Zero Mode Counting in F-Theory via CAP
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1 Zero Mode Counting in F-Theory via CAP Martin Bies String Pheno 2017 Martin Bies Zero Mode Counting in F-Theory via CAP 1 / 10
2 Overview Task 4 dim. F-theory compactification Count (anti)-chiral massless matter fields in 4d effective theory Martin Bies Zero Mode Counting in F-Theory via CAP 1 / 10
3 Overview Task 4 dim. F-theory compactification Count (anti)-chiral massless matter fields in 4d effective theory Structure Analyse physics (C. Mayrhofer, T. Weigand, M.B ) Martin Bies Zero Mode Counting in F-Theory via CAP 1 / 10
4 Overview Task 4 dim. F-theory compactification Count (anti)-chiral massless matter fields in 4d effective theory Structure Analyse physics (C. Mayrhofer, T. Weigand, M.B ) Compute sheaf cohomologies of non-pullback line bundles Martin Bies Zero Mode Counting in F-Theory via CAP 1 / 10
5 Overview Task 4 dim. F-theory compactification Count (anti)-chiral massless matter fields in 4d effective theory Structure Analyse physics (C. Mayrhofer, T. Weigand, M.B ) Compute sheaf cohomologies of non-pullback line bundles Developed and implemented algorithms with M. Barakat et al. ( project , , , , , ) Martin Bies Zero Mode Counting in F-Theory via CAP 1 / 10
6 Schematic Picture: Physics and Geometry of F-theory base B 3 Martin Bies Zero Mode Counting in F-Theory via CAP 2 / 10
7 Schematic Picture: Physics and Geometry of F-theory fibre π base B 3 Martin Bies Zero Mode Counting in F-Theory via CAP 2 / 10
8 Schematic Picture: Physics and Geometry of F-theory fibre π C R1 C R2 base B 3 Martin Bies Zero Mode Counting in F-Theory via CAP 2 / 10
9 Schematic Picture: Physics and Geometry of F-theory 6 E A B C D fibre π C R1 C R2 base B 3 Martin Bies Zero Mode Counting in F-Theory via CAP 2 / 10
10 Counting zero modes From Physics of F-Theory to Line Bundles a-th state in rep. R matter surface S a R = n a i P 1 i (C R ) i=1 Martin Bies Zero Mode Counting in F-Theory via CAP 3 / 10
11 Counting zero modes From Physics of F-Theory to Line Bundles n a-th state in rep. R matter surface SR a = a i P 1 i (C R ) i=1 G 4 -flux (complex) 2-cycle A in Y 4 Martin Bies Zero Mode Counting in F-Theory via CAP 3 / 10
12 Counting zero modes From Physics of F-Theory to Line Bundles n a-th state in rep. R matter surface SR a = a i P 1 i (C R ) i=1 G 4 -flux (complex) 2-cycle A in Y 4 Consequence SR a and A intersect in number of points in Y 4 Martin Bies Zero Mode Counting in F-Theory via CAP 3 / 10
13 Counting zero modes From Physics of F-Theory to Line Bundles n a-th state in rep. R matter surface SR a = a i P 1 i (C R ) i=1 G 4 -flux (complex) 2-cycle A in Y 4 Consequence SR a and A intersect in number of points in Y 4 π (SR a A) number of points in C R Martin Bies Zero Mode Counting in F-Theory via CAP 3 / 10
14 Counting zero modes From Physics of F-Theory to Line Bundles a-th state in rep. R matter surface S a R = Consequence n a i P 1 i (C R ) i=1 G 4 -flux (complex) 2-cycle A in Y 4 S a R and A intersect in number of points in Y 4 π (S a R A) number of points in C R L (S a R, A) := O C R (π (S a R A)) Pic (C R) Martin Bies Zero Mode Counting in F-Theory via CAP 3 / 10
15 Counting zero modes From Physics of F-Theory to Line Bundles a-th state in rep. R matter surface S a R = Consequence n a i P 1 i (C R ) i=1 G 4 -flux (complex) 2-cycle A in Y 4 S a R and A intersect in number of points in Y 4 π (S a R A) number of points in C R L (S a R, A) := O C R (π (S a R A)) Pic (C R) Zero Modes and Sheaf Cohomology chiral zero modes of S a R H0 (C R, L (S a R, A) O spin,c R ) anti-chiral zero modes of S a R H1 (C R, L (S a R, A) O spin,c R ) Martin Bies Zero Mode Counting in F-Theory via CAP 3 / 10
16 Generalities Of The Implementations In CAP Martin Bies Zero Mode Counting in F-Theory via CAP 4 / 10
17 Generalities Of The Implementations In CAP Why new algorithm? L (S a R, A) = O C R (D). Extend L (S a R, A) by zero outside of C R. Coherent sheaf on X Σ Martin Bies Zero Mode Counting in F-Theory via CAP 4 / 10
18 Generalities Of The Implementations In CAP Why new algorithm? L (S a R, A) = O C R (D). Extend L (S a R, A) by zero outside of C R. Coherent sheaf on X Σ Schematic Picture Martin Bies Zero Mode Counting in F-Theory via CAP 4 / 10
19 Generalities Of The Implementations In CAP Why new algorithm? L (S a R, A) = O C R (D). Extend L (S a R, A) by zero outside of C R. Coherent sheaf on X Σ Schematic Picture idea of mathematician G. Smith et al. (math/ , math/ , DOI: /OWR/2013/25) Martin Bies Zero Mode Counting in F-Theory via CAP 4 / 10
20 Generalities Of The Implementations In CAP Why new algorithm? L (S a R, A) = O C R (D). Extend L (S a R, A) by zero outside of C R. Coherent sheaf on X Σ Schematic Picture idea of mathematician G. Smith et al. (math/ , math/ , DOI: /OWR/2013/25) cohomcalg by R. Blumenhagen et al. ( , , , ) Martin Bies Zero Mode Counting in F-Theory via CAP 4 / 10
21 Generalities Of The Implementations In CAP Why new algorithm? L (S a R, A) = O C R (D). Extend L (S a R, A) by zero outside of C R. Coherent sheaf on X Σ Schematic Picture idea of mathematician G. Smith et al. (math/ , math/ , DOI: /OWR/2013/25) cohomcalg by R. Blumenhagen et al. ( , , , ) Combine to obtain algorithm which applies Martin Bies Zero Mode Counting in F-Theory via CAP 4 / 10
22 Generalities Of The Implementations In CAP Why new algorithm? L (S a R, A) = O C R (D). Extend L (S a R, A) by zero outside of C R. Coherent sheaf on X Σ Schematic Picture idea of mathematician G. Smith et al. (math/ , math/ , DOI: /OWR/2013/25) cohomcalg by R. Blumenhagen et al. ( , , , ) Combine to obtain algorithm which applies on more general toric spaces (than idea of G. Smith) Martin Bies Zero Mode Counting in F-Theory via CAP 4 / 10
23 Generalities Of The Implementations In CAP Why new algorithm? L (S a R, A) = O C R (D). Extend L (S a R, A) by zero outside of C R. Coherent sheaf on X Σ Schematic Picture idea of mathematician G. Smith et al. (math/ , math/ , DOI: /OWR/2013/25) cohomcalg by R. Blumenhagen et al. ( , , , ) Combine to obtain algorithm which applies on more general toric spaces (than idea of G. Smith) to all coherent sheaves (i.e. not only line bundles as cohomcalg) Martin Bies Zero Mode Counting in F-Theory via CAP 4 / 10
24 From Points to Coherent Sheaves How to encode O XΣ ( D)? Martin Bies Zero Mode Counting in F-Theory via CAP 5 / 10
25 From Points to Coherent Sheaves How to encode O XΣ ( D)? X Σ toric variety (without torus factor) with coordinate ring S Martin Bies Zero Mode Counting in F-Theory via CAP 5 / 10
26 From Points to Coherent Sheaves How to encode O XΣ ( D)? X Σ toric variety (without torus factor) with coordinate ring S divisor D = V (P 1,..., P n ) cut out by hom. polynomials Martin Bies Zero Mode Counting in F-Theory via CAP 5 / 10
27 From Points to Coherent Sheaves How to encode O XΣ ( D)? X Σ toric variety (without torus factor) with coordinate ring S divisor D = V (P 1,..., P n ) cut out by hom. polynomials A := ker (P 1,..., P n ) relations among the P i Martin Bies Zero Mode Counting in F-Theory via CAP 5 / 10
28 From Points to Coherent Sheaves How to encode O XΣ ( D)? X Σ toric variety (without torus factor) with coordinate ring S divisor D = V (P 1,..., P n ) cut out by hom. polynomials A := ker (P 1,..., P n ) relations among the P i R 2 Look at exact sequence S (e j ) A R 1 S (d i ) M 0, j=1 which defines M S-fpgrmod i=1 Martin Bies Zero Mode Counting in F-Theory via CAP 5 / 10
29 From Points to Coherent Sheaves How to encode O XΣ ( D)? X Σ toric variety (without torus factor) with coordinate ring S divisor D = V (P 1,..., P n ) cut out by hom. polynomials A := ker (P 1,..., P n ) relations among the P i R 2 Look at exact sequence S (e j ) A R 1 S (d i ) M 0, j=1 which defines M S-fpgrmod i=1 Answer M = O XΣ ( D) via the sheafification functor : S-fpgrmod CohX Σ, N Ñ Martin Bies Zero Mode Counting in F-Theory via CAP 5 / 10
30 From Points to Coherent Sheaves How to encode O XΣ ( D)? X Σ toric variety (without torus factor) with coordinate ring S divisor D = V (P 1,..., P n ) cut out by hom. polynomials A := ker (P 1,..., P n ) relations among the P i R 2 Look at exact sequence S (e j ) A R 1 S (d i ) M 0, j=1 which defines M S-fpgrmod Answer M = O XΣ ( D) via the sheafification functor : S-fpgrmod CohX Σ, N Ñ Use M S-fpgrmod as computer model for O XΣ ( D) i=1 Martin Bies Zero Mode Counting in F-Theory via CAP 5 / 10
31 Sketch of Algorithm in CAP Martin Bies Zero Mode Counting in F-Theory via CAP 6 / 10
32 Sketch of Algorithm in CAP Input and Output (smooth, complete) or (simplicial, projective) toric variety X Σ M S-fpgrmod h (X i Σ, M ) Martin Bies Zero Mode Counting in F-Theory via CAP 6 / 10
33 Sketch of Algorithm in CAP Input and Output (smooth, complete) or (simplicial, projective) toric variety X Σ M S-fpgrmod h (X i Σ, M ) Step-by-step 1 Use cohomcalg to compute (0 k dim Q (X Σ )) { } V k (X Σ ) := L Pic (X Σ ), h k (X Σ, L) = 0 Martin Bies Zero Mode Counting in F-Theory via CAP 6 / 10
34 Sketch of Algorithm in CAP Input and Output (smooth, complete) or (simplicial, projective) toric variety X Σ M S-fpgrmod h (X i Σ, M ) Step-by-step 1 Use cohomcalg to compute (0 k dim Q (X Σ )) { } V k (X Σ ) := L Pic (X Σ ), h k (X Σ, L) = 0 2 Find ideal I S (along idea of G. Smith) s.t. H i ( X Σ, M) = Ext i S (I, M) 0 Martin Bies Zero Mode Counting in F-Theory via CAP 6 / 10
35 Sketch of Algorithm in CAP Input and Output (smooth, complete) or (simplicial, projective) toric variety X Σ M S-fpgrmod h (X i Σ, M ) Step-by-step 1 Use cohomcalg to compute (0 k dim Q (X Σ )) { } V k (X Σ ) := L Pic (X Σ ), h k (X Σ, L) = 0 2 Find ideal I S (along idea of G. Smith) s.t. H i ( X Σ, M) = Ext i S (I, M) 0 3 Compute Q-dimension of Ext i S (I, M) 0 Martin Bies Zero Mode Counting in F-Theory via CAP 6 / 10
36 SU(5) U(1)-Tate model from Input and Output C 5 2 P 2 Q L 5 2 M and M defined by S ( 36) S ( 39) S ( 41) S ( 23) S ( 38) S ( 6) S ( 21) M 0 h (P 1 2 Q, M ) =? Martin Bies Zero Mode Counting in F-Theory via CAP 7 / 10
37 SU(5) U(1)-Tate model from Input and Output C 5 2 P 2 Q L 5 2 M and M defined by S ( 36) S ( 39) S ( 41) S ( 23) S ( 38) S ( 6) S ( 21) M 0 h (P 1 2 Q, M ) =? Apply Algorithm 1 Compute vanishing sets via cohomcalg: V 0 (P 2 Q ) = (, 1] Z, V 1 (P 2 Q ) = Z, V 2 (P 2 Q ) = [ 2, ) Z Martin Bies Zero Mode Counting in F-Theory via CAP 7 / 10
38 SU(5) U(1)-Tate model from Input and Output C 5 2 P 2 Q L 5 2 M and M defined by S ( 36) S ( 39) S ( 41) S ( 23) S ( 38) S ( 6) S ( 21) M 0 h (P 1 2 Q, M ) =? Apply Algorithm 1 V 0 (P 2 Q ) = (, 1] Z, V 1 (P 2 Q ) = Z, V 2 (P 2 Q ) = [ 2, ) Z 2 Use vanishing sets to find ideal I (along idea of G. Smith): I = B (44) Σ x0 44, x 1 44, x 2 44 Martin Bies Zero Mode Counting in F-Theory via CAP 7 / 10
39 SU(5) U(1)-Tate model from Input and Output C 5 2 P 2 Q L 5 2 M and M defined by S ( 36) S ( 39) S ( 41) S ( 23) S ( 38) S ( 6) S ( 21) M 0 h (P 1 2 Q, M ) =? Apply Algorithm 1 V 0 (P 2 Q ) = (, 1] Z, V 1 (P 2 Q ) = Z, V 2 (P 2 Q ) = [ 2, ) Z 2 I = B (44) Σ x0 44, x 1 44, x Compute presentation of Ext 1 S Ext 1 S ( B (44) Σ, M )0 : ( B (44) Σ, M )0 Martin Bies Zero Mode Counting in F-Theory via CAP 7 / 10
40 SU(5) U(1)-Tate model from Input and Output C 5 2 P 2 Q L 5 2 M and M defined by S ( 36) S ( 39) S ( 41) S ( 23) S ( 38) S ( 6) S ( 21) M 0 h (P 1 2 Q, M ) =? Apply Algorithm 1 V 0 (P 2 Q ) = (, 1] Z, V 1 (P 2 Q ) = Z, V 2 (P 2 Q ) = [ 2, ) Z 2 I = B (44) Σ x0 44, x 1 44, x Compute presentation of Ext 1 S Q Q Ext 1 S ( B (44) Σ, M )0 : ( B (44) Σ, M )0 0 Martin Bies Zero Mode Counting in F-Theory via CAP 7 / 10
41 SU(5) U(1)-Tate model from Input and Output C 5 2 P 2 Q L 5 2 M and M defined by S ( 36) S ( 39) S ( 41) S ( 23) S ( 38) S ( 6) S ( 21) M 0 h (P 1 2 Q, M ) =? Apply Algorithm 1 V 0 (P 2 Q ) = (, 1] Z, V 1 (P 2 Q ) = Z, V 2 (P 2 Q ) = [ 2, ) Z 2 I = B (44) Σ x0 44, x 1 44, x 2 44 ( 3 Q Q Ext 1 S B (44) Σ )0, M 0 ( ] 28 = dim Q [Ext 1 S B (44) Σ )0 (, M = h 1 P 2 Q, M ) Martin Bies Zero Mode Counting in F-Theory via CAP 7 / 10
42 Scan Over Moduli Space Martin Bies Zero Mode Counting in F-Theory via CAP 8 / 10
43 Scan Over Moduli Space Note Values of complex structure moduli enter definition of M Martin Bies Zero Mode Counting in F-Theory via CAP 8 / 10
44 Scan Over Moduli Space Note Values of complex structure moduli enter definition of M Smoothness of matter curves NOT required Martin Bies Zero Mode Counting in F-Theory via CAP 8 / 10
45 Scan Over Moduli Space Note Values of complex structure moduli enter definition of M Smoothness of matter curves NOT required Run computation for different choices of moduli Martin Bies Zero Mode Counting in F-Theory via CAP 8 / 10
46 Scan Over Moduli Space Note Values of complex structure moduli enter definition of M Smoothness of matter curves NOT required Run computation for different choices of moduli SU(5)xU(1)-Tate Model from (R = 5 2 ) ã 1,0 ã 2,1 ã 3,2 ã 4,3 h i (C R, L R ) M 1 (x 1 x 2 ) 4 x1 7 x2 10 x3 13 (22, 43) M 2 (x 1 x 2 ) x3 3 x1 7 x2 10 x3 13 (21, 42) M 3 x3 4 x1 7 x2 7 (x 1 + x 2 ) 3 x x 2 ) (11, 32) M 4 (x 1 x 2 ) 3 x 3 x1 7 x2 10 x3 13 (9, 30) M 5 x3 4 x1 7 x2 8 (x 1 + x 2 ) 2 x x 2 ) 2 (7, 28) M 6 x3 4 x1 7 x2 10 x3 8 (x 1 x 2 ) 5 (6, 27) M 7 x3 4 x1 7 x2 9 (x 1 + x 2 ) x x 2 ) 3 (5, 26) Martin Bies Zero Mode Counting in F-Theory via CAP 8 / 10
47 Summary and Conclusion Martin Bies Zero Mode Counting in F-Theory via CAP 9 / 10
48 Summary and Conclusion Have combined cohomcalg by R. Blumenhagen et al. ( , , , ) and idea of G. Smith et al. (math/ , math/ , DOI: /OWR/2013/25) Martin Bies Zero Mode Counting in F-Theory via CAP 9 / 10
49 Summary and Conclusion Have combined cohomcalg by R. Blumenhagen et al. ( , , , ) and idea of G. Smith et al. (math/ , math/ , DOI: /OWR/2013/25) Toolkit to compute sheaf cohomologies of all coherent sheaves on toric varieties (visit Martin Bies Zero Mode Counting in F-Theory via CAP 9 / 10
50 Summary and Conclusion Have combined cohomcalg by R. Blumenhagen et al. ( , , , ) and idea of G. Smith et al. (math/ , math/ , DOI: /OWR/2013/25) Toolkit to compute sheaf cohomologies of all coherent sheaves on toric varieties (visit Features: Count zero modes in 4d F-theory compactifications Matter curves need not be smooth, nor complete intersections! Of particular interest: hypercharge flux in F-theory GUTs (applications currently on their way) Martin Bies Zero Mode Counting in F-Theory via CAP 9 / 10
51 Summary and Conclusion Have combined cohomcalg by R. Blumenhagen et al. ( , , , ) and idea of G. Smith et al. (math/ , math/ , DOI: /OWR/2013/25) Toolkit to compute sheaf cohomologies of all coherent sheaves on toric varieties (visit Features: Count zero modes in 4d F-theory compactifications Matter curves need not be smooth, nor complete intersections! Of particular interest: hypercharge flux in F-theory GUTs (applications currently on their way) Further possible applications Quite generally zero mode counting in topological string, IIB or heterotic compactifications Martin Bies Zero Mode Counting in F-Theory via CAP 9 / 10
52 Summary and Conclusion Have combined cohomcalg by R. Blumenhagen et al. ( , , , ) and idea of G. Smith et al. (math/ , math/ , DOI: /OWR/2013/25) Toolkit to compute sheaf cohomologies of all coherent sheaves on toric varieties (visit Features: Count zero modes in 4d F-theory compactifications Matter curves need not be smooth, nor complete intersections! Of particular interest: hypercharge flux in F-theory GUTs (applications currently on their way) Further possible applications Quite generally zero mode counting in topological string, IIB or heterotic compactifications T-branes as coherent sheaves (Collinucci et al )... Martin Bies Zero Mode Counting in F-Theory via CAP 9 / 10
53 Thank you for your attention! Martin Bies Zero Mode Counting in F-Theory via CAP 10 / 10
54 From Divisors to Modules Input and Output C = V (g 1,..., g k ) X Σ D = V (f 1,..., f n ) Div(C) M s.t. supp( M) = C and M C = OC ( D) Martin Bies Zero Mode Counting in F-Theory via CAP 10 / 10
55 From Divisors to Modules Input and Output C = V (g 1,..., g k ) X Σ D = V (f 1,..., f n ) Div(C) M s.t. supp( M) = C and M C = OC ( D) Step 1: S(C) := S/ g 1,..., g k, π : S S(C) 0 S (C) (j) 0 j J ker (m) 0 S (C) (i) m = (π (f 1 ),..., π (f n )) S (C) i I A C ι S (C) Martin Bies Zero Mode Counting in F-Theory via CAP 10 / 10
56 From Divisors to Modules II Step 2: Extend by zero to coherent sheaf on X Σ j J S (j) ker (m) i I S (i) k K S (k) g 1. g k S (C) A B M = A B satisfies Supp( M) = C and M C = OC ( D) Martin Bies Zero Mode Counting in F-Theory via CAP 10 / 10
57 From Divisors to Modules III Input and Output C = V (g 1,..., g k ) X Σ D = V (f 1,..., f n ) Div(C) M s.t. supp( M) = C and M C = OC (+D) Strategy 1 Compute A C 2 Dualise via A C := Hom S(C) (S (C), A C ) 3 Extend by zero by considering A B M := A B satisfies Supp( M) = C and M C = OC (+D) Martin Bies Zero Mode Counting in F-Theory via CAP 10 / 10
58 An idea of the sheafification functor Martin Bies Zero Mode Counting in F-Theory via CAP 11 / 10
59 An idea of the sheafification functor Affine open cover Toric variety X Σ with Cox ring S { } Covered by affine opens U σ = Specm(S (x ˆσ )) σ Σ Martin Bies Zero Mode Counting in F-Theory via CAP 11 / 10
60 An idea of the sheafification functor Affine open cover Toric variety X Σ with Cox ring S { } Covered by affine opens U σ = Specm(S (x ˆσ )) Localising ( restricting) a module M S-fpgrmod M (x ˆσ ) is f.p. S (x ˆσ )-module σ Σ Martin Bies Zero Mode Counting in F-Theory via CAP 11 / 10
61 An idea of the sheafification functor Affine open cover Toric variety X Σ with Cox ring S { } Covered by affine opens U σ = Specm(S (x ˆσ )) Localising ( restricting) a module M S-fpgrmod M (x ˆσ ) is f.p. S (x ˆσ )-module Consequence M (x ˆσ ) coherent sheaf on U σ = Specm(S (x ˆσ )) σ Σ local sections: M (x ˆσ ) (D (f )) = M (x ˆσ ) S(x ˆσ ) (S (x ˆσ ) ) f Martin Bies Zero Mode Counting in F-Theory via CAP 11 / 10
62 Module M 5 from : Quality Check I ) ] =0 20 ( [ B (e) Σ, M 5 Ext 0 S dimq e Martin Bies Zero Mode Counting in F-Theory via CAP 10 / 10
63 Module M 5 from : Quality Check I ) ] =0 20 ( [ B (e) Σ, M 5 Ext 0 S dimq e Martin Bies Zero Mode Counting in F-Theory via CAP 10 / 10
64 Module M 5 from : Quality Check I ) ] =0 20 ( [ B (e) Σ, M 5 Ext 0 S dimq e Martin Bies Zero Mode Counting in F-Theory via CAP 10 / 10
65 Module M 5 from : Quality Check II ] 600 ) [ ( B (e) Σ, M 5 Ext 1 S = dimq e Martin Bies Zero Mode Counting in F-Theory via CAP 10 / 10
66 Module M 5 from : Quality Check II ] 600 ) [ ( B (e) Σ, M 5 Ext 1 S = dimq e Martin Bies Zero Mode Counting in F-Theory via CAP 10 / 10
67 How to determine the ideal I in step 2 of algorithm? Input M S-fpgrmod V k (X Σ ) = { L Pic (X Σ ), h k (X Σ, L) = 0 } Martin Bies Zero Mode Counting in F-Theory via CAP 10 / 10
68 How to determine the ideal I in step 2 of algorithm? Input M S-fpgrmod V k (X Σ ) = { L Pic (X Σ ), h k (X Σ, L) = 0 } Preparation p Cl (X Σ ) ample, m (p) = {m 1,..., m k } all monomials of degree p and I (p, e) = m e 1,..., me k Pick e = 0 and increase it until subsequent conditions are met Martin Bies Zero Mode Counting in F-Theory via CAP 10 / 10
69 How to determine the ideal I in step 2 of algorithm? Input M S-fpgrmod V k (X Σ ) = { L Pic (X Σ ), h k (X Σ, L) = 0 } How to find ideal I? ( ) Look at spectral sequence E p,q 2 Ext p+q O XΣ I (p, e), M Martin Bies Zero Mode Counting in F-Theory via CAP 10 / 10
70 How to determine the ideal I in step 2 of algorithm? Input M S-fpgrmod V k (X Σ ) = { L Pic (X Σ ), h k (X Σ, L) = 0 } How to find ideal I? ( ) Look at spectral sequence E p,q 2 Ext p+q O XΣ I (p, e), M Some objects E p,q 2 vanish as seen by V k (X Σ ) Martin Bies Zero Mode Counting in F-Theory via CAP 10 / 10
71 How to determine the ideal I in step 2 of algorithm? Input M S-fpgrmod V k (X Σ ) = { L Pic (X Σ ), h k (X Σ, L) = 0 } How to find ideal I? ( ) Look at spectral sequence E p,q 2 Ext p+q O XΣ I (p, e), M Some objects E p,q 2 vanish as seen by V k (X Σ ) Does E p,q 2 degenerate (on E 2 -sheet)? Is its limit (co)homology H m ( C 0) of complex of global sections of vector bundles? Martin Bies Zero Mode Counting in F-Theory via CAP 10 / 10
72 How to determine the ideal I in step 2 of algorithm? Input M S-fpgrmod V k (X Σ ) = { L Pic (X Σ ), h k (X Σ, L) = 0 } How to find ideal I? ( ) Look at spectral sequence E p,q 2 Ext p+q O XΣ I (p, e), M Some objects E p,q 2 vanish as seen by V k (X Σ ) Does E p,q 2 degenerate (on E 2 -sheet)? Is its limit (co)homology H m ( C 0) of complex of global sections of vector bundles? If no increase e until this is the case! Martin Bies Zero Mode Counting in F-Theory via CAP 10 / 10
73 How to determine the ideal I in step 2 of algorithm? Input M S-fpgrmod V k (X Σ ) = { L Pic (X Σ ), h k (X Σ, L) = 0 } How to find ideal I? ( ) Look at spectral sequence E p,q 2 Ext p+q O XΣ I (p, e), M Some objects E p,q 2 vanish as seen by V k (X Σ ) Does E p,q 2 degenerate (on E 2 -sheet)? Is its limit (co)homology H m ( C 0) of complex of global sections of vector bundles? If no increase e until this is the case! Long exact sequence: sheaf cohomology local cohomology Martin Bies Zero Mode Counting in F-Theory via CAP 10 / 10
74 How to determine the ideal I in step 2 of algorithm? Input M S-fpgrmod V k (X Σ ) = { L Pic (X Σ ), h k (X Σ, L) = 0 } How to find ideal I? ( ) Look at spectral sequence E p,q 2 Ext p+q O XΣ I (p, e), M Some objects E p,q 2 vanish as seen by V k (X Σ ) Does E p,q 2 degenerate (on E 2 -sheet)? Is its limit (co)homology H m ( C 0) of complex of global sections of vector bundles? If no increase e until this is the case! Long exact sequence: sheaf cohomology local cohomology Increase e further until H m ( C 0) = Ext m S (I (p, e), M) 0 Martin Bies Zero Mode Counting in F-Theory via CAP 10 / 10
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