We study aspects of the geometry and physics of type II string theory compactification on Calabi-Yau manifolds. In the second chapter we review some basic principles of the classical geometry of Calabi-Yau manifolds.
Chapter 2 Geometry and Physics of
Calabi-Yau Manifolds: the view from the worldsheet
Why Calabi Yau Manifolds?
In principle, such a tensor always exists if X is a complex manifold. *4• The more interesting question is whether a given real manifold X is also a complex manifold, ie whether one can find a globally defined complex structure f on X. The consequence of the Calabi-Yau condition is that the holonomy group of the manifold is at most SU(3).
- Non-linear Sigma Model
We can further show that the renormalization does not break the Ricci flattening of the metric. The rest of the massless spectrum can be obtained from the (R, R) ground states with spectral fluxes.
Moduli Spaces
- Families of Calabi-Yau Manifolds
- Families of CFT's
The space of allowed J;s, the classical Kahler structure moduli space, forms a cone called the Kahler cone. The moduli space of conformal field theories on X is the augmented moduli space of Calabi-Yau manifolds, .
Stringy Geometry
Given the base { ei} of H1•1, the space of modules of complex Kahler forms can be parameterized as Therefore, as far as conformal field theory is concerned, the moduli space of a complex structure is exact at the tree level.
Mirror Symmetry
For an appropriate choice of i's, there is a restriction in the moduli space of the complex structure, so that the classical correlation functions on Mx::(X) match those in Mc(X) [16]. In general, however, correlation functions are power-type in z/s whose coefficients are integers counting rational curves on X.
- View From Low Energies
- Massless Fields and Constraints from Supersymmetry
- Mirror Symmetry in Type II String Theory
- Effective Action and Singularities of Calabi-Yau Man- ifolds
- Application: Geometry for Physics of N
It is a consequence of Ricci's flatness that the holonomy group of Xi is a SU(3) subgroup of SU(4). We will consider the vector multiplet moduli space in type IIB theory, as we will have the added advantage that the result is exact.
SUSY Gauge Theories
Enhanced Gauge Symmetry in Type II String Theory
K3 is two complex dimensional Calabi-Yau manifold, and the resulting theory has two s1x-dimensional supersymmetries*13• The singularities that K3 can develop are extremely limited. In type IIA theory there is a D 2 brane that can wrap a vanishing 52, to obtain a particle in 6 dimensions*14• In type IIB, on the other hand, the "smallest" brane that can wrap ' n D 3- branes, leaving an almost tensionless string in six dimensions.
Solving Gauge Theories via Geometry
The intersection shapes of these spheres give the entries in the Cartan matrix of the corresponding gauge group (50(10) for D5. Thus it is clear that the modulus of the Kahler structure is related to the coordinates in the Coulomb branch of the gauge d = 4 theories in the view of type IIA.
Mirror Manifolds via Torie Geometry
Mirror symmetry, if it holds, must exchange the complex and Kahler moduli, and therefore must exchange the role of the two polyhedra [41]. The corners of the double polyhedron encoding the Kahler structure of the explosions on the type IIA side determine the complex structure of the mirror type IIB manifold. Thus the classical description of the complex structure moduli space of type IIB Calabi-Yau is exact.
Since these modules encode gauge theory behavior, we can read the exact solutions from the IIB manifold. In the first two cases we find exact solutions using the mirror construction by Batyrev [41].
Exact Solutions from Mirror Symmetry
- S0(7) with (3 - n)7 + (8 - 2n)8
- S0(9) with ( 4 - n)16 + (5 - n)9
- SO(ll) with ( 4 2n)32 + (7 - n)ll
This gives a description of the Calabi-Yau in some local coordinate patch with one residual C* action. It comes down to choosing a patch in which the relevant characteristics of the Calabi-Yau are most easily described. The analysis of the previous subsection can be repeated for 50(12) with r half hypermultiplications in the 32, (4-n-r) half hypermultiplications in the 32' and 8-n fundamentals.
Recall that on page IIB the field theory information is encoded in complex structural modules, which then determine the periodic integrals over the three Calabi-Yau cycles. For 50(9), Batyrev's construction gives a description of a mirror in which the content of the matter of the theory is seen in fibration above the lower sphere.
Summary and Concluding Remarks
In the other two asymptotically free cases n = 2, 3, we also get a curve that is not hyperelliptic. We can probably get an exact solution for any number of massive vectors and spinors by substitution. Calabi-Yau manifolds and groups of higher rank SO(N), N cannot be obtained by type IIA compactification on the Calabi-Yau triple or vice versa by breaking the adjunct E8 on the heterotic side.
The results presented may ultimately provide some insights into how to construct matter representations other than fundamentals and two index tensors from branes. Since our solutions agree with known field theory results, where available, the results of this paper can be seen as further confirmation of mirror symmetry and the duality between type IIA and heterotic strings.
- Calabi-Yau via D Brane Probes
- Introduction to Orbifolds (with Discrete Tor- sion)
- Closed String CFT on Orbifolds
- Generalities of "Ordinary" Orbifolds
- A Z 2 x Z 2 Example
- Discrete Torsion
- Z2 x Z 2 Theory with Torsion
- General Zm x Zn Case
- Interpretation of the Moduli
- Open Strings on Orbifolds
The quotient is taken in the usual sense to identify the points on the paths of the f action, a point x E M is identified with gx E M. Field theory on M/f is simply defined in terms of truncation of the theory on M to . One final note: in the case of an orbifold without torsion, the spectrum we calculated corresponded to the inflation of the orbifold.
In the "regular" orbifold, the CFT spectrum matches the spectrum on the orbifold magnification. In the language of D-branes, space-time itself appears indirectly, as the space of gauge theory modules that lives on the world volume of D-branes.
4. 7 Open String Orbifolds with Torsion
Moduli Spaces
- Resolution of Singularities in the Moduli Spaces
The open-string counterpart to this is the ability to smooth the singularities in the moduli spaces via superpotential distortions, or by turning on FI parameters. The deformations of the superpotential cause the complex structure of moduli space to change. When qs = qt = 1, there are distortions of the theory via W -t W + ~W that preserve supersymmetry and resolve singularities of moduli space.
Toric affine varieties have the property that the complex structure of the variety defines the space of possible Kahler structures on it. The way to describe the Kahler structure of the affine toric variety is to write it as a symptomatic coefficient, and this is what we will do next.
Discussion
The only certainty is that singularities of the geometry cannot be singularities in M-theory, simply because they are not singularities of type II string theory either. Finally, [53] also shows that there are still other solutions of orbifolds that we have not found in CFT. This is not so puzzling, since it is likely that these spaces are solutions of geometric orbifolds, but not in any way connected to any strict CFT orbifold.
Chapter 5 Mirror Symmetry, Brane
Configurations and Branes at Singularities
Background: Quantum Mirror Symmetry
Quantum mirror symmetry, as explained in [5], means not only that the BPS spectra of a type IIA theory on M and a type IIB theory on mirror W are the same, but that the equality of full theories, including interactions, also requires the equality of the moduli spaces of these facilities. By swapping the roles of M and W, i.e. the wrapping of D 3-branes on the fibers of M, we conclude that W is a fibration of T3 also with special Lagrange cycles, but the fibers of W are T-dual tori of T3 *1• Finally, the bases of both fibers must be. 1 The space of modules of flat connections on a torus is a dual torus that has the inverse metric of the original one.
In the rest of the chapter, we discuss T-duality and mirror symmetry for type II string theory on a certain class of Calabi-Yau spaces following [6]. In our view, the geometry will mainly serve as background, and we will study the gauge theories that live on D-branes that probe or enclose cycles in the manifolds, in accordance with the general discussion above.
Introduction to Mirror Symmetry, Brane Con- figurations and Branes on Singularities
One of the reasons why we are interested in these types of spaces is that the theory about sin is known. The theory of a D3 brane on C /f is then defined as a quotient of the theory on C by f*3. In the case of the conifold, we can deform either the complex structure of the space or its Kahler structure to obtain a smooth manifold.
The D3-brane theory constructed in [49] is the theory on the Kahler side of the conifold. We will find that the brane box is the natural dual of the blow-up of the orbifolded conifold and of the deformed generalized conifold.
The Geometries
- Conifold
- More General Singularities
In the next section, we will introduce the relevant geometries, namely the conifold singularities and the orbifold singularities and their generalizations. In the fourth section, we will discuss the gauge theories appearing on the D3 brane probes. There are many ways to display the small resolution of the conifold.
One can get a more physical interpretation of what we have done, which stems from the observation that the description of the conifold we have arrived at above is precisely the description of a Higgs branch of a particular linear sigma model. It is clear that the Qs are to be interpreted as the accusatives of the case fields under then U(l)s.
Hyperquotient Singularities
Another way to express the effect of the quotient is in terms of gauge invariant monomials. However, they will not contribute to the resolution of the singularity, but only change it by irrelevant pieces.). We will not attempt to specify the precise region of Kahler structure moduli space where the solution resides, which would be equivalent to choosing a triangulation of the toric diagram, because we do not need this information.
The correct interpretation of this is that we are exploring the region of the Kahler structure moduli space where the four cycles associated with this point in the toric diagram become large enough that they become in fact irrelevant to the local physics - the associated vectors can be dropped entirely. The conventional orbifold Oct We should find that the inflation of Ykt is mirrored by the deformation of Ckt. The blowout of the generalized conifolcl is Tu dual to NS branes separated along 67 (the interval). These, in turn, are tv double to the mirror, the deformation of the orbifoldecl conifold. Similarly, blowing up the orbifolcled conifold will dualize Tv into a box and then clualize Tu into the mirror, deforming the generalized conifold. Instead of the two extra scalars in the hypermultiplet, this time we see a vector from the RR 3 form on the . The Kahler (resp. complex structure) parameters of the geometric singularities correspond to positions of the NS branes in the double brane picture. We formulate rules for deriving the matter content of the gauge theories that live on boxes and diamonds. Along a baryonic branch of the gauge theory, which corresponds to the partial solution of the conifolds Ck1 to the orbifold singularities Oki, we recover the. This observation may be useful for investigating the non-perturbative quantum dynamics of these kinds of N = 1 gauge theories: that is, for every supersymmetric 2-cycle describing the dynamics of the interval theory embedded in M-theory, there must exist a supersymmetric mirror 3-cycle for brane box theory, also included in M-theory. Furthermore, one can expect that due to quantum corrections, the physics of gauge theories at the conifold point is not as special as in the classical description we have discussed. Morrison, “Calabi-Yau moduli space, mirror manifolds and space-time topology change in string theory,” Nucl.Mirror Symmetry
5. 7.1 The Mirror Branes
Summary
Bibliography
