Canonical Fibrations on Geodesics

There is a realisation of the canonical fibrations of flag manifolds that serves to introduce a twistor space. For this, assume that G is of adjoint type (i.e. has trivial centre) and let ΩG denote the infinite-dimensional manifold of based loops in G: the loop group. In fact ΩG is a Kähler manifold and may be viewed as a flag manifold G^C/P where G^C is the manifold of loops in G^C and P is the subgroup of those that extend holomorphically to the disc. We have various fibrations ρ_λ: ΩG → G given by evaluation at λ ∈ S¹ and in some ways ρ₋₁ behaves like a canonical fibration making ΩG into a universal twistor space for G. It is a theorem of Uhlenbeck that any harmonic map of S² into G is of the form ρ₋₁ ◦ Φ for some “super-horzontal” holomorphic map Φ : S² → ΩG.

The flag manifolds of G embed in ΩG as conjugacy classes of geodesics and we find a particular embedding of this kind using the canonical element. Indeed, our assumption that G be centre-free means that exp 2πξ = e for any canonical element ξ. Thus if F = G/H = G^C/P is a flag manifold with ξ the canonical element of p, we may define a map Γ: F → ΩG by setting

Γ(eH) = (e^√−1t → exp tξ)

and extending by equivariance. Moreover, if N is the inner symmetric space associated to F, we have a totally geodesic immersion γ : N → G defined by setting γ(x) equal to the element of G that generates the involution at x. We now have:

Γ: F → ΩG is a totally geodesic, holomorphic, isometric immersion and the following diagram commutes

where π₁ is a canonical fibration. Thus we have a realisation of the canonical fibrations as the trace of ρ₋₁ on certain conjugacy classes of geodesics.

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Reclaim Modernity: Beyond Markets, Beyond Machines (Mark Fisher & Jeremy Gilbert)

It is understandable that the mainstream left has traditionally been suspicious of anti-bureaucratic politics. The Fabian tradition has always believed – has been defined by its belief – in the development and extension of an enlightened bureaucracy as the main vehicle of social progress. Attacking ‘bureaucracy’ has been – since at least the 1940s – a means by which the Right has attacked the very idea of public service and collective action. Since the early days of Thatcherism, there has been very good reason to become nervous whenever someone attacks bureaucracy, because such attacks are almost invariably followed by plans not for democratisation, but for privatisation.

Nonetheless, it is precisely this situation that has produced a certain paralysis of the Left in the face of one of its greatest political opportunities, an opportunity which it can only take if it can learn to speak an anti-bureaucratic language with confidence and conviction. On the one hand, this is a simple populist opportunity to unite constituencies within both the public and private sectors: simple, but potentially strategically crucial. As workers in both sectors and as users of public services, the public dislike bureaucracy and apparent over-regulation. The Left misses an enormous opportunity if it fails to capitalise on this dislike and transform it into a set of democratic demands.

On the other hand, anti-bureaucratism marks one of the critical points of failure and contradiction in the entire neoliberal project. For the truth is that neoliberalism has not kept its promise in this regard. It has not reduced the interference of managerial mechanisms and apparently pointless rules and regulations in the working life of public-sector professionals, or of public-service users, or of the vast majority of workers in the private sector. In fact it has led in many cases to an enormous proliferation and intensification of just these processes. Targets, performance indicators, quantitative surveys and managerial algorithms dominate more of life today than ever before, not less. The only people who really suffer less regulation than they did in the past are the agents of finance capital: banks, traders, speculators and fund managers.

Where de-regulation is a reality for most workers is not in their working lives as such, but in the removal of those regulations which once protected their rights to secure work, and to a decent life outside of work (pensions, holidays, leave entitlements, etc.). The precarious labour market is not a zone of freedom for such workers, but a space in which the fact of precarity itself becomes a mechanism of discipline and regulation. It only becomes a zone of freedom for those who already have enough capital to be able to choose when and where to work, or to benefit from the hyper-mobility and enforced flexibility of contemporary capitalism.

Reclaiming Modernity Beyond Markets Beyond Machines

The Concern for Historical Materialism. Thought of the Day 53.0

The concern for historical materialism, in spite of Marx’s differentiation between history and pre-history, is that totalisation might not be historically groundable after all, and must instead be constituted in other ways: whether logically, transcendentally or naturally. The ‘Consciousness’ chapter of the Phenomenology, a blend of all three, becomes a transcendent(al) logic of phenomena – individual, universal, particular – and ceases to provide any genuine phenomenology of ‘the experience of consciousness’. Natural consciousness is not strictly speaking a standpoint (no real opposition), so it can offer no critical grounds of itself to confer synthetic unity upon the universal, that which is taken to a higher level in ‘Self-Consciousness’ (only to be retrospectively confirmed). Yet Hegel does just this from the outset. In ‘Perception’, we read that, ‘[o]n account of the universality [Allgemeinheit] of the property, I must … take the objective essence to be on the whole a community [Gemeinschaft]’. Universality always sides with community, the Allgemeine with the Gemeinschaft, as if the synthetic operation had taken place prior to its very operability. Unfortunately for Hegel, the ‘free matters’ of all possible properties paves the way for the ‘interchange of forces’ in ‘Force and the Understanding’, and hence infinity, life and – spirit. In the midst of the master-slave dialectic, Hegel admits that, ‘[i]n this movement we see repeated the process which represented itself as the play of forces, but repeated now in consciousness [sic].