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.

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Philosophical Identity of Derived Correspondences Between Smooth Varieties.

Let there be a morphism f : X → Y between varieties. Then all the information about f is encoded in the graph Γ_f ⊂ X × Y of f, which (as a set) is defined as

Γ_f = {(x, f(x)) : x ∈ X} ⊂ X × Y —– (1)

Now consider the natural projections p_X, p_Y from X × Y to the factors X, Y. Restricted to the subvariety Γ_f, p_X is an isomorphism (since f is a morphism). The fibres of p_Y restricted to Γ_f are just the fibres of f; so for example f is proper iff p_Y | Γ_f is.

If H(−) is any reasonable covariant homology theory (say singular homology in the complex topology for X, Y compact), then we have a natural push forward map

f_∗ : H(X) → H(Y)

This map can be expressed in terms of the graph Γ_f and the projection maps as

f_∗(α) = p_Y∗ (p_X^∗(α) ∪ [Γ_f]) —– (2)

where [Γ_f] ∈ H (X × Y) is the fundamental class of the subvariety [Γ_f]. Generalizing this construction gives us the notion of a “multi-valued function” or correspondence from X to Y, simply defined to be a general subvariety Γ ⊂ X × Y, replacing the assumption that p_X be an isomorphism with some weaker assumption, such as p_X |Γ_f, p_Y | Γ_f finite or proper. The right hand side of (2) defines a generalized pushforward map

Γ_∗ : H(X) → H(Y)

A subvariety Γ ⊂ X × Y can be represented by its structure sheaf O_Γ on X × Y. Associated to the projection maps p_X, p_Y, we also have pullback and pushforward operations on sheaves. The cup product on homology turns out to have an analogue too, namely tensor product. So, appropriately interpreted, (2) makes sense as an operation from the derived category of X to that of Y.

A derived correspondence between a pair of smooth varieties X, Y is an object F ∈ D^b(X × Y) with support which is proper over both factors. A derived correspondence defines a functor Φ_F by

Φ_F : D^b(X) → D^b(Y)
(−) ↦ Rp_Y∗(Lp_X^∗(−) ⊗^L F)

where (−) could refer to both objects and morphisms in D^b(X). F is sometimes called the kernel of the functor Φ_F.

The functor Φ_F is exact, as it is defined as a composite of exact functors. Since the projection p_X is flat, the derived pullback Lp_X^∗ is the same as ordinary pullback p_X^∗. Given derived correspondences E ∈ D^b(X × Y), F ∈ D^b(Y × Z), we obtain functors Φ_E : D^b(X) → D^b(Y), Φ_F : D^b(Y) → D^b(Z), which can then be composed to get a functor

Φ_F ◦ Φ_E : D^b(X) → D^b(Z)

which is a two-sided identity with respect to composition of kernels.

Philosophical Equivariance – Sewing Holonomies Towards Equal Trace Endomorphisms.

In d-dimensional topological field theory one begins with a category S whose objects are oriented (d − 1)-manifolds and whose morphisms are oriented cobordisms. Physicists say that a theory admits a group G as a global symmetry group if G acts on the vector space associated to each (d−1)-manifold, and the linear operator associated to each cobordism is a G-equivariant map. When we have such a “global” symmetry group G we can ask whether the symmetry can be “gauged”, i.e., whether elements of G can be applied “independently” – in some sense – at each point of space-time. Mathematically the process of “gauging” has a very elegant description: it amounts to extending the field theory functor from the category S to the category S_G whose objects are (d − 1)-manifolds equipped with a principal G-bundle, and whose morphisms are cobordisms with a G-bundle. We regard S as a subcategory of S_G by equipping each (d − 1)-manifold S with the trivial G-bundle S × G. In S_G the group of automorphisms of the trivial bundle S × G contains G, and so in a gauged theory G acts on the state space H(S): this should be the original “global” action of G. But the gauged theory has a state space H(S,P) for each G-bundle P on S: if P is non-trivial one calls H(S,P) a “twisted sector” of the theory. In the case d = 2, when S = S¹ we have the bundle P_g → S¹ obtained by attaching the ends of [0,2π] × G via multiplication by g. Any bundle is isomorphic to one of these, and P_g is isomorphic to P_g^‘ iff g′ is conjugate to g. But note that the state space depends on the bundle and not just its isomorphism class, so we have a twisted sector state space C_g = H(S,P_g) labelled by a group element g rather than by a conjugacy class.

We shall call a theory defined on the category S_G a G-equivariant Topological Field Theory (TFT). It is important to distinguish the equivariant theory from the corresponding “gauged theory”. In physics, the equivariant theory is obtained by coupling to nondynamical background gauge fields, while the gauged theory is obtained by “summing” over those gauge fields in the path integral.

An alternative and equivalent viewpoint which is especially useful in the two-dimensional case is that S_G is the category whose objects are oriented (d − 1)-manifolds S equipped with a map p : S → BG, where BG is the classifying space of G. In this viewpoint we have a bundle over the space Map(S,BG) whose fibre at p is H_p. To say that H_p depends only on the G-bundle p^∗EG on S pulled back from the universal G-bundle EG on BG by p is the same as to say that the bundle on Map(S,BG) is equipped with a flat connection allowing us to identify the fibres at points in the same connected component by parallel transport; for the set of bundle isomorphisms p^∗₀EG → p^∗₁EG is the same as the set of homotopy classes of paths from p₀ to p₁. When S = S¹ the connected components of the space of maps correspond to the conjugacy classes in G: each bundle P_g corresponds to a specific point p_g in the mapping space, and a group element h defines a specific path from p_g to p_hgh⁻¹ .

G-equivariant topological field theories are examples of “homotopy topological field theories”. Using Vladimir Turaev‘s two main results: first, an attractive generalization of the theorem that a two-dimensional TFT “is” a commutative Frobenius algebra, and, secondly, a classification of the ways of gauging a given global G-symmetry of a semisimple TFT.

Definition of the product in the G-equivariant closed theory. The heavy dot is the basepoint on S¹. To specify the morphism unambiguously we must indicate consistent holonomies along a set of curves whose complement consists of simply connected pieces. These holonomies are always along paths between points where by definition the fibre is G. This means that the product is not commutative. We need to fix a convention for holonomies of a composition of curves, i.e., whether we are using left or right path-ordering. We will take h(γ₁ ◦ γ₂) = h(γ₁) · h(γ₂).

A G-equivariant TFT gives us for each element g ∈ G a vector space C_g, associated to the circle equipped with the bundle p_g whose holonomy is g. The usual pair-of-pants cobordism, equipped with the evident G-bundle which restricts to p_g1 and p_g2 on the two incoming circles, and to p_g1g2 on the outgoing circle, induces a product

C_g1 ⊗ C_g2 → C_g1g2 —– (1)

making C := ⊕_g∈GC_g into a G-graded algebra. Also there is a trace θ: C1  → C defined by the disk diagram with one ingoing circle. The holonomy around the boundary of the disk must be 1. Making the standard assumption that the cylinder corresponds to the unit operator we obtain a non-degenerate pairing

C_g ⊗ C_g⁻¹ → C

A new element in the equivariant theory is that G acts as an automorphism group on C. That is, there is a homomorphism α : G → Aut(C) such that

α_h : C_g → C_hgh⁻¹ —– (2)

Diagramatically, α_h is defined by the surface in the immediately above figure. Now let us note some properties of α. First, if φ ∈ C_h then α_h(φ) = φ. The reason for this is diagrammatically in the below figure.

If the holonomy along path P₂ is h then the holonomy along path P₁ is 1. However, a Dehn twist around the inner circle maps P₁ into P₂. Therefore, α_h(φ) = α₁(φ) = φ, if φ ∈ C_h.

Next, while C is not commutative, it is “twisted-commutative” in the following sense. If φ₁ ∈ C_g1 and φ₂ ∈ C_g2 then

α_g2 (φ₁)φ₂ = φ₂φ₁ —– (3)

The necessity of this condition is illustrated in the figure below.

The trace of the identity map of C_g is the partition function of the theory on a torus with the bundle with holonomy (g,1). Cutting the torus the other way, we see that this is the trace of α_g on C₁. Similarly, by considering the torus with a bundle with holonomy (g,h), where g and h are two commuting elements of G, we see that the trace of α_g on C_h is the trace of α_h on C_g⁻¹. But we need a strengthening of this property. Even when g and h do not commute we can form a bundle with holonomy (g,h) on a torus with one hole, around which the holonomy will be c = hgh⁻¹g⁻¹. We can cut this torus along either of its generating circles to get a cobordism operator from C_c ⊗ C_h to C_h or from C_g⁻¹ ⊗ C_c to C_g⁻¹. If ψ ∈ C_{hgh⁻¹g⁻¹}. Let us introduce two linear transformations L_ψ, R_ψ associated to left- and right-multiplication by ψ. On the one hand, L_ψα_g : φ􏰀 ↦ ψ_αg(φ) is a map C_h → C_h. On the other hand R_ψα_h : φ ↦ α_h(φ)ψ is a map C_g⁻¹ → C_g⁻¹. The last sewing condition states that these two endomorphisms must have equal traces:

TrC_h 􏰌L_ψα_g􏰍 = TrC_g⁻¹ 􏰌R_ψα_h􏰍 —– (4)

(4) was taken by Turaev as one of his axioms. It can, however, be reexpressed in a way that we shall find more convenient. Let ∆_g ∈ C_g ⊗ C_g⁻¹ be the “duality” element corresponding to the identity cobordism of (S¹,P_g) with both ends regarded as outgoing. We have ∆_g = ∑ξ_i ⊗ ξⁱ, where ξ_i and ξⁱ ru􏰟n through dual bases of C_g and C_g⁻¹. Let us also write

∆_h = ∑η_i ⊗ ηⁱ ∈ C_h ⊗ C_h⁻¹. Then (4) is easily seen to be equivalent to

∑α_h(ξ_i)ξⁱ = 􏰟 ∑η_iα_g(ηⁱ) —– (5)

in which both sides are elements of C_{hgh⁻¹g⁻¹}.

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

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: S¹_ns, with the Möbius or “Neveu-Schwarz” structure, and S¹_r, with the trivial or “Ramond” structure. A topological-spin theory gives us state spaces C_ns and C_r, corresponding respectively to S¹_ns and S¹_r.

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 = S¹_ns,r × [0,1] which induce the identity maps of C_ns,r there are also cylinders A⁻_ns,r which connect S¹_ns,r to itself while interchanging the sheets. These cylinders A⁻_ns,r induce the grading automorphism on the state spaces. But because A⁻_ns ≅ 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 C_ns must be purely even. The space C_r 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 S¹_ns there is only one possibility, but if the boundaries are S¹_r 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 S¹_r ⊔ S¹_r, and use P to define a pairing C_r ⊗ C_r → C. We then choose a preferred cobordism Q in the other direction so that when we sew its right-hand outgoing S¹_r 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 C_r.

There is a unique spin structure on the pair-of-pants cobordism in the figure below, which restricts to S¹_ns on each boundary circle, and it makes C_ns into a commutative Frobenius algebra in the usual way.

If one incoming circle is S¹_ns and the other is S¹_r then the outgoing circle is S¹_r, and there are two possible spin structures, but the one obtained by removing a disc from the cylinder A⁺_r is preferred: it makes C_r into a graded module over C_ns. The chosen U-shaped cobordism P, with two incoming circles S¹_r, can be punctured to give us a pair of pants with an outgoing S¹_ns, and it induces a graded bilinear map C_r × C_r → C_ns which, composing with the trace on C_ns, gives a non-degenerate inner product on C_r. 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 : C_r ⊗ C_r → C_r ⊗ C_r 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 C_r is graded-symmetric with the usual sign; but if S is the graded symmetry then we see that the product on C_r 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 = (C_ns ⊕ C_r,θ_C) with the following property. Let {φ_k} be a basis for C_ns, with dual basis {φ^k} such that θ_C(φ_kφ^m) = δ^m_k, and let β_k and β^k be similar dual bases for C_r. Then the Euler elements χ_ns := ∑ φ_kφ^k and χ_r = ∑ β_kβ^k are independent of the choices of bases, and the condition we need on the algebra C is that χ_ns = χ_r. In particular, this condition implies that the vector spaces C_ns and C_r 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 C_ns. The Euler elements for the three even spin structures are equal to χ_e = χ_ns = χ_r. The Euler element χ_o corresponding to the odd spin structure, on the other hand, is given by χ_o = ∑(−1)^degβ_kβ_kβ^k.

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 C_ns is semisimple, i.e., is the algebra of functions on a finite set X. Then C_r is the space of sections of a vector bundle E on X, and it follows from the condition χ_ns = χ_r that the fibre at each point must have dimension 1. Thus the whole structure is determined by the Frobenius algebra C_ns together with a binary choice at each point x ∈ X of the grading of the fibre E_x 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 C_r 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 C_ns and C_r 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 ⊕ C_η where η² = 1 and η is odd. The Euler elements are χ_e = 1 and χ_o = −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 C_ns 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 H_S,L to the 1-manifold S with the spin bundle L, and is equipped with an additional map

Γ(S,L) ⊗ H_S,L → H_S,L

(σ,ψ) ↦ G_σψ,

where Γ(S,L) is the space of smooth sections of L, such that G_σ is real-linear in the section σ, and satisfies G²_σ = D_σ², where D_σ² 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 σ_i on S_i we have the intertwining property

G_σ1 ◦ U_Σ,L = U_Σ,L ◦ G_σ0

….

Symplectic Manifolds

The canonical example of the n-symplectic manifold is that of the frame bundle, so the question is whether this formalism can be generalized to other principal bundles, and distinguished from the quantization arising from symplectic geometry on the prototype manifold, the bundle of linear frames, a good place to motivate the formalism.

Let us start with an n-dimensional manifold M, and let π : LM → M be the space of linear frames over a base manifold M, the set of pairs (m,e_k), where m ∈ M and {e_k},k = 1,···,n is a linear frame at m. This gives LM dimension n(n + 1), with GL(n,R) as the structure group acting freely on the right. We define local coordinates on LM in terms of those on the manifold M – for a chart on M with coordinates {xⁱ}, let

qⁱ(m,e_k) = xⁱ ◦ π(m,e_k) = xⁱ(m)

π_jⁱ(m,e_k) = e^j ∂/∂xj

where {e^j} denotes the coframe dual to {e_j}. These coordinates are analogous to those on the cotangent bundle, except, instead of a single momentum coordinate, we now have a momentum frame. We want to place some kind of structure on LM, which is the prototype of n-symplectic geometry that is similar to symplectic geometry of the cotangent bundle T∗M. The structure equation for symplectic geometry

df= _| X dθ

gives Hamilton’s equations for the phase space of a particle, where θ is the canonical symplectic 2-form. There is a naturally defined Rⁿ-valued 1-form on LM, the soldering form, given by

θ(X) ≡ u⁻¹[π∗(X)] ∀X ∈ T_uLM

where the point u = (m,e_k) ∈ LM gives the isomorphism u : Rⁿ → T_π(u)M by ξⁱr_i → ξⁱe_i, where {r_i} is the standard basis of Rⁿ. The Rⁿ-valued 2-form dθ can be shown to be non-degenerate, that is,

X _| dθ = 0 ⇔ X = 0

where we mean that each component of X dθ is identically zero. Finally, since there is also a structure group on LM, there are also group transformation properties. Let ρ be the standard representation of GL(n, R) on Rn. Then it can be shown that the pullback of dθ under right translation by g ∈ GL (n,R) is R_g^∗ dθ = ρ(g−1) · dθ.

Thus, we have an Rⁿ-valued generalization of symplectic geometry, which motivates the following definition.

Let P be a principal fiber bundle with structure group G over an m-dimensional manifold M . Let ρ : G → GL(n, R) be a linear representation of G. An n-symplectic structure on P is a Rⁿ-valued 2-form ω on P that is (i) closed and non-degenerate, in the sense that

X _| ω = 0 ⇔ X = 0

for a vector field X on P, and (ii) ω is equivariant, such that under the right action of G, R_g^∗ ω = ρ(g−1) · ω. The pair (P, ω) is called an n-symplectic manifold.

Here, we have modeled n-symplectic geometry after the frame bundle by defining the general n-symplectic manifold as a principal bundle. There is no reason, however, to limit ourselves to this, since we can let P be any manifold with a group action defined on it. One example of this would be to look at the action of the conformal group on R⁴. Since this group is locally isomorphic to O(2, 4), which is not a subgroup of GL(4, R), then forming a O(2,4) bundle over R⁴ cannot be thought of as simply a reduction of the frame bundle.