# Conformal Field Theory and Virasoro Algebra. Note Quote.

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

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

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

Define a category B whose objects are the oriented submanifolds of X, and whose vector space of morphisms from Y to Z is OYZ = ExtH(X)(H(Y), H(Z)) – the cohomology, as usual, has complex coefficients, and H(Y) and H(Z) are regarded as H(X)-modules by restriction. The composition of morphisms is given by the Yoneda composition of Ext groups. With this definition, however, it will not be true that OYZ is dual to OZY. (To see this it is enough to consider the case when Y = Z is a point of X, and X is a product of odd-dimensional spheres; then OYZ is a symmetric algebra, and is not self-dual as a vector space.)

We can do better by defining a cochain complex O’YZ of morphisms by

O’YZ = BΩ(X)(Ω(Y), Ω(Z)) —– (1)

where Ω(X) denotes the usual de Rham complex of a manifold X, and BA(B,C), for a differential graded algebra A and differential graded A- modules B and C, is the usual cobar resolution

Hom(B, C) → Hom(A ⊗ B, C) → Hom(A ⊗ A ⊗ B, C) → · · ·  —– (2)

in which the differential is given by

dƒ(a1 ⊗ · · · ⊗ ak ⊗ b) = 􏰝a1 ƒ(a2 ⊗ · · · ⊗ ak ⊗ b) + ∑(-1)i ƒ(a1 ⊗ · · · ⊗ aiai+1 ⊗ ak ⊗ b) + (-1)k ƒ(a1 ⊗ · · · ⊗ ak-1 ⊗ akb) —– (3)

whose cohomology is ExtA(B,C). This is different from OYZ = ExtH(X)(H(Y), H(Z)), but related to it by a spectral sequence whose E2-term is OYZ and which converges to H(O’YZ) = ExtΩ(X)(Ω(Y), Ω(Z)). But more important is that H(O’YZ) is the homology of the space PYZ of paths in X which begin in Y and end in Z. To be precise, Hp(O’YZ) ≅ Hp+dZ(PYZ), where dZ is the dimension of Z. On the cochain complexes the Yoneda composition is associative up to cochain homotopy, and defines a structure of an A category B’. The corresponding composition of homology groups

Hi(PYZ) × Hj(PZW) → Hi+j−dZ(PYW) —— (4)

is the composition of the Gysin map associated to the inclusion of the codimension dZ submanifold M of pairs of composable paths in the product PYZ × PZW with the concatenation map M → PYW.

Now let’s attempt to fit the closed string cochain algebra C to this A category. C is equivalent to the usual Hochschild complex of the differential graded algebra Ω(X), whose cohomology is the homology of the free loop space LX with its degrees shifted downwards by the dimension dX of X, so that the cohomology Hi(C) is potentially non-zero for −dX ≤ i < ∞. There is a map Hi(X) → H−i(C) which embeds the ordinary cohomology ring of X to the Pontrjagin ring of the based loop space L0X, based at any chosen point in X.

The structure is, however, not a cochain-level open and closed theory, as we have no trace maps inducing inner products on H(O’YZ). When one tries to define operators corresponding to cobordisms it turns out to be possible only when each connected component of the cobordism has non-empty outgoing boundary.

# 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 Γf ⊂ X × Y of f, which (as a set) is defined as

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

Now consider the natural projections pX, pY from X × Y to the factors X, Y. Restricted to the subvariety Γf, pX is an isomorphism (since f is a morphism). The fibres of pY restricted to Γf are just the fibres of f; so for example f is proper iff pY | Γf 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 Γf and the projection maps as

f(α) = pY∗ (pX(α) ∪ [Γf]) —– (2)

where [Γf] ∈ H (X × Y) is the fundamental class of the subvariety [Γf]. 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 pX be an isomorphism with some weaker assumption, such as pXf, pY | Γf 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 OΓ on X × Y. Associated to the projection maps pX, pY, 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 ∈ Db(X × Y) with support which is proper over both factors. A derived correspondence defines a functor ΦF by

ΦF : Db(X) → Db(Y)
(−) ↦ RpY∗(LpX(−) ⊗L F)

where (−) could refer to both objects and morphisms in Db(X). F is sometimes called the kernel of the functor ΦF.

The functor ΦF is exact, as it is defined as a composite of exact functors. Since the projection pX is flat, the derived pullback LpX is the same as ordinary pullback pX. Given derived correspondences E ∈ Db(X × Y), F ∈ Db(Y × Z), we obtain functors Φ: Db(X) → Db(Y), Φ: Db(Y) → Db(Z), which can then be composed to get a functor

ΦF ◦ Φ: Db(X) → Db(Z)

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

# Indecomposable Objects – Part 1

An object X in a category C with an initial object is called indecomposable if X is not the initial object and X is not isomorphic to a coproduct of two noninitial objects. A group G is called indecomposable if it cannot be expressed as the internal direct product of two proper normal subgroups of G. This is equivalent to saying that G is not isomorphic to the direct product of two nontrivial groups.

A quiver Q is a directed graph, specified by a set of vertices Q0, a set of arrows Q1, and head and tail maps

h, t : Q1 → Q0

We always assume that Q is finite, i.e., the sets Q0 and Q1 are finite.

A (complex) representation of a quiver Q consists of complex vector spaces Vi for i ∈ Qand linear maps

φa : Vt(a) → Vh(a)

for a ∈ Q1. A morphism between such representations (V, φ) and (W, ψ) is a collection of linear maps fi : Vi → Wi for i ∈ Q0 such that the diagram

commutes ∀ a ∈ Q1. A representation of Q is finite-dimensional if each vector space Vi is. The dimension vector of such a representation is just the tuple of non-negative integers (dim Vi)i∈Q0.

Rep(Q) is the category of finite-dimensional representations of Q. This category is additive; we can add morphisms by adding the corresponding linear maps fi, the trivial representation in which each Vi = 0 is a zero object, and the direct sum of two representations is obtained by taking the direct sums of the vector spaces associated to each vertex. If Q is the one-arrow quiver, • → •, then the classification of indecomposable objects of Rep(Q), yields the objects E ∈ Rep(Q) which do not have a non-trivial direct sum decomposition E = A ⊕ B. An object of Rep(Q) is just a linear map of finite-dimensional vector spaces f: V1 → V2. If W = im(f) is a nonzero proper subspace of V2, then the splitting V2 = U ⊕ W, and the corresponding object of Rep(Q) splits as a direct sum of the two representations

V1 →ƒ W and 0 → W

Thus if an object f: V1 → V2 of Rep(Q) is indecomposable, the map f must be surjective. Similarly, if ƒ is nonzero, then it must also be injective. Continuing in this way, one sees that Rep(Q) has exactly three indecomposable objects up to isomorphism:

C → 0, 0 → C, C →id C

Every other object of Rep(Q) is a direct sum of copies of these basic representations.

# Game Theory and Finite Strategies: Nash Equilibrium Takes Quantum Computations to Optimality.

Finite games of strategy, within the framework of noncooperative quantum game theory, can be approached from finite chain categories, where, by finite chain category, it is understood a category C(n;N) that is generated by n objects and N morphic chains, called primitive chains, linking the objects in a specific order, such that there is a single labelling. C(n;N) is, thus, generated by N primitive chains of the form:

x0 →f1 x1 →f2 x1 → … xn-1 →fn xn —– (1)

A finite chain category is interpreted as a finite game category as follows: to each morphism in a chain xi-1 →fi xi, there corresponds a strategy played by a player that occupies the position i, in this way, a chain corresponds to a sequence of strategic choices available to the players. A quantum formal theory, for a finite game category C(n;N), is defined as a formal structure such that each morphic fundament fi of the morphic relation xi-1 →fi xis a tuple of the form:

fi := (Hi, Pi, Pˆfi) —– (2)

where Hi is the i-th player’s Hilbert space, Pi is a complete set of projectors onto a basis that spans the Hilbert space, and Pˆfi ∈ Pi. This structure is interpreted as follows: from the strategic Hilbert space Hi, given the pure strategies’ projectors Pi, the player chooses to play Pˆfi .

From the morphic fundament (2), an assumption has to be made on composition in the finite category, we assume the following tensor product composition operation:

fj ◦ fi = fji —– (3)

fji = (Hji = Hj ⊗ Hi, Pji = Pj ⊗ Pi, Pˆfji = Pˆfj ⊗ Pˆfi) —– (4)

From here, a morphism for a game choice path could be introduced as:

x0 →fn…21 xn —– (5)

fn…21 = (HG = ⊗i=n1 Hi, PG = ⊗i=n1 Pi, Pˆ fn…21 = ⊗i=n1fi) —– (6)

in this way, the choices along the chain of players are completely encoded in the tensor product projectors Pˆfn…21. There are N = ∏i=1n dim(Hi) such morphisms, a number that coincides with the number of primitive chains in the category C(n;N).

Each projector can be addressed as a strategic marker of a game path, and leads to the matrix form of an Arrow-Debreu security, therefore, we call it game Arrow-Debreu projector. While, in traditional financial economics, the Arrow-Debreu securities pay one unit of numeraire per state of nature, in the present game setting, they pay one unit of payoff per game path at the beginning of the game, however this analogy may be taken it must be addressed with some care, since these are not securities, rather, they represent, projectively, strategic choice chains in the game, so that the price of a projector Pˆfn…21 (the Arrow-Debreu price) is the price of a strategic choice and, therefore, the result of the strategic evaluation of the game by the different players.

Now, let |Ψ⟩ be a ket vector in the game’s Hilbert space HG, such that:

|Ψ⟩ = ∑fn…21 ψ(fn…21)|(fn…21⟩ —– (7)

where ψ(fn…21) is the Arrow-Debreu price amplitude, with the condition:

fn…21 |ψ(fn…21)|2 = D —– (8)

for D > 0, then, the |ψ(fn…21)|corresponds to the Arrow-Debreu prices for the game path fn…21 and D is the discount factor in riskless borrowing, defining an economic scale for temporal connections between one unit of payoff now and one unit of payoff at the end of the game, such that one unit of payoff now can be capitalized to the end of the game (when the decision takes place) through a multiplication by 1/D, while one unit of payoff at the end of the game can be discounted to the beginning of the game through multiplication by D.

In this case, the unit operator, 1ˆ = ∑fn…21 Pˆfn…21 has a similar profile as that of a bond in standard financial economics, with ⟨Ψ|1ˆ|Ψ⟩ = D, on the other hand, the general payoff system, for each player, can be addressed from an operator expansion:

πiˆ = ∑fn…21 πi (fn…21) Pˆfn…21 —– (9)

where each weight πi(fn…21) corresponds to quantities associated with each Arrow-Debreu projector that can be interpreted as similar to the quantities of each Arrow-Debreu security for a general asset. Multiplying each weight by the corresponding Arrow-Debreu price, one obtains the payoff value for each alternative such that the total payoff for the player at the end of the game is given by:

⟨Ψ|1ˆ|Ψ⟩ = ∑fn…21 πi(fn…21) |ψ(fn…21)|2/D —– (10)

We can discount the total payoff to the beginning of the game using the discount factor D, leading to the present value payoff for the player:

PVi = D ⟨Ψ|πiˆ|Ψ⟩ = D ∑fn…21 π (fn…21) |ψ(fn…21)|2/D —– (11)

, where π (fn…21) represents quantities, while the ratio |ψ(fn…21)|2/D represents the future value at the decision moment of the quantum Arrow- Debreu prices (capitalized quantum Arrow-Debreu prices). Introducing the ket

|Q⟩ ∈ HG, such that:

|Q⟩ = 1/√D |Ψ⟩ —– (12)

then, |Q⟩ is a normalized ket for which the price amplitudes are expressed in terms of their future value. Replacing in (11), we have:

PVi = D ⟨Q|πˆi|Q⟩ —– (13)

In the quantum game setting, the capitalized Arrow-Debreu price amplitudes ⟨fn…21|Q⟩ become quantum strategic configurations, resulting from the strategic cognition of the players with respect to the game. Given |Q⟩, each player’s strategic valuation of each pure strategy can be obtained by introducing the projector chains:

Cˆfi = ∑fn…i+1fi-1…1 Pˆfn…i+1 ⊗ Pˆfi ⊗ Pˆfi-1…1 —– (14)

with ∑fi Cˆfi = 1ˆ. For each alternative choice of the player i, the chain sums over all of the other choice paths for the rest of the players, such chains are called coarse-grained chains in the decoherent histories approach to quantum mechanics. Following this approach, one may introduce the pricing functional from the expression for the decoherence functional:

D (fi, gi : |Q⟩) = ⟨Q| Cˆfi Cgi|Q⟩  —– (15)

we, then have, for each player

D (fi, gi : |Q⟩) = 0, ∀ fi ≠ gi —– (16)

this is the usual quantum mechanics’ condition for an aditivity of measure (also known as decoherence condition), which means that the capitalized prices for two different alternative choices of player i are additive. Then, we can work with the pricing functional D(fi, fi :|Q⟩) as giving, for each player an Arrow-Debreu capitalized price associated with the pure strategy, expressed by fi. Given that (16) is satisfied, each player’s quantum Arrow-Debreu pricing matrix, defined analogously to the decoherence matrix from the decoherent histories approach, is a diagonal matrix and can be expanded as a linear combination of the projectors for each player’s pure strategies as follows:

Di (|Q⟩) = ∑fi D(fi, f: |Q⟩) Pˆfi —– (17)

which has the mathematical expression of a mixed strategy. Thus, each player chooses from all of the possible quantum computations, the one that maximizes the present value payoff function with all the other strategies held fixed, which is in agreement with Nash.

# Fréchet Spaces and Presheave Morphisms.

A topological vector space V is both a topological space and a vector space such that the vector space operations are continuous. A topological vector space is locally convex if its topology admits a basis consisting of convex sets (a set A is convex if (1 – t) + ty ∈ A ∀ x, y ∈ A and t ∈ [0, 1].

We say that a locally convex topological vector space is a Fréchet space if its topology is induced by a translation-invariant metric d and the space is complete with respect to d, that is, all the Cauchy sequences are convergent.

A seminorm on a vector space V is a real-valued function p such that ∀ x, y ∈ V and scalars a we have:

(1) p(x + y) ≤ p(x) + p(y),

(2) p(ax) = |a|p(x),

(3) p(x) ≥ 0.

The difference between the norm and the seminorm comes from the last property: we do not ask that if x ≠ 0, then p(x) > 0, as we would do for a norm.

If {pi}{i∈N} is a countable family of seminorms on a topological vector space V, separating points, i.e. if x ≠ 0, there is an i with pi(x) ≠ 0, then ∃ a translation-invariant metric d inducing the topology, defined in terms of the {pi}:

d(x, y) = ∑i=1 1/2i pi(x – y)/(1 + pi(x – y))

The following characterizes Fréchet spaces, giving an effective method to construct them using seminorms.

A topological vector space V is a Fréchet space iff it satisfies the following three properties:

• it is complete as a topological vector space;
• it is a Hausdorff space;
• its topology is induced by a countable family of seminorms {pi}{i∈N}, i.e., U ⊂ V is open iff for every u ∈ U ∃ K ≥ 0 and ε > 0 such that {v|pk(u – v) < ε ∀ k ≤ K} ⊂ U.

We say that a sequence (xn) in V converges to x in the Fréchet space topology defined by a family of seminorms iff it converges to x with respect to each of the given seminorms. In other words, xn → x, iff pi(xn – x) → 0 for each i.

Two families of seminorms defined on the locally convex vector space V are said to be equivalent if they induce the same topology on V.

To construct a Fréchet space, one typically starts with a locally convex topological vector space V and defines a countable family of seminorms pk on V inducing its topology and such that:

1. if x ∈ V and pk(x) = 0 ∀ k ≥ 0, then x = 0 (separation property);
2. if (xn) is a sequence in V which is Cauchy with respect to each seminorm, then ∃ x ∈ V such that (xn) converges to x with respect to each seminorm (completeness property).

The topology induced by these seminorms turns V into a Fréchet space; property (1) ensures that it is Hausdorff, while the property (2) guarantees that it is complete. A translation-invariant complete metric inducing the topology on V can then be defined as above.

The most important example of Fréchet space, is the vector space C(U), the space of smooth functions on the open set U ⊆ Rn or more generally the vector space C(M), where M is a differentiable manifold.

For each open set U ⊆ Rn (or U ⊂ M), for each K ⊂ U compact and for each multi-index I , we define

||ƒ||K,I := supx∈K |(∂|I|/∂xI (ƒ)) (x)|, ƒ ∈ C(U)

Each ||.||K,I defines a seminorm. The family of seminorms obtained by considering all of the multi-indices I and the (countable number of) compact subsets K covering U satisfies the properties (1) and (1) detailed above, hence makes C(U) into a Fréchet space. The sets of the form

|ƒ ∈ C(U)| ||ƒ – g||K,I < ε

with fixed g ∈ C(U), K ⊆ U compact, and multi-index I are open sets and together with their finite intersections form a basis for the topology.

All these constructions and results can be generalized to smooth manifolds. Let M be a smooth manifold and let U be an open subset of M. If K is a compact subset of U and D is a differential operator over U, then

pK,D(ƒ) := supx∈K|D(ƒ)|

is a seminorm. The family of all the seminorms  pK,D with K and D varying among all compact subsets and differential operators respectively is a separating family of seminorms endowing CM(U) with the structure of a complete locally convex vector space. Moreover there exists an equivalent countable family of seminorms, hence CM(U) is a Fréchet space. Let indeed {Vj} be a countable open cover of U by open coordinate subsets, and let, for each j, {Kj,i} be a countable family of compact subsets of Vj such that ∪i Kj,i = Vj. We have the countable family of seminorms

pK,I := supx∈K |(∂|I|/∂xI (ƒ)) (x)|, K ∈  {Kj,i}

inducing the topology. CM(U) is also an algebra: the product of two smooth functions being a smooth function.

A Fréchet space V is said to be a Fréchet algebra if its topology can be defined by a countable family of submultiplicative seminorms, i.e., a countable family {qi)i∈N of seminorms satisfying

qi(ƒg) ≤qi (ƒ) qi(g) ∀ i ∈ N

Let F be a sheaf of real vector spaces over a manifold M. F is a Fréchet sheaf if:

(1)  for each open set U ⊆ M, F(U) is a Fréchet space;

(2)  for each open set U ⊆ M and for each open cover {Ui} of U, the topology of F(U) is the initial topology with respect to the restriction maps F(U) → F(Ui), that is, the coarsest topology making the restriction morphisms continuous.

As a consequence, we have the restriction map F(U) → F(V) (V ⊆ U) as continuous. A morphism of sheaves ψ: F → F’ is said to be continuous if the map F(U) → F'(U) is open for each open subset U ⊆ M.

# Affine Schemes

Let us associate to any commutative ring A its spectrum, that is the topological space Spec A. As a set, Spec A consists of all the prime ideals in A. For each subset S A we define as closed sets in Spec A:

V(S) := {p ∈ Spec A | S ⊂ p} ⊂ Spec A

If X is an affine variety, defined over an algebraically closed field, and O(X) is its coordinate ring, we have that the points of the topological space underlying X are in one-to-one correspondence with the maximal ideals in O(X).

We also define the basic open sets in Spec A as

Uƒ := Spec A \ V(ƒ) = Spec Aƒ with ƒ ∈ A,

where Aƒ = A[ƒ-1] is the localization of A obtained by inverting the element ƒ. The collection of the basic open sets Uƒ, ∀ ƒ ∈ A forms a base for Zariski topology. Next, we define the structure sheaf OA on the topological space Spec A. In order to do this, it is enough to give an assignment

U ↦ OA(U) for each basic open set U = Uƒ in Spec A.

The assignment

Uƒ ↦ Aƒ

defines a B-sheaf on the topological space Spec A and it extends uniquely to a sheaf of commutative rings on Spec A, called the structure sheaf and denoted by OA. Moreover, the stalk at a point p ∈ Spec A, OA,p is the localization Ap of the ring at the prime p. While the differentiable manifolds are locally modeled, as ringed spaces, by (Rn, CRn), the schemes are geometric objects modeled by the spectrum of commutative rings.

Affine scheme is a locally ringed space isomorphic to Spec A for some commutative ring A. We say that X is a scheme if X = (|X|, OX) is a locally ringed space, which is locally isomorphic to affine schemes. In other words, for each x ∈ |X|, ∃ an open set Ux ⊂ |X| such that (Ux, OX|Ux) is an affine scheme. A morphism of schemes is just a morphism of locally ringed spaces.

There is an equivalence of categories between the category of affine schemes (aschemes) and the category of commutative rings (rings). This equivalence is defined on the objects by

(rings)op → (aschemes), A Spec A

In particular a morphism of commutative rings A → B contravariantly to a morphism Spec B → Spec A of the corresponding affine superschemes.

Since any affine variety X is completely described by the knowledge of its coordinate ring O(X), we can associate uniquely to an affine variety X, the affine scheme Spec O(X). A morphism between algebraic varieties determines uniquely a morphism between the corresponding schemes. In the language of categories, we say we have a fully faithful functor from the category of algebraic varieties to the category of schemes.

# Ringed Spaces (2)

Let |M| be a topological space. A presheaf of commutative algebras F on X is an assignment

U ↦ F(U), U open in |M|, F(U) is a commutative algebra, such that the following holds,

(1) If U ⊂ V are two open sets in |M|, ∃ a morphism rV, U: F(V) → F(U), called the restriction morphism and often denoted by rV, U(ƒ) = ƒ|U, such that

(i) rU, U = id,

(ii) rW, U = rV, U ○ rW, V

A presheaf ƒ is called a sheaf if the following holds:

(2) Given an open covering {Ui}i∈I of U and a family {ƒi}i∈I, ƒi ∈ F(Ui) such that ƒi|Ui ∩ Uj = ƒj|Ui ∩ Uj ∀ i, j ∈ I, ∃ a unique ƒ ∈ F(U) with ƒ|Ui = ƒi

The elements in F(U) are called sections over U, and with U = |M|, these are termed global sections.

The assignments U ↦ C(U), U open in the differentiable manifold M and U ↦ OX(U), U open in algebraic variety X are examples of sheaves of functions on the topological spaces |M| and |X| underlying the differentiable manifold M and the algebraic variety X respectively.

In the language of categories, the above definition says that we have defined a functor, F, from top(M) to (alg), where top(M) is the category of the open sets in the topological space |M|, the arrows given by the inclusions of open sets while (alg) is the category of commutative algebras. In fact, the assignment U ↦ F(U) defines F on the objects while the assignment

U ⊂ V ↦ rV, U: F(V) → F(U)

defines F on the arrows.

Let |M| be a topological space. We define a presheaf of algebras on |M| to be a functor

F: top(M)op → (alg)

The suffix “op” denotes as usual the opposite category; in other words, F is a contravariant functor from top(M) to (alg). A presheaf is a sheaf if it satisfies the property (2) of the above definition.

If F is a (pre)sheaf on |M| and U is open in |M|, we define F|U, the (pre)sheaf F restricted to U, as the functor F restricted to the category of open sets in U (viewed as a topological space itself).

Let F be a presheaf on the topological space |M| and let x be a point in |M|. We define the stalk Fx of F, at the point x, as the direct limit

lim F(U)

where the direct limit is taken ∀ the U open neighbourhoods of x in |M|. Fx consists of the disjoint union of all pairs (U, s) with U open in |M|, x ∈ U, and s ∈ F(U), modulo the equivalence relation: (U, s) ≅ (V, t) iff ∃ a neighbourhood W of x, W ⊂ U ∩ V, such that s|W = t|W.

The elements in Fx are called germs of sections.

Let F and G be presheaves on |M|. A morphism of presheaves φ: F → G, for each open set U in |M|, such that ∀ V ⊂ U, the following diagram commutes

Equivalently and more elegantly, one can also say that a morphism of presheaves is a natural transformation between the two presheaves F and G viewed as functors.

A morphism of sheaves is just a morphism of the underlying presheaves.

Clearly any morphism of presheaves induces a morphism on the stalks: φx: Fx → Gx. The sheaf property, i.e., property (2) in the above definition, ensures that if we have two morphisms of sheaves φ and ψ, such that φx = ψx ∀ x, then φ = ψ.

We say that the morphism of sheaves is injective (resp. surjective) if x is injective (resp. surjective).

On the notion of surjectivity, however, one should exert some care, since we can have a surjective sheaf morphism φ: F → G such that φU: F(U) → G(U) is not surjective for some open sets U. This strange phenomenon is a consequence of the following fact. While the assignment U ↦ ker(φ(U)) always defines a sheaf, the assignment

U ↦ im( φ(U)) = F(U)/G(U)

defines in general only a presheaf and not all the presheaves are sheaves. A simple example is given by the assignment associating to an open set U in R, the algebra of constant real functions on U. Clearly this is a presheaf, but not a sheaf.

We can always associate, in a natural way, to any presheaf a sheaf called its sheafification. Intuitively, one may think of the sheafification as the sheaf that best “approximates” the given presheaf. For example, the sheafification of the presheaf of constant functions on open sets in R is the sheaf of locally constant functions on open sets in R. We construct the sheafification of a presheaf using the étalé space, which we also need in the sequel, since it gives an equivalent approach to sheaf theory.

Let F be a presheaf on |M|. We define the étalé space of F to be the disjoint union ⊔x∈|M| Fx. Let each open U ∈ |M| and each s ∈ F(U) define the map šU: U ⊔x∈|U| Fx, šU(x) = sx. We give to the étalé space the finest topology that makes the maps š continuous, ∀ open U ⊂ |M| and all sections s ∈ F(U). We define Fet to be the presheaf on |M|:

U ↦ Fet(U) = {šU: U → ⊔x∈|U| Fx, šU(x) = sx ∈ Fx}

Let F be a presheaf on |M|. A sheafification of F is a sheaf F~, together with a presheaf morphism α: F → Fsuch that

(1) any presheaf morphism ψ: F → G, G a sheaf factors via α, i.e. ψ: F →α F~ → G,

(2) F and Fare locally isomorphic, i.e., ∃ an open cover {Ui}i∈I of |M| such that F(Ui) ≅ F~(Ui) via α.

Let F and G be sheaves of rings on some topological space |M|. Assume that we have an injective morphism of sheaves G → F such that G(U) ⊂ F(U) ∀ U open in |M|. We define the quotient F/G to be the sheafification of the image presheaf: U ↦ F(U)/G(U). In general F/G (U) ≠ F(U)/G(U), however they are locally isomorphic.

Ringed space is a pair M = (|M|, F) consisting of a topological space |M| and a sheaf of commutative rings F on |M|. This is a locally ringed space, if the stalk Fx is a local ring ∀ x ∈ |M|. A morphism of ringed spaces φ: M = (|M|, F) → N = (|N|, G) consists of a morphism |φ|: |M| → |N| of the topological spaces and a sheaf morphism φ*: ON → φ*OM, where φ*OM is a sheaf on |N| and defined as follows:

*OM)(U) = OM-1(U)) ∀ U open in |N|

Morphism of ringed spaces induces a morphism on the stalks for each

x ∈ |M|: φx: ON,|φ|(x) → OM,x

If M and N are locally ringed spaces, we say that the morphism of ringed spaces φ is a morphism of locally ringed spaces if φx is local, i.e. φ-1x(mM,x) = mN,|φ|(x), where mN,|φ|(x) and mM,x are the maximal ideals in the local rings ON,|φ|(x) and OM,x respectively.

# Ringed Spaces (1)

A ringed space is a broad concept in which we can fit most of the interesting geometrical objects. It consists of a topological space together with a sheaf of functions on it.

Let M be a differentiable manifold, whose topological space is Hausdorff and second countable. For each open set U ⊂ M , let C(U) be the R-algebra of smooth functions on U .

The assignment

U ↦ C(U)

satisfies the following two properties:

(1) If U ⊂ V are two open sets in M, we can define the restriction map, which is an algebra morphism:

rV, U : C(V) → C(U), ƒ ↦ ƒ|U

which is such that

i) rU, U = id

ii) rW, U = rV, U ○ rW, V

(2) Let {Ui}i∈I be an open covering of U and let {ƒi}i∈I, ƒi ∈ C(Ui) be a family such that ƒi|Ui ∩ Uj = ƒj| Ui ∩ Uj ∀ i, j ∈ I. In other words the elements of the family {ƒi}i∈I agree on the intersection of any two open sets Ui ∩ Uj. Then there exists a unique ƒ ∈ C(U) such that ƒ|Ui = ƒi.

Such an assignment is called a sheaf. The pair (M, C), consisting of the topological space M, underlying the differentiable manifold, and the sheaf of the C functions on M is an example of locally ringed space (the word “locally” refers to a local property of the sheaf of C functions.

Given two manifolds M and N, and the respective sheaves of smooth functions CM and CN, a morphism ƒ from M to N, viewed as ringed spaces, is a morphism |ƒ|: M → N of the underlying topological spaces together with a morphism of algebras,

ƒ*: CN(V) →  CM-1(V)), ƒ*(φ)(x) = φ(|ƒ|(x))

compatible with the restriction morphisms.

Notice that, as soon as we give the continuous map |ƒ| between the topological spaces, the morphism ƒ* is automatically assigned. This is a peculiarity of the sheaf of smooth functions on a manifold. Such a property is no longer true for a generic ringed space and, in particular, it is not true for supermanifolds.

A morphism of differentiable manifolds gives rise to a unique (locally) ringed space morphism and vice versa.

Moreover, given two manifolds, they are isomorphic as manifolds iff they are isomorphic as (locally) ringed spaces. In the language of categories, we say we have a fully faithful functor from the category of manifolds to the category of locally ringed spaces.

The generalization of algebraic geometry to the super-setting comes somehow more naturally than the similar generalization of differentiable geometry. This is because the machinery of algebraic geometry was developed to take already into account the presence of (even) nilpotents and consequently, the language is more suitable to supergeometry.

Let X be an affine algebraic variety in the affine space An over an algebraically closed field k and let O(X) = k[x1,…., xn]/I be its coordinate ring, where the ideal I is prime. This corresponds topologically to the irreducibility of the variety X. We can think of the points of X as the zeros of the polynomials in the ideal I in An. X is a topological space with respect to the Zariski topology, whose closed sets are the zeros of the polynomials in the ideals of O(X). For each open U in X, consider the assignment

U ↦ OX(U)

where OX(U) is the k-algebra of regular functions on U. By definition, these are the functions ƒ X → k that can be expressed as a quotient of two polynomials at each point of U ⊂ X. The assignment U ↦ OX(U) is another example of a sheaf is called the structure sheaf of the variety X or the sheaf of regular functions. (X, OX) is another example of a (locally) ringed space.

# Super Lie Algebra

A super Lie algebra L is an object in the category of super vector spaces together with a morphism [ , ] : L ⊗ L → L, often called the super bracket, or simply, the bracket, which satisfies the following conditions

Anti-symmetry,

[ , ] + [ , ] ○ cL,L = 0

which is the same as

[x, y] + (-1)|x||y|[y, x] = 0 for x, y ∈ L homogenous.

Jacobi identity,

[, [ , ]] + [, [ , ]] ○ σ + [, [ , ]] ○ σ2 = 0,

where σ ∈ S3 is a three-cycle, i.e. taking the first entity of [, [ , ]] to the second, and the second to the third, and then the third to the first. So, for x, y, z ∈ L homogenous, this reads

[x + [y, z]] + (-1)|x||y| + |x||z|[y, [z, x]] + (-1)|y||z| + |x||z|[z, [x, y]] = 0

It is important to note that in the super category, these conditions are modifications of the properties of the bracket in a Lie algebra, designed to accommodate the odd variables. We can immediately extend this definition to the case where L is an A-module for A a commutative superalgebra, thus defining a Lie superalgebra in the category of A-modules. In fact, we can make any associative superalgebra A into a Lie superalgebra by taking the bracket to be

[a, b] = ab – (-1)|a||b|ba,

i.e., we take the bracket to be the difference τ – τ ○ cA,A, where τ is the multiplication morphism on A.

A left A-module is a super vector space M with a morphism A ⊗ M → M, a ⊗ m ↦ am, of super vector spaces obeying the usual identities; that is, ∀ a, b ∈ A and x, y ∈ M, we have

a (x + y) = ax + ay

(a + b)x = ax + bx

(ab)x  = a(bx)

1x = x

A right A-module is defined similarly. Note that if A is commutative, a left A-module is also a right A-module if we define (the sign rule)

m . a = (-1)|m||a|a . m

for m ∈ M and a ∈ A. Morphisms of A-modules are defined in the obvious manner: super vector space morphisms φ: M → N such that φ(am) = aφ(m) ∀ a ∈ A and m ∈ M. So, we have the category of A-modules. For A commutative, the category of A-modules admits tensor products: for M1, M2 A-modules, M1 ⊗ M2 is taken as the tensor product of M1 as a right module with M2 as a left module.

Turning our attention to free A-modules, we have the notion of super vector kp|q over k, and so we define Ap|q := A ⊗ kp|q where

(Ap|q)0 = A0 ⊗ (kp|q)0 ⊕ A1 ⊗ (kp|q)1

(Ap|q)1 = A1 ⊗ (kp|q)0 ⊕ A0 ⊗ (kp|q)1

We say that an A-module M is free if it is isomorphic (in the category of A-modules) to Ap|q for some (p, q). This is equivalent to saying that M contains p even elements {e1, …, ep} and q odd elements {ε1, …, εq} such that

M0 = spanA0{e1, …, ep} ⊕ spanA11, …, εq}

M1 = spanA1{e1, …, ep} ⊕ spanA01, …, εq}

We shall also say M as the free module generated over A by the even elements e1, …, eand the odd elements ε1, …, εq.

Let T: Ap|q → Ar|s be a morphism of free A-modules and then write ep+1, …., ep+q for the odd basis elements ε1, …, εq. Then T is defined on the basis elements {e1, …, ep+q} by

T(ej) = ∑i=1p+q eitij

Hence T can be represented as a matrix of size (r + s) x (p + q)

T = (T1 T2 T3 T4)

where T1 is an r x p matrix consisting of even elements of A, T2 is an r x q matrix of odd elements, T3 is an s x p matrix of odd elements, and T4 is an s x q matrix of even elements. When we say that T is a morphism of super A-modules, it means that it must preserve parity, and therefore the parity of the blocks, T1 & T4, which are even and T2 & T3, which are odd, is determined. When we define T on the basis elements, the basis elements precedes the coordinates tij. This is important to keep the signs in order and comes naturally from composing morphisms. In other words if the module is written as a right module with T acting from the left, composition becomes matrix product in the usual manner:

(S . T)(ej) = S(∑i eitij) = ∑i,keksiktij

hence for any x ∈ Ap|q , we can express x as the column vector x = ∑eixi and so T(x) is given by the matrix product T x.