On generalized Sethi-Vafa-Witten formulas

Fullwood, James

doi:10.1063/1.3628633

Cited by 15 publications

(18 citation statements)

References 8 publications

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“…Theorem 3.15 (See [27] and [2,3,29,37]). Let L be a line bundle over a variety B and π ∶ X 0 = P[O B ⊕ L ⊗2 ⊕ L ⊗3 ] → B a projective bundle over B. LetQ(t) = ∑ a π * Q a t a be a formal power series in t such that Q a ∈ A * (B).…”

Section: Pushforward Formulasmentioning

confidence: 99%

Flopping and slicing: $\operatorname{SO}(4)$ and $\operatorname{Spin}(4)$-models

Esole

Kang

2019

Advances in Theoretical and Mathematical Physics

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We study the geometric engineering of gauge theories with gauge group Spin(4) and SO(4) using crepant resolutions of Weierstrass models. The corresponding elliptic fibrations realize a collision of singularities corresponding to two fibers with dual graphsÃ 1 . There are eight different ways to engineer such collisions using decorated Kodaira fibers. The Mordell-Weil group of the elliptic fibration is required to be trivial for Spin(4) and Z 2Z for SO(4).Each of these models have two possible crepant resolutions connected by a flop. We also compute a generating function for the Euler characteristic of such elliptic fibrations over a base of arbitrary dimensions. In the case of a threefold, we also compute the triple intersection numbers of the fibral divisors. In the case of Calabi-Yau threefolds, we also compute their Hodge numbers, and check the cancellations of anomalies in a six-dimensional supergravity theory.

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Section: Pushforward Formulasmentioning

confidence: 99%

Flopping and slicing: $\operatorname{SO}(4)$ and $\operatorname{Spin}(4)$-models

Esole

Kang

2019

Advances in Theoretical and Mathematical Physics

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“…As such, each blowup in the resolution satisfies the hypotheses of Lemma 2.1. Thus if Y is a small resolution of Y then pushing forward a class γ ∈ A * Y to A * B amounts to applying of formula ( † †) until one arrives at a class in A * P(E ), and then applying the pushforward formula for classes in a projective bundle which was first derived in [18].…”

Section: A Little Blowup Calculusmentioning

confidence: 99%

“…[16], computing ϕ * c str (Y ) amounts to computing ϕ * M str (Y ) = ϕ * (X · [Y sing ]). For this, we view X · [Y sing ] as a class in A * P(E ) and push it forward to the base via the pushforward formula for classes in a projective bundle first derived in [18]. To apply the pushforward formula of [18] to the case of a projective bundle of the form P(O ⊕ L 2 ⊕ L 3 ), we first expand X · [Y sing ] as a series in H:…”

Section: Stringy Chern Classesmentioning

confidence: 99%

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On stringy invariants of GUT vacua

Fullwood

Hoeij

2013

Communications in Number Theory and Physics

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We investigate aspects of certain stringy invariants of singular elliptic fibrations which arise in engineering Grand Unified Theories in F-theory. In particular, we exploit the small resolutions of the total space of these fibrations provided recently in the physics literature to compute "stringy characteristic classes", and find that numerical invariants obtained by integrating such characteristic classes are predetermined by the topology of the base of the elliptic fibration. Moreover, we derive a simple (dimension independent) formula for pushing forward powers of the exceptional divisor of a blowup, which one may use to reduce any integral (in the sense of Chow cohomology) on a small resolution of a singular elliptic fibration to an integral on the base. We conclude with a remark on the cohomology of small resolutions of GUT vacua, where we conjecture that certain simple formulas for their Hodge numbers may be given solely in terms of the first Chern class and Hodge numbers of the base.

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“…The purpose of this note is to compute formulas for the Euler characteristic for relative hypersurfaces such as Y X solely in terms of invariants of an arbitrary smooth base variety X. Such formulas are sometimes called 'Sethi-Vafa-Witten' formulas [1][2] [6] [8], since if X is a Fano threefold one may construct Y X → X such that the total space Y X is a Calabi-Yau elliptic fourfold for which Sethi, Vafa and Witten -who were motivated by constructing super-symmetric compactifications of M-and F-theory [14,Formula (2.12)] -derived a formula for the Euler characteristic of Y X in terms of the Chern classes of X, namely χ(Y X ) = X 12c 1 (X)c 2 (X) + 360c 1 (X) 3 .…”

Section: Introductionmentioning

confidence: 99%

On the Euler characteristic of a relative hypersurface

Fullwood

Helmer

2019

Journal of Mathematical Physics

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We derive a general formula for the Euler characteristic of a fibration of projective hypersurfaces in terms of invariants of an arbitrary base variety. When the general fiber is an elliptic curve, such formulas have appeared in the physics literature in the context of calculating D-brane charge for M-/F-theory and type-IIB compactifications of string vacua. While there are various methods for computing Euler characteristics of algebraic varieties, we prove a base-independent pushforward formula which reduces the computation of the Euler characteristic of relative hypersurfaces to simple algebraic manipulations of rational expressions determined by its divisor class in a projective bundle. We illustrate our methods by applying them to an explicit family of relative hypersurfaces whose fibers are of arbitrary dimension and degree.

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On generalized Sethi-Vafa-Witten formulas

Cited by 15 publications

References 8 publications

Flopping and slicing: $\operatorname{SO}(4)$ and $\operatorname{Spin}(4)$-models

Flopping and slicing: $\operatorname{SO}(4)$ and $\operatorname{Spin}(4)$-models

On stringy invariants of GUT vacua

On the Euler characteristic of a relative hypersurface

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