Conjectural Existence of the Categorial Complex Branes for Generalized Calabi-Yau.

Geometric Langlands Duality can be formulated as follows: Let C be a Riemann surface (compact, without boundary), G be a compact reductive Lie group, G_C be its complexification, and M_flat(G, C) be the moduli space of stable flat G_C-connections on C. The Langlands dual of G is another compact reductive Lie group ^LG defined by the condition that its weight and coweight lattices are exchanged relative to G. Let Bun(^LG, C) be the moduli stack of holomorphic ^LG-bundles on C. One of the statements of Geometric Langlands Duality is that the derived category of coherent sheaves on M_flat(G, C) is equivalent to the derived category of D-modules over Bun(^LG, C).

M_flat(G, C) is mirror to another moduli space which, roughly speaking, can be described as the cotangent bundle to Bun(^LG, C). The category of A-branes on T ^∗ Bun(^LG, C) (with the canonical symplectic form) is equivalent to the category of B-branes on a noncommutative deformation of T ^∗ Bun(^LG, C). The latter is the same as the category of (analytic) D-modules on Bun(^LG, C).

So, what exactly is, the relationship between A-branes and noncommutative B-branes. This relationship arises whenever the target space X is the total space of the cotangent bundle to a complex manifold Y. It is understood that the  symplectic form ω is proportional to the canonical symplectic form on T ^∗ Y. With the B-field vanishing, and Y as a complex, we regard ω as the real part of a holomorphic symplectic form Ω. If qⁱ are holomorphic coordinates on Y, and p_i are dual coordinates on the fibers of T ^∗ Y,  Ω can be written as

Ω = 1/ħdp_i ∧ dqⁱ = dΘ

Since ω (as well as Ω) is exact, the closed A-model of X is rather trivial: there are no nontrivial instantons, and the quantum cohomology ring is isomorphic to the classical one.

We would like to understand the category of A-branes on X = T ^∗ Y. The key observation is that ∃ a natural coisotropic A-brane on X well-defined up to tensoring with a flat line bundle on X. Its curvature 2-form is exact and given by

F = Im Ω

If we denote by I the natural almost complex structure on X coming from the complex structure on Y , we have F = ωI, and therefore the endomorphism ω⁻¹F = I squares to −1. Therefore any unitary connection on a trivial line bundle over X whose curvature is F defines a coisotropic A-brane. 

Now, what about the endomorphisms of the canonical coisotropic A-brane, i.e., the algebra of BRST-closed open string vertex operators? This is easy if Y is an affine space. If one covers Y with charts each of which is an open subset of Cⁿ, and then argues that the computation can be performed locally on each chart and the results “glued together”, one gets closer to the fact that the algebra in question is the cohomology of a certain sheaf of algebras, whose local structure is the same as for Y = Cⁿ. In general, the path integral defining the correlators of vertex operators does not have any locality properties in the target space. Each term in perturbation theory depends only on the infinitesimal neighbourhood of a point. This shows that the algebra of open-string vertex operators, regarded as a formal power series in ħ, is the cohomology of a sheaf of algebras, which is locally isomorphic to a similar sheaf for X = Cⁿ × Cⁿ.

Let us apply these observations to the canonical coisotropic A-brane on X = T ^∗ Y. Locally, we can identify Y with a region in Cⁿ by means of holomorphic coordinate functions q₁, . . . , q_n. Up to BRST-exact terms, the action of the A-model on a disc Σ 􏰠takes the form

S = 1/ħ ∫_∂Σ φ ^∗ (p_idqⁱ)

where φ is a map from Σ to X. This action is identical to the action of a particle on Y with zero Hamiltonian, except that qⁱ are holomorphic coordinates on Y rather than ordinary coordinates. The BRST-invariant open-string vertex operators can be taken to be holomorphic functions of p, q. Therefore quantization is locally straightforward and gives a noncommutative deformation of the algebra of holomorphic functions on T ^∗ Y corresponding to a holomorphic Poisson bivector

P = ħ∂/∂p_i ∧ ∂/∂qⁱ

One can write an explicit formula for the deformed product:

􏰋(f ⋆ g)(p, q) = exp(􏰋ħ/2(∂²/∂p_i∂q̃ⁱ  −  ∂²/∂q^i∂p̃_i )) f(p, q) g (p̃, q̃)|_{p̃ = p, q̃ = q}

This product is known as the Moyal-Wigner product, which is a formal power series in ħ that may have zero radius of convergence. To rectify the situation, one can restrict to functions which are polynomial in the fiber coordinates p_i. Such locally-defined functions on T ^∗ Y can be thought of as symbols of differential operators; the Moyal-Wigner product in this case reduces to the product of symbols and is a polynomial in ħ. Thus locally the sheaf of open-string vertex operators is modelled on the sheaf of holomorphic differential operators on Y (provided we restrict to operators polynomial in p_i).

Locally, there is no difference between the sheaf of holomorphic differential operators D(Y ) and the sheaf of holomorphic differential operatorsD(Y, L) on a holomorphic line bundle L over Y. Thus the sheaf of open-string vertex operators could be any of the sheaves D(Y, L). Moreover, the classical problem is symmetric under p_i → −p_i combined with the orientation reversal of Σ; if we require that quantization preserve this symmetry, then the algebra of open-string vertex operators must be isomorphic to its opposite algebra. It is known that the opposite of the sheaf D(Y, L) is the sheaf D(Y, L⁻¹ ⊗ K_Y), so symmetry under p_i → −p_i requires L to be a square root of the canonical line bundle K_Y. It does not matter which square root one takes, since they all differ by flat line bundles on Y, and tensoring L by a flat line bundle does not affect the sheaf D(Y, L). The conclusion is that the sheaf of open-string vertex operators for the canonical coisotropic A-brane α on X = T ^∗ Y is isomorphic to the sheaf of noncommutative algebras D(Y, K^1/2). One can use this fact to associate Y to any A-brane β on X a twisted D-module, i.e., a sheaf of modules over D(Y, K^1/2). Consider the A-model with target X on a strip Σ = I × R, where I is a unit interval, and impose boundary conditions corresponding to branes α and β on the two boundaries of Σ. Upon quantization of this model, one gets a sheaf on vector spaces on Y which is a module over the sheaf of open-string vertex operators inserted at the α boundary. A simple example is to take β to be the zero section of T ^∗ Y with a trivial line bundle. Then the corresponding sheaf is simply the sheaf of sections of K_Y^1/2, with a tautological action of D(Y, K_Y^1/2).

One can argue that the map from A-branes to (complexes of) D-modules can be extended to an equivalence of categories of A-branes on X and the derived category of D-modules on Y. The argument relies on the conjectural existence of the category of generalized complex branes for any generalized Calabi-Yau. One can regard the Geometric Langlands Duality as a nonabelian generalization. 

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

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

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

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

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

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

Introducing a binding process

A + B ↔ C —– (1)

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

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

Thus, there are binding processes

B + B̄ ↔ Z₁ + Z₂ + · · · .

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

B + B̄ ↔ 0

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

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

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

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

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

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

I(L₁, L₂) = #([L₁] ∩ [L₂]) —– (2)

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

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

∫C^{(p + 1)} —–(3)

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

∫_{R × X} C^{(p + 1)} ∧ ω_L —– (4)

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

∫_LC = ∫_XC ∧ ω_L —– (5)

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

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

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

(L − Λ)α = 0, α ∈ ⊕_p H^p(X, Q)

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

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

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

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

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

φ ≡ Re ε^t Γε|_M = Vol|_M —– (1)

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

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

Z(M) ≡ ∫_M φ —– (2)

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

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

φ|_V ≤ Vol|_V —– (3)

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

∫_M φ ≤ ∫_M Vol —– (4)

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

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

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

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

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

1. M is a p-dimensional complex submanifold of X.
2. The 2-form F is of type (1, 1), i.e., (E, ∇) is a holomorphic vector bundle on M.
3. In the supergravity limit, F satisfies the Hermitian Yang-Mills equation:ω|^p−1_M ∧ F = c · ω|^p_Mfor some real constant c.
4. F satisfies Im e^iφ(ω|_M + il_s²F)^p = 0 for some real constant φ, where l_s is the correction.

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

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

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

(3) Im e^iα Ω|_M = 0 for some real constant α.

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

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

(2)  The 2-form F annihilates ker ω_M.

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

(4)  Let r be the complex dimension of FM. r is even and that r + n = dim_R M. Let Ω be the holomorphic trivialization of K_X. One requires that Im e^iαΩ|_M ∧ F^r/2 = 0 for some real constant α.

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