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

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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 Ki 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 ZS 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 Zi 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) ∈ (ZS, ZS), 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 ωL ∈ H2n−p(X, R) to L in X, to rewrite (3) as an integral

R × X C(p + 1) ∧ ωL —– (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 ∧ ωL —– (5)

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

Embedding Branes in Minkowski Space-Time Dimensions To Decipher Them As Particles Or Otherwise

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The physics treatment of Dirichlet branes in terms of boundary conditions is very analogous to that of the “bulk” quantum field theory, and the next step is again to study the renormalization group. This leads to equations of motion for the fields which arise from the open string, namely the data (M, E, ∇). In the supergravity limit, these equations are solved by taking the submanifold M to be volume minimizing in the metric on X, and the connection ∇ to satisfy the Yang-Mills equations.

Like the Einstein equations, the equations governing a submanifold of minimal volume are highly nonlinear, and their general theory is difficult. This is one motivation to look for special classes of solutions; the physical arguments favoring supersymmetry are another. Just as supersymmetric compactification manifolds correspond to a special class of Ricci-flat manifolds, those admitting a covariantly constant spinor, supersymmetry for a Dirichlet brane will correspond to embedding it into a special class of minimal volume submanifolds. Since the physical analysis is based on a covariantly constant spinor, this special class should be defined using the spinor, or else the covariantly constant forms which are bilinear in the spinor.

The standard physical arguments leading to this class are based on the kappa symmetry of the Green-Schwarz world-volume action, in which one finds that the subset of supersymmetry parameters ε which preserve supersymmetry, both of the metric and of the brane, must satisfy

φ ≡ Re εt Γε|M = Vol|M —– (1)

In words, the real part of one of the covariantly constant forms on M must equal the volume form when restricted to the brane.

Clearly dφ = 0, since it is covariantly constant. Thus,

Z(M) ≡ ∫φ —– (2)

depends only on the homology class of M. Thus, it is what physicists would call a “topological charge”, or a “central charge”.

If in addition the p-form φ is dominated by the volume form Vol upon restriction to any p-dimensional subspace V ⊂ Tx X, i.e.,

φ|V ≤ Vol|V —– (3)

then φ will be a calibration in the sense of implying the global statement

φ ≤ ∫Vol —– (4)

for any submanifold M . Thus, the central charge |Z (M)| is an absolute lower bound for Vol(M).

A calibrated submanifold M is now one satisfying (1), thereby attaining the lower bound and thus of minimal volume. Physically these are usually called “BPS branes,” after a prototypical argument of this type due, for magnetic monopole solutions in nonabelian gauge theory.

For a Calabi-Yau X, all of the forms ωp can be calibrations, and the corresponding calibrated submanifolds are p-dimensional holomorphic submanifolds. Furthermore, the n-form Re eΩ for any choice of real parameter θ is a calibration, and the corresponding calibrated submanifolds are called special Lagrangian.

This generalizes to the presence of a general connection on M, and leads to the following two types of BPS branes for a Calabi-Yau X. Let n = dimR M, and let F be the (End(E)-valued) curvature two-form of ∇.

The first kind of BPS D-brane, based on the ωp calibrations, is (for historical reasons) called a “B-type brane”. Here the BPS constraint is equivalent to the following three requirements:

  1. M is a p-dimensional complex submanifold of X.
  2. The 2-form F is of type (1, 1), i.e., (E, ∇) is a holomorphic vector bundle on M.
  3. In the supergravity limit, F satisfies the Hermitian Yang-Mills equation:ω|p−1M ∧ F = c · ω|pMfor some real constant c.
  4. F satisfies Im e(ω|M + ils2F)p = 0 for some real constant φ, where ls is the correction.

The second kind of BPS D-brane, based on the Re eΩ calibration, is called an “A-type” brane. The simplest examples of A-branes are the so-called special Lagrangian submanifolds (SLAGs), satisfying

(1) M is a Lagrangian submanifold of X with respect to ω.

(2) F = 0, i.e., the vector bundle E is flat.

(3) Im e Ω|M = 0 for some real constant α.

More generally, one also has the “coisotropic branes”. In the case when E is a line bundle, such A-branes satisfy the following four requirements:

(1)  M is a coisotropic submanifold of X with respect to ω, i.e., for any x ∈ M the skew-orthogonal complement of TxM ⊂ TxX is contained in TxM. Equivalently, one requires ker ωM to be an integrable distribution on M.

(2)  The 2-form F annihilates ker ωM.

(3)  Let F M be the vector bundle T M/ ker ωM. It follows from the first two conditions that ωM and F descend to a pair of skew-symmetric forms on FM, denoted by σ and f. Clearly, σ is nondegenerate. One requires the endomorphism σ−1f : FM → FM to be a complex structure on FM.

(4)  Let r be the complex dimension of FM. r is even and that r + n = dimR M. Let Ω be the holomorphic trivialization of KX. One requires that Im eΩ|M ∧ Fr/2 = 0 for some real constant α.

Coisotropic A-branes carrying vector bundles of higher rank are still not fully understood. Physically, one must also specify the embedding of the Dirichlet brane in the remaining (Minkowski) dimensions of space-time. The simplest possibility is to take this to be a time-like geodesic, so that the brane appears as a particle in the visible four dimensions. This is possible only for a subset of the branes, which depends on which string theory one is considering. Somewhat confusingly, in the type IIA theory, the B-branes are BPS particles, while in IIB theory, the A-branes are BPS particles.

Hypersurfaces

Let (S, CS) and (M, CM) be manifolds of dimension k and n, respectively, with 1 ≤ k ≤ n. A smooth map : S → M is said to be an imbedding if it satisfies the following three conditions.

(I1) Ψ is injective.

(I2) At all points p in S, the associated (push-forward) linear map (Ψp) : Sp → MΨ(p) is injective.

(I3) ∀ open sets O1 in S, Ψ[O1] = [S] ∩ O2 for some open set O2 in M. (Equivalently, the inverse map Ψ−1 : Ψ[S] → S is continuous with respect to the relative topology on [S].)

Several comments about the definition are in order. First, given any point p in S, (I2) implies that (Ψp)[Sp] is a k-dimensional subspace of MΨ(p). So the condition cannot be satisfied unless k ≤ n. Second, the three conditions are independent of one another. For example, the smooth map Ψ : R → R2 defined by (s) = (cos(s), sin(s)) satisfies (I2) and (I3) but is not injective. It wraps R round and round in a circle. On the other hand, the smooth map : R → R defined by (s) = s3 satisfies (I1) and (I3) but is not an imbedding because (Ψ0) : R0 → R0 is not injective. (Here R0 is the tangent space to the manifold R at the point 0). Finally, a smooth map : S → M can satisfy (I1) and (I2) but still have an image that “bunches up on itself.” It is precisely this possibility that is ruled out by condition (I3). Consider, for example, a map : R → R2 whose image consists of part of the image of the curve y = sin(1/x) smoothly joined to the segment {(0, y) : y < 1}, as in the figure below. It satisfies conditions (I1) and (I2) but is not an imbedding because we can find an open interval O1 in R such that given any open set O2 in R2, Ψ[O1] ≠ O2 ∩ Ψ[R].

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Suppose(S, CS) and (M, CM) are manifolds with S ⊆ M. We say that (S, CS) is an imbedded submanifold of (M, CM) if the identity map id: S → M is an imbedding. If, in addition, k = n − 1 (where k and n are the dimensions of the two manifolds), we say that (S, CS) is a hypersurface in (M, CM). Let (S, CS) be a k-dimensional imbedded submanifold of the n-dimensional manifold (M, CM), and let p be a point in S. We need to distinguish two senses in which one can speak of “tensors at p.” There are tensors over the vector space Sp (call them S-tensors at p) and ones over the vector space Mp (call them M-tensors at p). So, for example, an S-vector ξ ̃a at p makes assignments to maps of the form f ̃: O ̃ → R where O ̃ is a subset of S that is open in the topology induced by CS, and f ̃ is smooth relative to CS. In contrast, an M-vector ξa at p makes assignments to maps of the form f : O → R where O is a subset of M that is open in the topology induced by CM, and f is smooth relative to CM. Our first task is to consider the relation between S-tensors at p and M-tensors there.

Let us say that ξa ∈ (Mp)a is tangent to S if ξa ∈ (idp)[(Sp)a]. (This makes sense. We know that (idp)[(Sp)a] is a k-dimensional subspace of (Mp)a; ξa either belongs to that subspace or it does not.) Let us further say that ηa in (Mp)a is normal to S if ηaξa =0 ∀ ξa ∈ (Mp)a that are tangent to S. Each of these classes of vectors has a natural vector space structure. The space of vectors ξa ∈ (Mp)a tangent to S has dimension k. The space of co-vectors ηa ∈ (Mp)a normal to S has dimension (n − k).

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/rs (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⟩= ∑ii⟩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 rs corresponding to a superposition of frequencies O(r−1s). 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 bb. 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−1s) 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)aω + C(ω)aω —– (2)

so that the full state cannot be both an a-vacuum (a|Ψ⟩ = 0) and a bb 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 bb he finds the expected eigenvalue, while if he measures the noncommuting operator aa 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 bb by postulate 1. The observer encounters it when its wavelength is much shorter: the field must be in the ground state aωaω = 0, by postulate 4, and so cannot be in an eigenstate of bb. But by postulate 2, the evolution of the mode outside the horizon is essentially free, so this is a contradiction.

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Figure: Eddington-Finkelstein coordinates, showing the infalling observer encountering the outgoing Hawking mode (shaded) at a time when its size is ω−1* ≪ rs. If the observer’s measurements are given by an eigenstate of aa, postulate 1 is violated; if they are given by an eigenstate of bb, 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,

SAB + SBC ≥ SB + SABC —– (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 SAB < SA. The absence of infalling drama means that SBC = 0 and so SABC = SA. Subadditivity then gives SA ≥ SB + SA, 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, SAB = SA − SB, so that we get from subadditivity that SA ≥ 2SB + SA.

Note that the measurement of Nb takes place entirely outside the horizon, while the measurement of Na (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 Nb the observer can infer something about what would have happened if Na 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 Nb 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 rs ln(rs/lP), 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.