Conformal Field Theory and Virasoro Algebra. Note Quote.

Realization of the Virasoro algebra

There are a few reasons why Conformal Field Theories (CFTs) are very interesting to study: The first is that at fixed points of Renormalization Group flows, or at second order phase transitions, a quantum field theory is scale invariant. Scale invariance is a weaker form of conformal invariance, and it turns out in all cases that we know of scale invariance of a quantum field theory actually ends up implying the larger symmetry of conformal invariance. The second reason is that the requirement that a theory is conformally invariant is so restrictive that many things can be solved for that would otherwise be intractable. As an example, conformal invariance fixes 2- and 3-point functions entirely. In an ordinary quantum field theory, especially one at strong coupling, these would be hard or impossible to calculate at all. A third reason is string theory. In string theory, the worldsheet theory describing the string’s excitations is a CFT, so if string theory is correct, then in some sense conformal invariance is really one of the most fundamental features of the elemental constituents of reality. And through string theory we have the most precise and best-understood gauge/gravity dualities (the AdS/CFT dualities) that also involve CFT’s.

A Conformal Field Theory (CFT) is a Quantum Field Theory (QFT) in which conformal rescaling of the metric acts by conjugation. For the family of morphisms Dg

D[ehg] = ec·α[h] L−1[h|B1] Dg L[h|B2] —– (1)

The analogous statement (conjugating the state on each boundary) is true for any Σ.

Here L is a linear operator depending only on the restriction of h to one of the boundaries of the annulus. All the dependence on the conformal rescaling away from the boundary is determined by a universal (independent of the particular Conformal Field Theory) functional α[h] ∈ R, which appears in an overall multiplicative factor ec·α[h]. The quantity c, called “Virasoro central charge”.

The corresponding operators L[h] form a semigroup, with a self-adjoint generator H. Then, since according to the axioms of QFT the spectrum of H is bounded below, we can promote this to a group action. This can be used to map any of the Hilbert spaces Hd to a single Hl for a fixed value of l, say l = 1. We will now do this and use the simpler notation H ≅ H1,

How do we determine the L[h]? First, we uniformize Σ – in other words, we find a complex diffeomorphism φ from our surface with boundary Σ to a constant curvature surface. We then consider the restriction of φ to each of the boundary components Bi, to get an element φi of Diff S1 × R+, where the R+ factor acts by an overall rescaling. We then express each φi as the exponential of an element li in the Lie algebra Diff S1, to find an appropriate projective representation of this Lie algebra on H.

Certain subtleties are in order here: The Lie algebra Diff S1 which appears is actually a subalgebra of a direct sum of two commuting algebras, which act independently on “left moving” and “right moving” factors in H. Thus, we can write H as a direct sum of irreps of this direct sum algebra,

H = ⊕iHL,i ⊗ HR,i —– (2)

Each of these two commuting algebras is a central extension of the Lie algebra Diff S1, usually called the Virasoro algebra or Vir.

Now, consider the natural action of Diff S1 on functions on an S1 parameterized by θ ∈ [0, 2π). After complexification, we can take the following set of generators,

ln = −ieinθ ∂/∂θ n ∈ Z —– (3)

which satisfy the relations

[lm, ln] = (m − n)lm+n —– (4)

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

[Lm, Ln] = (m − n)Lm+n + c/12 n(n2 − 1)δm+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 = Ln. In particular, L0 is self-adjoint and can be diagonalized. Take a “highest weight representation,” in which the spectrum of L0 is bounded below. The L0 eigenvector with the minimum eigenvalue, h, is by definition the “highest weight state”, or a state |h⟩, so that

L0|h⟩ = h|h⟩ —– (6)

and normalize it so that ⟨h|h⟩ = 1. Since this is a norm in a Hilbert space, we conclude that h ≥ 0, with equality only if L−1|h⟩ = 0. In fact, L−1|0⟩ = 0 can be related to the translation invariance of the vacuum. Rephrasing this in terms of local operators, instead of in terms of states, take Σ to be the infinite cylinder R × S1, or equivalently the punctured complex plane C with the complex coordinate z. In a CFT the component Tzz of the stress tensor can be expressed in terms of the Virasoro generators:

Tzz ≡ T(z) = ∑n∈Z Lnz−n−2 —– (7)

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

Tz̄z̄ ≡ T(z̄) = ∑n∈Zn−n−2 —– (8)

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

Closed String Algebra as a Graded-Commutative Algebra C: Cochain Complex Differentials: Part 2, Note Quote.

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The most general target category we can consider is a symmetric tensor category: clearly we need a tensor product, and the axiom HY1⊔Y2 ≅ HY1 ⊗ HY2 only makes sense if there is an involutory canonical isomorphism HY1 ⊗ HY2 ≅ HY2 ⊗ HY1 .

A very common choice in physics is the category of super vector spaces, i.e., vector spaces V with a mod 2 grading V = V0 ⊕ V1, where the canonical isomorphism V ⊗ W ≅ W ⊗ V is v ⊗ w ↦ (−1)deg v deg ww ⊗ v. One can also consider the category of Z-graded vector spaces, with the same sign convention for the tensor product.

In either case the closed string algebra is a graded-commutative algebra C with a trace θ : C → C. In principle the trace should have degree zero, but in fact the commonly encountered theories have a grading anomaly which makes the trace have degree −n for some integer n.

We define topological-spinc theories, which model 2d theories with N = 2 supersymmetry, by replacing “manifolds” with “manifolds with spinc structure”.

A spinc structure on a surface with a conformal structure is a pair of holomorphic line bundles L1, L2 with an isomorphism L1 ⊗ L2 ≅ TΣ of holomorphic line bundles. A spin structure is the particular case when L1 = L2. On a 1-manifold S a spinc structure means a spinc structure on a ribbon neighbourhood of S in a surface with conformal structure. An N = 2 superconformal theory assigns a vector space HS;L1,L2 to each 1-manifold S with spinc structure, and an operator

US0;L1,L2: HS0;L1,L2 → HS1;L1,L2

to each spinc-cobordism from S0 to S1. To explain the rest of the structure we need to define the N = 2 Lie superalgebra associated to a spin1-manifold (S;L1,L2). Let G = Aut(L1) denote the group of bundle isomorphisms L1 → L1 which cover diffeomorphisms of S. (We can identify this group with Aut(L2).) It has a homomorphism onto the group Diff+(S) of orientation-preserving diffeomorphisms of S, and the kernel is the group of fibrewise automorphisms of L1, which can be identified with the group of smooth maps from S to C×. The Lie algebra Lie(G) is therefore an extension of the Lie algebra Vect(S) of Diff+(S) by the commutative Lie algebra Ω0(S) of smooth real-valued functions on S. Let Λ0S;L1,L2 denote the complex Lie algebra obtained from Lie(G) by complexifying Vect(S). This is the even part of a Lie super algebra whose odd part is Λ1S;L1,L2 = Γ(L1) ⊕ Γ(L2). The bracket Λ1 ⊗ Λ1 → Λ0 is completely determined by the property that elements of Γ(L1) and of Γ(L2) anticommute among themselves, while the composite

Γ(L1) ⊗ Γ(L2) → Λ0 → VectC(S)

takes (λ12) to λ1λ2 ∈ Γ(TS).

In an N = 2 theory we require the superalgebra Λ(S;L1,L2) to act on the vector space HS;L1,L2, compatibly with the action of the group G, and with a similar intertwining property with the cobordism operators to that of the N = 1 case. For an N = 2 theory the state space always has an action of the circle group coming from its embedding in G as the group of fibrewise multiplications on L1 and L2. Equivalently, the state space is always Z-graded.

An N = 2 theory always gives rise to two ordinary conformal field theories by equipping a surface Σ with the spinc structures (C,TΣ) and (TΣ,C). These are called the “A-model” and the “B-model” associated to the N = 2 theory. In each case the state spaces are cochain complexes in which the differential is the action of the constant section of the trivial component of the spinc-structure.

Superconformal Spin/Field Theories: When Vector Spaces have same Dimensions: Part 1, Note Quote.

1-s2.0-S0001870802000592-gr7

A spin structure on a surface means a double covering of its space of non-zero tangent vectors which is non-trivial on each individual tangent space. On an oriented 1-dimensional manifold S it means a double covering of the space of positively-oriented tangent vectors. For purposes of gluing, this is the same thing as a spin structure on a ribbon neighbourhood of S in an orientable surface. Each spin structure has an automorphism which interchanges its sheets, and this will induce an involution T on any vector space which is naturally associated to a 1-manifold with spin structure, giving the vector space a mod 2 grading by its ±1-eigenspaces. A topological-spin theory is a functor from the cobordism category of manifolds with spin structures to the category of super vector spaces with its graded tensor structure. The functor is required to take disjoint unions to super tensor products, and additionally it is required that the automorphism of the spin structure of a 1-manifold induces the grading automorphism T = (−1)degree of the super vector space. This choice of the supersymmetry of the tensor product rather than the naive symmetry which ignores the grading is forced by the geometry of spin structures if the possibility of a semisimple category of boundary conditions is to be allowed. There are two non-isomorphic circles with spin structure: S1ns, with the Möbius or “Neveu-Schwarz” structure, and S1r, with the trivial or “Ramond” structure. A topological-spin theory gives us state spaces Cns and Cr, corresponding respectively to S1ns and S1r.

There are four cobordisms with spin structures which cover the standard annulus. The double covering can be identified with its incoming end times the interval [0,1], but then one has a binary choice when one identifies the outgoing end of the double covering over the annulus with the chosen structure on the outgoing boundary circle. In other words, alongside the cylinders A+ns,r = S1ns,r × [0,1] which induce the identity maps of Cns,r there are also cylinders Ans,r which connect S1ns,r to itself while interchanging the sheets. These cylinders Ans,r induce the grading automorphism on the state spaces. But because Ans ≅ A+ns by an isomorphism which is the identity on the boundary circles – the Dehn twist which “rotates one end of the cylinder by 2π” – the grading on Cns must be purely even. The space Cr can have both even and odd components. The situation is a little more complicated for “U-shaped” cobordisms, i.e., cylinders with two incoming or two outgoing boundary circles. If the boundaries are S1ns there is only one possibility, but if the boundaries are S1r there are two, corresponding to A±r. The complication is that there seems no special reason to prefer either of the spin structures as “positive”. We shall simply choose one – let us call it P – with incoming boundary S1r ⊔ S1r, and use P to define a pairing Cr ⊗ Cr → C. We then choose a preferred cobordism Q in the other direction so that when we sew its right-hand outgoing S1r to the left-hand incoming one of P the resulting S-bend is the “trivial” cylinder A+r. We shall need to know, however, that the closed torus formed by the composition P ◦ Q has an even spin structure. The Frobenius structure θ on C restricts to 0 on Cr.

There is a unique spin structure on the pair-of-pants cobordism in the figure below, which restricts to S1ns on each boundary circle, and it makes Cns into a commutative Frobenius algebra in the usual way.

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If one incoming circle is S1ns and the other is S1r then the outgoing circle is S1r, and there are two possible spin structures, but the one obtained by removing a disc from the cylinder A+r is preferred: it makes Cr into a graded module over Cns. The chosen U-shaped cobordism P, with two incoming circles S1r, can be punctured to give us a pair of pants with an outgoing S1ns, and it induces a graded bilinear map Cr × Cr → Cns which, composing with the trace on Cns, gives a non-degenerate inner product on Cr. At this point the choice of symmetry of the tensor product becomes important. Let us consider the diffeomorphism of the pair of pants which shows us in the usual case that the Frobenius algebra is commutative. When we lift it to the spin structure, this diffeomorphism induces the identity on one incoming circle but reverses the sheets over the other incoming circle, and this proves that the cobordism must have the same output when we change the input from S(φ1 ⊗ φ2) to T(φ1) ⊗ φ2, where T is the grading involution and S : Cr ⊗ Cr → Cr ⊗ Cr is the symmetry of the tensor category. If we take S to be the symmetry of the tensor category of vector spaces which ignores the grading, this shows that the product on the graded vector space Cr is graded-symmetric with the usual sign; but if S is the graded symmetry then we see that the product on Cr is symmetric in the naive sense.

There is an analogue for spin theories of the theorem which tells us that a two-dimensional topological field theory “is” a commutative Frobenius algebra. It asserts that a spin-topological theory “is” a Frobenius algebra C = (Cns ⊕ CrC) with the following property. Let {φk} be a basis for Cns, with dual basis {φk} such that θCkφm) = δmk, and let βk and βk be similar dual bases for Cr. Then the Euler elements χns := ∑ φkφk and χr = ∑ βkβk are independent of the choices of bases, and the condition we need on the algebra C is that χns = χr. In particular, this condition implies that the vector spaces Cns and Cr have the same dimension. In fact, the Euler elements can be obtained from cutting a hole out of the torus. There are actually four spin structures on the torus. The output state is necessarily in Cns. The Euler elements for the three even spin structures are equal to χe = χns = χr. The Euler element χo corresponding to the odd spin structure, on the other hand, is given by χo = ∑(−1)degβkβkβk.

A spin theory is very similar to a Z/2-equivariant theory, which is the structure obtained when the surfaces are equipped with principal Z/2-bundles (i.e., double coverings) rather than spin structures.

It seems reasonable to call a spin theory semisimple if the algebra Cns is semisimple, i.e., is the algebra of functions on a finite set X. Then Cr is the space of sections of a vector bundle E on X, and it follows from the condition χns = χr that the fibre at each point must have dimension 1. Thus the whole structure is determined by the Frobenius algebra Cns together with a binary choice at each point x ∈ X of the grading of the fibre Ex of the line bundle E at x.

We can now see that if we had not used the graded symmetry in defining the tensor category we should have forced the grading of Cr to be purely even. For on the odd part the inner product would have had to be skew, and that is impossible on a 1-dimensional space. And if both Cns and Cr are purely even then the theory is in fact completely independent of the spin structures on the surfaces.

A concrete example of a two-dimensional topological-spin theory is given by C = C ⊕ Cη where η2 = 1 and η is odd. The Euler elements are χe = 1 and χo = −1. It follows that the partition function of a closed surface with spin structure is ±1 according as the spin structure is even or odd.

The most common theories defined on surfaces with spin structure are not topological: they are 2-dimensional conformal field theories with N = 1 supersymmetry. It should be noticed that if the theory is not topological then one does not expect the grading on Cns to be purely even: states can change sign on rotation by 2π. If a surface Σ has a conformal structure then a double covering of the non-zero tangent vectors is the complement of the zero-section in a two-dimensional real vector bundle L on Σ which is called the spin bundle. The covering map then extends to a symmetric pairing of vector bundles L ⊗ L → TΣ which, if we regard L and TΣ as complex line bundles in the natural way, induces an isomorphism L ⊗C L ≅ TΣ. An N = 1 superconformal field theory is a conformal-spin theory which assigns a vector space HS,L to the 1-manifold S with the spin bundle L, and is equipped with an additional map

Γ(S,L) ⊗ HS,L → HS,L

(σ,ψ) ↦ Gσψ,

where Γ(S,L) is the space of smooth sections of L, such that Gσ is real-linear in the section σ, and satisfies G2σ = Dσ2, where Dσ2 is the Virasoro action of the vector field σ2 related to σ ⊗ σ by the isomorphism L ⊗C L ≅ TΣ. Furthermore, when we have a cobordism (Σ,L) from (S0,L0) to (S1,L1) and a holomorphic section σ of L which restricts to σi on Si we have the intertwining property

Gσ1 ◦ UΣ,L = UΣ,L ◦ Gσ0

….

The Natural Theoretic of Electromagnetism. Thought of the Day 147.0

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In Maxwell’s theory, the field strength F = 1/2Fμν dxμ ∧ dxν is a real 2-form on spacetime, and thence a natural object at the same time. The homogeneous Maxwell equation dF = 0 is an equation involving forms and it has a well-known local solution F = dA’, i.e. there exists a local spacetime 1-form A’ which is a potential for the field strength F. Of course, if spacetime is contractible, as e.g. for Minkowski space, the solution is also a global one. As is well-known, in the non-commutative Yang-Mills theory case the field strength F = 1/2FAμν TA ⊗ dxμ ∧ dxν is no longer a spacetime form. This is a somewhat trivial remark since the transformation laws of such field strength are obtained as the transformation laws of the curvature of a principal connection with values in the Lie algebra of some (semisimple) non-Abelian Lie group G (e.g. G = SU(n), n 2 ≥ 2). However, the common belief that electromagnetism is to be intended as the particular case (for G =U(1)) of a non-commutative theory is not really physically evident. Even if we subscribe this common belief, which is motivated also by the tremendous success of the quantized theory, let us for a while discuss electromagnetism as a standalone theory.

From a mathematical viewpoint this is a (different) approach to electromagnetism and the choice between the two can be dealt with on a physical ground only. Of course the 1-form A’ is defined modulo a closed form, i.e. locally A” = A’ + dα is another solution.

How can one decide whether the potential of electromagnetism should be considered as a 1-form or rather as a principal connection on a U(1)-bundle? First of all we notice that by a standard hole argument (one can easily define compact supported closed 1-forms, e.g. by choosing the differential of compact supported functions which always exist on a paracompact manifold) the potentials A and A’ represent the same physical situation. On the other hand, from a mathematical viewpoint we would like the dynamical field, i.e. the potential A’, to be a global section of some suitable configuration bundle. This requirement is a mathematical one, motivated on the wish of a well-defined geometrical perspective based on global Variational Calculus.

The first mathematical way out is to restrict attention to contractible spacetimes, where A’ may be always chosen to be global. Then one can require the gauge transformations A” = A’ + dα to be Lagrangian symmetries. In this way, field equations select a whole equivalence class of gauge-equivalent potentials, a procedure which solves the hole argument problem. In this picture the potential A’ is really a 1-form, which can be dragged along spacetime diffeomorphism and which admits the ordinary Lie derivatives of 1-forms. Unfortunately, the restriction to contractible spacetimes is physically unmotivated and probably wrong.

Alternatively, one can restrict electromagnetic fields F, deciding that only exact 2-forms F are allowed. That actually restricts the observable physical situations, by changing the homogeneous Maxwell equations (i.e. Bianchi identities) by requiring that F is not only closed but exact. One should in principle be able to empirically reject this option.

On non-contractible spacetimes, one is necessarily forced to resort to a more “democratic” attitude. The spacetime is covered by a number of patches Uα. On each patch Uα one defines a potential A(α). In the intersection of two patches the two potentials A(α) and A(β) may not agree. In each patch, in fact, the observer chooses his own conventions and he finds a different representative of the electromagnetic potential, which is related by a gauge transformation to the representatives chosen in the neighbour patch(es). Thence we have a family of gauge transformations, one in each intersection Uαβ, which obey cocycle identities. If one recognizes in them the action of U(1) then one can build a principal bundle P = (P, M, π; U(1)) and interpret the ensuing potential as a connection on P. This leads way to the gauge natural formalism.

Anyway this does not close the matter. One can investigate if and when the principal bundle P, in addition to the obvious principal structure, can be also endowed with a natural structure. If that were possible then the bundle of connections Cp (which is associated to P) would also be natural. The problem of deciding whether a given gauge natural bundle can be endowed with a natural structure is quite difficult in general and no full theory is yet completely developed in mathematical terms. That is to say, there is no complete classification of the topological and differential geometric conditions which a principal bundle P has to satisfy in order to ensure that, among the principal trivializations which determine its gauge natural structure, one can choose a sub-class of trivializations which induce a purely natural bundle structure. Nor it is clear how many inequivalent natural structures a good principal bundle may support. Though, there are important examples of bundles which support at the same time a natural and a gauge natural structure. Actually any natural bundle is associated to some frame bundle L(M), which is principal; thence each natural bundle is also gauge natural in a trivial way. Since on any paracompact manifold one can choose a global Riemannian metric g, the corresponding tangent bundle T(M) can be associated to the orthonormal frame bundle O(M, g) besides being obviously associated to L(M). Thence the natural bundle T(M) may be also endowed with a gauge natural bundle structure with structure group O(m). And if M is orientable the structure can be further reduced to a gauge natural bundle with structure group SO(m).

Roughly speaking, the task is achieved by imposing restrictions to cocycles which generate T(M) according to the prescription by imposing a privileged class of changes of local laboratories and sets of measures. Imposing the cocycle ψ(αβ) to take its values in O(m) rather than in the larger group GL(m). Inequivalent gauge natural structures are in one-to-one correspondence with (non isometric) Riemannian metrics on M. Actually whenever there is a Lie group homomorphism ρ : GU(m) → G for some s onto some given Lie group G we can build a natural G-principal bundle on M. In fact, let (Uα, ψ(α)) be an atlas of the given manifold M, ψ(αβ) be its transition functions and jψ(αβ) be the induced transition functions of L(M). Then we can define a G-valued cocycle on M by setting ρ(jψ(αβ)) and thence a (unique up to fibered isomorphisms) G-principal bundle P(M) = (P(M), M, π; G). The bundle P(M), as well as any gauge natural bundle associated to it, is natural by construction. Now, defining a whole family of natural U(1)-bundles Pq(M) by using the bundle homomorphisms

ρq: GL(m) → U(1): J ↦ exp(iq ln det|J|) —– (1)

where q is any real number and In denotes the natural logarithm. In the case q = 0 the image of ρ0 is the trivial group {I}; and, all the induced bundles are trivial, i.e. P = M x U(1).

The natural lift φ’ of a diffeomorphism φ: M → M is given by

φ'[x, e]α = [φ(x), eiq ln det|J|. e]α —– (2)

where J is the Jacobin of the morphism φ. The bundles Pq(M) are all trivial since they allow a global section. In fact, on any manifold M, one can define a global Riemannian metric g, where the local sections glue together.

Since the bundles Pq(M) are all trivial, they are all isomorphic to M x U(1) as principal U(1)-bundles, though in a non-canonical way unless q = 0. Any two of the bundles Pq1(M) and Pq2(M) for two different values of q are isomorphic as principal bundles but the isomorphism obtained is not the lift of a spacetime diffeomorphism because of the two different values of q. Thence they are not isomorphic as natural bundles. We are thence facing a very interesting situation: a gauge natural bundle C associated to the trivial principal bundle P can be endowed with an infinite family of natural structures, one for each q ∈ R; each of these natural structures can be used to regard principal connections on P as natural objects on M and thence one can regard electromagnetism as a natural theory.

Now that the mathematical situation has been a little bit clarified, it is again a matter of physical interpretation. One can in fact restrict to electromagnetic potentials which are a priori connections on a trivial structure bundle P ≅ M x U(1) or to accept that more complicated situations may occur in Nature. But, non-trivial situations are still empirically unsupported, at least at a fundamental level.

Gauge Fixity Towards Hyperbolicity: General Theory of Relativity and Superpotentials. Part 1.

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Gravitational field is described by a pseudo-Riemannian metric g (with Lorentzian signature (1, m-1)) over the spacetime M of dimension dim(M) = m; in standard General Relativity, m = 4. The configuration bundle is thence the bundle of Lorentzian metrics over M, denoted by Lor(M) . The Lagrangian is second order and it is usually chosen to be the so-called Hilbert Lagrangian:

LH: J2Lor(m) → ∧om(M)

LH: LH(gαβ, Rαβ)ds = 1/2κ (R – 2∧)√g ds —– (1)

where

R = gαβ Rαβ denotes the scalar curvature, √g the square root of the absolute value of the metric determinant and ∧ is a real constant (called the cosmological constant). The coupling constant 1/2κ which is completely irrelevant until the gravitational field is not coupled to some other field, depends on conventions; in natural units, i.e. c = 1, h = 1, G = 1, dimension 4 and signature ( + , – , – , – ) one has κ = – 8π.

Field equations are the well known Einstein equations with cosmological constant

Rαβ – 1/2 Rgαβ = -∧gαβ —— (2)

Lagrangian momenta is defined by:

pαβ = ∂LH/∂gαβ = 1/2κ (Rαβ – 1/2(R – 2∧)gαβ)√g

Pαβ = ∂LH/∂Rαβ = 1/2κ gαβ√g —– (3)

Thus the covariance identity is the following:

dα(LHξα) = pαβ£ξgαβ + Pαβ£ξRαβ —– (4)

or equivalently,

α(LHξα) = pαβ£ξgαβ + PαβεξΓεαβ – δεβ£ξΓλαλ) —– (5)

where ∇ε denotes the covariant derivative with respect to the Levi-Civita connection of g. Thence we have a weak conservation law for the Hilbert Lagrangian

Div ε(LH, ξ) = W(LH, ξ) —– (6)

Conserved currents and work forms have respectively the following expressions:

ε(LH, ξ) = [Pαβ£ξΓεαβ – Pαε£ξΓλαλ – LHξε]dsε = √g/2κ(gαβgεσ – gσβgεα) ∇α£ξgβσdsε – √g/2κξεRdsε = √g/2κ[(3/2Rαλ – (R – 2∧)δαλλ + (gβγδαλ – gα(γδβ)λβγξλ]dsα —– (7)

W(LH, ξ) = √g/κ(Rαβ – 1/2(R – 2∧)gαβ)∇(αξβ)ds —– (8)

As any other natural theory, General Relativity allows superpotentials. In fact, the current can be recast into the form:

ε(LH, ξ) = ε'(LH, ξ) + Div U(LH, ξ) —– (9)

where we set

ε'(LH, ξ) = √g/κ(Rαβ – 1/2(R – 2∧)δαββ)dsα

U(LH, ξ) = 1/2κ ∇[βξα] √gdsαβ —– (10)

The superpotential (10) generalizes to an arbitrary vector field ξ, the well known Komar superpotential which is originally derived for timelike Killing vectors. Whenever spacetime is assumed to be asymptotically fiat, then the superpotential of Komar is known to produce upon integration at spatial infinity ∞ the correct value for angular momentum (e.g. for Kerr-Newman solutions) but just one half of the expected value of the mass. The classical prescriptions are in fact:

m = 2∫ U(LH, ∂t, g)

J = ∫ U(LH, ∂φ, g) —– (11)

For an asymptotically flat solution (e.g. the Kerr-Newman black hole solution) m coincides with the so-called ADM mass and J is the so-called (ADM) angular momentum. For the Kerr-Newman solution in polar coordinates (t, r, θ, φ) the vector fields ∂t and ∂φ are the Killing vectors which generate stationarity and axial symmetry, respectively. Thence, according to this prescription, U(LH, ∂φ) is the superpotential for J while 2U(LH, ∂t) is the superpotential for m. This is known as the anomalous factor problem for the Komar potential. To obtain the expected values for all conserved quantities from the same superpotential, one has to correct the superpotential (10) by some ad hoc additional boundary term. Equivalently and alternatively, one can deduce a corrected superpotential as the canonical superpotential for a corrected Lagrangian, which is in fact the first order Lagrangian for standard General Relativity. This can be done covariantly, provided that one introduces an extra connection Γ’αβμ. The need of a reference connection Γ’ should be also motivated by physical considerations, according to which the conserved quantities have no absolute meaning but they are intrinsically relative to an arbitrarily fixed vacuum level. The simplest choice consists, in fact, in fixing a background metric g (not necessarily of the correct Lorentzian signature) and assuming Γ’ to be the Levi-Civita connection of g. This is rather similar to the gauge fixing à la Hawking which allows to show that Einstein equations form in fact an essentially hyperbolic PDE system. Nothing prevents, however, from taking Γ’ to be any (in principle torsionless) connection on spacetime; also this corresponds to a gauge fixing towards hyperbolicity.

Now, using the term background for a field which enters a field theory in the same way as the metric enters Yang-Mills theory, we see that the background has to be fixed once for all and thence preserved, e.g. by symmetries and deformations. A background has no field equations since deformations fix it; it eventually destroys the naturality of a theory, since fixing the background results in allowing a smaller group of symmetries G ⊂ Diff(M). Accordingly, in truly natural field theories one should not consider background fields either if they are endowed with a physical meaning (as the metric in Yang-Mills theory does) or if they are not.

On the contrary we shall use the expression reference or reference background to denote an extra dynamical field which is not endowed with a direct physical meaning. As long as variational calculus is concerned, reference backgrounds behave in exactly the same way as other dynamical fields do. They obey field equations and they can be dragged along deformations and symmetries. It is important to stress that such a behavior has nothing to do with a direct physical meaning: even if a reference background obeys field equations this does not mean that it is observable, i.e. it can be measured in a laboratory. Of course, not any dynamical field can be treated as a reference background in the above sense. The Lagrangian has in fact to depend on reference backgrounds in a quite peculiar way, so that a reference background cannot interact with any other physical field, otherwise its effect would be observable in a laboratory….

Spinorial Algebra

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Superspace is to supersymmetry as Minkowski space is to the Lorentz group. Superspace provides the most natural geometrical setting in which to describe supersymmetrical theories. Almost no physicist would utilize the component of Lorentz four-vectors or higher rank tensor to describe relativistic physics.

In a field theory, boson and fermions are to be regarded as diffeomorphisms generating two different vector spaces; the supersymmetry generators are nothing but sets of linear maps between these spaces. We can thus include a supersymmetric theory in a more general geometrical framework defining the collection of diffeomorphisms,

φi : R → RdL, i = 1,…, dL —– (1)

ψαˆ : R → RdR, i = 1,…, dR —– (2)

where the one-dimensional dependence reminds us that we restrict our attention to mechanics. The free vector spaces generated by {φi}i=1dL and {ψαˆ}αˆdR are respectively VL and VR, isomorphic to RdL and RdR. For matrix representations in the following, the two integers are restricted to the case dL = dR = d. Four different linear mappings can act on VL and VR

ML : VL → VR, MR : VR → VL

UL : VL → VL, UR : VR → VR —– (3)

with linear map space dimensions

dimML = dimMR = dRdL = d2,

dimUL = dL2 = d2, dimUR = dR2 = d2 —– (4)

as a consequence of linearity. To relate this construction to a general real (≡ GR) algebraic structure of dimension d and rank N denoted by GR(d,N), two more requirements need to be added.

Defining the generators of GR(d,N) as the family of N + N linear maps

LI ∈ {ML}, I = 1,…, N

RK ∈ {MR}, K = 1,…, N —– (5)

such that ∀ I, K = 1,…, N, we have

LI ◦ RK + LK ◦ RI = −2δIKIVR

RI ◦ LK + RK ◦ LI = −2δIKIVL —– (6)

where IVL and IVR are identity maps on VL and VR. Equations (6) will later be embedded into a Clifford algebra but one point has to be emphasized, we are working with real objects.

After equipping VL and VR with euclidean inner products ⟨·,·⟩VL and ⟨·,·⟩VR, respectively, the generators satisfy the property

⟨φ, RI(ψ)⟩VL = −⟨LI(φ), ψ⟩VR, ∀ (φ, ψ) ∈ VL ⊕ VR —— (7)

This condition relates LI to the hermitian conjugate of RI, namely RI, defined as usual by

⟨φ, RI(ψ)⟩VL = ⟨RI(φ), ψ⟩VR —– (8)

such that

RI = RIt = −LI —– (9)

The role of {UL} and {UR} maps is to connect different representations once a set of generators defined by conditions (6) and (7) has been chosen. Notice that (RILJ)ij ∈ UL and (LIRJ)αˆβˆ ∈ UR. Let us consider A ∈ {UL} and B ∈ {UR} such that

A : φ → φ′ = Aφ

B : ψ → ψ′ = Bψ —– (10)

with Vas an example,

⟨φ, RI(ψ)⟩VL → ⟨Aφ, RI B(ψ)⟩VL

= ⟨φ,A RI B(ψ)⟩VL

= ⟨φ, RI (ψ)⟩VL —– (11)

so a change of representation transforms the generators in the following manner:

LI → LI = BLIA

RI → RI = ARIB —– (12)

In general (6) and (7) do not identify a unique set of generators. Thus, an equivalence relation has to be defined on the space of possible sets of generators, say {LI, RI} ∼ {LI, RI} iff ∃ A ∈ {UL} and B ∈ {UR} such that L′ = BLIA and R′ = ARIB.

Moving on to how supersymmetry is born, we consider the manner in which algebraic derivations are defined by

δεφi = iεI(RI)iαˆψαˆ

δεψαˆ = −εI(LI)αˆiτφi —– (13)

where the real-valued fields {φi}i=1dL and {ψαˆ}αˆ=1dR can be interpreted as bosonic and fermionic respectively. The fermionic nature attributed to the VR elements implies that ML and MR generators, together with supersymmetry transformation parameters εI, anticommute among themselves. Introducing the dL + dR dimensional space VL ⊕ VR with vectors

Ψ = (ψ φ) —– (14)

(13) reads

δε(Ψ) = (iεRψ εL∂τφ) —– (15)

such that

ε1, δε2]Ψ = iε1Iε2J (RILJτφ LIRJτψ) – iε2Jε1I (RJLIτφ LJRIτψ) = – 2iε1Iε2IτΨ —– (16)

utilizing that we have classical anticommuting parameters and that (6) holds. From (16) it is clear that δε acts as a supersymmetry generator, so that we can set

δQΨ := δεΨ = iεIQIΨ —– (17)

which is equivalent to writing

δQφi = i(εIQIψ)i

δQψαˆ = i(εIQIφ)αˆ —– (18)

with

Q1 = (0LIH RI0) —– (19)

where H = i∂τ. As a consequence of (16) a familiar anticommutation relation appears

{QI, QJ} = − 2iδIJH —– (20)

confirming that we are about to recognize supersymmetry, and once this is achieved, we can associate to the algebraic derivations (13), the variations defining the scalar supermultiplets. However, the choice (13) is not unique, for this is where we could have a spinorial one,

δQξαˆ = εI(LI)αˆiFi

δQFi = − iεI(RI)iαˆτξαˆ —– (21)

Is There a Philosophy of Bundles and Fields? Drunken Risibility.

The bundle formulation of field theory is not at all motivated by just seeking a full mathematical generality; on the contrary it is just an empirical consequence of physical situations that concretely happen in Nature. One among the simplest of these situations may be that of a particle constrained to move on a sphere, denoted by S2; the physical state of such a dynamical system is described by providing both the position of the particle and its momentum, which is a tangent vector to the sphere. In other words, the state of this system is described by a point of the so-called tangent bundle TS2 of the sphere, which is non-trivial, i.e. it has a global topology which differs from the (trivial) product topology of S2 x R2. When one seeks for solutions of the relevant equations of motion some local coordinates have to be chosen on the sphere, e.g. stereographic coordinates covering the whole sphere but a point (let us say the north pole). On such a coordinate neighbourhood (which is contractible to a point being a diffeomorphic copy of R2) there exists a trivialization of the corresponding portion of the tangent bundle of the sphere, so that the relevant equations of motion can be locally written in R2 x R2. At the global level, however, together with the equations, one should give some boundary conditions which will ensure regularity in the north pole. As is well known, different inequivalent choices are possible; these boundary conditions may be considered as what is left in the local theory out of the non-triviality of the configuration bundle TS2.

Moreover, much before modem gauge theories or even more complicated new field theories, the theory of General Relativity is the ultimate proof of the need of a bundle framework to describe physical situations. Among other things, in fact, General Relativity assumes that spacetime is not the “simple” Minkowski space introduced for Special Relativity, which has the topology of R4. In general it is a Lorentzian four-dimensional manifold possibly endowed with a complicated global topology. On such a manifold, the choice of a trivial bundle M x F as the configuration bundle for a field theory is mathematically unjustified as well as physically wrong in general. In fact, as long as spacetime is a contractible manifold, as Minkowski space is, all bundles on it are forced to be trivial; however, if spacetime is allowed to be topologically non-trivial, then trivial bundles on it are just a small subclass of all possible bundles among which the configuration bundle can be chosen. Again, given the base M and the fiber F, the non-unique choice of the topology of the configuration bundle corresponds to different global requirements.

A simple purely geometrical example can be considered to sustain this claim. Let us consider M = S1 and F = (-1, 1), an interval of the real line R; then ∃ (at least) countably many “inequivalent” bundles other than the trivial one Mö0 = S1 X F , i.e. the cylinder, as shown

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Furthermore the word “inequivalent” can be endowed with different meanings. The bundles shown in the figure are all inequivalent as embedded bundles (i.e. there is no diffeomorphism of the ambient space transforming one into the other) but the even ones (as well as the odd ones) are all equivalent among each other as abstract (i.e. not embedded) bundles (since they have the same transition functions).

The bundles Mön (n being any positive integer) can be obtained from the trivial bundle Mö0 by cutting it along a fiber, twisting n-times and then glueing again together. The bundle Mö1 is called the Moebius band (or strip). All bundles Mön are canonically fibered on S1, but just Mö0 is trivial. Differences among such bundles are global properties, which for example imply that the even ones Mö2k allow never-vanishing sections (i.e. field configurations) while the odd ones Mö2k+1 do not.

Causal Isomorphism as a Diffeomorphism. Some further Rumination on Philosophy of Science. Thought of the Day 82.0

Let (M, gab) and (M′, g′ab) be (temporally oriented) relativistic spacetimes that are both future- and past-distinguishing, and let φ : M → M′ be a ≪-causal isomorphism. Then φ is a diffeomorphism and preserves gab up to a conformal factor; i.e. φ⋆(g′ab) is conformally equivalent to gab.

Under the stated assumptions, φ must be a homeomorphism. If a spacetime (M, gab) is not just past and future distinguishing, but strongly causal, then one can explicitly characterize its (manifold) topology in terms of the relation ≪. In this case, a subset O ⊆ M is open iff, ∀ points p in O, ∃ points q and r in O such that q ≪ p ≪ r and I+(q) ∩ I(r) ⊆ O (Hawking and Ellis). So a ≪-causal isomorphism between two strongly causal spacetimes must certainly be a homeomorphism. Then one invokes a result of Hawking, King, and McCarthy that asserts, in effect, that any continuous ≪-causal isomorphism must be smooth and must preserve the metric up to a conformal factor.

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The following example shows that the proposition fails if the initial restriction on causal structure is weakened to past distinguishability or to future distinguishability alone. We give the example in a two-dimensional version to simplify matters. Start with the manifold R2 together with the Lorentzian metric

gab = (d(at)(db)x) − (sinh2t)(dax)(dbx)

where t, x are global projection coordinates on R2. Next, form a vertical cylinder by identifying the point with coordinates (t, x) with the one having coordinates (t, x + 2). Finally, excise two closed half lines – the sets with respective coordinates {(t, x): x = 0 and t ≥ 0} and {(t, x): x = 1 and t ≥ 0}. Figure shows, roughly, what the null cones look like at every point. (The future direction at each point is taken to be the “upward one.”) The exact form of the metric is not important here. All that is important is the indicated qualitative behavior of the null cones. Along the (punctured) circle C where t = 0, the vector fields (∂/∂t)a and (∂/∂x)a both qualify as null. But as one moves upward or downward from there, the cones close. There are no closed timelike (or null) curves in this spacetime. Indeed, it is future distinguishing because of the excisions. But it fails to be past distinguishing because I(p) = I(q) for all points p and q on C. For all points p there, I(p) is the entire region below C. Now let φ be the bijection of the spacetime onto itself that leaves the “lower open half” fixed but reverses the position of the two upper slabs. Though φ is discontinuous along C, it is a ≪-causal isomorphism. This is the case because every point below C has all points in both upper slabs in its ≪-future.

Causal Isomorphism as Homeomorphism, or Diffeomorphism or a Conformal Isometry? Drunken Risibility.

Let (M, gab) and (M′, g′ab) be (temporally oriented) relativistic spacetimes.

We say that a bijection φ : M → M′ between their underlying point sets is a ≪-causal isomorphism if, ∀ p and q in M,

p ≪ q ⇐⇒ φ(p) ≪ φ(q).

Then we can ask the following: Does a ≪-causal isomorphism have to be a homeomorphism? A diffeomorphism? A conformal isometry? (We know in advance that a causal isomorphism need not be a (full) isometry because conformally equivalent metrics gab and Ω2gab on a manifold M determine the same relation ≪. The best one can ask for is that it be a conformal isometry – i.e. that it be a diffeomorphism that preserves the metric up to a conformal factor.) Without further restrictions on (M, gab) and (M′, g′ab), the answer is certainly “no” to all three questions. Unless the “causal structure” of a spacetime (i.e., the structure determined by ≪) is reasonably well behaved, it provides no useful information at all. For example, let us say that a spacetime is causally degenerate if p ≪ q for all points p and q. Any bijection between two causally degenerate spacetimes qualifies, trivially, as a ≪-causal isomorphism. But we can certainly find causally degenerate spacetimes whose underlying manifolds have different topologies. But a suitably “rolled-up” version of Minkowski spacetime is also causally degenerate, and the latter has the manifold structure S1 × R3.

There is a hierarchy of “causality conditions” that is relevant here. Hawking and Ellis impose, with varying degrees of stringency, the requirement that there exist no closed, or “almost closed,” timelike curves. Here are three.

Chronology: There do not exist smooth closed timelike curves. (Equivalently, for all p, it is not the case that p ≪ p.)

Future (respectively, past) distinguishablity: ∀ points p, and all sufficiently small open sets O containing p, no smooth future-directed (respectively, past-directed) timelike curve that starts at p, and leaves O, ever returns to O.

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Strong causality: For all points p, and all sufficiently small open sets O containing p, no smooth future-directed timelike curve that starts in O, and leaves O, ever returns to O.

It is clear that strong causality implies both future distinguishability and past distinguishability, and that each of the distinguishability conditions (alone) implies chronology.

The names “future distinguishability” and “past distinguishability” are easily explained. For any p, let I+(p) be the set {q: p ≪ q} and let I(p) be the set {q : q ≪ p}. It turns out that future distinguishability is equivalent to the requirement that, ∀ p and q,

I+(p) = I+(q) =⇒ p = q.

And the counterpart requirement with I+ replaced by I is equivalent to past distinguishability.

Unique Derivative Operator: Reparametrization. Metric Part 2.

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Moving on from first part.

Suppose ∇ is a derivative operator, and gab is a metric, on the manifold M. Then ∇ is compatible with gab iff ∇a gbc = 0.

Suppose γ is an arbitrary smooth curve with tangent field ξa and λa is an arbitrary smooth field on γ satisfying ξnnλa = 0. Then

ξnn(gabλaλb) = gabλaξnnλb + gabλbξnnλa + λaλbξnngab

= λaλbξnngab

Suppose first that ∇ngab = 0. Then it follows immediately that ξnngabλaλb = 0. So ∇ is compatible with gab. Suppose next that ∇ is compatible with gab. Then ∀ choices of γ and λa (satisfying ξnnλa =0), we have λaλbξnngab = 0. Since the choice of λa (at any particular point) is arbitrary and gab is symmetric, it follows that ξnngab = 0. But this must be true for arbitrary ξa (at any particular point), and so we have ∇ngab = 0.

Note that the condition of compatibility is also equivalent to ∇agbc = 0. Hence,

0 = gbnaδcn = gbna(gnrgrc) = gbngnragrc + gbngrcagnr

= δbragrc + gbngrcagnr = ∇agbc + gbngrcagnr.

So if ∇agbc = 0,it follows immediately that ∇agbc = 0. Conversely, if ∇agbc =0, then gbngrcagnr = 0. And therefore,

0 = gpbgscgbngrcagnr = δnpδrsagnr = ∇agps

The basic fact about compatible derivative operators is the following.

Suppose gab is a metric on the manifold M. Then there is a unique derivative operator on M that is compatible with gab.

It turns out that if a manifold admits a metric, then it necessarily satisfies the countable cover condition. And then it guarantees the existence of a derivative operator.) We do prove that if M admits a derivative operator ∇, then it admits exactly one ∇′ that is compatible with gab.

Every derivative operator ∇′ on M can be realized as ∇′ = (∇, Cabc), where Cabc is a smooth, symmetric field on M. Now

∇′agbc = ∇agbc + gnc Cnab + gbn Cnac = ∇agbc + Ccab + Cbac. So ∇′ will be compatible with gab (i.e., ∇′agbc = 0) iff

agbc = −Ccab − Cbac —– (1)

Thus it suffices for us to prove that there exists a unique smooth, symmetric field Cabc on M satisfying equation (1). To do so, we write equation (1) twice more after permuting the indices:

cgab = −Cbca − Cacb,

bgac = −Ccba − Cabc

If we subtract these two from the first equation, and use the fact that Cabc is symmetric in (b, c), we get

Cabc = 1/2 (∇agbc − ∇bgac − ∇cgab) —– (2)

and, therefore,

Cabc = 1/2 gan (∇ngbc − ∇bgnc − ∇cgnb) —– (3)

This establishes uniqueness. But clearly the field Cabc defined by equation (3) is smooth, symmetric, and satisfies equation (1). So we have existence as well.

In the case of positive definite metrics, there is another way to capture the significance of compatibility of derivative operators with metrics. Suppose the metric gab on M is positive definite and γ : [s1, s2] → M is a smooth curve on M. We associate with γ a length

|γ| = ∫s1s2 gabξaξb ds,

where ξa is the tangent field to γ. This assigned length is invariant under reparametrization. For suppose σ : [t1, t2] → [s1, s2] is a diffeomorphism we shall write s = σ(t) and ξ′a is the tangent field of γ′ = γ ◦ σ : [t1, t2] → M. Then

ξ′a = ξads/dt

We may as well require that the reparametrization preserve the orientation of the original curve – i.e., require that σ (t1) = s1 and σ (t2) = s2. In this case, ds/dt > 0 everywhere. (Only small changes are needed if we allow the reparametrization to reverse the orientation of the curve. In that case, ds/dt < 0 everywhere.) It

follows that

|γ’| = ∫t1t2 (gabξ′aξ′b)1/2 dt = ∫t1t2 (gabξaξb)1/2 ds/dt

= ∫s1s2 (gabξaξb)1/2 ds = |γ|

Let us say that γ : I → M is a curve from p to q if I is of the form [s1, s2], p = γ(s1), and q = γ(s2). In this (positive definite) case, we take the distance from p to q to be

d(p,q)=g.l.b. |γ|:γ is a smooth curve from p to q.

Further, we say that a curve γ : I → M is minimal if, for all s ∈ I, ∃ an ε > 0 such that, for all s1, s2 ∈ I with s1 ≤ s ≤ s2, if s2 − s1 < ε and if γ′ = γ|[s1, s2] (the restriction of γ to [s1, s2]), then |γ′| = d(γ(s1), γ(s2)) . Intuitively, minimal curves are “locally shortest curves.” Certainly they need not be “shortest curves” outright. (Consider, for example, two points on the “equator” of a two-sphere that are not antipodal to one another. An equatorial curve running from one to the other the “long way” qualifies as a minimal curve.)

One can characterize the unique derivative operator compatible with a positive definite metric gab in terms of the latter’s associated minimal curves. But in doing so, one has to pay attention to parametrization.

Let us say that a smooth curve γ : I → M with tangent field ξa is parametrized by arc length if ∀ ξa, gabξaξb = 1. In this case, if I = [s1, s2], then

|γ| = ∫s1s2 (gabξaξb)1/2 ds = ∫s1s2 1.ds = s2 – s1

Any non-trivial smooth curve can always be reparametrized by arc length.