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## Anomaly Equations and Intersection Theory (2012)

Venue: | JHEP |

Citations: | 18 - 2 self |

### Citations

2092 |
Principles of algebraic geometry
- Griffiths, Harris
- 1994
(Show Context)
Citation Context ...X. We do not consider such X, i.e., we assume all singularities of X can be resolved by blowing up rational curves throughout this paper. 2Some standard references on intersection theory are [21] and =-=[22]-=-. 3We are aware of an effort to carefully reconstruct the six-dimensional effective action based on this approach that is under way [27]. – 2 – a smooth resolution of an elliptically fibered manifold ... |

1331 |
Representation Theory
- Fulton, Harris
(Show Context)
Citation Context ...heory on X. We do not consider such X, i.e., we assume all singularities of X can be resolved by blowing up rational curves throughout this paper. 2Some standard references on intersection theory are =-=[21]-=- and [22]. 3We are aware of an effort to carefully reconstruct the six-dimensional effective action based on this approach that is under way [27]. – 2 – a smooth resolution of an elliptically fibered ... |

251 |
Superstrings With Torsion,” Nucl
- Strominger
- 1986
(Show Context)
Citation Context ... for our purposes. 6Determining the abelian gauge symmetry is subtle in a wide variety of string constructions, including heterotic, orbifold, intersecting brane, fractional brane and F-theory models =-=[3, 8, 9, 32, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55]-=-. The subtlety comes from the fact that abelian vector fields in the spectrum that are naively massless can be lifted at the linear level by coupling to Stückelberg fields. – 11 – is encoded in globa... |

212 |
Some comments on string dynamics
- Witten
(Show Context)
Citation Context ... into a four-cycle [13]—or equivalently, when X has codimension-two singularities in the base whose resolution involves blow-ups of the base [14]—one obtains an exotic theory with tensionless strings =-=[9, 15, 16, 17, 18, 19, 20]-=- by compactifying F-theory on X. We do not consider such X, i.e., we assume all singularities of X can be resolved by blowing up rational curves throughout this paper. 2Some standard references on int... |

158 |
Geometric singularities and enhanced gauge symmetries,” Nucl. Phys. B481
- Intriligator, Kachru, et al.
- 1996
(Show Context)
Citation Context ...e thought of as the submanifolds the seven-branes wrap. The type of degeneration determines the nature of the seven-brane and tells us the non-abelian gauge group we get in the six-dimensional theory =-=[8, 9, 10, 11, 12, 35]-=-. The codimension-two singularities can be thought of as intersecting points of the seven-branes. These contain information on the local matter we obtain [11, 12, 35, 36, 37, 38, 39, 40, 41]. A beauti... |

140 |
Tseytlin, “Green-Schwarz string
- Drukker, Gross, et al.
- 2000
(Show Context)
Citation Context ...inite class of theories to known string vacua, and hope to gain insight from it. 10We note that in ten dimensions, the anomaly constraints are strong enough to impose string universality on their own =-=[2, 6, 71, 72]-=-. – 26 – N # SU(N) models # satisfy Kodaira 13-24 1 1 12 2 2 11 2 2 10 2 2 9 3 3 8 15 14 7 16 16 6 48 47 5 23 16 4 207 154 3 10100 262 2 ∼ 5× 107 176 Table 4: The table of numbers of non-anomalous T =... |

139 |
Constraints on low-dimensional string compactifications,” Nucl. Phys. B480
- Sethi, Vafa, et al.
- 1996
(Show Context)
Citation Context ...ly [10, 11, 12], although not quite in the language that we have used. An interesting fact is that the equation (1.5) can be translated into a threefold analogue [67] of the Sethi-Vafa-Witten formula =-=[68]-=- for elliptically fibered Calabi-Yau threefolds with various fiber types. We note that the Sethi-Vafa-Witten formula was originally derived for elliptically fibered Calabi-Yau fourfolds in Weierstrass... |

135 |
Model Building with F-theory,” arXiv:0802.2969
- Donagi, Wijnholt
(Show Context)
Citation Context ...nd foremost I would like to thank Wati Taylor for his guidance, support and en12An incomplete list of references on the structure of four-dimensional F-theory backgrounds is given in the bibliography =-=[25, 52, 55, 58, 79, 80, 81, 82, 83, 84]-=-. A nice review of this subject and further references can be found in [85]. – 28 – couragement throughout the process of writing this paper. I would also like to thank Allan Adams, Koushik Balasubram... |

134 |
Axial vector vertex in spinor electrodynamics, Phys. Rev
- Adler
- 1969
(Show Context)
Citation Context ...es 42 1. Introduction The massless spectrum of six-dimensional supergravity theories with minimal supersymmetry is severely restricted due to the existence of gravitational, gauge and mixed anomalies =-=[1, 2]-=-. This implies that the massless spectrum must satisfy anomaly equations that come from a generalized version of the Green-Schwarz factorization condition [3, 4, 5], originally formulated in ten-dimen... |

120 |
Anomaly Cancellation
- Green, Schwarz
- 1984
(Show Context)
Citation Context ... implies that the massless spectrum must satisfy anomaly equations that come from a generalized version of the Green-Schwarz factorization condition [3, 4, 5], originally formulated in ten-dimensions =-=[6]-=-. These equations involve two types of data of the theory: – 1 – 1. The anomaly coefficients, which we schematically denote by {b}. 2. The spectrum of the theory, which we schematically denote by S. T... |

110 |
Open p-branes,” Phys
- Strominger
- 1996
(Show Context)
Citation Context ... into a four-cycle [13]—or equivalently, when X has codimension-two singularities in the base whose resolution involves blow-ups of the base [14]—one obtains an exotic theory with tensionless strings =-=[9, 15, 16, 17, 18, 19, 20]-=- by compactifying F-theory on X. We do not consider such X, i.e., we assume all singularities of X can be resolved by blowing up rational curves throughout this paper. 2Some standard references on int... |

90 |
Some comments on string dynamics,” hep-th/9507121
- Witten
(Show Context)
Citation Context ... into a four-cycle [13]—or equivalently, when X has codimension-two singularities in the base whose resolution involves blow-ups of the base [14]—one obtains an exotic theory with tensionless strings =-=[9, 15, 16, 17, 18, 19, 20]-=- by compactifying F-theory on X. We do not consider such X, i.e., we assume all singularities of X can be resolved by blowing up rational curves throughout this paper. 2Some standard references on int... |

89 | A Note on the Green-Schwarz mechanism in open string theories,” Phys
- Sagnotti
- 1992
(Show Context)
Citation Context ...avitational, gauge and mixed anomalies [1, 2]. This implies that the massless spectrum must satisfy anomaly equations that come from a generalized version of the Green-Schwarz factorization condition =-=[3, 4, 5]-=-, originally formulated in ten-dimensions [6]. These equations involve two types of data of the theory: – 1 – 1. The anomaly coefficients, which we schematically denote by {b}. 2. The spectrum of the ... |

82 | Loop-corrected compactifications of the heterotic string with line bundles
- Blumenhagen, Honecker, et al.
(Show Context)
Citation Context ... for our purposes. 6Determining the abelian gauge symmetry is subtle in a wide variety of string constructions, including heterotic, orbifold, intersecting brane, fractional brane and F-theory models =-=[3, 8, 9, 32, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55]-=-. The subtlety comes from the fact that abelian vector fields in the spectrum that are naively massless can be lifted at the linear level by coupling to Stückelberg fields. – 11 – is encoded in globa... |

79 |
Lectures on F-theory compactifications and model building
- Weigand
- 2010
(Show Context)
Citation Context ...s on the structure of four-dimensional F-theory backgrounds is given in the bibliography [25, 52, 55, 58, 79, 80, 81, 82, 83, 84]. A nice review of this subject and further references can be found in =-=[85]-=-. – 28 – couragement throughout the process of writing this paper. I would also like to thank Allan Adams, Koushik Balasubramanian, Federico Bonetti, Frederik Denef, Jacques Distler, Mboyo Esole, Iña... |

68 |
Eleven-dimensional supergravity compactified on Calabi-Yau threefolds,
- Cadavid, Ceresole, et al.
- 1995
(Show Context)
Citation Context ...able 3: Six-dimensional (1,0) supersymmetry multiplets and their descendants in five dimensions when compactified on a circle. Let us first identify the massless fields of the five-dimensional theory =-=[59, 60, 61]-=-. Mtheory compactified on the fully resolved manifold X̂ has h2(X̂) = h1,1(X̂) massless vector fields coming from descending the three-form on the harmonic two-forms of X̂ . Among these, one vector fi... |

66 | Semisimple Lie Algebras And Their Representations,” Menlo Park
- Cahn
(Show Context)
Citation Context ... that is convenient for our purposes. A.1 Some Lie Algebra In this section, we review some relevant Lie algebra. Almost all of what is discussed in this section can be found in standard texts such as =-=[87, 88]-=-. For a given Lie group G and its Lie algebra g, let us define the generators of Cartan sub-algebra {Ti}. Let us normalize the Cartan generators so that trTiTj = δij (A.5) where the trace is taken in ... |

62 |
Phase Transitions
- Witten
- 1996
(Show Context)
Citation Context ...onal to Z ′ and B′α. The terms that involve B cannot receive corrections for the following reason. The corrections to these Chern-Simons terms come from oneloop integrals of five-dimensional fermions =-=[13]-=-. The only way that terms involving B could receive corrections on the F-theory side is if some six-dimensional fermion in a short multiplet couples to the tensor field B in a way that reduces to ψ̄Bµ... |

58 | Low-energy analysis - Ferrara, Minasian, et al. |

53 |
Compactifications of F theory on Calabi-Yau threefolds
- Morrison, Vafa
- 1996
(Show Context)
Citation Context ...mpare all non-anomalous models with all known string vacua for a given gauge group. All six-dimensional (1, 0) vacua constructed in string theory that we are aware of satisfy the Kodaira constraint11 =-=[8, 9]-=-: ∑ κ νκ(j · bκ) ≤ −12j · a . (5.1) Recall that a, bκ are anomaly coefficients from (2.4) and (2.13) and j is the SO(1, T ) unit vector that parametrizes the vacuum expectation value of the scalars of... |

50 |
F-theory GUTs with
- Dolan, Marsano, et al.
(Show Context)
Citation Context ... for our purposes. 6Determining the abelian gauge symmetry is subtle in a wide variety of string constructions, including heterotic, orbifold, intersecting brane, fractional brane and F-theory models =-=[3, 8, 9, 32, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55]-=-. The subtlety comes from the fact that abelian vector fields in the spectrum that are naively massless can be lifted at the linear level by coupling to Stückelberg fields. – 11 – is encoded in globa... |

48 |
Small E(8) instantons and tensionless noncritical strings," Nucl. Phys. B474
- Ganor, Hanany
- 1996
(Show Context)
Citation Context |

46 | The N=1 effective action of F-theory compactifications
- Grimm
(Show Context)
Citation Context ... A convenient method to investigate the abelian sector of F-theory backgrounds is to use M-theory/F-theory duality [7], and take an intersection theory-based2 approach in the dual M-theory background =-=[10, 13, 23, 24, 25, 26]-=-.3 Once {b}(X) and S(X) are identified for X, we may translate the anomaly equations to identities of the geometric data of X. The main result of this paper is that the anomaly equations imply the fol... |

46 | G4 flux, chiral matter and singularity resolution
- Krause, Mayrhofer, et al.
(Show Context)
Citation Context ... six-dimensional theory [8, 9, 10, 11, 12, 35]. The codimension-two singularities can be thought of as intersecting points of the seven-branes. These contain information on the local matter we obtain =-=[11, 12, 35, 36, 37, 38, 39, 40, 41]-=-. A beautiful fact is that the geometrical data of an elliptically fibered Calabi-Yau threefold can be encoded in an integral lattice. More precisely, geometric data such as the canonical divisor clas... |

45 | Anomaly cancellation in six-dimensions
- Erler
- 1994
(Show Context)
Citation Context |

43 |
Enhanced Gauge Symmetry
- Katz, Morrison, et al.
- 1996
(Show Context)
Citation Context ... A convenient method to investigate the abelian sector of F-theory backgrounds is to use M-theory/F-theory duality [7], and take an intersection theory-based2 approach in the dual M-theory background =-=[10, 13, 23, 24, 25, 26]-=-.3 Once {b}(X) and S(X) are identified for X, we may translate the anomaly equations to identities of the geometric data of X. The main result of this paper is that the anomaly equations imply the fol... |

42 |
On type IIB vacua with varying coupling constant,” Nucl. Phys. B558
- Berglund, Klemm, et al.
- 1999
(Show Context)
Citation Context |

42 |
On hypercharge flux and exotics
- Dudas, Palti
- 2010
(Show Context)
Citation Context ...nvolved. There has, however, been beautiful work [53] in which constraints on “hypercharge fluxes” on F-theory SU(5) GUT models with U(1) symmetries—referred to as “generalized Dudas-Palti relations” =-=[86]-=-—are derived by four-dimensional anomaly cancellation. The generalized Dudas-Palti relations provide a good handle on F-theory GUT models with U(1) symmetries [53, 54]. It would be interesting to expa... |

41 |
Heterotic phase transitions and singularities of the gauge dyonic string
- Duff, Lu, et al.
(Show Context)
Citation Context |

38 |
K3 surfaces and string duality," hep-th/9611137
- Aspinwall
(Show Context)
Citation Context ...F : Rational curves fibered over a curve that shrink to zero size in the limit X̂ → X. These curves are all rational curves; they are topologically P1’s as only these types of curves can shrink in X̂ =-=[62]-=-. We index the curves of type I by r and denote them cr, and index the curves of type F by ρ and denote them χρ. We use gρ to denote the genus of the curve χρ is fibered over. The curve a type F curve... |

37 | Matter from geometry
- Katz, Vafa
- 1997
(Show Context)
Citation Context ... six-dimensional theory [8, 9, 10, 11, 12, 35]. The codimension-two singularities can be thought of as intersecting points of the seven-branes. These contain information on the local matter we obtain =-=[11, 12, 35, 36, 37, 38, 39, 40, 41]-=-. A beautiful fact is that the geometrical data of an elliptically fibered Calabi-Yau threefold can be encoded in an integral lattice. More precisely, geometric data such as the canonical divisor clas... |

34 |
Anomaly Free Chiral Theories
- Green, Schwarz, et al.
- 1985
(Show Context)
Citation Context ...avitational, gauge and mixed anomalies [1, 2]. This implies that the massless spectrum must satisfy anomaly equations that come from a generalized version of the Green-Schwarz factorization condition =-=[3, 4, 5]-=-, originally formulated in ten-dimensions [6]. These equations involve two types of data of the theory: – 1 – 1. The anomaly coefficients, which we schematically denote by {b}. 2. The spectrum of the ... |

33 | Group representations and the Euler characteristic of elliptically fibered Calabi-Yau threefolds
- Grassi, Morrison
(Show Context)
Citation Context ...e algebra, we draw the corresponding tree of resolved rational curves and label the linearly independent curves as αI and label the monodromy invariant combinations of rational curves γJ according to =-=[11]-=-. The curves χI that M2 branes wrap to give root vectors are identified with αI . The four cycles CJ dual to non-abelian gauge field components of the coroot basis elements TJ of the Cartan are obtain... |

33 | Anomalies and the Euler characteristic of elliptic Calabi-Yau threefolds,” arXiv:1109.0042 [hep-th
- Grassi, Morrison
(Show Context)
Citation Context ...less spectrum of this theory S(X), they must satisfy the anomaly equations: fi({b}(X)) = Fi(S(X)) (1.2) Therefore, the anomaly equations imply non-trivial identities involving the geometric data of X =-=[10, 11, 12]-=-.1 In order to get to the geometric identities, we must first be able to identify {b}(X) and S(X), given a manifold X. {b}(X) are identified readily when all the massless vector fields of the theory a... |

30 |
Lie Algebras In
- Georgi
- 1995
(Show Context)
Citation Context ... that is convenient for our purposes. A.1 Some Lie Algebra In this section, we review some relevant Lie algebra. Almost all of what is discussed in this section can be found in standard texts such as =-=[87, 88]-=-. For a given Lie group G and its Lie algebra g, let us define the generators of Cartan sub-algebra {Ti}. Let us normalize the Cartan generators so that trTiTj = δij (A.5) where the trace is taken in ... |

29 |
Tensor and vector multiplets in six-dimensional supergravity
- Ferrara, Riccioni, et al.
- 1998
(Show Context)
Citation Context ... the theory. Note that in the presence of U(1)’s, the anomaly polynomial factorization condition is generalized in the form (2.4) due to the fact that the field strength is gauge invariant on its own =-=[31]-=-. Under linear redefinitions of the U(1)’s bij transforms as a bilinear, i.e., Fi =M j i F ′ j , b ′ ij =M k i M l jbkl, M ∈ GL(VA,R) . (2.5) The λκ’s are normalization factors that are fixed by deman... |

29 |
Hypercharge Flux, Exotics, and Anomaly Cancellation in F- theory GUTs
- Marsano
(Show Context)
Citation Context |

25 |
Calabi-Yau fourfolds for M theory and F theory compactifications,” Nucl
- Klemm, Lian, et al.
- 1998
(Show Context)
Citation Context ...straint—has also been studied previously [10, 11, 12], although not quite in the language that we have used. An interesting fact is that the equation (1.5) can be translated into a threefold analogue =-=[67]-=- of the Sethi-Vafa-Witten formula [68] for elliptically fibered Calabi-Yau threefolds with various fiber types. We note that the Sethi-Vafa-Witten formula was originally derived for elliptically fiber... |

24 | Sums over topological sectors and quantization of FayetIliopoulos parameters
- Hellerman, Sharpe
- 2011
(Show Context)
Citation Context ...m of the infinite classes of theories with different U(1) charge assignments. Since 11See, for example, section 4 of [75]. – 27 – U(1) charges of quantum gravity theories are expected to be quantized =-=[76, 77, 78]-=-, the anomaly equations (2.19) imply that the magnitude of some bii must be unbounded for any infinite class of U(1) charge assignments. If a generalized version of the Kodaira constraint bounds the m... |

23 | String Universality
- Kumar, Taylor
(Show Context)
Citation Context ...cations. In particular, we focus on explaining why identifying the abelian anomaly coefficients in F-theory is an important step in investigating the question of string universality in six dimensions =-=[28]-=-. 2. 6D (1, 0) Theories and Anomaly Cancellation In this section we review six-dimensional theories with N = (1, 0) supersymmetry and the dynamics of the massless spectrum. In section 2.1 we present a... |

21 | D5 elliptic fibrations: non-Kodaira fibers and new orientifold limits of F-theory,” arXiv:1110.6177 [hep-th
- Esole, Fullwood, et al.
(Show Context)
Citation Context ... six-dimensional theory [8, 9, 10, 11, 12, 35]. The codimension-two singularities can be thought of as intersecting points of the seven-branes. These contain information on the local matter we obtain =-=[11, 12, 35, 36, 37, 38, 39, 40, 41]-=-. A beautiful fact is that the geometrical data of an elliptically fibered Calabi-Yau threefold can be encoded in an integral lattice. More precisely, geometric data such as the canonical divisor clas... |

21 |
A Note on G-Fluxes for F-theory Model Building
- Marsano, Saulina, et al.
(Show Context)
Citation Context |

20 | Global aspects of the space - Kumar, Morrison, et al. |

19 |
Generalized Green-Schwarz mechanism
- Sadov
- 1996
(Show Context)
Citation Context ...avitational, gauge and mixed anomalies [1, 2]. This implies that the massless spectrum must satisfy anomaly equations that come from a generalized version of the Green-Schwarz factorization condition =-=[3, 4, 5]-=-, originally formulated in ten-dimensions [6]. These equations involve two types of data of the theory: – 1 – 1. The anomaly coefficients, which we schematically denote by {b}. 2. The spectrum of the ... |

19 |
class identities from tadpole matching
- Aluffi, Esole, et al.
(Show Context)
Citation Context ... note that the Sethi-Vafa-Witten formula was originally derived for elliptically fibered Calabi-Yau fourfolds in Weierstrass form, and has been extended to more general elliptically fibered fourfolds =-=[67, 69, 70]-=-. While the equations we have derived are aesthetically pleasing, we do not yet have much insight into how much they add to what we already know about the geometry of Calabi-Yau threefolds. Understand... |

19 | G-flux in F-theory and algebraic cycles
- Braun, Collinucci, et al.
(Show Context)
Citation Context ...nd foremost I would like to thank Wati Taylor for his guidance, support and en12An incomplete list of references on the structure of four-dimensional F-theory backgrounds is given in the bibliography =-=[25, 52, 55, 58, 79, 80, 81, 82, 83, 84]-=-. A nice review of this subject and further references can be found in [85]. – 28 – couragement throughout the process of writing this paper. I would also like to thank Allan Adams, Koushik Balasubram... |

18 |
6D supergravity without tensor multiplets
- Kumar, Park, et al.
(Show Context)
Citation Context ...r of non-anomalous theories with N = 2 are very large and have not been computed precisely. An example that sheds light on this question can be found in non-abelian theories with no tensor multiplets =-=[74]-=-. When T = 0, all the anomaly equations simplify as all the anomaly coefficients become numbers. Due to this fact, there is a systematic way of constructing all non-anomalous models given the gauge gr... |

14 | Calculating the Mordell-Weil rank of elliptic threefolds and the cohomology of singular hypersurfaces
- Hulek, Kloosterman
(Show Context)
Citation Context ...e fact that abelian vector fields in the spectrum that are naively massless can be lifted at the linear level by coupling to Stückelberg fields. – 11 – is encoded in global data of the full manifold =-=[9, 56, 57]-=-. In order to understand the abelian sector of the theory, it turns out to be more convenient to take an intersection theory-based approach in the M-theory dual of the F-theory background [10, 13, 23,... |

14 |
N=2 Heterotic Superstring and its Dual Theory
- Antoniadis, Ferrara, et al.
- 1996
(Show Context)
Citation Context ...able 3: Six-dimensional (1,0) supersymmetry multiplets and their descendants in five dimensions when compactified on a circle. Let us first identify the massless fields of the five-dimensional theory =-=[59, 60, 61]-=-. Mtheory compactified on the fully resolved manifold X̂ has h2(X̂) = h1,1(X̂) massless vector fields coming from descending the three-form on the harmonic two-forms of X̂ . Among these, one vector fi... |

14 |
Mordell-Weil lattices and Galois representation
- Shioda
- 1989
(Show Context)
Citation Context ... for any TI,κ and Bα. Equation (3.35) is the threefold analog of the map Shioda used to map rational sections of an elliptically fibered surface to points in the Néron-Severi lattice of that surface =-=[64]-=-.9 Once the rational sections were mapped to a lattice, a number-valued inner-product on the sections could be defined. In our case there is an H2(B) vector-valued inner-product on type S cycles. It i... |

14 |
GUTs and exceptional branes in F-theory.
- Beasley, Heckman, et al.
- 2009
(Show Context)
Citation Context ...nd foremost I would like to thank Wati Taylor for his guidance, support and en12An incomplete list of references on the structure of four-dimensional F-theory backgrounds is given in the bibliography =-=[25, 52, 55, 58, 79, 80, 81, 82, 83, 84]-=-. A nice review of this subject and further references can be found in [85]. – 28 – couragement throughout the process of writing this paper. I would also like to thank Allan Adams, Koushik Balasubram... |

13 | F-theory fluxes, Chirality and Chern-Simons theories
- Grimm, Hayashi
(Show Context)
Citation Context ... A convenient method to investigate the abelian sector of F-theory backgrounds is to use M-theory/F-theory duality [7], and take an intersection theory-based2 approach in the dual M-theory background =-=[10, 13, 23, 24, 25, 26]-=-.3 Once {b}(X) and S(X) are identified for X, we may translate the anomaly equations to identities of the geometric data of X. The main result of this paper is that the anomaly equations imply the fol... |

13 |
Mapping 6D
- Kumar, Morrison, et al.
(Show Context)
Citation Context ...onal theory with N = (1, 0) supersymmetry [7, 8, 9]. There is an elegant picture of how the geometry of the internal Calabi-Yau threefold is encoded in the non-abelian sector of the low-energy theory =-=[11, 12, 30, 34]-=-. In particular, there is a precise geometric meaning of the non-abelian anomaly coefficients of the sixdimensional theory. In this section, we clarify the geometric data encoded in the abelian sector... |

13 | Arithmetic on elliptic threefolds
- Wazir
- 2004
(Show Context)
Citation Context ...e fact that abelian vector fields in the spectrum that are naively massless can be lifted at the linear level by coupling to Stückelberg fields. – 11 – is encoded in global data of the full manifold =-=[9, 56, 57]-=-. In order to understand the abelian sector of the theory, it turns out to be more convenient to take an intersection theory-based approach in the M-theory dual of the F-theory background [10, 13, 23,... |

11 |
New branches of string compactifications and their F theory duals,” Nucl.Phys. B492
- Aldazabal, Font, et al.
- 1997
(Show Context)
Citation Context |

10 |
M theory on a Calabi-Yau manifold,” Phys
- Ferrara, Khuri, et al.
- 1996
(Show Context)
Citation Context ...able 3: Six-dimensional (1,0) supersymmetry multiplets and their descendants in five dimensions when compactified on a circle. Let us first identify the massless fields of the five-dimensional theory =-=[59, 60, 61]-=-. Mtheory compactified on the fully resolved manifold X̂ has h2(X̂) = h1,1(X̂) massless vector fields coming from descending the three-form on the harmonic two-forms of X̂ . Among these, one vector fi... |

10 | Mordell-Weil lattice via string junctions
- Fukae, Yamada, et al.
(Show Context)
Citation Context ...tically fibered K3 manifold M under the Shioda map— which are two-cycles—are also dual to the abelian vector fields of the eight-dimensional supergravity theory obtained by compacifying F-theory on M =-=[7, 62, 65, 66]-=-. – 22 – Furthermore, if we normalize the gauge fields A′i so that the charge of the hypermultiplet coming from branes wrapping cr is cr · Si, we can equate A ′ i and Ai. Then due to the normalization... |

9 | Constraints on 6D supergravity theories with Abelian gauge symmetry
- Park, Taylor
(Show Context)
Citation Context ...r purposes. The lowercase greek letters κ, λ, · · · are used to denote the simple non-abelian gauge group factors; lowercase 4A more detailed treatment of the contents of this section can be found in =-=[29]-=-. – 4 – Multiplet Field Content Gravity (gµν , ψ + µ , B + µν) Tensor (φ, χ−, B−µν) Vector (Aµ, λ +) Hyper (4ϕ,ψ−) Table 1: Six-dimensional (1,0) supersymmetry multiplets. The signs on the fermions in... |

9 |
Global F-theory models: instantons and gauge dynamics
- Cvetič, Garcia-Etxebarria, et al.
(Show Context)
Citation Context |

8 | Lie groups, Calabi-Yau threefolds
- Aspinwall, Katz, et al.
- 2000
(Show Context)
Citation Context ... [9, 56, 57]. In order to understand the abelian sector of the theory, it turns out to be more convenient to take an intersection theory-based approach in the M-theory dual of the F-theory background =-=[10, 13, 23, 26]-=-. By recovering the low-energy data of the six-dimensional F-theory background using this duality, we can identify the geometric meaning of the abelian anomaly coefficients by comparing the coefficien... |

8 |
Evidence for heterotic / heterotic duality, Nucl. Phys. B465
- Duff, Minasian, et al.
- 1996
(Show Context)
Citation Context |

7 |
Massive U(1)s and heterotic five-branes on K3” Nucl.Phys. B748
- Honecker
- 2006
(Show Context)
Citation Context |

6 |
Monopoles, duality, and string theory,” Int
- Polchinski
- 2004
(Show Context)
Citation Context ...m of the infinite classes of theories with different U(1) charge assignments. Since 11See, for example, section 4 of [75]. – 27 – U(1) charges of quantum gravity theories are expected to be quantized =-=[76, 77, 78]-=-, the anomaly equations (2.19) imply that the magnitude of some bii must be unbounded for any infinite class of U(1) charge assignments. If a generalized version of the Kodaira constraint bounds the m... |

5 |
Evidence for F theory,” Nucl. Phys
- Vafa
- 1996
(Show Context)
Citation Context ... For an elliptically fibered Calabi-Yau threefold X with a section and a smooth resolution, we can obtain a corresponding consistent six-dimensional supergravity theory by compactifying F-theory on X =-=[7, 8, 9]-=-. The resulting low-energy effective theory is guaranteed to be non-anomalous as the the background is consistent. Hence if we denote the anomaly coefficients of this theory {b}(X), and the massless s... |

5 |
String junctions and nonsimply connected gauge groups
- Guralnik
- 2001
(Show Context)
Citation Context ...tically fibered K3 manifold M under the Shioda map— which are two-cycles—are also dual to the abelian vector fields of the eight-dimensional supergravity theory obtained by compacifying F-theory on M =-=[7, 62, 65, 66]-=-. – 22 – Furthermore, if we normalize the gauge fields A′i so that the charge of the hypermultiplet coming from branes wrapping cr is cr · Si, we can equate A ′ i and Ai. Then due to the normalization... |

5 |
On generalized Sethi-Vafa-Witten formulas,” [arXiv:1103.6066 [math.AG
- Fullwood
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Citation Context ... note that the Sethi-Vafa-Witten formula was originally derived for elliptically fibered Calabi-Yau fourfolds in Weierstrass form, and has been extended to more general elliptically fibered fourfolds =-=[67, 69, 70]-=-. While the equations we have derived are aesthetically pleasing, we do not yet have much insight into how much they add to what we already know about the geometry of Calabi-Yau threefolds. Understand... |

3 |
On the algebras of BPS states
- Phys
- 1996
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Citation Context ...ctor or hypermultiplets under the vector multiplets coming from shrinking rational curves of type F can be obtained by considering the algebra of BPS states in the Calabi-Yau manifold as described in =-=[63]-=-. Some of the multiplets that are uncharged under the vector fields in the Cartan sub-algebra can be charged under vector fields that come from shrinking type F rational curves. Let us summarize what ... |

3 |
String universality in ten dimensions,” Phys
- Adams, DeWolfe, et al.
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Citation Context ...inite class of theories to known string vacua, and hope to gain insight from it. 10We note that in ten dimensions, the anomaly constraints are strong enough to impose string universality on their own =-=[2, 6, 71, 72]-=-. – 26 – N # SU(N) models # satisfy Kodaira 13-24 1 1 12 2 2 11 2 2 10 2 2 9 3 3 8 15 14 7 16 16 6 48 47 5 23 16 4 207 154 3 10100 262 2 ∼ 5× 107 176 Table 4: The table of numbers of non-anomalous T =... |

2 | An analysis of fluxes by duality
- Aspinwall
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1 |
Six-dimensional (1,0) action for F-theory via M-theory on Calabi-Yau threefolds
- Bonetti, Grimm
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Citation Context .... 2Some standard references on intersection theory are [21] and [22]. 3We are aware of an effort to carefully reconstruct the six-dimensional effective action based on this approach that is under way =-=[27]-=-. – 2 – a smooth resolution of an elliptically fibered manifold X with a section: π(S1 · S2) · π(S3 · S4) + π(S1 · S3) · π(S2 · S4) + π(S1 · S4) · π(S2 · S3) = ∑ r 4∏ k=1 (cr · Sk) + ∑ ρ (2gρ − 2) 4∏ ... |