“Buildings broadcast a message. Good and bad architecture can lift, or subdue a message… aesthetic ugliness promotes ugly behavior,” says 35-year-old Paul Joseph Watson, a commentator on Infowars, in a video titled “Why Modern Architecture SUCKS.” Watson refers to modernist architects — those who designed buildings after World War II, like Ernő Goldfinger, Owen Luder and John Bancroft — as “the social justice warriors of their time” who actively “rebelled against beauty.” By creating large concrete tower blocks — often with the intention of building social housing for the poor — Watson believes they attempted to “socially engineer society” like the Soviet Union.
He’s also far from the only critic to complain about the legacy of brutalism, a style of modern architecture that emerged in the 1950s and 1960s in the U.K., but was developed largely by French architects like Le Corbusier. Brutalist buildings were characterized by simple, block-like structures that often featured exposed concrete and were constructed in the belief that architects should design buildings with their function in mind first and foremost. As a result, brutalist architects would usually prioritize public space over monuments to gawk at. “Many Brutalist buildings expressed a progressive or even utopian vision of communal living and public ownership,” writes Felix Torkar in Jacobin magazine. (To that end, brutalist buildings were often favored by European governments as social housing for impoverished communities.) “The battle to protect them is also a fight to defend this social inheritance.”
There are a few reasons why Conformal Field Theories (CFTs) are very interesting to study: The first is that at fixed points of Renormalization Group flows, or at second order phase transitions, a quantum field theory is scale invariant. Scale invariance is a weaker form of conformal invariance, and it turns out in all cases that we know of scale invariance of a quantum field theory actually ends up implying the larger symmetry of conformal invariance. The second reason is that the requirement that a theory is conformally invariant is so restrictive that many things can be solved for that would otherwise be intractable. As an example, conformal invariance fixes 2- and 3-point functions entirely. In an ordinary quantum field theory, especially one at strong coupling, these would be hard or impossible to calculate at all. A third reason is string theory. In string theory, the worldsheet theory describing the string’s excitations is a CFT, so if string theory is correct, then in some sense conformal invariance is really one of the most fundamental features of the elemental constituents of reality. And through string theory we have the most precise and best-understood gauge/gravity dualities (the AdS/CFT dualities) that also involve CFT’s.
A Conformal Field Theory (CFT) is a Quantum Field Theory (QFT) in which conformal rescaling of the metric acts by conjugation. For the family of morphisms Dg,
D[ehg] = ec·α[h] L−1[h|B1] Dg L[h|B2] —– (1)
The analogous statement (conjugating the state on each boundary) is true for any Σ.
Here L is a linear operator depending only on the restriction of h to one of the boundaries of the annulus. All the dependence on the conformal rescaling away from the boundary is determined by a universal (independent of the particular Conformal Field Theory) functional α[h] ∈ R, which appears in an overall multiplicative factor ec·α[h]. The quantity c, called “Virasoro central charge”.
The corresponding operators L[h] form a semigroup, with a self-adjoint generator H. Then, since according to the axioms of QFT the spectrum of H is bounded below, we can promote this to a group action. This can be used to map any of the Hilbert spaces Hd to a single Hl for a fixed value of l, say l = 1. We will now do this and use the simpler notation H ≅ H1,
How do we determine the L[h]? First, we uniformize Σ – in other words, we find a complex diffeomorphism φ from our surface with boundary Σ to a constant curvature surface. We then consider the restriction of φ to each of the boundary components Bi, to get an element φi of Diff S1 × R+, where the R+ factor acts by an overall rescaling. We then express each φi as the exponential of an element li in the Lie algebra Diff S1, to find an appropriate projective representation of this Lie algebra on H.
Certain subtleties are in order here: The Lie algebra Diff S1 which appears is actually a subalgebra of a direct sum of two commuting algebras, which act independently on “left moving” and “right moving” factors in H. Thus, we can write H as a direct sum of irreps of this direct sum algebra,
H = ⊕iHL,i ⊗ HR,i —– (2)
Each of these two commuting algebras is a central extension of the Lie algebra Diff S1, usually called the Virasoro algebra or Vir.
Now, consider the natural action of Diff S1 on functions on an S1 parameterized by θ ∈ [0, 2π). After complexification, we can take the following set of generators,
ln = −ieinθ ∂/∂θ n ∈ Z —– (3)
which satisfy the relations
[lm, ln] = (m − n)lm+n —– (4)
The Virasoro algebra is the universal central extension of this, with generators Ln with n ∈ Z, c ∈ R, and the relations
The parameter c is again the Virasoro central charge. It is to be noted that the central extension is required in any non-trivial unitary CFT. Unitarity and other QFT axioms require the Virasoro representation to act on a Hilbert space, so that L−n = L†n. In particular, L0 is self-adjoint and can be diagonalized. Take a “highest weight representation,” in which the spectrum of L0 is bounded below. The L0 eigenvector with the minimum eigenvalue, h, is by definition the “highest weight state”, or a state |h⟩, so that
L0|h⟩ = h|h⟩ —– (6)
and normalize it so that ⟨h|h⟩ = 1. Since this is a norm in a Hilbert space, we conclude that h ≥ 0, with equality only if L−1|h⟩ = 0. In fact, L−1|0⟩ = 0 can be related to the translation invariance of the vacuum. Rephrasing this in terms of local operators, instead of in terms of states, take Σ to be the infinite cylinder R × S1, or equivalently the punctured complex plane C∗ with the complex coordinate z. In a CFT the component Tzz of the stress tensor can be expressed in terms of the Virasoro generators:
Tzz ≡ T(z) = ∑n∈Z Lnz−n−2 —– (7)
The component Tz̄z̄ is antiholomorphic and can be similarly expressed in terms of the generators L̄n of the second copy of the Virasoro algebra:
Tz̄z̄ ≡ T(z̄) = ∑n∈Z L̄nz̄−n−2 —– (8)
The mixed component Tzz̄ = Tz̄z is a c-number which vanishes for a flat metric. The state corresponding to T(z) is L−2|0⟩.
For the upcoming conference on “The Intellectual Geography of Albert Camus” on the 3rd of May, 2019, at the Alliance Française, New Delhi. Watch this space..
Imagine the real world extending into the fictive milieu, or its mirror image, the fictive world territorializing the real leaving it to portend such an intercourse consequent to an existential angst. Such an imagination now moves along the coordinates of hyperreality, where it collaterally damages meaning in a violent burst of EX/IM-plosion. This violent burst disturbs the idealized truth overridden by a hallucinogenic madness prompting iniquities calibrated for an unpleasant future. This invading dissonant realism slithers through the science fiction of Richard Morgan before it culminates in human characteristics of expediency. Such expediencies abhor fixation to being in the world built on deluded principles, which in my reading is Camus’ recommendation of confrontation with the absurd. This paper attempts to unravel the hyperreal as congruent on the absurd in a fictitious landscape of “existentialism meets the intensity of a relatable yet cold future”.
What I purport to do in this paper is pick up two sci-fi works of Richard Morgan, the first of which also happens to be the first of the Takeshi Kovacs Trilogy, Altered Carbon, while the second is Market Forces,a brutal journey into the heart of conflict investment by way of conscience elimination. Thereafter a conflation with Camus’ absurdity unravels the very paradoxical ambiguity underlying absurdism as a human condition. The paradoxical ambiguity is as a result of Camus’ ambivalence towards the neo-Platonist conception of the ultimate unifying principle, while accepting Plotinus’ principled pattern or steganography, but rejecting its culmination.
Richard Morgan’s is a parody, a commentary, or even en epic fantasy overcharged almost to the point of absurdity and bordering extropianism. If at all there is a semblance of optimism in the future as a result of Moore’s Law of dense hardware realizable through computational extravagance, it is spectacularly offset by complexities of software codes resulting in a disconnect that Morgan brilliantly transposes on to a society in a dystopian ethic underlining his plot pattern recognitions. This offsetting disconnect between the physical and mental, between the tangible and the intangible is the existential angst writ large on the societal maneuvered by the powers that be…..to be continued…
Define a category B whose objects are the oriented submanifolds of X, and whose vector space of morphisms from Y to Z is OYZ = Ext∗H∗(X)(H∗(Y), H∗(Z)) – the cohomology, as usual, has complex coefficients, and H∗(Y) and H∗(Z) are regarded as H∗(X)-modules by restriction. The composition of morphisms is given by the Yoneda composition of Ext groups. With this definition, however, it will not be true that OYZ is dual to OZY. (To see this it is enough to consider the case when Y = Z is a point of X, and X is a product of odd-dimensional spheres; then OYZ is a symmetric algebra, and is not self-dual as a vector space.)
We can do better by defining a cochain complex O’YZ of morphisms by
O’YZ = BΩ(X)(Ω(Y), Ω(Z)) —– (1)
where Ω(X) denotes the usual de Rham complex of a manifold X, and BA(B,C), for a differential graded algebra A and differential graded A- modules B and C, is the usual cobar resolution
Hom(B, C) → Hom(A ⊗ B, C) → Hom(A ⊗ A ⊗ B, C) → · · · —– (2)
in which the differential is given by
dƒ(a1 ⊗ · · · ⊗ ak ⊗ b) = a1 ƒ(a2 ⊗ · · · ⊗ ak ⊗ b) + ∑(-1)i ƒ(a1 ⊗ · · · ⊗ aiai+1 ⊗ ak ⊗ b) + (-1)k ƒ(a1 ⊗ · · · ⊗ ak-1 ⊗ akb) —– (3)
whose cohomology is ExtA(B,C). This is different from OYZ = Ext∗H∗(X)(H∗(Y), H∗(Z)), but related to it by a spectral sequence whose E2-term is OYZ and which converges to H∗(O’YZ) = ExtΩ(X)(Ω(Y), Ω(Z)). But more important is that H∗(O’YZ) is the homology of the space PYZ of paths in X which begin in Y and end in Z. To be precise, Hp(O’YZ) ≅ Hp+dZ(PYZ), where dZ is the dimension of Z. On the cochain complexes the Yoneda composition is associative up to cochain homotopy, and defines a structure of an A∞ category B’. The corresponding composition of homology groups
Hi(PYZ) × Hj(PZW) → Hi+j−dZ(PYW) —— (4)
is the composition of the Gysin map associated to the inclusion of the codimension dZ submanifold M of pairs of composable paths in the product PYZ × PZW with the concatenation map M → PYW.
Now let’s attempt to fit the closed string cochain algebra C to this A∞ category. C is equivalent to the usual Hochschild complex of the differential graded algebra Ω(X), whose cohomology is the homology of the free loop space LX with its degrees shifted downwards by the dimension dX of X, so that the cohomology Hi(C) is potentially non-zero for −dX ≤ i < ∞. There is a map Hi(X) → H−i(C) which embeds the ordinary cohomology ring of X to the Pontrjagin ring of the based loop space L0X, based at any chosen point in X.
The structure is, however, not a cochain-level open and closed theory, as we have no trace maps inducing inner products on H∗(O’YZ). When one tries to define operators corresponding to cobordisms it turns out to be possible only when each connected component of the cobordism has non-empty outgoing boundary.