Black Hole Entropy in terms of Mass. Note Quote.

If M-theory is compactified on a d-torus it becomes a D = 11 – d dimensional theory with Newton constant

G_D = G₁₁/L_d = l⁹₁₁/L_d —– (1)

A Schwartzschild black hole of mass M has a radius

R_s ~ M^(1/(D-3)) G_D^(1/(D-3)) —– (2)

According to Bekenstein and Hawking the entropy of such a black hole is

S = Area/4G_D —– (3)

where Area refers to the D – 2 dimensional hypervolume of the horizon:

Area ~ R_s^D-2 —– (4)

Thus

S ~ 1/G_D (MG_D)^(D-2)/(D-3) ~ M^(D-2)/(D-3) G_D^1/(D-3) —– (5)

From the traditional relativists’ point of view, black holes are extremely mysterious objects. They are described by unique classical solutions of Einstein’s equations. All perturbations quickly die away leaving a featureless “bald” black hole with ”no hair”. On the other hand Bekenstein and Hawking have given persuasive arguments that black holes possess thermodynamic entropy and temperature which point to the existence of a hidden microstructure. In particular, entropy generally represents the counting of hidden microstates which are invisible in a coarse grained description. An ultimate exact treatment of objects in matrix theory requires a passage to the infinite N limit. Unfortunately this limit is extremely difficult. For the study of Schwarzchild black holes, the optimal value of N (the value which is large enough to obtain an adequate description without involving many redundant variables) is of order the entropy, S, of the black hole.

Considering the minimum such value for N, we have

N_min(S) = MR_s = M(MG_D)^1/D-3 = S —– (6)

We see that the value of N_min in every dimension is proportional to the entropy of the black hole. The thermodynamic properties of super Yang Mills theory can be estimated by standard arguments only if S ≤ N. Thus we are caught between conflicting requirements. For N >> S we don’t have tools to compute. For N ~ S the black hole will not fit into the compact geometry. Therefore we are forced to study the black hole using N = N_min = S.

Matrix theory compactified on a d-torus is described by d + 1 super Yang Mills theory with 16 real supercharges. For d = 3 we are dealing with a very well known and special quantum field theory. In the standard 3+1 dimensional terminology it is U(N) Yang Mills theory with 4 supersymmetries and with all fields in the adjoint repersentation. This theory is very special in that, in addition to having electric/magnetic duality, it enjoys another property which makes it especially easy to analyze, namely it is exactly scale invariant.

Let us begin by considering it in the thermodynamic limit. The theory is characterized by a “moduli” space defined by the expectation values of the scalar fields φ. Since the φ also represents the positions of the original DO-branes in the non compact directions, we choose them at the origin. This represents the fact that we are considering a single compact object – the black hole- and not several disconnected pieces.

The equation of state of the system, defined by giving the entropy S as a function of temperature. Since entropy is extensive, it is proportional to the volume ∑³ of the dual torus. Furthermore, the scale invariance insures that S has the form

S = constant T³∑³ —– (7)

The constant in this equation counts the number of degrees of freedom. For vanishing coupling constant, the theory is described by free quanta in the adjoint of U(N). This means that the number of degrees of freedom is ~ N².

From the standard thermodynamic relation,

dE = TdS —– (8)

and the energy of the system is

E ~ N²T⁴∑³ —– (9)

In order to relate entropy and mass of the black hole, let us eliminate temperature from (7) and (9).

S = N²∑³⁽(E/N²∑³))^3/4 —– (10)

Now the energy of the quantum field theory is identified with the light cone energy of the system of DO-branes forming the black hole. That is

E ≈ M²/N R —– (11)

Plugging (11) into (10)

S = N²∑³(M²R/N²∑³)^3/4 —– (12)

This makes sense only when N << S, as when N >> S computing the equation of state is slightly trickier. At N ~ S, this is precisely the correct form for the black hole entropy in terms of the mass.

Catastrophe, Gestalt and Thom’s Natural Philosophy of 3-D Space as Underlying All Abstract Forms – Thought of the Day 157.0

The main result of mathematical catastrophe theory consists in the classification of unfoldings (= evolutions around the center (the germ) of a dynamic system after its destabilization). The classification depends on two sorts of variables:

(a) The set of internal variables (= variables already contained in the germ of the dynamic system). The cardinal of this set is called corank,

(b) the set of external variables (= variables governing the evolution of the system). Its cardinal is called codimension.

The table below shows the elementary catastrophes for Thom:

The A-unfoldings are called cuspoids, the D-unfoldings umbilics. Applications of the E-unfoldings have only been considered in A geometric model of anorexia and its treatment. By loosening the condition for topological equivalence of unfoldings, we can enlarge the list, taking in the family of double cusps (X₉) which has codimension 8. The unfoldings A₃(the cusp) and A₅ (the butterfly) have a positive and a negative variant A₊₃, A_-3, A₊₅, A_-5.

We obtain Thorn’s original list of seven “catastrophes” if we consider only unfoldings up to codimension 4 and without the negative variants of A₃ and A₅.

Thom argues that “gestalts” are locally con­stituted by maximally four disjoint constituents which have a common point of equilibrium, a common origin. This restriction is ultimately founded in Gibb’s law of phases, which states that in three-dimensional space maximally four independent systems can be in equilibrium. In Thom’s natural philosophy, three-dimensional space is underlying all abstract forms. He, therefore, presumes that the restriction to four constituents in a “gestalt” is a kind of cognitive universal. In spite of the plausibility of Thom’s arguments there is a weaker assumption that the number of constituents in a gestalt should be finite and small. All unfoldings with codimension (i.e. number of external variables) smaller than or equal to 5 have simple germs. The unfoldings with corank (i.e. number of internal variables) greater than two have moduli. As a matter of fact the most prominent semantic archetypes will come from those unfoldings considered by René Thom in his sketch of catastrophe theoretic semantics.

F-Theory Compactifications on Calabi-Yau Manifolds Capture Nonperturbative Physics of String Theory. Note Quote.

The distinct string theory and their strong-coupling limit. The solid line (-) denotes toroidal compactification, the dashed line (–) denotes K3 compactifications and the dotted line (…) denotes Y₃ compactifications. The fine-dotted line (…) denotes Y₄ compactifications while the horizontal bar (-) indicates a string-string duality. The theories marked with a ‘U’ (‘8’) have a U-duality (8-duality); the strong-coupling limit of the theories marked by ‘M’ (‘F’) are controlled by M-theory (F-theory).

The type-IIB theory in 10 spacetime dimensions is believed to have an exact SL(2, Z) quantum symmetry which acts on the complex scalar τ = e^-2φ + iφ’, where φ and φ’ are the two scalar fields of type-lIB theory. This fact led Vafa to propose that the type-lIB string could be viewed as the toroidal compactification of a twelve-dimensional theory, called F-theory, where T is the complex structure modulus of a two-torus T² and the Kähler-class modulus is frozen. Apart from having a geometrical interpretation of the SL(2, Z) symmetry this proposal led to the construction of new, nonperturbative string vacua in lower space-time dimensions. In order to preserve the SL(2, Z) quantum symmetry the compactification manifold cannot be arbitrary but has to be what is called an elliptic fibration. That is, the manifold is locally a fibre bundle with a two-torus T² over some base B but there are a finite number of singular points where the torus degenerates. As a consequence nontrivial closed loops on B can induce a SL(2, Z) transformation of the fibre. This implies that the dilaton is not constant on the compactification manifold, but can have SL(2, Z) monodromy. It is precisely this fact which results in nontrivial (nonperturbative) string vacua inaccessible in string perturbation theory.

F-theory can be compactified on elliptic Calabi-Yau manifolds and each of such compactifications is conjectured to capture the nonperturbative physics of an appropriate string vacua. One finds:

• The lIB string in D = 10 can be viewed as F-theory compactified on T² with a frozen Kähler modulus.
• F-theory compactified on an elliptic K3 yields an 8-dimensional vacuum with 16 supercharges which is quantum equivalent to the heterotic string compactified on T².
• F-theory compactified on an elliptic Calabi-Yau threefold has 8 unbroken supercharges and is quantum equivalent to the heterotic string compactified on K3.
• The heterotic string compactified on a Calabi-Yau threefold Y₃ is quantum equivalent to F-theory compactified on an elliptic Calabi-Yau four-fold. Calabi-Yau fourfolds are Calabi-Yau manifolds of complex dimension four and holonomy group SU(4).

From God’s Perspective, There Are No Fields…Justified Newtonian, Unjustified Relativistic Claim. Note Quote.

Electromagnetism is a relativistic theory. Indeed, it had been relativistic, or Lorentz invariant, before Einstein and Minkowski understood that this somewhat peculiar symmetry of Maxwell’s equations was not accidental but expressive of a radically new structure of time and space. Minkowski spacetime, in contrast to Newtonian spacetime, doesn’t come with a preferred space-like foliation, its geometric structure is not one of ordered slices representing “objective” hyperplanes of absolute simultaneity. But Minkowski spacetime does have an objective (geometric) structure of light-cones, with one double-light-cone originating in every point. The most natural way to define a particle interaction in Minkowski spacetime is to have the particles interact directly, not along equal-time hyperplanes but along light-cones

In other words, if z_i(τ_􏱁i)  and z_j(τ_j􏱁) denote the trajectories of two charged particles, it wouldn’t make sense to say that the particles interact at “equal times” as it is in Newtonian theory. It would however make perfectly sense to say that the particles interact whenever

(z^μ_i z^μ_j)(z^μ_i z^μ_j) = (z_i – z_j)² = 0 —– (1)

For an observer finding himself in a universe guided by such laws it might then seem like the effects of particle interactions were propagating through space with the speed of light. And this observer may thus insist that there must be something in addition to the particles, something moving or evolving in spacetime and mediating interactions between charged particles. And all this would be a completely legitimate way of speaking, only that it would reflect more about how things appear from a local perspective in a particular frame of reference than about what is truly and objectively going on in the physical world. From “Gods perspective” there are no fields (or photons, or anything of that kind) – only particles in spacetime interacting with each other. This might sound hypothetical, but, it actually is not entirely fictitious. for such a formulation of electrodynamics actually exists and is known as Wheeler-Feynman electrodynamics, or Wheeler-Feynman Absorber Theory. There is a formal property of field equations called “gauge invariance” which makes it possible to look at things in several different, but equivalent, ways. Because of gauge invariance, this theory says that when you push on something, it creates a disturbance in the gravitational field that propagates outward into the future. Out there in the distant future the disturbance interacts with chiefly the distant matter in the universe. It wiggles. When it wiggles it sends a gravitational disturbance backward in time (a so-called “advanced” wave). The effect of all of these “advanced” disturbances propagating backward in time is to create the inertial reaction force you experience at the instant you start to push (and cancel the advanced wave that would otherwise be created by you pushing on the object). So, in this view fields do not have a real existence independent of the sources that emit and absorb them. It is defined by the principle of least action.

Wheeler–Feynman electrodynamics and Maxwell–Lorentz electrodynamics are for all practical purposes empirically equivalent, and it may seem that the choice between the two candidate theories is merely one of convenience and philosophical preference. But this is not really the case since the sad truth is that the field theory, despite its phenomenal success in practical applications and the crucial role it played in the development of modern physics, is inconsistent. The reason is quite simple. The Maxwell–Lorentz theory for a system of N charged particles is defined, as it should be, by a set of mathematical equations. The equation of motion for the particles is given by the Lorentz force law, which is

The electromagnetic force F on a test charge at a given point and time is a certain function of its charge q and velocity v, which can be parameterized by exactly two vectors E and B, in the functional form:

describing the acceleration of a charged particle in an electromagnetic field. The electromagnetic field, represented by the field-tensor F^μν, is described by Maxwell’s equations. The homogenous Maxwell equations tell us that the antisymmetric tensor F^μν (a 2-form) can be written as the exterior derivative of a potential (a 1-form) A^μ(x), i.e. as

F^μν = ∂^μ A^ν – ∂^ν A^μ —– (2)

The inhomogeneous Maxwell equations couple the field degrees of freedom to matter, that is, they tell us how the charges determine the configuration of the electromagnetic field. Fixing the gauge-freedom contained in (2) by demanding ∂_μA^μ(x) = 0 (Lorentz gauge), the remaining Maxwell equations take the particularly simple form:

□ A^μ = – 4π j^μ —– (3)

where

□ = ∂_μ∂^μ

is the d’Alembert operator and j^μ the 4-current density.

The light-cone structure of relativistic spacetime is reflected in the Lorentz-invariant equation (3). The Liénard–Wiechert field at spacetime point x depends on the trajectories of the particles at the points of intersection with the (past and future) light-cones originating in x. The Liénard–Wiechert field (the solution of (3)) is singular precisely at the points where it is needed, namely on the world-lines of the particles. This is the notorious problem of the electron self-interaction: a charged particle generates a field, the field acts back on the particle, the field-strength becomes infinite at the point of the particle and the interaction terms blow up. Hence, the simple truth is that the field concept for managing interactions between point-particles doesn’t work, unless one relies on formal manipulations like renormalization or modifies Maxwell’s laws on small scales. However, we don’t need the fields and by taking the idea of a relativistic interaction theory seriously, we can “cut the middle man” and let the particles interact directly. The status of the Maxwell equation’s (3) in Wheeler–Feynman theory is now somewhat analogous to the status of Laplace’s equation in Newtonian gravity. We can get to the Gallilean invariant theory by writing the force as the gradient of a potential and having that potential satisfy the simplest nontrivial Galilean invariant equation, which is the Laplace equation:

∆V(x, t) = ∑_iδ(x – x_i(t)) —– (4)

Similarly, we can get the (arguably) simplest Lorentz invariant theory by writing the force as the exterior derivative of a potential and having that potential satisfy the simplest nontrivial Lorentz invariant equation, which is (3). And as concerns the equation of motion for the particles, the trajectories, if, are parametrized by proper time, then the Minkowski norm of the 4-velocity is a constant of motion. In Newtonian gravity, we can make sense of the gravitational potential at any point in space by conceiving its effect on a hypothetical test particle, feeling the gravitational force without gravitating itself. However, nothing in the theory suggests that we should take the potential seriously in that way and conceive of it as a physical field. Indeed, the gravitational potential is really a function on configuration space rather than a function on physical space, and it is really a useful mathematical tool rather than corresponding to physical degrees of freedom. From the point of view of a direct interaction theory, an analogous reasoning would apply in the relativistic context. It may seem (and historically it has certainly been the usual understanding) that (3), in contrast to (4), is a dynamical equation, describing the temporal evolution of something. However, from a relativistic perspective, this conclusion seems unjustified.

Conformal Field Theory and Virasoro Algebra. Note Quote.

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 D_g, 

D_[e^hg] = e^c·α[h] L⁻¹[h|_B1] D_g 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 e^c·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 H_d to a single H_l 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 B_i, to get an element φ_i of Diff S¹ × R⁺, where the R⁺ factor acts by an overall rescaling. We then express each φ_i as the exponential of an element l_i 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 = ⊕_iH_L,i ⊗ H_R,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,

l_n = −ie^inθ ∂/∂θ n ∈ Z —– (3)

which satisfy the relations

[l_m, l_n] = (m − n)l_m+n —– (4)

The Virasoro algebra is the universal central extension of this, with generators L_n with n ∈ Z, c ∈ R, and the relations

[L_m, L_n] = (m − n)L_m+n + c/12 n(n² − 1)δ_m+n,0 —– (5)

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 T_zz of the stress tensor can be expressed in terms of the Virasoro generators:

T_zz ≡ T(z) = ∑_n∈Z L_nz⁻ⁿ⁻² —– (7)

The component T_z̄z̄ is antiholomorphic and can be similarly expressed in terms of the generators L̄_n of the second copy of the Virasoro algebra:

T_z̄z̄ ≡ T(z̄) = ∑_n∈Z L̄_nz̄⁻ⁿ⁻² —– (8)

The mixed component T_zz̄ = T_z̄z is a c-number which vanishes for a flat metric. The state corresponding to T(z) is L₋₂|0⟩.

Albert Camus Reads Richard Morgan: Unsaid Existential Absurdism…(Abstract/Blurb)

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 Operators Corresponding to Cobordisms Only Iff Each Connected Component of the Cobordism has Non-empty Outgoing Boundary. Drunken Risibility.

 

Define a category B whose objects are the oriented submanifolds of X, and whose vector space of morphisms from Y to Z is O_YZ = 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 O_YZ is dual to O_ZY. (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 O_YZ 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 = B_Ω(X)(Ω(Y), Ω(Z)) —– (1)

where Ω(X) denotes the usual de Rham complex of a manifold X, and B_A(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 ⊗ · · · ⊗ a_k ⊗ b) = 􏰝a₁ ƒ(a₂ ⊗ · · · ⊗ a_k ⊗ b) + ∑(-1)ⁱ ƒ(a₁ ⊗ · · · ⊗ a_ia_i+1 ⊗ a_k ⊗ b) + (-1)^k ƒ(a₁ ⊗ · · · ⊗ a_k-1 ⊗ a_kb) —– (3)

whose cohomology is Ext_A(B,C). This is different from O_YZ = Ext^∗_H^∗(X)(H^∗(Y), H^∗(Z)), but related to it by a spectral sequence whose E₂-term is O_YZ and which converges to H^∗(O’_YZ) = Ext_Ω(X)(Ω(Y), Ω(Z)). But more important is that H^∗(O’_YZ) is the homology of the space P_YZ of paths in X which begin in Y and end in Z. To be precise, H_p(O’_YZ) ≅ H_p+dZ(P_YZ), 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

H_i(P_YZ) × H_j(P_ZW) → H_i+j−dZ(P_YW) —— (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 P_YZ × P_ZW with the concatenation map M → P_YW.

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 d_X of X, so that the cohomology H_i(C) is potentially non-zero for −d_X ≤ i < ∞. There is a map H_i(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. 

Graviton Fields Under Helicity Rotations. Thought of the Day 156.0

Einstein described gravity as equivalent to curves in space and time, but physicists have long searched for a theory of gravitons, its putative quantum-scale source. Though gravitons are individually too weak to detect, most physicists believe the particles roam the quantum realm in droves, and that their behavior somehow collectively gives rise to the macroscopic force of gravity, just as light is a macroscopic effect of particles called photons. But every proposed theory of how gravity particles might behave faces the same problem: upon close inspection, it doesn’t make mathematical sense. Calculations of graviton interactions might seem to work at first, but when physicists attempt to make them more exact, they yield gibberish – an answer of “infinity.” This is the disease of quantized gravity. With regard to the exchange particles concept in the quantum electrodynamics theory and the existence of graviton, let’s consider a photon that is falling in the gravitational field, and revert back to the behavior of a photon in the gravitational field. But when we define the graviton relative to the photon, it is necessary to explain the properties and behavior of photon in the gravitational field. The fields around a “ray of light” are electromagnetic waves, not static fields. The electromagnetic field generated by a photon is much stronger than the associated gravitational field. When a photon is falling in the gravitational field, it goes from a low layer to a higher layer density of gravitons. We should assume that the graviton is not a solid sphere without any considerable effect. Graviton carries gravity force, so it is absorbable by other gravitons; in general; gravitons absorb each other and combine. This new view on graviton shows, identities of graviton changes, in fact it has mass with changeable spin.

When we derive various supermultiplets of states, at the noninteracting level, these states can easily be described in terms of local fields. But, at the interacting level, there are certain ambiguities that withdraw as a result of different field representations describing the same massless free states. So the proper choice of the field representation may be subtle. The supermultiplets can then be converted into supersymmetric actions, quadratic in the fields. For selfdual tensor fields, the action must be augmented by a duality constraint on the corresponding field strength. For the graviton field,

The linearized Einstein equation for g_μν = η_μν + κh_μν implies that (for D ≥ 3)

R_μν ∝ ∂²h_μν + ∂_μ∂_νh – ∂_μ∂^ρh_νρ – ∂_ν∂^ρh_ρμ = 0 —– (1)

where h ≡ h_μμ and R_μν is the Ricci tensor. To analyze the number of states implied by this equation, one may count the number of plane-wave solutions with given momentum q^μ. It then turns out that there are D arbitrary solutions, corresponding to the linearized gauge invariance h_μν → h_μν + ∂_μξ_ν + ∂_νξ_μ, which can be discarded. Many other components vanish and the only nonvanishing ones require the momentum to be lightlike. Thee reside in the fields h_ij, where the components i, j are in the transverse (D-2) dimensional subspace. In addition, the trace of h_ij must be zero. Hence, the relevant plane-wave solutions are massless and have polarizations (helicities) characterized by a symmetric traceless 2-rank tensor. This tensor comprises 1/2D(D-3), which transform irreducibly under the SO(D-2) helicity group of transverse rotations. For the special case of D = 6 spacetime dimensions, the helicity group is SO(4), which factorizes into two SU(2) groups. The symmetric traceless representation then transforms as a doublet under each of the SU(2) factors and it is thus denoted by (2,2). As for D = 3, there are obviously no dynamic degrees of freedom associated with the gravitational field. When D = 2 there are again no dynamic degrees of freedom, but here (1) should be replaced by R_μν = 1/2g_μνR.

Ricci-Flat Metric & Diffeomorphism – Does there Exist a Possibility of a Complete Construction of a Metric if the Surface isn’t a Smooth Manifold? Note Quote.

Using twistors, the Gibbons-Hawking ansatz is generalized to investigate 4n-dimensional hyperkähler metrics admitting an action of the n-torus Tⁿ. The hyperkähler case could further admit a tri-holomorphic free action of the n-torus Tⁿ. It turns out that the metric may be written in coordinates adapted to the torus action, in a form similar to the Gibbons-Hawking ansatz in dimension 4, and such that the non-linear Einstein equations reduce to a set of linear equations (essentially saying that certain functions on Euclidean 3-space are harmonic). In the case of 8-manifolds (n = 2), the solutions can be described geometrically, in terms of arrangements of 3-dimensional linear subspaces in Euclidean 6-space.

There are in fact many explicit examples known of metrics on non-compact manifolds with SU(n) or Sp(2n) holonomy. The other holonomy groups automatically yielding Ricci-flat metrics are the special holonomy groups G₂ in dimension 7 and Spin(7) in dimension 8. Until fairly recently only three explicit examples of complete metrics (in dimension 7) with G₂-holonomy and one explicit example (in dimension 8) with Spin(7)-holonomy were known. The G₂-holonomy examples are asymptotically conical and live on the bundle of self-dual two-forms over S⁴, the bundle of self-dual two-forms over CP², and the spin bundle of S³ (topologically R⁴ × S³), respectively. The metrics are of cohomogeneity one with respect to the Lie groups SO(5), SU(3) and SU(2) × SU(2) × SU(2) respectively. A cohomogeneity-one metric has a Lie group acting via isometries, with general (principal) orbits of real codimension one. In particular, if the metric is complete, then X is the holomorphic cotangent bundle of projective n-space T^∗CPⁿ, and the metric is the Calabi hyperkähler metric.

The G₂-holonomy examples are all examples in which a Lie group G acts with low codimension orbits. This is a general feature of explicit examples of Einstein metrics. The simplest case of such a situation would be when there is a single orbit of a group action, in which case the metric manifold is homogeneous. For metrics on homogeneous manifolds, the Einstein condition may be expressed purely algebraically. Moreover, all homogeneous Ricci-flat manifolds are flat, and so no interesting metrics occur. Then what about cohomogeneity one with respect to G, i.e., the orbits of G are codimension one in general? Here, the Einstein condition reduces to a system of non-linear ordinary differential equations in one variable, namely the parameter on the orbit space. In the Ricci-flat case, Cheeger-Gromoll theorem implies that the manifold has at most one end. In the non-compact case, the orbit space is R₊ and there is just one singular orbit. Geometrically, if the principal orbit is of the form G/K, the singular orbit (the bolt) is G/H for some subgroup H ⊃ K; if G is compact, a necessary and sufficient condition for the space to be a smooth manifold is that H/K is diffeomorphic to a sphere. In many cases, this is impossible because of the form of the group G, and so any metric constructed will not be complete.

Complicated Singularities – Why Should the Discriminant Locus Change Under Dualizing?

Consider the surface S ⊆ (C^∗)² defined by the equation z₁ + z₂ + 1 = 0. Define the map log : (C^∗)² → R² by log(z₁, z₂) = (log|z₁|, log|z₂|). Then log(S) can be seen as follows. Consider the image of S under the absolute value map.

The line segment r₁ + r₂ = 1 with r₁, r₂ ≥ 0 is the image of {(−a, a−1)|0 < a < 1} ⊆ S; the ray r₂ = r₁ + 1 with r₁ ≥ 0 is the image of {(−a, a−1)|a < 0} ⊆ S; and the ray r₁ = r₂ + 1 is the image of {(−a, a−1)|a > 1} ⊆ S. The map S → |S| is one-to-one on the boundary of |S| and two-to-one in the interior, with (z₁, z₂) and (z̄₁, z̄₂) mapping to the same point in |S|. Taking the logarithm of this picture, we obtain the amoeba of S, log(S) as depicted below.

Now consider S = S × {0} ⊆ Y = (C^∗)² × R = T² × R³. We can now obtain a six-dimensional space X, with a map π : X → Y, an S¹-bundle over Y\S degenerating over S, so that π⁻¹(S) →^≅ S. We then have a T³-fibration on X, f : X → R³, by composing π with the map (log, id) : (C^∗)² × R → R³ = B. Clearly the discriminant locus of f is log(S) × {0}. If b is in the interior of log(S) × {0}, then f⁻¹(b) is obtained topologically by contracting two circles {p₁} × S¹ and {p₂} × S¹ on T³ = T² × S¹ to points. These are the familiar conical singularities seen in the special Lagrangian situation.

If b ∈ ∂(log(S) × {0}), then f⁻¹(b) has a slightly more complicated singularity, but only one. Let us examine how the “generic” singular fiber fits in here. In particular, for b in the interior of log(S) × {0}, locally this discriminant locus splits B into two regions, and these regions represent two different possible smoothings of f⁻¹(b).

Assume now that f : X → B is a special Lagrangian fibration with topology and discriminant locus ∆ being an amoeba. Let b ∈ Int(∆), and set M = f⁻¹(b). Set M^o = M\{x₁, x₂}, where x₁, x₂ are the two conical singularities of M. Suppose that the tangent cones to these two conical singularities, C₁ and C₂, are both cones of the form M⁰. Then the links of these cones, Σ₁ and Σ₂, are T²’s, and one expects that topologically these can be described as follows. Note that M^o ≅ (T²\{y₁, y₂}) × S¹ where y₁, y₂ are two points in T². We assume that the link Σ_i takes the form γ_i × S¹, where γ_i is a simple loop around y_i. If these assumptions hold, then to see how M can be smoothed, we consider the restriction maps in cohomology

H¹(M^o, R) → H¹(Σ₁, R) ⊕ H¹(Σ₂, R)

The image of this map is two-dimensional. Indeed, if we write a basis eⁱ₁, eⁱ₂ of H¹(Σ_i, R) where eⁱ₁ is Poincaré dual to [γ_i] × pt and eⁱ₂ is Poincaré dual to pt × S¹, it is not difficult to see the image of the restriction map is spanned by {(e¹₁, e²₁)} and {(e¹₂, −e²₂)}. Now this model of a topological fibration is not special Lagrangian, so in particular we don’t know exactly how the tangent cones to M at x₁ and x₂ are sitting inside C³, and thus can’t be compared directly with an asymptotically conical smoothing. So to make a plausibility argument, choose new bases fⁱ₁, fⁱ₂ of H¹(Σ_i, R) so that if M(a,0,0), M(0,a,0) and M(0,0,a) are the three possible smoothings of the two singular tangent cones at the singular points x₁, x₂ of M. Then Y(M_i^(a,0,0)) = πafⁱ₁, Y(M_i^(0,a,0)) = πafⁱ₂, and Y(M_i(0,0,a)) = −πa(fⁱ₁ + fⁱ₂).

Suppose that in this new basis, the image of the restriction map is spanned by the pairs (f¹₁, rf²₂) and (rf¹₂, f²₁) for r > 0, r ≠ 1. Then, there are two possible ways of smoothing M, either by gluing in M₁^(a,0,0) and M₂^(0,ra,0) at the singular points x₁ and x₂ respectively, or by gluing in M₁^(0,ra,0) and M₂^(a,0,0) at x₁ and x₂ respectively. This could correspond to deforming M to a fiber over a point on one side of the discriminant locus of f or the other side. This at least gives a plausibility argument for the existence of a special Lagrangian fibration of the topological type given by f. To date, no such fibrations have been constructed, however.

On giving a special Lagrangian fibration with codimension one discriminant and singular fibers with cone over T² singularities, one is just forced to confront a codimension one discriminant locus in special Lagrangian fibrations. This leads inevitably to the conclusion that a “strong form” of the Strominger-Yau-Zaslow conjecture cannot hold. In particular, one is forced to conclude that if f : X → B and f’ : X’ → B are dual special Lagrangian fibrations, then their discriminant loci cannot coincide. Thus one cannot hope for a fiberwise definition of the dualizing process, and one needs to refine the concept of dualizing fibrations. Let us see why the discriminant locus must change under dualizing. The key lies in the behaviour of the positive and negative vertices, where in the positive case the critical locus of the local model of the fibration is a union of three holomorphic curves, while in the negative case the critical locus is a pair of pants. In a “generic” special Lagrangian fibration, we expect the critical locus to remain roughly the same, but its image in the base B will be fattened out. In the negative case, this image will be an amoeba. In the case of the positive vertex, the critical locus, at least locally, consists of a union of three holomorphic curves, so that we expect the discriminant locus to be the union of three different amoebas. The figure below shows the new discriminant locus for these two cases.

Now, under dualizing, positive and negative vertices are interchanged. Thus the discriminant locus must change. This is all quite speculative, of course, and underlying this is the assumption that the discriminant loci are just fattenings of the graphs. However, it is clear that a new notion of dualizing is necessary to cover this eventuality.