- About this Journal ·
- Abstracting and Indexing ·
- Advance Access ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents
Advances in High Energy Physics
Volume 2011 (2011), Article ID 404691, 18 pages
Discrete Wilson Lines in F-Theory
School of Theoretical Physics, Dublin Institute for Advanced Studies, 10 Burlington Road, Dublin 4, Ireland
Received 15 May 2011; Accepted 3 July 2011
Academic Editor: Yang-Hui He
Copyright © 2011 Volker Braun. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
F-theory models are constructed where the -brane has a nontrivial fundamental group. The base manifolds used are a toric Fano variety and a smooth toric threefold coming from a reflexive polyhedron. The discriminant locus of the elliptically fibered Calabi-Yau fourfold can be chosen such that one irreducible component is not simply connected (namely, an Enriques surface) and supports a non-Abelian gauge theory.
F-theory  is a way to use geometry as a tool to understand certain compactifications of string theory that are otherwise not entirely geometric . It uses an auxiliary elliptically fibered Calabi-Yau fourfold, not to be confused with the space-time manifold to study string theory in a regime away from any known weakly coupled perturbative description. Recently [3, 4] a particular model building Ansatz has been suggested where the GUT gauge group arises from a 7-brane wrapped on a contractible del Pezzo surface. Various models [5–10] and more have been constructed along these lines.
One key feature of this Ansatz is that the scales of gravity and gauge physics can be decoupled as one can decompactify the Calabi-Yau manifold without changing the del Pezzo surface. However, the price one has to pay for this is that the usual way of GUT symmetry breaking in string theory, namely, the Hosotani mechanism [11, 12] using discrete Wilson lines, no longer works: all del Pezzo surfaces are simply connected. Alternatives have been developed [3, 4, 13], but require one to give a vacuum expectation value to fields locally and not just make global nontrivial identifications. Turning on fields locally then affects the running of the coupling constants and, potentially, defocuses the gauge coupling unification [14, 15].
In this paper I will advocate for a different Ansatz for GUT model building and symmetry breaking in F-theory, namely, by wrapping the GUT 7-brane on a nonsimply connected divisor in the base of the elliptic fibration. This allows one to choose a globally nontrivial identification of the gauge bundle while keeping it locally trivial, breaking the GUT gauge group by the usual Hosotani mechanism. For what it is worth, this setup also implies that there is no gravity/gauge theory decoupling limit.
Of course this raises the question of whether there are any such divisors in threefolds that are suitable as bases for elliptically fibered Calabi-Yau manifolds. In this paper I will answer this question and work out a rather simple example of an Enriques surface embedded into a toric threefold associated to a reflexive 3-dimensional polytope. There is nothing particularly unique about this example. It just combines the most simple surface with fundamental group and the class of threefolds we are most used to work with. All toric geometry computations used in this paper were done using [16–18].
2. Base Threefold
An Enriques surface is a free quotient of a surface by a freely acting holomorphic involution and is probably the best-known example of a complex surface with fundamental group . Its first Chern class is the torsion element in , so it admits a Ricci-flat metric but has no covariantly constant spinor (equivalently, no covariantly constant -form). Some, but not all, surfaces can be realized  as quartics in . Somewhat unfortunately, the locus of quartic s and the locus of surfaces with an Enriques involution do not intersect in the moduli space of smooth surfaces. In other words, no smooth quartic in carries an Enriques involution. Therefore, out of necessity one is forced to look at singular (birational) models and then resolve these singularities. This will be the central theme in the following.
To explicitly construct and resolve the singularities, I will make extensive use of toric geometry. However, before delving into these technical details let me first give an overview. The basic idea is to look at the following action on : The fixed point set of are the two disjoint rational curves and the fixed points of are the north and south poles on these (4 points altogether). A sufficiently generic invariant quartic is then a (singular) Enriques surface on the quotient (this construction is rather similar to the way to construct nonsimply connected Calabi-Yau threefolds [20–24], except that I will not be looking at sections of the anticanonical bundle (which would be Calabi-Yau)). The fastest way to see this is to note that the would-be -form is projected out by .
Here is where this paper essentially begins, because so far we only have a singular Enriques surface in an even more singular ambient space. Clearly, one wants to resolve the singularities. The first step is to resolve the curves of singularities, for which there is a unique crepant resolution. Then one has to deal with the remaining 4 singularities. By a happy coincidence, the above quotient of is itself a toric variety. Hence, the methods of toric geometry can be applied and allow us to construct partial and complete resolutions explicitly as toric varieties.
2.2. Toric Geometry
As a warm-up, I will first review some basic notions of toric geometry. The defining data is a rational polyhedral fan in a lattice , where is the complex dimension of the variety. A fan is a finite set of cones , closed under taking faces. Often, the fan will be the cones over the faces of a polytope. This is called the face fan of the polytope.
Amongst the different, but equivalent, ways to define the corresponding complex algebraic variety from the fan data, I will use the Cox homogeneous coordinate  description in the following. The basic idea is to associate one complex-valued homogeneous coordinate to each ray (one-dimensional cone) of the fan. Then one has to remove a codimension-2 or higher algebraic subset and mod out generalized homogeneous rescalings. This construction will be reviewed and applied in much more detail in Section 2.4. For now, let us just consider as an example. Its fan consists of the cones where , , are a basis for . There are 4 one-dimensional rays satisfying a unique linear relation, which translates into 4 homogeneous coordinates with the usual identification A map of fans is a map of ambient lattices such that every cone of the domain maps into a cone of the range fan. Any such fan morphism defines a morphism of toric varieties in a covariantly functorial way. For the purposes of this paper (except for A), we will only consider the case where the lattice map is the identity. In this case the domain fan is simply a subdivision of the range fan. The toric map corresponding to a subdivision of a cone is the blow-up along a toric subvariety of dimension equal to .
A toric divisor (all divisors in this paper will be Cartier divisors, even though we will be working with auxiliary singular varieties where not all divisors are Cartier) is a formal linear combination of the codimension-one subvarieties corresponding to the one-dimensional cones of the fan. There are two basic constructions associated to such a toric divisor that will be important in the following. (i) Every coefficient can be thought of as the value of a function on the generating lattice point of the -th one-cone. If every cone is simplicial, then there is a uniquely defined continuous function on the fan with the above property. The pull-back of the function on the fan corresponds to the pull-back of the divisor by the toric map. (ii) The divisor also defines a polytope where is the dual lattice. The global sections are in one-to-one correspondence with the integral lattice points and can easily be counted for any given divisor.
A particularly relevant divisor is the anticanonical divisor . Given a lattice polytope , we can construct its face fan and the polytope . If is again a lattice polytope, then is called reflexive.
Finally, note that . Hence, the line bundles on are classified by a single integer, their first Chern class. The toric divisors, on the other hand, are defined by 4 integers. Clearly, there is no one-to-one correspondence between divisors and the isomorphism class of the associated line bundle . To make this into a bijection, one must mod out linear equivalence of divisors. That is, one has to identify the piecewise linear functions modulo linear functions. In particular, one can easily see that on .
2.3. Three Birational Models
We now begin with the core of this paper and define the base threefold of the elliptically fibered Calabi-Yau fourfold. In fact, I will choose a smooth toric variety as the base manifold, containing a nonsimply connected divisor . However, directly analyzing will be overly complicated. In particular, contains exceptional divisors that do not intersect the divisor we are interested in. Therefore, to better understand , I will blow down these additional exceptional divisors. This will produce a singular variety containing the same divisor . Finally, I will blow down two more curves in to obtain a (even more singular) three-dimensional variety . The blown-down divisor is the most suitable one to compute the fundamental groups. To summarize, I am going to define successive blow-ups of three-dimensional toric varieties. Both of the maps are toric morphisms defined in the obvious way by combining cones of the fan into bigger cones, discarding all rays that are no longer part of the more coarse (blown-down) fan.
I now define the fans corresponding to the toric varieties. Let me start by the rays . The most singular variety has see Figure 1. The convex hull of these four points is a tetrahedron, but not a minimal lattice simplex. In addition to the origin (which is an interior point), it contains the two points and along two different edges. The variety will be the maximal crepant partial resolution of , that is, the (in this case unique) maximal triangulation of the convex hull . Hence, one must add the additional integral points to the ray generators, Neither the variety nor its maximal crepant partial resolution is smooth, related to the fact that the polytope is not reflexive. One again needs to add rays to resolve all singularities; however these time the generators are necessarily outside of . One particular choice I am going to make is the rays generated by the 18 points listed in Table 1.
The 8 facets of the convex hull are given by the inequalities which is, therefore, a reflexive polytope.
Finally, to completely specify the toric varieties , , and , let me define the generating cones of the respective fans: (i) is the face fan of the polytope , see Figure 1; (ii) is the unique maximal subdivision of ; (iii) is a maximal subdivision of the face fan of the polytope , see Figure 2.
As there are many different maximal subdivisions of the face fan, this alone does not uniquely specify the fan . For concreteness, I will fix the one listed in Table 3. Note that not all combinatorial symmetries of the graph in 3 are actually symmetries of the fan.
2.4. Homogeneous Coordinates
For future reference, let me list the toric Chow groups : Since all three toric varieties have at most orbifold singularities, the Hodge numbers are and if .
The appearance of torsion in the Chow group slightly complicates the Cox homogeneous coordinate  construction of the toric varieties, so let me spell out the details. In general, a simplicial (that is, with at most orbifold singularities) -dimensional toric variety can be written as a geometric quotient where is the number of rays in the fan . The exceptional set is the variety defined by the irrelevant ideal. A more catchy way of remembering is that it forbids homogeneous coordinates from vanishing simultaneously if and only if their product is a monomial in the Stanley-Reisner ideal. The latter is It remains to describe the groups in the denominator of (2.14). For the most singular variety , one finds and for the intermediate blow-up
2.5. A Nonsimply Connected Divisor
I am now going to define a divisor in the same linear system as the toric divisor ( denotes the toric divisor associated to the -th ray) that is, as the zero set of a sufficiently generic section of the line bundle . (By ~, we will always denote rational equivalence of divisors. That is, means that there is a one-parameter family of divisors interpolating between and . Equivalently, the Chow cycle defined by and is the same.) A basis for the sections is corresponding to the points of the polytope Note that the fan is precisely the normal fan of the Newton polytope . In this sense, is the “natural” ambient toric variety for the surface .
For explicitness, let me fix once and for all a linear combination of the monomials as the defining equation of the divisor . I will select the vertices of the Newton polytope and define This surface is known to be an Enriques surface since it projects out the potential -form as mentioned in Section 2.1. In fact, this example has been known for some time, see Remark 3.6 in .
2.6. Kähler Cone and Canonical Divisors
The content of this subsection is not necessary for the understanding of the paper, but I would like to pause for a moment and mention how the “Fermat quartic” in (2.21) fails to define a surface. In other words, how does the divisor differ from the anticanonical divisor of ? Comparing with , see (2.8), one might have thought that they were linearly equivalent.
Similarly to the quartic , one can also define a Calabi-Yau variety in as the zero locus of a section of the anticanonical bundle (after resolution of singularities, the anticanonical divisor will be a smooth 2-dimensional Calabi-Yau manifold, that is, again a surface). The available sections are Note that this differs from the sections of , see (2.19). Therefore, the two divisors are not linearly equivalent. Nevertheless, and are very close to being linearly equivalent. In fact, it is easy to see that they are in the same rational divisor class since the rational divisor class group is one-dimensional, . (The toric divisor class group of a variety is often written as . I will not use this notation in the following, but opt for instead.) However, their difference is a 2-torsion element in the (integral) divisor class The same is true on the crepant partial resolution, where is again a 2-torsion element in .
On the final smooth resolution the divisor class group is torsion-free. However, the last blow-up is not crepant, so Therefore, the divisors and are no longer in the same rational equivalence class.
Finally, let me describe the Kähler cones of these varieties. First, let me remind the reader that the Kähler cone of a toric variety is an open rational polyhedral cone in the rational divisor class group corresponding to the cone of convex piecewise linear support functions on the fan. For the two singular varieties, one obtains As the anticanonical class is rationally equivalent to , we see the following. (i) is a (singular) Fano variety. (ii) is not Fano, but the anticanonical class is on the boundary of the Kähler cone. In other words, is nef but not ample.
On the smooth blow-up , the Kähler cone is rather complicated, and we will refrain from listing it explicitly. It is spanned by the origin and 169 rays and has 20 facets (that is, 14-dimensional faces). The anticanonical divisor as well as sits on the boundary of the Kähler cone, that is, is nef but not ample. However, each satisfies a different subset of 16 out of the 20 facet equations, so they lie on different faces of the Kähler cone.
2.7. Pull-Back Divisors
By the usual dictionary of toric geometry, the toric divisor corresponds to a continuous piecewise linear function on . Explicitly, the function is The pull-back of this toric divisor by the toric morphisms and is simply given by the pull-back of the piecewise linear function. Therefore, What is the exceptional set of the first blow-up ? Recall that it corresponds to the subdivisions along the 2-cones see Figure 1. Therefore, is the blow-up along two disjoint rational curves of -singularities in . A standard intersection computation in the Chow group  yields that each curve intersects the divisor in two points. Therefore, the proper transform of is blown up in four points. The final blow-up does not further subdivide the 2-skeleton and, therefore, corresponds to the blow-up of points in . Any sufficiently generic divisor misses these blow-up points, and, therefore, the surfaces and are isomorphic. To summarize, (i) is a singular Enriques surface with four -orbifold points; (ii) and are the same smooth Enriques surface after blowing up the orbifold points; (iii)Since the blow-up at a point does not change the fundamental group, we find that
It is important to remember that the actual divisor is a fixed subvariety defined as the zero locus of an equation. To relate this defining equation before and after the blow-up, it is instructional to write the first blow-up map explicitly in terms of its action on homogeneous coordinates. One finds Note that this map is well defined on the equivalence classes (2.17) thanks to the identifications (2.16). One sees that, for example, the section corresponds to the section under the pull-back. I leave the analogous expression for as an exercise to the reader.
To summarize, the equation for the divisor determines the equation satisfied by the proper transforms and on the blow-ups. They are
3. Elliptic Fibration
So far, I have constructed (i)a three-dimensional (singular) Fano variety , (ii)a quasismooth divisor in with , (iii)a smooth three-dimensional toric variety , corresponding to a maximal subdivision of a reflexive polytope, (iv)a smooth divisor in with . This divisor is a smooth Enriques surface.
I will now proceed and construct four-dimensional elliptically fibered Calabi-Yau varieties , over and whose discriminant contains and , respectively.
3.1. Weierstrass Models
Ideally, one would like to classify all elliptic fibrations over the base manifold. Unfortunately it is not known how to do so in this generality. It is known, however, that there exists a Weierstrass model (not necessarily over the same base) which is a (in general) different elliptic fibration [28, 29], at least assuming that the base is smooth and the discriminant is a normal crossing divisor. The Weierstrass model and the original elliptic fibration are birational to each other, but apart from that their relationship is arduous at best.
Having said this, let us define the elliptically fibered variety in the most unimaginative way possible as a (global) Calabi-Yau Weierstrass model on a base variety with coordinates . The remaining coordinates, , , and , are sections The defining data of the Weierstrass model is the choice of coefficients in the Weierstrass equation, that is, the choice of sections
To engineer gauge theories on 7-branes wrapped on a divisor , one needs suitable singularities. In addition, the singularity must be of the correct split or nonsplit type as in Tate’s algorithm . For this purpose it is convenient to parametrize the Weierstrass (technically, the singularity appears after blowing down all fiber components of the Weierstrass model not intersecting the zero section, but we will not dwell on this) model by polynomials (that is, sections of suitable line bundles) , , , , as The degree of vanishing of the then determines (except for a few special cases that will be of no relevance for us) the low-energy effective gauge theory, see [31, 32]. For everything to be globally defined, the need to be sections of
3.2. Weierstrass Model on the Singular Base
To engineer a gauge theory coming from a 7-brane wrapped on the divisor , one needs a split singularity . This translates into vanishing to degree on . In other words, must be divisible by , where is the defining equation for the divisor as given in (2.33). Put yet differently, The number of sections is tabulated in Table 2. Note how the rows repeat with periodicity 2. This again follows from the fact that and differ by 2-torsion in the divisor class group, see (2.24). Hence, there are plenty sections available for , and one can easily find an elliptic fibration with a split over .
3.3. Weierstrass Model on the Smooth Base
Let me now turn to the smooth threefold and construct a suitable singularity over the smooth divisor . The main difference is that now, after resolving the singularity, the anticanonical divisor is “smaller” than , by which I mean that there are strictly less sections available for the Weierstrass model. See Figure 3 for details. Note that, if one always imposes the maximal degree of vanishing such that there are still nonzero sections, one can at most implement a split singularity leading to a low-energy gauge theory.
Having being dealt this lemon, let me try to make some lemonade. As in the previous subsection, I will write for the defining equation, see (2.33). The split singularity corresponds to a factorized form A basis for all sections can, of course, be written as in terms of homogeneous monomials in the 18 homogeneous coordinates . To save a tree I will now switch to inhomogeneous coordinates for the coordinate patch, say, corresponding to the cone . This amounts to replacing the homogeneous coordinates with In this patch, and the sections of the relevant line bundles are For simplicity I will choose to be the sum of the monomials corresponding to the vertices of the Newton polyhedron, that is, For all purposes in the following, this choice is generic. By construction, the discriminant then factorizes as where the remainder (in fact, is a polynomial consisting of 1083 monomials in , , and ) defines a new divisor . In particular, the homology class splits as Using the explicit equations, one can check  that (i) is an irreducible divisor; (ii)Neither nor vanishes at a generic point of . Hence it supports Kodaira fibers in the Weierstrass model; (iii) is smooth; (iv) is not smooth, for example, is a singular point; (v)The curve is not a complete intersection.
Let me further investigate the intersection curve . One component (in the patch) is given by the surprisingly simple expression Therefore, are good normal coordinates. Taylor expanding along the normal directions for a generic point , we see that and share the same tangent plane but do not osculate to any higher degree. Therefore, the degree of vanishing of the discriminant jumps from 5 to 7 along the intersection locus , corresponding to worsening of the singularity to an singularity.
In this paper I have constructed F-theory models with, a priori, gauge theory on a singular Fano threefold and a gauge theory on the blown-up smooth threefold. In both cases the non-Abelian gauge theory comes from a 7-brane wrapped on an Enriques surface, which has fundamental group . Therefore, in both cases one can switch on a discrete Wilson line and break the gauge group below the compactification scale in the usual manner.
The fact that the only partially resolved base allows for a higher rank gauge group on the 7-brane than its smooth blow-up is curious: one might be tempted to interpret the Kähler deformation as the usual Higgs mechanism; however the singular points are disjoint from the 7-brane. In any case, there must be further physical degrees of freedom associated to the singularities in the base, and it would be nice to have a more concise F-theory dictionary for them.
A. Fibrations of the Base
The base manifolds , , and are fibered in an interesting manner which I will describe in this appendix. The map to the 2-dimensional base is given by the -lattice projection This defines a toric morphism of toric varieties if and only if every cone of the domain fan is mapped into a cone of the range fan. It is easy to see that the rays and map to the rays of the fan of , and the rays map to the crepant resolution . See Figure 4 for a graphical representation of the fans of and .
However, consistently mapping the rays of the fans is not enough to define a toric morphism. Checking all higher-dimensional cones with respect to the lattice homomorphism , one finds that (i)the variety is not fibered; (ii)the variety is a -fibrations over ; (iii)the smooth threefold is a -fibration over , but not over the crepant resolution .
It is, perhaps, vexing that the resolved threefold is not a fibration over the resolved base . However, a closer investigation reveals that one can flop 4 offending curves, corresponding to the 4 bistellar flips of the fan . The flopped threefold is then a fibration over the resolved base . Of course the flopped threefold is then only birational to , and no longer a direct blow-up. However, it supports essentially the same elliptic fibration as as constructed in 3.
- C. Vafa, “Evidence for -theory,” Nuclear Physics B, vol. 469, no. 3, pp. 403–415, 1996.
- A. Sen, “-theory and orientifolds,” Nuclear Physics B, vol. 475, no. 3, pp. 562–578, 1996.
- C. Beasley, J. J. Heckman, and C. Vafa, “GUTs and exceptional branes in F-theory. I,” Journal of High Energy Physics, vol. 2009, no. 1, article 058, 2009.
- C. Beasley, J. J. Heckman, and C. Vafa, “GUTs and exceptional branes in F-theory. II. Experimental predictions,” Journal of High Energy Physics, vol. 2009, no. 1, article 059, 2009.
- R. Donagi and M. Wijnholt, “Model building with F-theory,” http://arxiv.org/abs/0802.2969.
- J. Marsano, N. Saulina, and S. Schäfer-Nameki, “F-theory compactifications for supersymmetric GUTs,” Journal of High Energy Physics, vol. 2009, no. 8, article 030, 2009.
- R. Blumenhagen, T. W. Grimm, B. Jurke, and T. Weigand, “Global F-theory GUTs,” Nuclear Physics B, vol. 829, no. 1-2, pp. 325–369, 2010.
- C. M. Chen, J. Knapp, M. Kreuzer, and C. Mayrhofer, “Global SO(10) F-theory guts,” Journal of High Energy Physics, vol. 2010, no. 10, article 057, 2010.
- C.-M. Chen and Y.-C. Chung, “Flipped SU(5) GUTs from Singularity in F-theory,” Journal of High Energy Physics, vol. 2001, no. 3, article 49, 2011.
- R. Blumenhagen, “Basics of F-theory from the type IIB perspective,” Fortschritte der Physik, vol. 58, no. 7-9, pp. 820–826, 2010.
- Y. Hosotani, “Dynamical mass generation by compact extra dimensions,” Physics Letters B, vol. 126, no. 5, pp. 309–313, 1983.
- Y. Hosotani, “Dynamics of nonintegrable phases and gauge symmetry breaking,” Annals of Physics, vol. 190, no. 2, pp. 233–253, 1989.
- R. Blumenhagen, V. Braun, T. W. Grimm, and T. Weigand, “GUTs in type IIB orientifold compactifications,” Nuclear Physics B, vol. 815, no. 1-2, pp. 1–94, 2009.
- R. Donagi and M. Wijnholt, “Breaking GUT groups in F-theory,” http://arxiv.org/abs/0808.2223.
- R. Blumenhagen, “Gauge coupling unification in F-theory grand unified theories,” Physical Review Letters, vol. 102, no. 7, Article ID 071601, 4 pages, 2009.
- V. Braun and A. Novoseltsev, “Toric geometry in the Sage CAS,” to appear.
- W. Stein, et al., “Sage Mathematics Software (Version 4.5.3),” The Sage Development Team, 2010, http://www.sagemath.org/.
- G.-M. Greuel, G. Pfister, and H. Schönemann, “Singular 3.0,” a computer algebra system for polynomial computations, Centre for Computer Algebra, University of Kaiserslautern, 2005, http://www.singular.uni-kl.de/.
- W. Nahm and K. Wendland, “A hiker's guide to . Aspects of superconformal field theory with central charge ,” Communications in Mathematical Physics, vol. 216, no. 1, pp. 85–138, 2001.
- V. Batyrev and M. Kreuzer, “Integral cohomology and mirror symmetry for Calabi-Yau 3-folds,” in Mirror Symmetry. V, vol. 38 of AMS/IP Stud. Adv. Math., pp. 255–270, American Mathematical Society, Providence, RI, USA, 2006.
- B. Nill, “Gorenstein toric Fano varieties,” Manuscripta Mathematica, vol. 116, no. 2, pp. 183–210, 2005.
- V. Braun, “On free quotients of complete intersection Calabi-Yau manifolds,” Journal of High Energy Physics, vol. 2011, no. 4, article 5, 2011.
- V. Braun, P. Candelas, and R. Davies, “A three-generation Calabi-Yau manifold with small Hodge numbers,” Fortschritte der Physik, vol. 58, no. 4-5, pp. 467–502, 2010.
- P. Candelas and A. Constantin, “Completing the web of —quotients of complete intersection Calabi-Yau manifolds,” http://arxiv.org/abs/1010.1878.
- D. A. Cox, “The homogeneous coordinate ring of a toric variety,” Journal of Algebraic Geometry, vol. 4, no. 1, pp. 17–50, 1995.
- W. Fulton, Introduction to Toric Varieties, vol. 131 of Annals of Mathematics Studies, Princeton University Press, Princeton, NJ, USA, 1993, The William H. Roever Lectures in Geometry.
- M. Oka, “Finiteness of fundamental group of compact convex integral polyhedra,” Kodai Mathematical Journal, vol. 16, no. 2, pp. 181–195, 1993.
- N. Nakayama, “Local structure of an elliptic fibration,” in Higher Dimensional Birational Geometry (Kyoto, 1997), vol. 35 of Adv. Stud. Pure Math., pp. 185–295, Math. Soc. Japan, Tokyo, Japan, 2002.
- N. Nakayama, “Global structure of an elliptic fibration,” Publications of the Research Institute for Mathematical Sciences, vol. 38, no. 3, pp. 451–649, 2002.
- J. Tate, “Algorithm for determining the type of a singular fiber in an elliptic pencil,” in Modular Functions of One Variable, IV (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), vol. 476 of Lecture Notes in Math., pp. 33–52, Springer, Berlin, Germany, 1975.
- M. Bershadsky, K. Intriligator, S. Kachru, D. R. Morrison, V. Sadov, and C. Vafa, “Geometric singularities and enhanced gauge symmetries,” Nuclear Physics B, vol. 481, no. 1-2, pp. 215–252, 1996.
- K.-S. Choi, “ vacua in F-theory,” Nuclear Physics B, vol. 842, no. 1, pp. 1–32, 2011.