Topic
Clifford module
About: Clifford module is a research topic. Over the lifetime, 40 publications have been published within this topic receiving 347 citations. The topic is also known as: representation of a Clifford algebra.
Papers
Posted Content•
TL;DR: In this paper, the authors classify extended Poincare Lie super algebras and Lie algesas of any signature (p,q) into two classes: (1) Lie super algebra and (2) Z_2-graded Lie algebra.
Abstract: We classify extended Poincare Lie super algebras and Lie algebras of any signature (p,q), that is Lie super algebras and Z_2-graded Lie algebras g = g_0 + g_1, where g_0 = so(V) + V is the (generalized) Poincare Lie algebra of the pseudo Euclidean vector space V = R^{p,q} of signature (p,q) and g_1 = S is the spinor so(V)-module extended to a g_0-module with kernel V. The remaining super commutators {g_1,g_1} (respectively, commutators [g_1, g_1]) are defined by an so(V)-equivariant linear mapping vee^2 g_1 -> V (respectively, wedge^2 g_1 -> V). Denote by P^+(n,s) (respectively, P^-(n,s)) the vector space of all such Lie super algebras (respectively, Lie algebras), where n = p + q = dim V and s = p - q is the signature. The description of P^+-(n,s) reduces to the construction of all so(V)-invariant bilinear forms on S and to the calculation of three Z_2-valued invariants for some of them.
This calculation is based on a simple explicit model of an irreducible Clifford module S for the Clifford algebra Cl_{p,q} of arbitrary signature (p,q). As a result of the classification, we obtain the numbers L^+-(n,s) = \dim P^+-(n,s) of independent Lie super algebras and algebras, which take values 0,1,2,3,4 or 6. Due to Bott periodicity, L^+-(n,s) may be considered as periodic functions with period 8 in each argument. They are invariant under the group Gamma generated by the four reflections with respect to the axes n=-2, n=2, s-1 = -2 and s-1 = 2. Moreover, the reflection (n,s) -> (-n,s) with respect to the axis s=0 interchanges L^+ and L^- : L^+(-n,s) = L^-(n,s).
55 citations
TL;DR: In this article, the expectation value of the one-particle projector (s) in the reduced matrix model and matrix quantum mechanics in general is studied and the relevance of this with regard to the spacetime structure is discussed.
Abstract: We study the expectation value of (the product) of the one-particle projector (s) in the reduced matrix model and matrix quantum mechanics in general. This quantity is given by the nonabelian Berry phase: we discuss the relevance of this with regard to the spacetime structure. The case of the USp matrix model is examined from this respect. Generalizing our previous work, we carry out the complete computation of this quantity which takes into account both the nature of the degeneracy of the fermions and the presence of the space time points belonging to the antisymmetric representation. We find the singularities as those of the SU(2) Yang monopole connection as well as the pointlike singularities in 9+1 dimensions coming from its SU(8) generalization. The former type of singularities, which extend to four of the directions lying in the antisymmetric representations, may be regarded as seeds of our four dimensional spacetime structure and is not shared by the IIB matrix model. From a mathematical viewpoint, these connections can be generalizable to arbitrary odd space dimensions due to the nontrivial nature of the eigenbundle and the Clifford module structure.
20 citations
TL;DR: In this paper, it was shown that Khovanov homology with Z/2 Z coefficients is invariant under Conway mutation, and a strategy to prove Baldwin and Levine's conjecture that δ-graded knot Floer homology is mutation invariant.
17 citations
TL;DR: In this paper, the diffeological version of the Clifford algebra of a Diffeological finite dimensional vector space is considered and the notion of a diffeology algebra is introduced.
Abstract: We consider the diffeological version of the Clifford algebra of a diffeological finite dimensional vector space; we start by commenting on the notion of a diffeological algebra (which is t...
11 citations
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TL;DR: In this article, the Verlinde ring has been used to define an analytic index of the Dirac operator of a Clifford module bundle, which is used for Borel-Weil theory in terms of non-commutative geometry.
Abstract: Let $T$ be a circle and $LT$ be its loop group. Let $\mathcal{M}$ be an infinite dimensional manifold equipped with a nice $LT$-action. We construct an analytic $LT$-equivariant index for $\mathcal{M}$, and justify it in terms of noncommutative geometry. More precisely, we construct a Hilbert space $\mathcal{H}$ consisting of "$L^2$-sections of a Clifford module bundle" and a "Dirac operator" $\mathcal{D}$ which acts on $\mathcal{H}$. Then, we define an analytic index of $\mathcal{D}$ valued in the representation group of $LT$, so called Verlinde ring. We also define a "twisted crossed product $LT\ltimes_\tau C_0(\mathcal{M})$," although we cannot define each concept "function algebra for $\mathcal{M}$ vanishing at infinity," "function from $LT$ to a $C^*$-algebra vanishing at infinity," and a Haar measure on $LT$. Moreover we combine all of them in terms of spectral triples and verify that the triple has an infinite spectral dimension. Lastly, we add some applications including Borel-Weil theory for $LT$.
9 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2021 | 4 |
| 2020 | 3 |
| 2019 | 4 |
| 2017 | 3 |
| 2016 | 2 |
| 2015 | 1 |