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, GC be its complexification, and Mflat(G, C) be the moduli space of stable flat GC-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 Mflat(G, C) is equivalent to the derived category of D-modules over Bun(LG, C).

Mflat(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 pi are dual coordinates on the fibers of T Y,  Ω can be written as

Ω = 1/ħdpi ∧ 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, . . . , qn. Up to BRST-exact terms, the action of the A-model on a disc Σ 􏰠takes the form

S = 1/ħ ∫∂Σ φ (pidqi)

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∂/∂pi ∧ ∂/∂qi

One can write an explicit formula for the deformed product:

􏰋(f ⋆ g)(p, q) = exp(􏰋ħ/2(∂2/∂pi∂q̃i  −  ∂2/∂qi∂p̃i )) 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 pi. 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 pi).

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 pi → −pi 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 ⊗ KY), so symmetry under pi → −pi requires L to be a square root of the canonical line bundle KY. 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, K1/2). 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, K1/2). 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 KY1/2, with a tautological action of D(Y, KY1/2).

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. 

Underlying the Non-Perturbative Quantum Geometry of the Quartic Gauge Couplings in 8D.

A lot can be learned by simply focussing on the leading singularities in the moduli space of the effective theory. However, for the sake of performing really non-trivial quantitative tests of the heterotic/F-theory duality, we should try harder in order to reproduce the exact functional form of the couplings ∆eff(T) from K3 geometry. The hope is, of course, to learn something new about how to do exact non-perturbative computations in D-brane physics.

More specifically, the issue is to eventually determine the extra contributions to the geometric Green’s functions. Having a priori no good clue from first principles how to do this, the results of the previous section, together with experience with four dimensional compactifications with N = 2 supersymmetry, suggest that somehow mirror symmetry should be a useful tool.

The starting point is the observation that threshold couplings of similar structure appear also in four dimensional, N = 2 supersymmetric compactifications of type II strings on Calabi-Yau threefolds. More precisely, these coupling functions multiply operators of the form TrFG2 (in contrast to quartic operators in d = 8), and can be written in the form

(4d)eff ∼ ln[λα1(1-λ)α2(λ’)3] + γ(λ) —– (1)

which is similar to Green’s function

eff (λ) = ∆N-1prime form (λ) + δ(λ)

It is to be noted that a Green’s function is in general ambiguous up to the addition of a finite piece, and it is this ambiguous piece to which we can formally attribute those extra non-singular, non-perturbative corrections.

The term δ(λ) contributes to dilation flat coordinate. The dilation S is a period associated with the CY threefold moduli space, and like all period integrals, it satisfies a system of linear differential equations. This differential equation may then be translated back into geometry, and this then would hopefully give us a clue about what the relevant quantum geometry is that underlies those quartic gauge couplings in eight dimensions.

The starting point is the families of singular K3 surfaces associated with which are the period integrals that evaluate to the hypergeometric functions. Generally, period integrals satisfy the Picard-Fuchs linear differential equations.

The four-dimensional theories are obtained by compactifying the type II strings on CY threefolds of special type, namely they are fibrations of the K3 surfaces over Pl. The size of the P1 yields then an additional modulus, whose associated fiat coordinate is precisely the dilaton S (in the dual, heterotic language; from the type II point of view, it is simply another geometric modulus). The K3-fibered threefolds are then associated with enlarged PF systems of the form:

LN(z, y) = θzz – 2θy) – z(θz + 1/2N)(θz + 1/2 – 1/2N)

L2(y) = θy2 – 2y(2θy +1)θy —– (2)

For perturbative, one-loop contributions on the heterotic side (which capture the full story in d = 8, in contrast to d = 4), we need to consider only the weak coupling limit, which corresponds to the limit of large base space: y ∼ e-S → 0. Though we might now be tempted to drop all the θ≡ y∂y terms in the PF system, we better note that the θy term in LN(z, y) can a non-vanishing contribution, namely in particular when it hits the logarithmic piece of the dilaton period, S = -In[y] + γ. As a result one finds that the piece , that we want to compute satisfies in the limit y → 0 the following inhomogenous differential equation

LN . (γϖ0)(z) = ϖ0(z) —– (3)

We now apply the inverse of this strategy to our eight dimensional problem. Since we know from the perturbative heterotic calculation what the exact answer for δ must be, we can work backwards and see what inhomogenous differential equation the extra contribution δ(λ) obeys. It satisfies

LN⊗2 . (δϖ0)(z) = ϖ02(z) —– (4)

whose homogenous operator

LN⊗2(z) = θz3 – z(θz + 1 – 1/N)(θz + 1/2)(θz + 1/N) —– (5)

is the symmetric square of the K3 Picard-Fuchs operator. This means that its solution space is given by the symmetric square of the solution space of LN(z), i.e.,

LN⊗2 . (ϖ02, ϖ0ϖ1, ϖ12) = 0 —– (6)

Even though the inhomogenous PF equation (4) concisely captures the extra corrections in the eight-dimensional threshold terms, the considerations leading to this equation have been rather formal and it would be clearly desirable to get a better understanding of what it mathematically and physically means.

Note that in the four dimensional situation, the PF operator LN(z), which figures as homogenous piece in (3), is by construction associated with the K3 fiber of the threefold. By analogy, the homogenous piece of equation (4) should then tell us something about the geometry that is relevant in the eight dimensional situation. Observing that the solution space (6) is given by products of the K3 periods, it is clear what the natural geometrical object is: it must be the symmetric square Sym2(K3) = (K3 x K3)/Ζ2. Being a hyperkähler manifold, its periods (not subject to world-sheet instanton corrections) indeed enjoy the factorization property exhibited by (6).

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Formal similarity of the four and eight-dimensional string compactifications: the underlying quantum geometry that underlies the quadratic or quartic gauge couplings appears to be given by three- or five-folds, which are fibrations of K3 or its symmetric square, respectively. The perturbative computations on the heterotic side are supposdly reproduced by the mirror maps on these manifolds in the limit where the base Pl‘s are large.

The occurrence of such symmetric products is familiar in D-brane physics. The geometrical structure that is relevant to us is however not just the symmetric square of K3, but rather a fibration of it, in the limit of large base space – this is precisely what the content of the inhomogenous PF equation (4) is. It is however not at all obvious to us why this particular structure of a hyperkähler-fibered five-fold would underlie the non-perturbative quantum geometry of the quartic gauge couplings in eight dimensions.