Anthony Bak

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ALGEBRAIC GEOMETRY AND STRING THEORY
Geometry and Physics Intertwined


0. INTRODUCTION

I am interested in the interplay between Geometry and Physics, particularly Algebraic Geometry and High Energy Physics. My view is that mathematics is more than a analytical tool for solving problems; it is a guide to the fundamental structures and principles of the natural world. I am currently working on two research projects in this area:

I.
Geometric Langlands and Quantum Field Theory. I am exploring the connection between the classical limit of the Geometric Langlands Conjecture and the full conjecture in the case of a hyperelliptic curve where many of the relevant moduli spaces can be explicitly calculated.
II.
Algebraic Geometry and Heterotic String Theory. A major deficit in the construction of Heterotic String theories is the lack of models with realistic particle spectrum. I propose to examine some promising candidate models building on techniques I developed previously [3].
Although both projects arise from different problems in physics many similar objects and techniques appear but under different contexts. These mathematical concepts include derived categories, Fourier-Mukai and other integral transforms, Abelian varieties (Tori), degenerations of Abelian varieties to singular objects, stability conditions, elementary transforms (appearing in (II) as types of Hecke transforms) and moduli spaces of sheaves.

I developed some familiarity with these objects through past work related to (II) [2,4] which applied algebraic geometry to build a heterotic string theory model. In (I) a more fundamental connection between Mathematics and Physics is explored (Geometric Langlands on the Math side and a duality between Field Theories on the Physics side).



I BACKGROUND



Geometric Langlands: The Geometric Langlands Conjecture (GLC) can be viewed as a non-abelian extension of Class Field Theory and has recently generated much interest in the physics community where it is viewed as an isomorphism between different 4-dimensional conformal topological field theories [22]. In physics this isomorphism is called ``electric-magnetic'' duality and should be viewed as a non-commutative generalization of electric-magnetic duality in Maxwell's equations. In [22], a quantum field theoretical argument for the conjecture is given, and a claim is made that important objects in the Geometric Langlands program, such as Hecke eigensheaves, Fourier-Mukai transforms, and $ \mathcal{D}$ -modules, arise naturally from the physics, giving new meaning and interpretation to these mathematical constructions. Mathematically GLC will associate to a $ G$ -local system on a curve C a particular sheaf called a Hecke eigensheaf on the moduli space $ Bun_G$ of principal $ G$ -bundles.

On the math side some progress has been made on the ``classical limit'' of this quantum conjecture [14]. This limit describes a duality between Hitchin's integrable systems for a reductive group and for its Langlands dual group. In [14], the authors prove the duality for a Zariski open set in the Hitchin base parameterizing the non-singular spectral covers. We hope to pass from the full GLC through these classical limit results and then lift back to GLC in the case of a specific example where all of the relevant quantities can be calculated directly.



Heterotic String Theory: The standard model of particle physics is a theory that includes three of the four fundamental physical forces (electromagnetism, weak force and the strong force), all of the detected elementary particles and has been verified experimentally to astounding precision. It is however, an incomplete theory in that it leaves off one of the fundamental forces (gravity), does not explain the apparent unification of the fundamental forces at high energies and has no explanation for other parameters appearing in the theory such as particle mass. String Theory is an attempt to fix these deficits.

Two decades ago it was shown that a physically realistic string theory can be realized with a Calabi-Yau threefold as the compact portion of space-time [18]. On this Calabi-Yau a Heterotic String theory compactification is specified by giving a vector bundle with special properties determined by physics. In the last decade, many examples of Calabi-Yau manifolds have been produced, but until recently [8] there were no examples of a Heterotic String with a phenomenologically correct particle spectrum.

Key ingredients in the construction of potential models are applications of powerful results from algebraic geometry including new examples of non-simply connected Calabi-Yaus, Fourier-Mukai transforms and the spectral construction, computational techniques for calculating vector bundle cohomology, the Serre construction for vector bundles among others.

New experimental results out of LHC may potentially verify some of the predictions of string theory including perhaps some signs of supersymmetry and the discovery of the Higgs particle. However more string theory models with the phenomenologically correct low energy particle spectrum are needed if string theory is to be a grand unified theory for the standard model.

We propose to search several promising and newly discovered families of non-simply connected Calabi-Yau for such constructions. For physics, large fundamental groups give more flexibility when breaking the grand unified group to the standard model group. Some of the examples we propose to examine have some of the largest known fundamental groups of all Calabi-Yau.




II GEOMETRIC LANGLANDS AND QUANTUM FIELD THEORY

The simplest case of GLC is the well known result that a flat line bundle on C extends uniquely to a bundle on $ Pic^0(C)$ . GLC states that we should to attach to every rank 1 local system $ \mathcal {L}$ on $ C$ (a line bundle with flat connection) a certain sheaf called a Hecke eigensheaf of $ \mathcal {L}$ on $ Bun_{\mathbb{C}^*} \cong Pic(C)$ . Let $ \phi:\pi_1(C) \rightarrow \mathbb{C}^*$ be the homomorphism associated to $ \mathcal {L}$ . Then $ \phi$ factors through the abelianization of $ \pi_1(C)$ , $ H_1(C,\mathbb{Z})$ . By Poincaré duality we have $ H_1(C,Z) \cong H^1(C,\mathbb{Z})$ which is just $ \pi_1(Pic^0(C))$ . Hence we get a homomorphism $ \pi_1(Pic^0(C)) \rightarrow \mathbb{C}^*$ . This gives a line bundle on $ Pic^0(C)$ that can be extended to all of $ Pic$ (see for example [15]).

To generalize this idea to the full GLC we will need to take G to be any complex reductive group and T a maximal torus. There is a notion of Langlands duality for groups interchanging the characters and cocharaters of the groups. Two groups, $ G$ and $ G'$ are Langlands Dual if $ cochar_G \cong char_{G'}$ in which case we write $ G' = {}^L{G}$ .

There is a moduli space $ Hecke$ classifying quadruples $ (V,V',x,\beta)$ where $ V$ and $ V'$ are $ G$ -bundles and $ \beta$ is a isomorphism over $ C\setminus x$ of $ V$ and $ V'$ . Viewing the triple $ (V,V',x) \subset Bun \times Bun \times C$ we have the natural induced projections on $ Hecke$ : $ p:Hecke\rightarrow Bun$ and $ q:Hecke\rightarrow Bun \times C$ with $ p(V,V',x,\beta) = V$ and $ q(V,V',x,\beta) = (V',x)$ . To the highest weight $ \lambda$ of a finite dimensional irreducible representation of $ {}^L{G}$ we can associate a perverse sheave $ \mathcal {P}_\lambda$ . This gives a functor from the derived category of perverse sheaves on $ Bun_G$ to those on $ Bun_G \times C$ called a Hecke functor via the integral transform

$\displaystyle H_\lambda(\mathcal {F}) = q_{!}(p^* \mathcal {F} \otimes \mathcal {P}_\lambda)$      

Writing $ \mathcal {L}$ for a $ {}^L{G}$ local system on C we have for each irreducible representation of $ {}^L{G}$ , $ V_\lambda$ the local system $ V_{\lambda,\mathcal {L}} = \mathcal {L} \times_G V_\lambda$ . We call a sheaf a Hecke eigensheaf with eigenvalue $ \mathcal {L}$ if
$\displaystyle H_\lambda(\mathcal {F}) \cong V_{\lambda,\mathcal {L}} \boxtimes \mathcal {F}$      

where the isomorphism is compatible with the tensor product structure on the category of representations of $ {}^L{G}$ . Now we have statement of the conjecture

Conjecture 1   (Geometric Langlands Conjecture) Let $ \mathcal {L}$ be a $ {}^L{G}$ local system on C. Then there exists a non-zero Hecke eigensheaf $ A_\mathcal {L}$ on $ Bun_G$ with eigenvalue $ \mathcal {L}$ .

Recall that a G-local system V is a G-bundle along with a holomorphic flat connection on V. Given appropriate stability conditions there is a moduli stack of G-local systems, $ Loc_G$ . Using this we can recast the conjecture into the language of derived categories.

Conjecture 2   (GLC)There is an equivalence of derived categories
$\displaystyle D^b(Loc_G) \rightarrow D^b(Bun_{{}^L{G}},\mathcal {D})$      

taking structure sheaves of points in $ Loc_G$ to automorphic $ \mathcal{D}$ -modules on $ Bun_{{}^L{G}}$ .

Much progress has been made to understand this conjecture and the conjecture is known for Gl(n) [23]. However, a deficit of the work is that the proof is indirect and does not construct non abelian Hecke eigensheaves. We propose to investigate the GLC through the so called classical limit defined in [14].

First recall that a Higgs $ G$ -bundle is $ G$ -principal bundle V along with a section $ \psi$ of $ ad(V)\otimes K_C$ , where $ ad(V)$ is adjoint bundle associated to $ V$ . There is a moduli stack of Higgs G-bundles denoted $ Higgs_G$ . The classical limit is found by specializing both sides of the GLC [14].

Theorem 1   (Classical Geometric Langlands)
$\displaystyle D^b(Higgs_G,\mathcal{O}) \cong D^b(Higgs_{{}^L{G}},\mathcal{O})$      

What we would like to work out the classical Langlands conjecture for a well chosen example and try to deform the result back to the full GLC.



Proposal: I am working out the correspondence in the case of a hyperelliptic curve of any genus with no marked points with $ G=SL(2)$ . There we should be able to work out explicit descriptions of the relevant moduli spaces. In particular we have an explicit description of the moduli space $ Bun_G$ as the intersection of two quadrics in $ \mathbb{P}^5$ [11].

We start on the left hand side of GLC and specialize to the classical case. In the classical case we attempt to work out explicitly the Hecke correspondences in a purely algebraic manner (one of the deficits of the constructions in [14] are their use of non-algebraic constructions in the proof). Knowing the Hecke correspondences we will try to lift to right hand side of the statement of GLC by showing explicit deformation extension order by order in the deformation parameter to move away from the specialized fiber (a similar approach was attempted by Arinkin in the case of $ \mathbb{P}^1$ with some marked points).

Doing this process we hope to gain insight into the connection between the classical case and the full GLC. The hope would be to show that one could do this in general, thereby proving GLC. A different transcendental approach is being pursued by Donagi and Pantev using recent advances in Non Abelian Hodge Theory (see for example [24]-[29]) and it will be interesting to compare these two approaches for the hyperelliptic $ G=SL(2)$ case.





III ALGEBRAIC GEOMETRY AND HETEROTIC STRING THEORY
After some simplifications [13] to define a Heterotic theory we get a well defined problem in algebraic geometry. The task is to find a Calabi-Yau threefold X with Kähler form $ \omega$ and $ \mathcal {E}$ a $ SU(4)$ or $ SU(5)$ vector bundle on X with $ rk(\mathcal{E}) = 4$ or $ 5$ . The bundle $ \mathcal {E}$ should be Mumford polystable with respect to $ \omega$ and have chern classes satisfying $ c_1(\mathcal{E}) = 0$ , $ c_2(T_X) - c_2(\mathcal{E}) =$     {class of effective curve} and $ c_3(\mathcal{E}) = 6$ . These numerical requirements are quite rigid, and the search for bundles satisfying the constraints typically requires detailed analysis of the particular threefold in question using the techniques described above.



Spectral Approach to Bundle Building: The Basic tool I use to construct bundles is the spectral construction which relies on the machinery of Fourier-Mukai transforms. Fourier-Mukai transform relates spaces of sheaves on one algebraic variety to the space of sheaves on another related variety. Certain types of vector bundles have particularly simple Fourier-Mukai transforms (they are just line bundles supported on smooth curves in the dual variety). We exploit this fact to build physically interesting bundles.

Mukai originally used the Fourier-Mukai transform to study the moduli of bundles on surfaces [30] and it has been used extensively to study moduli spaces of bundles on Elliptic Fibrations [17], [35]. Recently it has been used to give very short proofs of Atiyah's classification of bundles on elliptic curves [21] and [34]. For Calabi-Yaus it has been especially fruitful, for instance, [10] provides criteria for a Calabi-Yau fibration $ X \rightarrow B$ and a connected component of its moduli space of stable sheaves to have equivalent derived categories. The proof relies heavily on the machinery of Fourier-Mukai transforms.

Let $ X$ and $ Y$ be algebraic varieties and consider an object $ \mathcal{Q} \in D^b(X \times Y)$ , the Fourier-Mukai transform with kernel $ \mathcal{Q}$ is a map $ D^b(X) \rightarrow D^b(Y)$ given by S$ _{\mathcal{Q}}(\mathcal {F})= p_{2\ast} (p_1^\ast \mathcal{F} \otimes \mathcal{Q})$ , where all maps are in the derived category. We will also consider a relative Fourier-Mukai transform. We'll take the kernel to be the Poincaré sheaf $ \mathcal{P}$ on Y, the relative moduli space of degree 0 line bundles on $ X$ .

A key property of the Fourier-Mukai transform is that it descends to a linear map on cohomology s$ _{\mathcal{Q}}: H^\ast(X,\mathbb{Q}) \rightarrow H^\ast(Y,\mathbb{Q})$ . In the case that S$ _\mathcal{Q}$ is an equivalence, s$ _\mathcal{Q}$ is an isomorphism of vector spaces.

The spectral construction uses Fourier-Mukai transforms to build bundles with specified Chern classes over abelian variety fibered varieties $ X \rightarrow B$ [12]. It relies on the result that for a single abelian variety $ A$ , with $ \widehat{A}$ its dual, the Fourier-Mukai transform of a sheaf $ \mathcal{F}$ on A with finite support is a flat vector bundle on $ \widehat{A}$ .

In the relative case, if we can construct a dual fibration $ \widehat{X} \rightarrow B$ , with relative universal sheaf $ \mathcal{P}$ , then, given a curve $ \mathfrak{s}$ in $ X$ , finite over B, and a vector bundle $ \mathcal{N}$ on $ \mathfrak{s}$ , the Fourier-Mukai transform S$ _\mathcal{P}(\mathcal{N})$ is a vector bundle on $ \widehat{X}$ , flat on the fibers of $ \widehat{X} \rightarrow B$ . In many cases this construction reduces the hard problem of building a bundle on $ \widehat{X}$ to the simpler problem of building a line bundle on the spectral curve. Using the descent of the Fourier-Mukai transform to a linear map on cohomology, we can construct a spectral sheaf whose Fourier-Mukai transform yields a vector bundle with specified Chern classes. We note in passing that in general s$ _\mathcal{P}$ is not a diagonal matrix and there is mixing of the degree of the cohomology classes.

In general we expect that we will have to combine the spectral construction with other bundle building techniques such as the Serre Construction and other extension methods in order to meet the requirements from physics.



Past Results:

I have carried out the above analysis in detail for the $ d=8$ class Gross-Popescu Calabi-Yaus. These examples arise from explicit constructions of the moduli of $ (1,d)$ -polarized abelian varieties with level structure for $ d=6,7,8$ , originally considered in some detail in [20]. For each $ d$ we have an abelian surface fibration, $ V_d \rightarrow B$ . The fibration is flat but not smooth. We will use quotients of $ V_8$ by a freely acting group $ G=\mathbb{Z}_8 \times \mathbb{Z}_8$ as our target variety $ \widehat{V}_8$ on which we examine the moduli of vector bundles.

For the $ d=8$ and $ G=\mathbb{Z}_8 \times \mathbb{Z}_8$ case we've shown that:

Theorem 2   There is a dual fibration $ \widehat{V}_d \rightarrow B$ and a universal Poincaré sheaf, $ \mathcal{P}$ , on the fiber product $ V_d \times_B \widehat{V}_d$ such that on the smooth fibers $ A \times \widehat{A} \hookrightarrow V_d \times_B \widehat{V}_d$ it restricts the usual Poincaré line bundle.

The main difficulty in the above results are the singular fibers on which we need to extend the Poincaré sheaf. In all the cases we consider the singular fibers are elliptic translation scrolls and the primary technical result to make the extension is

Proposition 3   There is a projective small resolution $ \widetilde{V_d \times_B V_d} \stackrel{q}{\longrightarrow} V_d \times_B V_d$ and an extension of the natural multiplication map on the smooth fibers to the singular fibers of the small resolution, $ \widetilde{V_d \times_B V_d} \stackrel{m}{\longrightarrow} V_d$ .

The type of singularities considered here are generic in 1-parameter families; they form a Zariski open set on the boundary of the moduli of abelian surfaces. We expect the result to generalize to many abelian surface fibrations not considered here. With proposition 3 we are able to define the Poincaré sheaf $ \mathcal{P}$ on $ V_d \times \widehat{V}_d$ . My research is the first to extend these techniques and constructions to the case of abelian surface fibrations.

One of the nice features of this construction is that we are able to characterize stability in terms of geometric data regarding the spectral cover. Consider a spectral curve $ C \hookrightarrow V$ with a line bundle $ \mathcal {L}$ supported on $ C$ . Associated to this data via the Fourier-Mukai transform is a vector bundle $ \mathcal {E}$ or rank $ r$ on the dual fibration. Write $ C_{r'}$ for the spectral curve associate with $ \wedge^{r'} \mathcal {E}$ .

Definition 4   We will say that the spectral data is $ r'-$ irreducible (resp. $ r'-$ reducible) if $ C_{r'}$ is irreducible (resp. reducible) and is absolutely irreducible (resp. reducible) if $ C_{r'}$ is irreducible (reducible) for all $ 0 < r' <r$ .

The following theorem will characterize the stable bundles in our construction.

Theorem 5   Let $ \mathcal {E}$ be given as the Fourier-Mukai transform of a line bundle supported on a curve such that $ \mathcal {E}$ is absolutely irreducible (see definition 4), and let $ H_0$ be a fixed ample Divisor. Write $ D_k = H_0 + k\widehat{A}$ , then for $ k>>0$ we have that $ \mathcal {E}$ is stable with respect $ D_k$ .

Using Gromov Witten theory we are able to find spectral curves that meet these requirements and so can explicitly build stable bundles using the spectral construction. Unfortunately we are unable to build a physically viable bundle and have under mild conditions the following ``no-go'' theorem [4]

Theorem 6   Let $ V$ be a stable holomorphic vector bundle on $ X$ with respect to an ample class $ D = H_0 + k\widehat{A}$ , where $ k \gg 0$ , $ H_0$ is some fixed ample class on $ X$ , and $ \widehat{A}$ is the class of the Abelian fiber. Then its Chern character cannot satisfy the weak heterotic conditions.



New Prospects: We would like to make a systematic survey of possible Calabi-Yau on which it is possible to build physically realistic Heterotic Models. We expect the bundle building techniques we developed while completing [3] will apply with some modifications to other families of abelian surfaces.

The primary difficulty in extending these techniques is an understanding of what happens on the singular fibers of the fibration. To make a complete description we need a detailed understanding of Fourier-Mukai transforms and stable bundles on these degenerations as well as a global understanding of the kernel sheaf used in the Fourier-Mukai transform.

For physical reasons we want to consider non-simply connected Calabi-Yaus. Most of the known Calabi-Yau varieties are simply connected while those considered here have some of the largest known fundamental groups. It is notoriously difficult to find such Calabi-Yaus; In a survey of 473,800,776 [5] found only 32 families with nonzero $ Hom(\pi_i(X),\mathbb{Q}/\mathbb{Z})$ . Having a nontrivial fundamental group gives more flexibility when building physical models. For this reason we propose examining several families with non-trivial fundamental groups as they are expected to be candidates for the so called ``peaks'' in the string landscape.

The first set of families we would like to consider are the rest of the Gross-Popescu Calabi-Yaus. These are very closely related to the ones already examined in some detail above. They arise from explicit constructions of the moduli of $ (1,d)$ -polarized abelian varieties with level structure for $ d=6,7,8$ . For each $ d$ we have an abelian surface fibration, $ V_d \rightarrow B$ . The fibration is flat but not smooth. We will use quotients of $ V_d$ by a freely acting group $ G$ as our target variety $ \widehat{V}_d$ on which we examine the moduli of vector bundles. This means that all have fundamental groups of order bounded by $ d^2$ , not necessarily abelian [7].

Another promising family of Abelian surfaces are the Saito manifolds considered in [38]. These do not necessarily have a natural uniform relative polarization and we expect the need to further refine the spectral construction to handle this case.

Once we find a physically viable model we can make further calculations. It becomes a straightforward computation to determine relevant physical quantities testing our model. For instance, the tri-linear couplings will give predictions that can be matched with Yukawa couplings in the standard model. Potentially, these Calabi-Yaus can improve on the calculations in [9] since large fundamental groups allows for a more careful tuning of the texture parameter.


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