Calabi-Yau Spaces in String Theory

Transcription

1 Habilitationsschrift Calabi-Yau Spaces in String Theory Johanna Knapp Institut fu r Theoretische Physik Technische Universita t Wien Wiedner Hauptstraße Wien O sterreich Wien, September 05

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3 Abstract String Theory is a quantum theory which unifies the fundamental forces of nature. One of its most striking properties is the requirement of a ten-dimensional space-time. String compactifications establish a connection between ten-dimensional string theory and our four-dimensional world. The idea is that the extra dimensions are small and curled up so that they are not accessible at low energies. Consistency requirements constrain the mathematical structure of the six-dimensional compact space to be Calabi-Yau. The geometry and topology of the Calabi-Yau space determine the physics in four dimensions. Furthermore the structure of the extra dimensions reveals remarkable properties of string theory itself. This thesis focuses on the interplay between the mathematics of Calabi-Yau spaces and the physics of string theory.

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5 Contents Introduction 5 Definitions and Constructions 7. Definitions Topological classification of CYs Constructions in Toric Geometry D-branes on Calabi-Yaus and Mirror Symmetry. Mirror Symmetry D-branes Mirror Symmetry for D-branes Calabi-Yaus in F-theory and Model Building 9 4. Calabi-Yaus, F-theory and Phenomenology Global F-theory Models Calabi-Yaus and the Gauged Linear Sigma Models 4 5. Gauged Linear Sigma Models New Calabi-Yaus from non-abelian Gauged Linear Sigma Models Outlook 7 Summary of the attached articles Introduction Understanding the fundamental forces of nature is intrinsically tied to two success stories of modern theoretical physics: Quantum Field Theory and General Relativity. The former describes the quantum interactions of fundamental particles, the latter is Einstein s Theory of Gravity. The Standard Model of Particle Physics is a unified theory that accounts the strong, weak and electromagnetic forces as gauge theories with different gauge groups. Einstein Gravity connects gravitational interactions to the curvature of space-time. Both theories have been confirmed in experiments to very high precision. Despite the success there are still many unanswered questions concerning the fundamental forces of nature. One class of open problems is to understand the early universe and the physics of black holes. This concerns physics at very high energies (Planck scale 0 9 GeV) and very short distances (Planck Length 0 5 m). In this regime a theory of quantum gravity is required. Another puzzle is concerned with dark matter and the cosmological constant. Currently we understand only 5% of what the universe is made of. The missing 95% are made of about one quarter of dark matter and about 70% dark energy. The latter may be explained by a cosmological constant. However, there is currently no explanation for its smallness. Dark matter may be due to new types of elementary particles. Given the success of unification of forces in the Standard Model is seems natural to look for a quantum theory that unifies all the known fundamental forces, and, hopefully, answers some or all of these open questions. This is a hard question, also because particle interactions and gravity require different mathematical technologies. 5

6 Currently string theory is the only known physical theory that achieves such a unification of forces at the quantum level. The idea of string theory is that elementary particles get replaced by one-dimensional extended objects, the fundamental strings. All particles and interactions then come from vibrations and interactions of strings. Closed strings account for gravity, while open strings account for gauge interactions. This naturally achieves unification. Despite being more complex than originally expected - each complication leading to new fascinating insights - string theory is currently still the only candidate for a Theory of Everything. String theory makes fundamental qualitative predictions about space-time and particle physics which may be accessible to experimental particle physics and astronomy: consistency requires that string theory is a supersymmetric theory which lives in a ten-dimensional spacetime. Supersymmetry postulates new elementary particles that are related to the known ones by a new, currently unobserved, symmetry. These supersymmetric particles may present a solution to the dark matter problem. Concerning the extra dimensions, the most straightforward way to make contact with our four-dimensional world is to divide up the ten dimensions into four large ones and six extra dimensions which are small and compact and have yet to be observed. This is called string compactification. The nature of the six-dimensional internal space is constrained by supersymmetry. This leads to the condition that the extra dimensions are a Calabi-Yau (CY) space, i.e. a Kähler manifold, possibly with singularities, with vanishing Ricci curvature. While string theory in ten dimensions is almost unique, this is not true for string compactifications to four dimensions. Even though the number of CY spaces of (complex) dimension three, referred to as CY threefolds, is conjectured to be finite, it turns out to be very large. The mathematical properties of the internal space have great impact on the physics in four dimensions: they encode crucial data such as particle spectra, coupling constants, etc. In view of making testable predictions from string theory it is therefore essential to understand what is going on in the extra dimensions. The possibility of millions of different CY threefolds, which upon adding further ingredients such as D-branes and gauge fluxes multiplies to the famous number of possible string compactifications [] has been named the landscape problem. The existence of a large zoo of CY compactifications has unveiled remarkable properties of string theory and spectacularly confirmed its self-consistency. This particularly applies to string dualities whose essence is that compactifications on different internal spaces may lead to the same physics in four dimensions. The most prominent example is mirror symmetry which relates one CY to a different one its mirror. Furthermore, there can be topological transitions connecting different CYs and therefore different string compactifications. This fact led to the bold conjecture by Miles Reid [] (Reid s fantasy) that all CYs are connected by topology changing transitions. In the realm of mathematics, many of these dualities and transitions have been formulated in terms of equivalences of categories associated to the CY. Therefore research on CYs provides the perfect setting for fruitful interactions between mathematicians and physicists. Even though the study of CY spaces has been an integral part of string theory from the very beginning there are still many unanswered questions. In the following we will discuss a This is indeed a problem. However, the large number of string vacua may provide a possible, albeit not very pretty, solution to the cosmological constant problem: if the number of possible vacua is so large, it is at least not completely unlikely that we live in a universe where the cosmological constant is small. Anthropic reasoning suggests that only universes with small cosmological constant can form life-supporting structure []. 6

7 selection of recent topics in string theory where CY spaces play a crucial role. If not stated otherwise the main focus will be on compact CY threefolds in the context of compactifications in type II string theory. In the following sections we give background information to explain the context of the attached publications [4 8]. In section we give the mathematical definition of a CY space and explain how CYs can be constructed and classified using methods of toric geometry. Section focuses on mirror symmetry. In particular mirror symmetry for open strings and their boundary conditions, called D-branes, is still not well-understood in the context of compact CYs. In section 4 we discuss CYs in the context of F-theory, which is a non-perturbative formulation of string theory and recent applications in the construction of Grand Unified Theories in the context of string theory. Section 5 focuses on gauged linear sigma models (GLSMs). These supersymmetric gauge theories provide a physics approach to construct and analyze new types of CY spaces which cannot be described by the standard methods. After a brief outlook on future research in section 6 we give a brief summary of the attached articles in section 7. Definitions and Constructions In this section we summarize the most important definitions and constructions of CYs. In view of string theory applications we focus mostly on CYs of complex dimension three.. Definitions To give a definition of a CY, we follow [9]. Let (X, J, g) be a complex Riemannian manifold X of complex dimension m with complex structure J and metric g. Definition. g is hermitean if g(v, w) = g(jv, Jw) for all vector fields v, w in X. Definition. We can define a real (, )-form by ω(v, w) = g(jv, w). Definition. A hermitean metric on a complex manifold (X, J) is called Kähler if one of the three equivalent definitions holds. dω = 0. J = 0, where is the Levi-Civitá connection of g.. ω = 0. The triple (X, J, g) is called Kähler manifold, ω is called Kähler form. Since J and ω are constant due to this definition, it follows that the holonomy group Hol(g) must preserve the constant J, ω on R m. Proposition. A metric on X is Kähler with respect to a complex structure J on X if and only if Hol(g) U(m) O(m). Locally, one can construct ω as ω = i φ, where φ : X R is called Kähler potential. Proposition. Let (X, J) be a compact complex manifold and let g, g be Kähler metrics on X with Kähler forms ω, ω H (X, R). Then there exists a smooth function φ in X such that ω = ω + i φ. This function is unique up to addition of a constant. 7

8 Definition 4. Let (X, J, g) be a Kähler manifold with Ricci curvature R ab. The Ricci form ρ is defined by ρ ac = J b a R bc. ρ is a closed, real (, )-form. To give an explicit expression for ρ, let (z,..., z m ) be holomorphic coordinates on a neighborhood U X. W define a smooth f : U (0, ) by ω m = f ( )m(m )/ i m m! m dz... dz m d z... d z m (.) Then one can show that ρ = (log f) on U. ρ is the curvature two-form of the connection K of the canonical bundle K X. To see this, remember that the canonical bundle is a holomorphic line bundle defined by K X = m,0 T X; g induces a metric on K X and hence the connection K. The curvature on K is a closed two-form with values in the Lie algebra u() R. This is the Ricci form. With [ρ] H (X, R), characteristic class theory implies [ρ] = πc (K X ) = πc (X) c (X) H (X, Z), (.) where c is the first Chern class. Now we can define a Calabi-Yau manifold as follows. Definition 5. A CY m-fold is a quadruple (X, J, g, Ω) such that (X, J) is a compact m- dimensional complex manifold, g a Kähler metric on (X, J) with holonomy group Hol(g) = SU(m) and Ω a non-zero constant (m, 0)-form on X, called holomorphic volume form, which satisfies where ω is the Kähler form of g. ω m m! = ( )m(m )/ One need not put Ω into the definition because: ( ) i m Ω Ω, (.) Lemma. Let (X, J, g) be compact Kähler manifold with Hol(g) = SU(m). Then X admits a holomorphic volume form Ω, unique up to a change of phase Ω e iθ Ω, such that (X, J, g, Ω) is a CY manifold. Since Ω is constant (and therefore holomorphic), K X admits a non-vanishing holomorphic section. Therefore, (X, J) has trivial canonical bundle, and thus c (X) = 0. This can be used as an alternative definition of a CY. Moreover, the connection K on K X must be flat. Since the curvature is the Ricci-form ρ, ρ 0. Therefore g is Ricci-flat. This serves as yet another definition of a CY: Proposition. Let (X, J, g) be a Kähler m-fold with Hol(g) SU(m). Then g is Ricci-flat. Conversely, let (X, J, g) be a Ricci-flat Kähler m-fold. If X is simply-connected or K X is trivial, then Hol(g) SU(m). The name Calabi-Yau stems from a conjecture due to Calabi in 954 and its proof by Yau in 976. Conjecture. (Calabi 54) Let (X, J) be a compact, complex manifold, and g be a Kähler metric on X, with Kähler form ω. Suppose ρ is a real closed (, )-form on X with [ρ ] = πc (X). Then there exists a unique Kähler metric g on X with Kähler form ω such that [ω ] = [ω] H (X, R), and the Ricci-form of g is ρ. 8

9 Yau s proof of Calabi s conjecture is not constructive. The metric on a CY is usually not known explicitly, except for trivial examples like the elliptic curve. There are several generalizations to the definitions given above. For example, the CY need not be a smooth manifold but can be an orbifold. This will play a role in the forthcoming sections. Furthermore there is a notion of a non-compact CY. This has been discussed extensively in the physics literature. However, we will only focus on compact CYs in this work. Due to the more general concepts we will use the term CY space instead of CY manifold.. Topological classification of CYs For physics applications, the threefold case is the most important one. CYs can be distinguished by their topological invariants. These invariants appear in physical quantities of string compactifications (e.g. couplings). Theorem. (Wall [0]) A CY threefold X is topologically classified by the following data.. The Hodge numbers h, and h,, counting the dimensions of the Kähler and complex structure moduli spaces, respectively.. The second Chern class c (X). Given an integral basis of (, )-forms J a, a =,..., h,, this is defined by c J a = c (X) J a. (.4). The triple intersection numbers κ abc = The Hodge diamond of a CY threefold looks as follows X X h, 0 h, h, 0 h, J a J b J c. (.5). (.6) In section 4 we will also be concerned with CY fourfolds. The non-trivial Hodge numbers are h,, h,, h,, h, which satisfy the relation h, = ( + h, + h, h, ). See e.g. [] for further details. Roughly speaking, the Kähler moduli parametrize the size of the CY and complex structure moduli determine its shape. Physical quantities in string compactifications depend on these moduli unless there is a mechanism that stabilizes them. Moduli spaces have an intricate mathematical structure. Depending on the position in the moduli space, different mathematical an physical methods are required to study CY compactifications. This will play an important role in the forthcoming sections. 9

10 . Constructions in Toric Geometry A large class of CYs can be systematically constructed by employing the combinatorial technology of toric geometry. Standard references are [, ]. We will mostly follow the review article [4]. Definition 6. A toric variety Y of dimension n is defined as the quotient Y = Cr Z (C ) r n G, (.7) where (C ) r n is an algebraic r n-torus, G is a finite abelian group and Z C r is an exceptional set containing the information about which coordinates of C r are not allowed to vanish simultaneously in order to get a well-defined quotient. Example. Take (z, z, z ) C and λ C. Then CP can be written as CP = C {z = z = z = 0} (z, z, z ) (λz, λz, λ ). (.8) The geometric information is most conveniently encoded in terms of cones, fans and lattice polytopes. Definition 7. A fan Σ is a collection of strongly convex rational polyhedral cones where all faces and intersections of pairs of cones also belong to Σ. The rational property means that the rays spanning the cones go through points of an integer lattice. There is actually a pair of integer lattices, denoted as the N- and M-lattice. To define these, note that Y contains an n-torus (C ) n = T as a dense open subset whose action extends to Y. We parametrize T by (t,..., t n ). Definition 8. The character group is defined as M = {χ : T C } with χ m (t) = t m... t mn n t m for m M. Definition 9. The set of one-parameter subgroups is defined as N = {λ : C T } with λ(τ) = (τ u,..., τ un ) with τ C, u N. M and N are integer lattices M, N Z n. The fan Σ and its cones are defined in the real extension N R of N. M and N are dual due to (χ λ)(τ) = χ(λ(τ)) = τ χ,λ χ m, λ u = m u m M, u N. (.9) Divisors on Y are defined by χ m = 0. They can be decomposed in terms of irreducible divisors D j as r div(χ m ) = a j D j a j (m) Z. (.0) j= The coefficients a j (m) are unique and there exists a map m a j (m) = m, v j v j N. (.) Thus, there is a vector v j in the N-lattice for every irreducible divisor D j. The v j are the primitive generators of the one-dimensional cones ρ j in Σ. The convex hull of v j defines a 0

11 lattice polytope = conv{v j } N R. Locally, one can write D j = {z j = 0}, where z j is a local section of a line bundle. The D j are called toric divisors. The {z j } can be viewed as global homogeneous coordinates (z :... : z r ) 0. Take λ C, then (λ q z :... : λ qr z r ) (z :... : z r ) describe the same point in T if r j= q jv j = 0 for v j N. A subset of the z j is only allowed to vanish simultaneously if and only if there is a cone σ Σ containing all the corresponding rays ρ j. The exceptional set Z = I Z I is the union of Z I with minimal index sets I of rays for which there is no cone that contains them. Theorem. Y is compact if and only if the fan Σ is complete, i.e. if it covers the whole N-lattice: Σ = Σ σ = N R. Theorem. Y is non-singular if and only if all cones are simplicial and basic, i.e. if all cones σ Σ are generated by a subset of a lattice basis on N. Note that the homogeneous weights q j can be arranged into an (r n) n-matrix Q ij, where the r n lines encode the C -actions and the columns correspond to the divisors D j in Y. We will see in section 5 that Q ij encodes the gauge charges of an abelian gauged linear sigma model. The weights Q ij contain all the information to reconstruct the M- and N-lattices (if G is trivial). The largest known class of CYs are hypersurfaces and complete intersections in toric ambient spaces. Definition 0. Given a divisor D = j a jd j we define a polytope D = {m M R : m σ, v j a j }, σ Σ (n), (.) where Σ (n) is the set of n-dimensional cones. This is a convex lattice polytope whose lattice points provide global sections of the line bundle O(D). Equations for hypersurfaces are sections of line bundles O(D) given by the Laurent polynomial f = m D M c m χ m = m D M c m j z m,v j j. (.) Definition. Given a polytope D M we define the polar polytope D by D = {y N R : x, y x D } (.4) Theorem 4. (Batyrev) D is CY if and only if D M R is polar to = D N R. A lattice polytope that is polar to another lattice polytope is called reflexive. We denote the CY associated to a reflexive lattice polytope by X. For a CY hypersurface in a toric variety Y D = j D j. Theorem 5. (Batyrev) Given a pair of reflexive lattice polytopes (, ), the Hodge numbers of the associated CYs (X, X ) are h, (X ) = h dim, (X ) = l( ) dim l (θ ) + l (θ )l (θ) codim(θ )= codim(θ )= (.5) where θ and θ is a dual pair of faces of and. Furthermore, l(θ) is the number of lattice points of a face θ and l (θ) is the number of its interior lattice points.

12 For the case of complete intersections the concept of polar pairs of reflexive lattice polytopes can be generalized as follows: = r =,..., r conv =,..., r conv ( n, m ) δ nm (.6) = r Here r is the codimension of the Calabi-Yau and the defining equations f i = 0 are sections of O(D i ), where D i is the divisor associated to i. The decomposition of the M-lattice polytope M R into a Minkowski sum = r is dual to a nef (numerically effective) partition of the vertices of a reflexive polytope N R such that the convex hulls i conv of the respective vertices and 0 N only intersect at the origin. The nef-property means that the restriction of the line bundles associated to the divisors specified by the N- lattice points to any algebraic curve of the variety are non-negative. A combinatorial formula for the Hodge numbers of complete intersection CYs is available [5]. For smooth toric varieties the topological information besides the Hodge numbers can be extracted from the intersection ring of the toric variety. We assume that X is non-singular. Definition. The intersection ring A (Y ) is defined by A (Y ) = Z[D,..., D r ]/ R, j m, v j D j (.7) The two ideals to be divided out take into account linear and non-linear relations between the divisors. The linear relations have the form j m, v j D j, where m M form a set of basis vectors in the M-lattice. The non-linear relations are denoted by R = R I where the R I are of the form R I = D j... D jk = 0. They come from the exceptional set Z = Z I. The non-linear ideal is called Stanley-Reisner ideal. The total Chern class of the tangent bundle T Y of a toric variety is c(t Y ) = r ( + D j ) (.8) j= To compute the Chern class of a CY hypersurface given by a divisor D = j D j in Y we can make use of the restriction formula that relates the intersection form on the hypersurface divisor to the intersection form on Y : D j... D jn D = D j... D jn D Y (.9) This allows us to compute the intersection ring of D from the intersection ring of Y. In the intersection ring restriction to D amounts to computing the ideal quotient of A (Y ) with the ideal generated by D. Note that this does not yield the full intersection ring of the CY hypersurface. We only get the part coming from the toric ambient space. By adjunction the Chern class for the hypersurface specified by D is c(d) = r ( + D j )/( + D). (.0) j= The Minkowski sum A + B of two sets A, B is defined as follows: A + B = {a + b a A, b B}.

13 In order to be able to calculate all the full topological data from the intersection ring we miss one more ingredient: the Mori cone. The Mori cone is defined to be the dual of the Kähler cone. We need the information about the Kähler cone in order to be able to compute the volumes of divisors. By definition the volumes are positive inside the Kähler cone. The Mori cone is generated by the linear relations l (),..., l (k) (a =,..., k), where k = r n if the fan Σ associated to is simplicial and the toric variety is smooth. Otherwise the number of Mori generators can be larger. The Mori cone L is then defined as follows: L = R 0 l () R 0 l (k). For the calculation of the Mori cone we also require a maximal triangulation of. Each such triangulation corresponds to a different desingularization of the toric variety and leads in general to topologically distinct CYs. Given such a triangulation the Mori generators can be determined [6]. The relations r i= l(a) i D i = 0 define a special basis for the linear relations defining the intersection ring. Assembling the Mori vectors into a k r-matrix, the columns of the matrix encode inequalities for the values of the Kähler parameters. Solving these inequalities yields a basis K i of the Kähler cone such that the Kähler form of Y can be written as J = i r ik i with r i > 0. Note that this prescription computes the Kähler cone of the toric variety Y and not of the CY itself. The combinatorics behind toric geometry makes it possible to systematically construct large classes of CY manifolds. Software to construct and analyze CYs includes PALP [7,8] to construct and analyze lattice polytopes, TOPCOM [9] to compute triangulations of polytopes and the multi-purpose computer algebra packages SAGE [0], Singular [] or Macaulay []. The largest known class is given by the complete list of 47,800,776 reflexive lattice polytopes in four dimensions [] by Kreuzer and Skarke. Each polytope corresponds to a CY hypersurface in the four-dimensional toric ambient space specified by the polytope. There are 0,08 distinct pairs of Hodge numbers. The calculation of the other topological numbers is more involved due to the large number of triangulations of the lattice polytopes in the list. Recently they have been determined for examples up to h, 6 [4]. Therefore it is currently not clear how many topologically distinct CY threefolds arise from the Kreuzer-Skarke list. A full classification of CY threefolds which are complete intersections in toric ambient spaces is presently not known. Beyond toric constructions, several examples exist, most of which are still closely related to toric geometry. Two large classes within this setup are free quotients of CYs in toric spaces by discrete groups [5 8] and CYs which are obtained from toric examples by topological transitions [9 ]. In section 5 we discuss the non-abelian gauged linear sigma model as a new means to construct exotic CYs that are not hypersurfaces or complete intersections in toric ambient spaces. D-branes on Calabi-Yaus and Mirror Symmetry Mirror symmetry is one of the most spectacular and well-studied string dualities. It states that string theory compactified on a CY produces the same 4D physics as string theory on a different CY, called the mirror. This duality is one of the most impressive examples of the intricate mathematical structure behind string theory. Mirror symmetry is therefore also of interest to mathematicians. In this section we list the key features of mirror symmetry with special focus on mirror symmetry for open strings. The boundary conditions of open strings are called D-branes. The study of D-branes in the context of mirror symmetry has a overlap with many current research areas in mathematics.

14 . Mirror Symmetry Mirror symmetry for CYs was first observed in the seminal paper [] for the quintic and its mirror and has since then become a field of research of its own in both mathematics and physics. A detailed account on all the various aspects and viewpoints would go beyond the scope of this summary. Standard references are the books [4] and [5]. We will restrict ourselves to highlight some of the most important facts abut mirror symmetry. Type II string theory has an N = (, ) superconformal field theory on the world sheet of the string. The left-moving sector of the superconformal algebra is generated by the worldsheet energy-momentum tensor T (z), two supercharges G ± (z) and the U() R-current J(z). A second copy comes from the right-moving sector: { T ( z), Ḡ± ( z), J( z)}. The superconformal algebra is invariant under the mirror automorphism [6] (T (z), T ( z)) (T (z), T ( z)) (G ± (z), Ḡ± ( z)) (G ± (z), Ḡ ( z)) (J(z), J( z)) (J(z), J( z)). (.) The isomorphism extends to the rings of chiral and antichiral primary fields denoted by (c, c), (c, a), (a, c) and (a, a) where the two entries correspond to the left- and right-moving sectors. Under mirror symmetry they are exchanged as follows (c, c) (c, a) (a, a) (a, c). (.) The superconformal algebra can be realized in terms of a supersymmetric field theory. Depending on the on the position in the moduli space of the CY, different field theories are accurate. Near a large radius point, where the CY has a geometric description, one can realize the symmetry algebra by a non-linear sigma model. In certain non-geometric regimes, the CY can be described in terms of a supersymmetric Landau-Ginzburg model. At a generic points in the moduli space it is usually not clear which supersymmetric field theory describes the CY. Mirror symmetry is best understood using topological string theory [7, 8]. Topological string theory is a subsector of the full string theory obtained by the topological twist. T (z) T (z) ± J(z) J(z) ±J(z). (.) For the right-moving sector we can either perform the twist with the same signs or with opposite signs, which amounts to two inequivalent redefinitions of the theory, called the A- twist and the B-twist, respectively. The corresponding theories are referred to as A-model and B-model. Mirror symmetry exchanges the A- and the B-model. By redefining the energy-momentum tensor, the spins of the fields and symmetry generators get modified. In particular, one of the supercharges becomes a scalar which is identified as a BRST operator. The twisted theories turn out to be topological field theories, i.e. the energy-momentum tensor becomes exact with respect to the BRST operator. Furthermore the conformal anomaly vanishes due to the topological twist. This makes it possible to define the topological string on target spaces of dimension other than 0. In particular one can define topological sigma models with Kähler target spaces. Supersymmetric localization using the BRST operators Q A and Q B in the A- and B-model shows that the path integral for the A-model localizes on holomorphic maps from the worldsheet to the target space, whereas the B-model localizes on constant maps. Anomaly cancellation in the B-model furthermore requires the target space to be CY. One further finds that the A-model only depends on the 4

15 Kähler parameters of the CY. The B-model only depends on the complex structure moduli. Hence, mirror symmetry exchanges the Kähler and complex structure moduli spaces. Thus, CYs that are mirror to each other have their non-trivial Hodge numbers exchanged. For CYs which are complete intersections in toric ambient spaces mirror symmetry amounts to exchanging the M- and N-lattice polytopes discussed in the previous section. In Batyrev s formula for the Hodge numbers (.5) the exchange of Hodge numbers under mirror symmetry is manifest. Mirror symmetry maps classical quantities into quantum corrected ones. In the topological A-model all quantum corrections are due to instanton corrections. The crucial quantity to compute is the prepotential. In the topological A-model it is the generating function of holomorphic maps of spheres into the CY X. Near a large radius point this quantity looks as follows. F A (t) = c(t) + ñ β q Area(β). (.4) β H (X,Z) Here q = e πit, where t is the Kähler parameter 4, c(t) is a cubic polynomial in t which encodes the topological data that characterizes the CY, β is the image of the holomorphic sphere in X and ñ β counts the number of these maps. These are the Gromov-Witten invariants. Note that they are rational numbers. Taking into account multi-coverings one arrives at integers, as one would expect from a counting problem. In the four-dimensional effective action the prepotential and its higher genus cousins compute F-terms involving vector multiplets [9]. In the B-model the prepotential is a purely classical quantity. We choose a symplectic basis of periods ϖ i = γ i H ( X) Ω, i = 0,..., h, ( X) of the CY X mirror to X, where γ is a basis of three-cycles and Ω is the holomorphic three-form. Restricting again to the one-parameter case, the prepotential is given by a certain linear combination of periods F B (z) = (ϖ (z)ϖ 0 (z) ϖ (z)ϖ (z)), (.5) where z is the complex structure modulus of X and ϖ0 is a power series in z called the fundamental period and ϖ i (z) also contain log-powers up to log i z. Note that the periods will only be of this form at a large complex structure point in the complex structure moduli space that is mirror to a large volume point in the Kähler moduli space of X. The periods satisfy a linear differential equation called Picard-Fuchs equation L P F ϖ i = 0. (.6) Under the mirror map z(t) with t(z) = ϖ ϖ 0 the A- and B-model prepotentials are related as follows F A (t) = ϖ 0 (z(t)) F B (z(t)). (.7) This means that we can compute quantum corrected quantities in the A-model by calculating purely classical ones in the B-model on the mirror CY. The Picard-Fuchs equation can be derived from variation of Hodge structures and the special geometry of the complex structure moduli space [40 4]. Higher genus amplitudes are encoded in the holomorphic anomaly equation [44]. This makes mirror symmetry an important tool to compute quantum corrections in string compactifications. In practice, these rather involved calculations are 4 This holds for CYs with h, =. The generalization to the case with more than one Kähler parameter is straight forward. 5

16 only doable for a small number of moduli and for CYs which are hypersurfaces and complete intersections in toric ambient spaces. One-parameter and two-parameter models have been discussed in [45 48]. While mirror symmetry at the level of superconformal field theory holds independently of the geometry of the CY, explicit calculations are mostly restricted to certain special points in the moduli space of a CY. The complex structure moduli space of a CY is globally defined for all values of the complex structure parameters. The quantities computed for different regions in the moduli space are related by coordinate transformations. The Kähler moduli space however decomposes into chambers corresponding to topologically distinct CYs. The (quantum-corrected) physical quantities in the different chambers are related via analytic continuation. Using the gauged linear sigma model (GLSM) to be discussed in section 5 one can interpolate between the various regions of the Kähler moduli space. Under mirror symmetry all these regions map to different points in the complex structure moduli space of the mirror CY [49]. Away from large radius points, mirror symmetry is not well understood. In some regions in the Kähler moduli space one can describe the CY in terms of a supersymmetric Landau-Ginzburg model. There one does not have a geometric interpretation of the CY as a hypersurface or complete intersection in some ambient space. Using the GLSM there is a mirror construction due to Hori and Vafa [50] that maps the large radius point of a geometric CY to the Landau-Ginzburg point of the mirror. Complete intersection CYs and models with more than one Kähler parameter have regions that are hybrids of geometric and Landau- Ginzburg descriptions. The knowledge about such hybrid models and how mirror symmetry works for them is very limited. Explicit mirror constructions are best understood in the toric setting where both the Hori- Vafa mirror construction and the mirror construction due to Batyrev and Borisov based on reflexive lattice polytopes work [5]. Further mirror constructions that are not necessarily restricted to geometric regions in the moduli space are due to Berglund and Hübsch [5] and Greene and Plesser [5]. Beyond the toric framework not much is known. There are several mirror constructions in the mathematics literature which rely on the existence of toric degenerations to construct mirrors, (see e.g. [9, 0, 5, 54]). The most general statement of mirror symmetry is in terms of equivalences of suitably chosen categories related to the CY. In order to understand this within a physics framework one has to consider open strings and D-branes.. D-branes D-branes are boundary conditions for the open string. The presence of a boundary breaks translation invariance and consequently also supersymmetry. In order to maintain some control over the theory, it is desirable to maintain at least some amount of supersymmetry. In N = (, ) superconformal theories there are two inequivalent choices of boundary conditions which preserve one half of the supersymmetry. They are called A-type and B-type boundary conditions. These are also compatible with the A-twist and the B-twist, respectively. At the level of the superconformal algebra, A-type boundary conditions correspond to the following constraints on the currents at the boundary z = z [55] T (z) = T ( z) G + (z) = ±Ḡ ( z) G (z) = ±Ḡ+ ( z) J(z) = J( z). (.8) B-type boundary conditions are given by T (z) = T ( z) G + (z) = ±Ḡ+ ( z) G (z) = ±Ḡ ( z) J(z) = J( z). (.9) 6

17 The worldsheet boundary conditions get mapped into non-perturbative objects in the target space the D-branes. Depending on the position in the moduli space and the type of supersymmetry preserved, different mathematical and physical methods are required to characterize D-branes. The situation is best understood near a large-radius/large complex structure point where the theory can be described in terms of a non-linear sigma model. A standard reference for D-branes in CY threefolds is [56]. A topological A-brane wraps a Lagrangian submanifold L. This means that the Kähler class ω restricted to L is zero: ω L = 0, the gauge connection F on L is flat, i.e. F = 0. Furthermore anomaly cancellation requires that the Maslov class is trivial. If we do not restrict to the topological string, additional stability conditions have to be imposed which entail that A-branes are special Lagrangian. This means that Ree iθ Ω L = 0 with Ω the holomorphic threeform and θ a phase. B-type D- branes wrap holomorphic cycles in the CY. More generally they are coherent sheaves, which have to be stable in the non-topological setting. In a mathematical framework D-branes are objects in suitably chosen derived categories. A-branes are objects in the Fukaya category associated to the CY. B-branes are objects in the derived category of coherent sheaves. The physical properties of D-branes translate into the mathematical structure of categories in a straight forward manner. Open strings stretching between D-branes, for instance, correspond to morphisms between corresponding objects in the category. Bound states of D-branes are formed through tachyon condensation which is captured by the triangulated structure of the categories. D-brane categories are also A -categories. This structure accounts for Ward identities of open string amplitudes [57]. The categorical language also makes is possible to treat single D-branes (i.e. the boundary preserving sector) and intersecting branes (i.e. the boundary changing sector) on equal footing. Furthermore various charges and gradings of D-branes can be easily incorporated. While B-branes are well-studied, A-branes on compact CY threefolds are a notoriously different subjects. By now the mathematics of the Fukaya category is quite well-understood, however at a rather abstract level. Explicit examples of A-branes on CY threefolds are still quite rare. Away from the large radius point, B-type D-branes have a nice description in topological Landau-Ginzburg theories. In this case D-branes are matrix factorizations of the Landau- Ginzburg potential [58, 59]. Given a supersymmetric Landau-Ginzburg B-model with holomorphic potential W (x,..., x N ), a matrix factorization is a square matrix Q with polynomial entries satisfying the condition Q = W. (.0) The advantage of the description of D-branes in terms of matrix factorization is that it is well-suited for working with explicit examples. For details on the many uses of matrix factorizations we refer to [60]. Matrix factorizations are also of great importance, even if the moduli space does not have a Landau-Ginzburg point. Also boundary conditions in gauged linear sigma models (GLSMs) are identified to be matrix factorizations [6, 6]. The GLSM is the UV completion of the superconformal field theories describing string theory in CY spaces. Therefore matrix factorizations are the high energy description of D-branes throughout the Kähler moduli space. We will come back to D-branes in GLSMs in section 5... Recently, matrix factorizations have also have found applications in the context of F-theory [6]. D-branes come with extra moduli parametrizing possible deformations of the D-brane. In general one cannot deform a D-brane without breaking supersymmetry. Therefore simultaneous open-closed deformations are typically obstructed. This generates a potential the effective superpotential W eff which encodes the obstruction to the deformation of a 7

18 D-brane. The effective superpotential can be interpreted in various ways W eff is the effective superpotential in 4D N = compactifications of string theory. W eff encodes the obstruction to open/closed deformations of D-branes. W eff is the generating function of open string disk amplitudes. The effective superpotential is one of the key quantities in string compactifications with D- branes. It can be computed using topological strings and mirror symmetry.. Mirror Symmetry for D-branes Mirror symmetry extends to the open string sector. The homological mirror conjecture [64] states that the Fukaya category associated to a CY X is equivalent to the bounded derived category of coherent sheaves of the mirror CY X. This statement is true for the geometric setting and has since been generalized in many directions. Under mirror symmetry A-branes and B-branes get exchanged. Like for closed strings, the open/closed topological B-model is a purely classical theory. Special geometry also extends to the open string sector [65 67] and the effective superpotential can be computed by solving extended Picard-Fuchs equations. There are two ways to compute the effective superpotential, corresponding to an on-shell and an off-shell calculation. Since the effective superpotential encodes the obstructions to the deformations of the brane, its critical set, i.e. the F-term equations, will fix the open moduli in terms of the closed ones 5. The different solutions correspond to the different vacua of the theory. Having discrete set of vacua one can compute the domain-wall tension defined as T = W eff (vac ) W eff (vac ). This is the actual physical quantity one has to compute. Its mirror is the generating function of open Gromov-Witten invariants [68]. Since the result only depends on the closed string moduli, one does not have to derive an open mirror map. This approach was used in [4, 66, 69]. The off-shell approach computes the effective superpotential without making explicit use of the F-terms. The result depends on the open and closed moduli and one also gets a mirror map for open strings. Computing the F-terms and inserting them back into the effective potential, one recovers the domain-wall tension. Note however, that the off-shell effective superpotential is only defined up to field redefinitions, corresponding to gauge transformations or, mathematically speaking, to A -morphisms of the associated D-brane category. The off-shell approach to compute the effective superpotential was used in [67, 70 79]. As this is relevant for the present work, we give a few more details on the on-shell approach to compute W eff. In the one-parameter case the domain-wall tension in the A-model for a Lagrangian L α (with α denoting the flat connection) is defined as T A,α (t) = t + T class,α + D H (X,L α,z) ñ D q Area(D), (.) where H (X, L α, Z) is the relative homology group labeling the classes D of the image of the holomorphic disks ending on the Lagrangian. The classical term contains topological information about the Lagrangian and is independent of the Kähler parameter t at large 5 Note that not all brane moduli are obstructed. Those associated to D0- and D6-branes will not be constrained by deformations of the complex structure. Lacking obstructions, they will either not appear at all in the effective superpotential, or the F-terms will not put any constraints on them. 8

19 radius. The ñ D are the open Gromov-Witten invariants. In the B-model the domainwall tension is T B,α = W α (z) W 0 (z), (.) where W is the effective superpotential which is the restriction of the holomorphic Chern- Simons functional with hermitean connection α to the D-brane and z is the complex structure modulus at the large complex structure point. W 0 (z) denotes the effective superpotential with respect to some reference connection. To compute the domainwall tension in the B-model one uses that T B,α = ν Cα (Ω) = Ω, (.) where ν Cα is the truncated normal function [66]. It is defined as an integral of the holomorphic threeform over a three-chain Γ. The choice of three-chain amounts to a choice of B-brane. The key observation is that the normal function satisfies an inhomogeneous Picard-Fuchs equation L P F T B,α (z) = f α (z), (.4) where the differential operator is the same as for the closed string case 6. Using mirror symmetry one can then compute the instanton-corrected domainwall tension in the A-model Γ T A,α (t) = ϖ 0 (z(t)) T B,α (z(t)), (.5) where z(t) is the closed string mirror map. A generalization to higher-genus open string amplitudes is also known [80]. At a Landau-Ginzburg point one can compute the effective superpotential using the deformation theory of matrix factorizations [5, 60, 8, 8]. Since the open/closed mirror map at the Landau-Ginzburg point is not known one first has to rewrite the the effective potential in terms of the large complex structure variable z. Equivalently one can also transport the D-brane itself using [6], extract the corresponding three-chain and compute the D-brane tension as described above. The latter approach was used in [4,66]. So far this has only been applied for one-parameter CYs. A generalization to the two-parameter case for the effective potentials computed in [5] is still missing [8]. 4 Calabi-Yaus in F-theory and Model Building F-theory is a non-perturbative formulation of type II string theory that is related to CY fourfolds. It plays a prominent role in the context of string dualities. Recently it has been used to construct Grand Unified Theories. This has lead to significant progress in string phenomenology and model building. This section gives a short overview over the basics of F-theory and its recent applications in phenomenology. 4. Calabi-Yaus, F-theory and Phenomenology F-theory [84] is a strong coupling limit of type IIB string theory, more specifically type IIB with D/D7 branes 7 and O/O7 orientifold planes. To explain the basics, we will mostly 6 This is no longer true for the off-shell case, where one has a system of homogeneous Picard-Fuchs equations that depends on both open and closed moduli. 7 In contrast the previous section, we also count the three non-compact spatial dimensions when referring to a Dp-brane. 9

20 follow the review article [85]. In perturbative string theory the backreaction of the branes on the geometry is typically neglected, which is justified if it goes to zero at asymptotic distances from the brane. This is not true for D7-branes. The backreaction of the D7 is included in the following way. The Ramond-Ramond field sourced by the D7-brane is an 8-form field C 8. In ten dimensions this is dual to a zero-form C 0, which is related to the axion in the compactification to four dimensions. In 4D supergravity C 0 is combines with the dilaton φ to the axio-dilaton τ = C 0 + ie φ g s = e φ, (4.) where g s is the string coupling constant. The Ramond-Ramond fields of a p-brane obey a Poisson equation in the spatial directions normal to the brane. For a seven-brane at z = z 0 the solution to the Poisson equation has logarithmic behavior τ(z) = πi log z z 0 λ e φ = π log z z 0 λ, (4.) where λ is an integration constant. This shows that the string coupling is varying and may become large so that perturbation theory breaks down. Due to the singularity at z = z 0, τ(z) is subject to monodromies. If one circles the D7-brane in the z-plane τ transforms as τ τ +. In order to be consistent with S-duality of type IIB, τ has to transform under the S-duality group SL(, Z): τ aτ + b cτ + d a, b, c, d Z, ad bc =. (4.) Since SL(, Z) is the modular group of a two-torus, one can give the varying axio-dilaton a geometric interpretation by viewing it as the complex structure of a torus. Type IIB string theory with varying coupling is then replaced by F-theory on an elliptically fibered CY fourfold. The advantage is that the SL(, Z)-invariance is automatically built into this formulation. The extra two dimensions should not be viewed as space-time dimensions but rather as the parameter space of the string-coupling. Hence, there is no contradiction with the ten-dimensional space-time required by an anomaly-free perturbative string theory. Due to its non-perturbative nature, there is no worldsheet formulation and no action for F-theory. In order to construct F-theory geometries we need to describe elliptic fibrations. Every elliptic curve can be brought to Weierstrass form by a suitable coordinate transformation: y = x + f xz + g z 6, (4.4) where f, g distinguish between different elliptic curves. In an elliptic fibration f, g get promoted to functions of the coordinates y i of the base manifold: The elliptic fiber degenerates at zeros of the discriminant y = x + f(y i )xz + g(y i )z 6. (4.5) = 4f + 7g. (4.6) Degenerations of elliptic curves have been classified by Kodaira [86]. The singularities are connected to simply-laced Lie groups of type ADE. The degeneration loci of the elliptic fibration correspond to the positions of the seven-branes. At this locus there is a gauge 0

21 degenerate ell. fiber elliptic fiber GUT brane S Threefold Base B (non CY) Figure : F-theory GUTs symmetry with gauge group determined by the singularity type of the degenerate elliptic fiber [87, 88]. Triggered by [89 9] F-theory has come into focus as a means for constructing Grand Unified Theories (GUTs) and from there string models beyond the Standard Model. In contrast to conventional GUT theories the GUT is not in four space-time dimensions but in eight. It is located on a divisor S of the base manifold B of an elliptically fibered CY fourfold at the locus where the elliptic fiber becomes singular, see Figure. The theory can then be described locally as a supersymmetric gauge theory on the on S. The symmetry can then be broken further to arrive at a standard model-like theory on four dimensions. Constructing GUTs from F-theory has several advantages compared to other methods within string theory. In the heterotic string it is easy to construct higher rank gauge groups from the E 8 E 8 group. However, since both gauge and gravitational degrees of freedom come from the closed string sector, is is difficult to decouple gravity from gauge theory. This decoupling is easily achieved in a type IIA/B setting where the gauge degrees of freedom are localized on the world-volume of D-branes. However, higher rank gauge groups are realized by stacks of branes and O-planes and it seems more natural to directly construct the Standard Model from D-brane configurations than taking the detour via a GUT model. F-theory achieves in many respects the best of both worlds as it combines Grand Unification with localized gauge degrees of freedom. In order to realize a specific GUT from a Weierstrass model it is practical to use Tate s algorithm [9, 9]. A general Weierstrass model can be locally written in Tate form P W = x y + xyza + x z a + yz a + xz 4 a 4 + z 6 a 6, (4.7) where the a n (y i, w) are sections of the anticanonical bundle K n B recover the Weierstrass model one defines of the base manifold. To β = a + 4a β 4 = a a + a 4 β 6 = a + 4a 6. (4.8) The functions f, g of the Weierstrass model are then: f = 48 (β 4β 4 ) g = 864 ( β + 6β β 4 6β 6 ) (4.9)