Resolving F-Theory Models

In F-theory, the standard approach to handling singular geometries is to replace them with smooth ones via crepant resolutions. This process preserves the Calabi–Yau condition and ensures the correct encoding of physical data. However, several important caveats apply:

Not all singularities admit crepant resolutions, rather some singularities are obstructed from being resolved without violating the Calabi–Yau condition. No algorithm is known to the authors that determines whether a given singularity admits a crepant resolution.
Likewise, no general algorithm is known for computing a crepant resolution of a given singular geometry. In practice, one applies all known resolution techniques, guided by mathematical structure and physical expectations. A particularly prominent strategy is a sequence of blowups, which we will discuss below.

After applying a resolution strategy, one obtains a partially resolved model. For the reasons stated above, we do not currently verify whether the model has been fully resolved—i.e., whether all resolvable singularities have been removed crepantly to the extent possible:

is_partially_resolved — Method

is_partially_resolved(m::AbstractFTheoryModel)

Return true if resolution techniques were applied to the F-theory model, thereby potentially resolving its singularities. Otherwise, return false.

julia> B3 = projective_space(NormalToricVariety, 3)
Normal toric variety

julia> w = torusinvariant_prime_divisors(B3)[1]
Torus-invariant, prime divisor on a normal toric variety

julia> t = literature_model(arxiv_id = "1109.3454", equation = "3.1", base_space = B3, defining_classes = Dict("w" => w), completeness_check = false)
Construction over concrete base may lead to singularity enhancement. Consider computing singular_loci. However, this may take time!

Global Tate model over a concrete base -- SU(5)xU(1) restricted Tate model based on arXiv paper 1109.3454 Eq. (3.1)

julia> is_partially_resolved(t)
false

julia> t2 = blow_up(t, ["x", "y", "x1"]; coordinate_name = "e1")
Partially resolved global Tate model over a concrete base -- SU(5)xU(1) restricted Tate model based on arXiv paper 1109.3454 Eq. (3.1)

julia> is_partially_resolved(t2)
true

Experimental

This function is part of the experimental code in Oscar. Please read here for more details.

source

We can execute individual blowups with the following functions:

blow_up — Method

blow_up(m::AbstractFTheoryModel, ideal_gens::Vector{String}; coordinate_name::String = "e")

Resolve an F-theory model by blowing up a locus in the ambient space.

julia> B3 = projective_space(NormalToricVariety, 3)
Normal toric variety

julia> w = torusinvariant_prime_divisors(B3)[1]
Torus-invariant, prime divisor on a normal toric variety

julia> t = literature_model(arxiv_id = "1109.3454", equation = "3.1", base_space = B3, defining_classes = Dict("w" => w), completeness_check = false)
Construction over concrete base may lead to singularity enhancement. Consider computing singular_loci. However, this may take time!

Global Tate model over a concrete base -- SU(5)xU(1) restricted Tate model based on arXiv paper 1109.3454 Eq. (3.1)

julia> blow_up(t, ["x", "y", "x1"]; coordinate_name = "e1")
Partially resolved global Tate model over a concrete base -- SU(5)xU(1) restricted Tate model based on arXiv paper 1109.3454 Eq. (3.1)

Here is an example for a Weierstrass model.

julia> B2 = projective_space(NormalToricVariety, 2)
Normal toric variety

julia> b = torusinvariant_prime_divisors(B2)[1]
Torus-invariant, prime divisor on a normal toric variety

julia> w = literature_model(arxiv_id = "1208.2695", equation = "B.19", base_space = B2, defining_classes = Dict("b" => b), completeness_check = false)
Construction over concrete base may lead to singularity enhancement. Consider computing singular_loci. However, this may take time!

Weierstrass model over a concrete base -- U(1) Weierstrass model based on arXiv paper 1208.2695 Eq. (B.19)

julia> blow_up(w, ["x", "y", "x1"]; coordinate_name = "e1")
Partially resolved Weierstrass model over a concrete base -- U(1) Weierstrass model based on arXiv paper 1208.2695 Eq. (B.19)

Experimental

This function is part of the experimental code in Oscar. Please read here for more details.

source

blow_up — Method

blow_up(m::AbstractFTheoryModel, I::MPolyIdeal; coordinate_name::String = "e")

Resolve an F-theory model by blowing up a locus in the ambient space.

julia> B3 = projective_space(NormalToricVariety, 3)
Normal toric variety

julia> w = torusinvariant_prime_divisors(B3)[1]
Torus-invariant, prime divisor on a normal toric variety

julia> t = literature_model(arxiv_id = "1109.3454", equation = "3.1", base_space = B3, defining_classes = Dict("w" => w), completeness_check = false)
Construction over concrete base may lead to singularity enhancement. Consider computing singular_loci. However, this may take time!

Global Tate model over a concrete base -- SU(5)xU(1) restricted Tate model based on arXiv paper 1109.3454 Eq. (3.1)

julia> x1, x2, x3, x4, x, y, z = gens(cox_ring(ambient_space(t)))
7-element Vector{MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}}:
 x1
 x2
 x3
 x4
 x
 y
 z

julia> blow_up(t, ideal([x, y, x1]); coordinate_name = "e1")
Partially resolved global Tate model over a concrete base -- SU(5)xU(1) restricted Tate model based on arXiv paper 1109.3454 Eq. (3.1)

Experimental

This function is part of the experimental code in Oscar. Please read here for more details.

source

blow_up — Method

blow_up(m::AbstractFTheoryModel, I::AbsIdealSheaf; coordinate_name::String = "e")

Resolve an F-theory model by blowing up a locus in the ambient space. For this method, the blowup center is encoded by an ideal sheaf.

julia> B3 = projective_space(NormalToricVariety, 3)
Normal toric variety

julia> w = torusinvariant_prime_divisors(B3)[1]
Torus-invariant, prime divisor on a normal toric variety

julia> t = literature_model(arxiv_id = "1109.3454", equation = "3.1", base_space = B3, defining_classes = Dict("w" => w), completeness_check = false)
Construction over concrete base may lead to singularity enhancement. Consider computing singular_loci. However, this may take time!

Global Tate model over a concrete base -- SU(5)xU(1) restricted Tate model based on arXiv paper 1109.3454 Eq. (3.1)

julia> x1, x2, x3, x4, x, y, z = gens(cox_ring(ambient_space(t)))
7-element Vector{MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}}:
 x1
 x2
 x3
 x4
 x
 y
 z

julia> blowup_center = ideal_sheaf(ambient_space(t), ideal([x, y, x1]))
Sheaf of ideals
  on normal, simplicial toric variety
with restrictions
   1: Ideal (x_5_1, x_4_1, x_1_1)
   2: Ideal (1)
   3: Ideal (x_5_3, x_4_3, x_1_3)
   4: Ideal (x_5_4, x_4_4, x_1_4)
   5: Ideal (1)
   6: Ideal (1)
   7: Ideal (1)
   8: Ideal (1)
   9: Ideal (1)
  10: Ideal (1)
  11: Ideal (1)
  12: Ideal (1)

julia> blow_up(t, blowup_center; coordinate_name = "e1")
Partially resolved global Tate model over a concrete base -- SU(5)xU(1) restricted Tate model based on arXiv paper 1109.3454 Eq. (3.1)

Experimental

This function is part of the experimental code in Oscar. Please read here for more details.

source

Typically, an entire sequence of blowups is needed to resolve an F-Theory model. If known, it can therefore be advantageous to inform the model of the resolution:

add_resolution — Method

add_resolution(m::AbstractFTheoryModel, centers::Vector{Vector{String}}, exceptionals::Vector{String})

Add a known resolution for a model.

julia> m = literature_model(arxiv_id = "1109.3454", equation = "3.1")
Assuming that the first row of the given grading is the grading under Kbar

Global Tate model over a not fully specified base -- SU(5)xU(1) restricted Tate model based on arXiv paper 1109.3454 Eq. (3.1)

julia> add_resolution(m, [["x", "y"], ["y", "s", "w"], ["s", "e4"], ["s", "e3"], ["s", "e1"]], ["s", "w", "e3", "e1", "e2"])

julia> length(resolutions(m))
2

Experimental

This function is part of the experimental code in Oscar. Please read here for more details.

source

Subsequently, the following method can be used to execute the entire sequence of blowups all at once:

resolve — Method

resolve(m::AbstractFTheoryModel, index::Int)

Resolve a model with the index-th resolution that is known.

julia> B3 = projective_space(NormalToricVariety, 3)
Normal toric variety

julia> w = torusinvariant_prime_divisors(B3)[1]
Torus-invariant, prime divisor on a normal toric variety

julia> t = literature_model(arxiv_id = "1109.3454", equation = "3.1", base_space = B3, defining_classes = Dict("w" => w), completeness_check = false)
Construction over concrete base may lead to singularity enhancement. Consider computing singular_loci. However, this may take time!

Global Tate model over a concrete base -- SU(5)xU(1) restricted Tate model based on arXiv paper 1109.3454 Eq. (3.1)

julia> t2 = resolve(t, 1)
Partially resolved global Tate model over a concrete base -- SU(5)xU(1) restricted Tate model based on arXiv paper 1109.3454 Eq. (3.1)

julia> cox_ring(ambient_space(t2))
Multivariate polynomial ring in 12 variables over QQ graded by
  x1 -> [1 0 0 0 0 0 0]
  x2 -> [0 1 0 0 0 0 0]
  x3 -> [0 1 0 0 0 0 0]
  x4 -> [0 1 0 0 0 0 0]
  x -> [0 0 1 0 0 0 0]
  y -> [0 0 0 1 0 0 0]
  z -> [0 0 0 0 1 0 0]
  e1 -> [0 0 0 0 0 1 0]
  e4 -> [0 0 0 0 0 0 1]
  e2 -> [-1 -3 -1 1 -1 -1 0]
  e3 -> [0 4 1 -1 1 0 -1]
  s -> [2 6 -1 0 2 1 1]

julia> w2 = 2 * torusinvariant_prime_divisors(B3)[1]
Torus-invariant, non-prime divisor on a normal toric variety

julia> t3 = literature_model(arxiv_id = "1109.3454", equation = "3.1", base_space = B3, defining_classes = Dict("w" => w2), completeness_check = false)
Construction over concrete base may lead to singularity enhancement. Consider computing singular_loci. However, this may take time!

Global Tate model over a concrete base -- SU(5)xU(1) restricted Tate model based on arXiv paper 1109.3454 Eq. (3.1)

julia> t4 = resolve(t3, 1)
Partially resolved global Tate model over a concrete base -- SU(5)xU(1) restricted Tate model based on arXiv paper 1109.3454 Eq. (3.1)

Experimental

This function is part of the experimental code in Oscar. Please read here for more details.

source

Analyzing the resolved Fiber Structure

The following is currently only supported for Global Tate Models.

After resolution, one typically studies the structure of the (resolved) fibers to extract intersection numbers and representation theory information. The following method computes the fiber components and their intersection graph:

analyze_fibers — Method

analyze_fibers(model::GlobalTateModel, centers::Vector{<:Vector{<:Integer}})

Determine the fiber of a (singular) global Tate model over a particular base locus.

Warning

This method may run for very long time and is currently not tested as part of the regular OSCAR CI due to its excessive run times.

Experimental

This function is part of the experimental code in Oscar. Please read here for more details.

source