifold $M$ and let $D_0$ and $D_1$ be two Dirac operators (compatible with the Clifford action). The proof of the Atiyah-Singer index theorem ...
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Classification of real Clifford algebras
$\DeclareMathOperator\Cl{Cl}$Let $V$ be a real vector space of dimension $p+q$. Let $Q$ be a non-degenerate quadratic form on $V$ of signature $(p,q)$ where $p$ is the number of positive eigenvalues, ...
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On Dirac/ Clifford matrices
Let $(\eta^{\mu\nu})=\operatorname{diag}(+1,-1,-1,-1)$.
The Dirac matrices $\gamma^\mu$, $\mu=0,1,2,3$ satisfy by definition
$$\{\gamma^\mu,\gamma^\nu\}=2\eta^{\mu\nu}\tag{1}\label{1}$$
where $\{A,B\}=...
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Unitary operators with the same inner product as vectors
Suppose we have a set of real unit vectors $v_1,\ldots,v_m \in \mathbb{R}^n$. We can always find a set of unitary operators $U_1,\ldots,U_m$ acting on $\mathbb{C}^N$ (for $N$ that is possibly much ...
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Uniqueness of spinor representation
$\DeclareMathOperator\SU{SU}\DeclareMathOperator\SO{SO}\DeclareMathOperator\GL{GL}$I asked a similar question on math stack exchange here, but I wonder if it may be better received here.
Let $n$ be ...
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Why Representation of Clifford algebra are constant for an orthonormal frame?
Let $e_\alpha$ be a basis of the tangent bundle $TM$ and $ \rho: T_x M \rightarrow \operatorname{End}\left( W\right)$ a representation of a Clifford algebra.
In this text Field theory from a bundle ...
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Shouldn't $\mathrm{End}_{C(TM)}(E)$ be defined differently in Heat Kernels and Dirac Operators?
The first four chapters of the book lead up to the proof of theorem 4.1. Its main consequence is that it provides the local index theorem for Dirac operators. The statement of theorem 4.1 involves a ...
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Clifford modules - How is the grading on $\mathrm{Hom}_{C(V)}(S,E)$ defined?
From the book Heat Kernels and Dirac Operators:
Proposition 3.27. If $V$ is an even-dimensional real Euclidean vector space,
then every finite-dimensional $\mathbb{Z}_2$-graded complex module $E$ of ...
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Group of invertible elements in a degree 4 central simple algebra with symplectic involution with norm in a center
Let $A$ be a central simple algebra of degree 4 (i.e. dimension 16) over a field $F$ with $\mathrm{char}(F) \neq 2$. It is known that any such algebra is a tensor product $D_1 \otimes D_2$ of two ...
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Literature on Clifford modules
I encountered Clifford modules in the book Heat Kernels and Dirac Operators. I am particularly interested in the definition of the isomorphism
$$\mathrm{End}(E)\cong C(V)\otimes \mathrm{End}_{C(V)}(E)$...
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Action of volume form on spinors in odd dimension
We know that for a smooth orientable manifold of dimension $2n, i^n$ times the volume form acts as identity on the positive spinors and acts as minus identity on the negative spinors via Clifford ...
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