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⟩.

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Grothendieckian Construction of K-Theory with a Bundle that is Topologically Trivial and Class that is Torsion.

All relativistic quantum theories contain “antiparticles,” and allow the process of particle-antiparticle annihilation. This inspires a physical version of the Grothendieck construction of K-theory. Physics uses topological K-theory of manifolds, whose motivation is to organize vector bundles over a space into an algebraic invariant, that turns out to be useful. Algebraic K-theory started from K_i defined for i, with relations to classical constructions in algebra and number theory, followed by Quillen’s homotopy-theoretic definition ∀ i. The connections to algebra and number theory often persist for larger values of i, but in ways that are subtle and conjectural, such as special values of zeta- and L-functions.

One could also use the conserved charges of a configuration which can be measured at asymptotic infinity. By definition, these are left invariant by any physical process. Furthermore, they satisfy quantization conditions, of which the prototype is the Dirac condition on allowed electric and magnetic charges in Maxwell theory.

There is an elementary construction which, given a physical theory T, produces an abelian group of conserved charges K(T). Rather than considering the microscopic dynamics of the theory, all that is needed to be known is a set S of “particles” described by T, and a set of “bound state formation/decay processes” by which the particles combine or split to form other particles. These are called “binding processes.” Two sets of particles are “physically equivalent” if some sequence of binding processes convert the one to the other. We then define the group K(T) as the abelian group Z^S of formal linear combinations of particles, quotiented by this equivalence relation.

Suppose T contains the particles S = {A,B,C}.

If these are completely stable, we could clearly define three integral conserved charges, their individual numbers, so K(T) ≅ Z³.

Introducing a binding process

A + B ↔ C —– (1)

with the bidirectional arrow to remind us that the process can go in either direction. Clearly K(T) ≅ Z² in this case.

One might criticize this proposal on the grounds that we have assumed that configurations with a negative number of particles can exist. However, in all physical theories which satisfy the constraints of special relativity, charged particles in physical theories come with “antiparticles,” with the same mass but opposite charge. A particle and antiparticle can annihilate (combine) into a set of zero charge particles. While first discovered as a prediction of the Dirac equation, this follows from general axioms of quantum field theory, which also hold in string theory.

Thus, there are binding processes

B + B̄ ↔ Z₁ + Z₂ + · · · .

where B̄ is the antiparticle to a particle B, and Z_i are zero charge particles, which must appear by energy conservation. To define the K-theory, we identify any such set of zero charge particles with the identity, so that

B + B̄ ↔ 0

Thus the antiparticles provide the negative elements of K(T).

Granting the existence of antiparticles, this construction of K-theory can be more simply rephrased as the Grothendieck construction. We can define K(T) as the group of pairs (E, F) ∈ (Z^S, Z^S), subject to the relations (E, F) ≅ (E+B, F +B) ≅ (E+L, F +R) ≅ (E+R, F +L), where (L, R) are the left and right hand side of a binding process (1).

Thinking of these as particles, each brane B must have an antibrane, which we denote by B̄. If B wraps a submanifold L, one expects that B̄ is a brane which wraps a submanifold L of opposite orientation. A potential problem is that it is not a priori obvious that the orientation of L actually matters physically, especially in degenerate cases such as L a point.

Now, let us take X as a Calabi-Yau threefold for definiteness. A physical A-brane, which are branes of the A-model topological string and thereby a TQFT shadow of the D-branes of the superstring, is specified by a pair (L, E) of a special Lagrangian submanifold L with a flat bundle E. The obvious question could be: When are (L₁, E₁) and (L₂, E₂) related by a binding process? A simple heuristic answer to this question is given by the Feynman path integral. Two configurations are connected, if they are connected by a continuous path through the configuration space; any such path (or a small deformation of it) will appear in the functional integral with some non-zero weight. Thus, the question is essentially topological. Ignoring the flat bundles for a moment, this tells us that the K-theory group for A-branes is H₃(Y, Z), and the class of a brane is simply (rank E)·[L] ∈ H₃(Y, Z). This is also clear if the moduli space of flat connections on L is connected.

But suppose it is not, say π₁(L) is torsion. In this case, we need deeper physical arguments to decide whether the K-theory of these D-branes is H₃(Y, Z), or some larger group. But a natural conjecture is that it will be K₁(Y), which classifies bundles on odd-dimensional submanifolds. Two branes which differ only in the choice of flat connection are in fact connected in string theory, consistent with the K-group being H₃(Y, Z). For Y a simply connected Calabi-Yau threefold, K₁(Y) ≅ H₃(Y, Z), so the general conjecture is borne out in this case

There is a natural bilinear form on H₃(Y, Z) given by the oriented intersection number

I(L₁, L₂) = #([L₁] ∩ [L₂]) —– (2)

It has symmetry (−1)ⁿ. In particular, it is symplectic for n = 3. Furthermore, by Poincaré duality, it is unimodular, at least in our topological definition of K-theory.

D-branes, which are extended objects defined by mixed Dirichlet-Neumann boundary conditions in string theory, break half of the supersymmetries of the type II superstring and carry a complete set of electric and magnetic Ramond-Ramond charges. The product of the electric and magnetic charges is a single Dirac unit, and that the quantum of charge takes the value required by string duality. Saying that a D-brane has RR-charge means that it is a source for an “RR potential,” a generalized (p + 1)-form gauge potential in ten-dimensional space-time, which can be verified from its world-volume action that contains a minimal coupling term,

∫C^{(p + 1)} —–(3)

where C^{(p + 1)} denotes the gauge potential, and the integral is taken over the (p+1)-dimensional world-volume of the brane. For p = 0, C⁽¹⁾ is a one-form or “vector” potential (as in Maxwell theory), and thus the D0-brane is an electrically charged particle with respect to this 10d Maxwell theory. Upon further compactification, by which, the ten dimensions are R⁴ × X, and a Dp-brane which wraps a p-dimensional cycle L; in other words its world-volume is R × L where R is a time-like world-line in R⁴. Using the Poincaré dual class ω_L ∈ H^2n−p(X, R) to L in X, to rewrite (3) as an integral

∫_{R × X} C^{(p + 1)} ∧ ω_L —– (4)

We can then do the integral over X to turn this into the integral of a one-form over a world-line in R⁴, which is the right form for the minimal electric coupling of a particle in four dimensions. Thus, such a wrapped brane carries a particular electric charge which can be detected at asymptotic infinity. Summarizing the RR-charge more formally,

∫_LC = ∫_XC ∧ ω_L —– (5)

where C ∈ H∗(X, R). In other words, it is a class in H_p(X, R).

In particular, an A-brane (for n = 3) carries a conserved charge in H₃(X, R). Of course, this is weaker than [L] ∈ H₃(X, Z). To see this physically, we would need to see that some of these “electric” charges are actually “magnetic” charges, and study the Dirac-Schwinger-Zwanziger quantization condition between these charges. This amounts to showing that the angular momentum J of the electromagnetic field satisfies the quantization condition J = ħn/2 for n ∈ Z. Using an expression from electromagnetism, J⃗ = E⃗ × B⃗ , this is precisely the condition that (2) must take an integer value. Thus the physical and mathematical consistency conditions agree. Similar considerations apply for coisotropic A-branes. If X is a genuine Calabi-Yau 3-fold (i.e., with strict SU(3) holonomy), then a coisotropic A-brane which is not a special Lagrangian must be five-dimensional, and the corresponding submanifold L is rationally homologically trivial, since H⁵(X, Q) = 0. Thus, if the bundle E is topologically trivial, the homology class of L and thus its K-theory class is torsion.

If X is a torus, or a K3 surface, the situation is more complicated. In that case, even rationally the charge of a coisotropic A-brane need not lie in the middle-dimensional cohomology of X. Instead, it takes its value in a certain subspace of ⊕_p H^p(X, Q), where the summation is over even or odd p depending on whether the complex dimension of X is even or odd. At the semiclassical level, the subspace is determined by the condition

(L − Λ)α = 0, α ∈ ⊕_p H^p(X, Q)

where L and Λ are generators of the Lefschetz SL(2, C) action, i.e., L is the cup product with the cohomology class of the Kähler form, and Λ is its dual.

Black Hole Complementarity: The Case of the Infalling Observer

The four postulates of black hole complementarity are:

Postulate 1: The process of formation and evaporation of a black hole, as viewed by a distant observer, can be described entirely within the context of standard quantum theory. In particular, there exists a unitary S-matrix which describes the evolution from infalling matter to outgoing Hawking-like radiation.

Postulate 2: Outside the stretched horizon of a massive black hole, physics can be described to good approximation by a set of semi-classical field equations.

Postulate 3: To a distant observer, a black hole appears to be a quantum system with discrete energy levels. The dimension of the subspace of states describing a black hole of mass M is the exponential of the Bekenstein entropy S(M).

We take as implicit in postulate 2 that the semi-classical field equations are those of a low energy effective field theory with local Lorentz invariance. These postulates do not refer to the experience of an infalling observer, but states a ‘certainty,’ which for uniformity we label as a further postulate:

Postulate 4: A freely falling observer experiences nothing out of the ordinary when crossing the horizon.

To be more specific, we will assume that postulate 4 means both that any low-energy dynamics this observer can probe near his worldline is well-described by familiar Lorentz-invariant effective field theory and also that the probability for an infalling observer to encounter a quantum with energy E ≫ 1/r_s (measured in the infalling frame) is suppressed by an exponentially decreasing adiabatic factor as predicted by quantum field theory in curved spacetime. We will argue that postulates 1, 2, and 4 are not consistent with one another for a sufficiently old black hole.

Consider a black hole that forms from collapse of some pure state and subsequently decays. Dividing the Hawking radiation into an early part and a late part, postulate 1 implies that the state of the Hawking radiation is pure,

|Ψ⟩= ∑_i |ψ_i⟩E ⊗|i⟩_L —– (1)

Here we have taken an arbitrary complete basis |i⟩_L for the late radiation. We use postulates 1, 2, and 3 to make the division after the Page time when the black hole has emitted half of its initial Bekenstein-Hawking entropy; we will refer to this as an ‘old’ black hole. The number of states in the early subspace will then be much larger than that in the late subspace and, as a result, for typical states |Ψ⟩ the reduced density matrix describing the late-time radiation is close to the identity. We can therefore construct operators acting on the early radiation, whose action on |Ψ⟩ is equal to that of a projection operator onto any given subspace of the late radiation.

To simplify the discussion, we treat gray-body factors by taking the transmission coefficients T to have unit magnitude for a few low partial waves and to vanish for higher partial waves. Since the total radiated energy is finite, this allows us to think of the Hawking radiation as defining a finite-dimensional Hilbert space.

Now, consider an outgoing Hawking mode in the later part of the radiation. We take this mode to be a localized packet with width of order r_s corresponding to a superposition of frequencies O(r⁻¹_s). Note that postulate 2 allows us to assign a unique observer-independent s lowering operator b to this mode. We can project onto eigenspaces of the number operator b^†b. In other words, an observer making measurements on the early radiation can know the number of photons that will be present in a given mode of the late radiation.

Following postulate 2, we can now relate this Hawking mode to one at earlier times, as long as we stay outside the stretched horizon. The earlier mode is blue-shifted, and so may have frequency ω_* much larger than O(r⁻¹_s) though still sub-Planckian.

Next consider an infalling observer and the associated set of infalling modes with lowering operators a. Hawking radiation arises precisely because

b = ∫₀^∞ dω B(ω)a_ω + C(ω)a^†_ω —– (2)

so that the full state cannot be both an a-vacuum (a|Ψ⟩ = 0) and a b^†b eigenstate. Here again we have used our simplified gray-body factors.

The application of postulates 1 and 2 has thus led to the conclusion that the infalling observer will encounter high-energy modes. Note that the infalling observer need not have actually made the measurement on the early radiation: to guarantee the presence of the high energy quanta it is enough that it is possible, just as shining light on a two-slit experiment destroys the fringes even if we do not observe the scattered light. Here we make the implicit assumption that the measurements of the infalling observer can be described in terms of an effective quantum field theory. Instead we could simply suppose that if he chooses to measure b^†b he finds the expected eigenvalue, while if he measures the noncommuting operator a^†a instead he finds the expected vanishing value. But this would be an extreme modification of the quantum mechanics of the observer, and does not seem plausible.

Figure below gives a pictorial summary of our argument, using ingoing Eddington-Finkelstein coordinates. The support of the mode b is shaded. At large distance it is a well-defined Hawking photon, in a predicted eigenstate of b^†b by postulate 1. The observer encounters it when its wavelength is much shorter: the field must be in the ground state a^†_ωa_ω = 0, by postulate 4, and so cannot be in an eigenstate of b^†b. But by postulate 2, the evolution of the mode outside the horizon is essentially free, so this is a contradiction.

Figure: Eddington-Finkelstein coordinates, showing the infalling observer encountering the outgoing Hawking mode (shaded) at a time when its size is ω⁻¹_* ≪ r_s. If the observer’s measurements are given by an eigenstate of a^†a, postulate 1 is violated; if they are given by an eigenstate of b^†b, postulate 4 is violated; if the result depends on when the observer falls in, postulate 2 is violated.

To restate our paradox in brief, the purity of the Hawking radiation implies that the late radiation is fully entangled with the early radiation, and the absence of drama for the infalling observer implies that it is fully entangled with the modes behind the horizon. This is tantamount to cloning. For example, it violates strong subadditivity of the entropy,

S_AB + S_BC ≥ S_B + S_ABC —– (3)

Let A be the early Hawking modes, B be outgoing Hawking mode, and C be its interior partner mode. For an old black hole, the entropy is decreasing and so S_AB < S_A. The absence of infalling drama means that S_BC = 0 and so S_ABC = S_A. Subadditivity then gives S_A ≥ S_B + S_A, which fails substantially since the density matrix for system B by itself is thermal.

Actually, assuming the Page argument, the inequality is violated even more strongly: for an old black hole the entropy decrease is maximal, S_AB = S_A − S_B, so that we get from subadditivity that S_A ≥ 2S_B + S_A.

Note that the measurement of N_b takes place entirely outside the horizon, while the measurement of N_a (real excitations above the infalling vacuum) must involve a region that extends over both sides of the horizon. These are noncommuting measurements, but by measuring N_b the observer can infer something about what would have happened if N_a had been measured instead. For an analogy, consider a set of identically prepared spins. If each is measured along the x-axis and found to be +1/2, we can infer that a measurement along the z-axis would have had equal probability to return +1/2 and −1/2. The multiple spins are needed to reduce statistical variance; similarly in our case the observer would need to measure several modes N_b to have confidence that he was actually entangled with the early radiation. One might ask if there could be a possible loophole in the argument: A physical observer will have a nonzero mass, and so the mass and entropy of the black hole will increase after he falls in. However, we may choose to consider a particular Hawking wavepacket which is already separated from the streched horizon by a finite amount when it is encountered by the infalling observer. Thus by postulate 2 the further evolution of this mode is semiclassical and not affected by the subsequent merging of the observer with the black hole. In making this argument we are also assuming that the dynamics of the stretched horizon is causal.

Thus far the asymptotically flat discussion applies to a black hole that is older than the Page time; we needed this in order to frame a sharp paradox using the entanglement with the Hawking radiation. However, we are discussing what should be intrinsic properties of the black hole, not dependent on its entanglement with some external system. After the black hole scrambling time, almost every small subsystem of the black hole is in an almost maximally mixed state. So if the degrees of freedom sampled by the infalling observer can be considered typical, then they are ‘old’ in an intrinsic sense. Our conclusions should then hold. If the black hole is a fast scrambler the scrambling time is r_s ln(r_s/l_P), after which we have to expect either drama for the infalling observer or novel physics outside the black hole.

We note that the three postulates that are in conflict – purity of the Hawking radiation, absence of infalling drama, and semiclassical behavior outside the horizon — are widely held even by those who do not explicitly label them as ‘black hole complementarity.’ For example, one might imagine that if some tunneling process were to cause a shell of branes to appear at the horizon, an infalling observer would just go ‘splat,’ and of course Postulate 4 would not hold.