Euler Characteristics of Crepant Resolutions of Weierstrass Models

We show that the Weak Gravity Conjecture (WGC) implies a nontrivial upper bound on the volumes of the minimal-volume cycles in certain homology classes that admit no calibrated representatives. In compactification of type IIB string theory on an orientifold X of a Calabi-Yau threefold, we consider a homology class [Σ] ∈ H 4 (X, Z) represented by a union Σ ∪ of holomorphic and antiholomorphic cycles. The instanton form of the WGC applied to the axion charge [Σ] implies an upper bound on the action of a non-BPS Euclidean D3-brane wrapping the minimal-volume representative Σ min of [Σ]. We give an explicit example of an orientifold X of a hypersurface in a toric variety, and a hyperplane H ⊂ H 4 (X, Z), such that for any [Σ] ∈ H that satisfies the WGC, the minimal volume obeys Vol(Σ min ) Vol(Σ ∪ ): the holomorphic and antiholomorphic components recombine to form a much smaller cycle. In particular, the sub-Lattice WGC applied to X implies large recombination, no matter how sparse the sublattice. Non-BPS instantons wrapping Σ min are then more important than would be predicted from a study of BPS instantons wrapping the separate components of Σ ∪ . Our analysis hinges on a novel computation of effective divisors in X that are not inherited from effective divisors of the toric variety.

Section: A Geometry Of the Examplementioning

confidence: 99%

Minimal surfaces and weak gravity

Demirtaş

Long

McAllister

et al. 2020

“…A fiber resolution (and the associated smooth fibration) giving such codimension two fibers is called flat (the fiber dimension stays at complex one dimension). An in-depth analysis of the possible fiber types can be found in [28,46,65,66,[82][83][84][85]. Here we will consider a simple example.…”

Section: Codimension Two: Flat and Non-flat Resolutionsmentioning

confidence: 99%

Fibers add flavor. Part I. Classification of 5d SCFTs, flavor symmetries and BPS states

Apruzzi

Lawrie

Lin

et al. 2019

164

278

We propose a graph-based approach to 5d superconformal field theories (SCFTs) based on their realization as M-theory compactifications on singular elliptic Calabi-Yau threefolds. Fieldtheoretically, these 5d SCFTs descend from 6d N = (1, 0) SCFTs by circle compactification and mass deformations. We derive a description of these theories in terms of graphs, so-called Combined Fiber Diagrams, which encode salient features of the partially resolved Calabi-Yau geometry, and provides a combinatorial way of characterizing all 5d SCFTs that descend from a given 6d theory. Remarkably, these graphs manifestly capture strongly coupled data of the 5d SCFTs, such as the superconformal flavor symmetry, BPS states, and mass deformations.The capabilities of this approach are demonstrated by deriving all rank one and rank two 5d SCFTs. The full potential, however, becomes apparent when applied to theories with higher rank. Starting with the higher rank conformal matter theories in 6d, we are led to the discovery of previously unknown flavor symmetry enhancements and new 5d SCFTs.

“…(S i C · S j D · S k E ) Xr can be computed by pushing it forward to intersection ring of B as outlined in [9]. The first type of pushforward map whose properties we need to understand is the pushforward f * associated to a blowup f : Y ′ → Y .…”

Section: Computation Of Triple Intersection Numbersmentioning

confidence: 99%

“…where the label "h or e" indicates that the curve will be labeled either as h or as e depending on the value of k. The 4 − k blowups on S 3 C and S 1 C correspond to 4 − k hypers in the vector representation. 8 − 2k blowups out of 16 − 4k blowups on S 4 C have been paired into 4 − k pairs denoted by z i , w i where i = 1, · · · , 4 − k, and correspond 9 More precisely, a smoothing of self-glued F 4r−16 6 equals a ruled surface of genus 2r − 7 and degree 4r − 8.…”

Section: So(2r + 1) R ≥ 4 On −4 Curvementioning

confidence: 99%

“…Reference [3] provided a description ofX T for all T that are rank one 6d SCFTs. This was done by explicitly performing resolutions of X T using methods described in detail in [9][10][11][12][13][14][15]. In this paper, we extend the work of [3] and provide a description of X T for T that are 6d SCFTs of any arbitrary rank (see [16][17][18] for related analyses of F-theory models involving collisions of elliptic fibers in cases of semi-simple, as opposed to simple, gauge algebras.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Classifying 5d SCFTs via 6d SCFTs: arbitrary rank

Bhardwaj

Jefferson

2019

Self Cite

111

175

According to a conjecture, all 5d SCFTs should be obtainable by rankpreserving RG flows of 6d SCFTs compactified on a circle possibly twisted by a background for the discrete global symmetries around the circle. For a 6d SCFT admitting an F-theory construction, its untwisted compactification admits a dual M-theory description in terms of a "parent" Calabi-Yau threefold which captures the Coulomb branch of the compactified 6d SCFT. The RG flows to 5d SCFTs can then be identified with a sequence of flop transitions and blowdowns of the parent Calabi-Yau leading to "descendant" Calabi-Yau threefolds which describe the Coulomb branches of the resulting 5d SCFTs. An explicit description of parent Calabi-Yaus is known for untwisted compactifications of rank one 6d SCFTs. In this paper, we provide a description of parent Calabi-Yaus for untwisted compactifications of arbitrary rank 6d SCFTs. Since 6d SCFTs of arbitrary rank can be viewed as being constructed out of rank one SCFTs, we accomplish the extension to arbitrary rank by identifying a prescription for gluing together Calabi-Yaus associated to rank one 6d SCFTs. 1 lbhardwaj@fas.harvard.edu 2 patrickjefferson@fas.harvard.edu 42 4.8 e 6 , e 7 , e 8 and f 4 48 4.9 su(3) on −3 50 -i -5 Gluing rules 50 5.1 Gluing of su(m), m ≥ 1 or m =6 and su(n), n ≥ 1 50 5.2 Gluing of sp(m), m ≥ 1 and su(n), n ≥ 1 52 5.3 Gluing of sp(m), m ≥ 1 and so(2r), r ≥ 4 52 5.4 Gluing of sp(m), m ≥ 1 and so(2r + 1), r ≥ 4 53 5.5 Gluing of sp(m), m ≥ 1 and so(7) 53 5.6 Gluing of sp(m), m ≥ 1 and g 2 54 5.7 Gluings of sp(0) = E-string 54 5.7.1 Simply laced 55 5.7.2 Non-simply laced 66 6 Future work 76 A User's guide for Mathematica notebook Pushforward.nb 77 • The set of compact holomorphic surfaces S i .