On Flux Quantization in F-Theory

On Flux Quantization in F-Theory Raffaele Savelli MPI - Munich Bad Honnef, March 2011 Based on work with A. Collinucci, arxiv: 1011.6388

Motivations

Motivations The recent attempts to find UV-completions of F-theory inspired GUT-models need several global consistency checks.

Motivations The recent attempts to find UV-completions of F-theory inspired GUT-models need several global consistency checks. For instance, Freed-Witten anomalies and tadpoles have to vanish in any consistent compactification.

Motivations The recent attempts to find UV-completions of F-theory inspired GUT-models need several global consistency checks. For instance, Freed-Witten anomalies and tadpoles have to vanish in any consistent compactification. In particular, FW anomaly cancellation sheds light on crucial issues like quantization of fluxes and integrality of tadpoles.

Motivations The recent attempts to find UV-completions of F-theory inspired GUT-models need several global consistency checks. For instance, Freed-Witten anomalies and tadpoles have to vanish in any consistent compactification. In particular, FW anomaly cancellation sheds light on crucial issues like quantization of fluxes and integrality of tadpoles. A better understanding of how to treat fluxes is relevant for problems like moduli stabilization and generation of chiral matter.

Plan of the talk

Plan of the talk Brief introduction to F-theory and to its duality with M- theory.

Plan of the talk Brief introduction to F-theory and to its duality with M- theory. Review of flux quantization in M-theory.

Plan of the talk Brief introduction to F-theory and to its duality with M- theory. Review of flux quantization in M-theory. Analysis of the F-theory consequences thereof in smooth fourfold compactifications.

Plan of the talk Brief introduction to F-theory and to its duality with M- theory. Review of flux quantization in M-theory. Analysis of the F-theory consequences thereof in smooth fourfold compactifications. Analysis of the 7-branes gauge flux quantization rules in the case of fourfolds with symplectic-type singularities.

Plan of the talk Brief introduction to F-theory and to its duality with M- theory. Review of flux quantization in M-theory. Analysis of the F-theory consequences thereof in smooth fourfold compactifications. Analysis of the 7-branes gauge flux quantization rules in the case of fourfolds with symplectic-type singularities. Summary and outlook.

F vs M - theory F-theory is a geometric way of taking into account the back-reaction of 7-brane solutions of type IIB theory: T 2 π :CY B projection CY i : B CY 0-section B Kähler It is formulated in 12 dimensions. The 2 additional directions describe an auxiliary torus fibered over B with complex modulus τ = C 0 + ie φ axio-dilaton Supersymmetry and equation of motion for the axio-dilaton force the internal space to be an elliptically fibered Calabi-Yau, with the 7-branes being the degeneration loci of the fiber. τ varies holomorphically along B with transitions in SL(2, Z)

Focus on the case of base space of complex dimension 3, B 3 F-theory on R 1,3 CY 4 N =1 D =4 gauge theory To deal with issues about fluxes and tadpoles, we use M/F-theory duality F-theory can be obtained from M-theory on elliptic Calabi-Yau in the limit of vanishing fiber volume V F. Denef arxiv: 0803.1194 reduce M-theory to type IIA along one non-trivial cycle of T 2, A 1 T-dualize to type IIB along the other non-trivial cycle, A 2 send V to 0 to obtain IIB string theory on R 1,3 B 3 with varying τ

Focus on the case of base space of complex dimension 3, B 3 F-theory on R 1,3 CY 4 N =1 D =4 gauge theory To deal with issues about fluxes and tadpoles, we use M/F-theory duality F-theory can be obtained from M-theory on elliptic Calabi-Yau in the limit of vanishing fiber volume V F. Denef arxiv: 0803.1194 reduce M-theory to type IIA along one non-trivial cycle of T 2, A 1 T-dualize to type IIB along the other non-trivial cycle, A 2 send V to 0 to obtain IIB string theory on R 1,3 B 3 with varying τ A 7-brane is a divisor of In order not to break 4D Lorentz, the flux must have one and only one leg along giving rise to both bulk and brane type fluxes in IIB: T 2 B 3 on which some 1-cycle of the fiber collapses. flux lines along collapsing 1-cycle F 2 brane flux otherwise F 3,H 3 bulk flux G 4

The quantum theory of M2 propagating in CY 4 may suffer from a global anomaly E. Witten hep-th/9609122 pfaff D φ exp 2πi φ C 3 path integral measure with a sign ambiguity M2 ( 1) R c 2 (CY 4) ( 1) R 2G 4 integrals made on the 4-cycle swept by bringing M2 along a non-trivial loop

The quantum theory of M2 propagating in CY 4 may suffer from a global anomaly E. Witten hep-th/9609122 pfaff D φ exp 2πi φ C 3 path integral measure with a sign ambiguity M2 ( 1) R c 2 (CY 4) ( 1) R 2G 4 integrals made on the 4-cycle swept by bringing M2 along a non-trivial loop shift in the quantization of G 4 [G 4 ]+ c 2(CY 4 ) 2 H 4 (CY 4, Z)

The quantum theory of M2 propagating in CY 4 may suffer from a global anomaly E. Witten hep-th/9609122 pfaff D φ exp 2πi φ C 3 path integral measure with a sign ambiguity M2 ( 1) R c 2 (CY 4) ( 1) R 2G 4 integrals made on the 4-cycle swept by bringing M2 along a non-trivial loop We ll find: shift in the quantization of G 4 [G 4 ]+ c 2(CY 4 ) 2 H 4 (CY 4, Z)

The quantum theory of M2 propagating in CY 4 may suffer from a global anomaly E. Witten hep-th/9609122 pfaff D φ exp 2πi φ C 3 path integral measure with a sign ambiguity M2 ( 1) R c 2 (CY 4) ( 1) R 2G 4 integrals made on the 4-cycle swept by bringing M2 along a non-trivial loop We ll find: shift in the quantization of G 4 [G 4 ]+ c 2(CY 4 ) 2 H 4 (CY 4, Z) a half -quantized gauge flux on the 7-branes may arise, depending on the geometry of CY 4 topological obstruction to make vanish! F 2

The quantum theory of M2 propagating in CY 4 may suffer from a global anomaly E. Witten hep-th/9609122 pfaff D φ exp 2πi φ C 3 path integral measure with a sign ambiguity M2 ( 1) R c 2 (CY 4) ( 1) R 2G 4 integrals made on the 4-cycle swept by bringing M2 along a non-trivial loop We ll find: shift in the quantization of G 4 [G 4 ]+ c 2(CY 4 ) 2 F 3 and H 3 are integrally quantized H 4 (CY 4, Z) a half -quantized gauge flux on the 7-branes may arise, depending on the geometry of CY 4 topological obstruction to make vanish! F 2

The quantum theory of M2 propagating in CY 4 may suffer from a global anomaly E. Witten hep-th/9609122 pfaff D φ exp 2πi φ C 3 path integral measure with a sign ambiguity M2 ( 1) R c 2 (CY 4) ( 1) R 2G 4 integrals made on the 4-cycle swept by bringing M2 along a non-trivial loop We ll find: shift in the quantization of G 4 [G 4 ]+ c 2(CY 4 ) 2 expected F 3 and H 3 are integrally quantized H 4 (CY 4, Z) a half -quantized gauge flux on the 7-branes may arise, depending on the geometry of CY 4 topological obstruction to make vanish! Notice: cancellation between left and right movers for closed strings (and the S-dual D1-branes) makes their pfaff well-defined! F 2

Complete answer for smooth CY 4 Description by a non-singular Weierstrass hypersurface Y 2 = X 3 + fxz 4 + gz 6 in M 5 B 3 with fiber W P 2 2,3,1(X, Y, Z) Let α be the Poincaré dual of the 0-section by adjunction we can express c(cy 4 ) in terms of c(b 3 ) c 2 (CY 4 ) = 12 α c 1 (B 3 )+c 2 (B 3 ) + 11c 2 1(B 3 ) T 2 α =1 while c 1 (B 3 ) and c 2 (B 3 ) are pulled-back from the base c 2 (CY 4 ) has either two or no legs along the fiber!

it cannot induce any shift in the D7 gauge flux quantization Sen s weak coupling limit confirms this expectation: D7-brane has no obstructed deformation moduli!

it cannot induce any shift in the D7 gauge flux quantization Sen s weak coupling limit confirms this expectation: D7-brane has no obstructed deformation moduli! if odd, it implies the presence of a Lorentz-violating G 4 Such backgrounds necessarily generate F-theory vacua without Poincaré invariance in 4D!

it cannot induce any shift in the D7 gauge flux quantization Sen s weak coupling limit confirms this expectation: D7-brane has no obstructed deformation moduli! if odd, it implies the presence of a Lorentz-violating G 4 Such backgrounds necessarily generate F-theory vacua without Poincaré invariance in 4D! Claim: c 2 (CY 4 ) is always an even class. The proof reduces to show that c 2 (B 3 )+c 2 1(B 3 ) is an even class of B 3 Basic facts in algebraic topology can be used to find that: c 2 + c 2 1 is even for any smooth, complex variety of dim. at most 3

Non-abelian singularities

Non-abelian singularities No Weierstrass representation of c 2 CY 4 after blow-up possibly odd only on vertical 4-cycles of the exceptional divisors. Possibly half-quantized 7-brane gauge fluxes along Cartan directions of the gauge group.

Non-abelian singularities No Weierstrass representation of c 2 CY 4 after blow-up possibly odd only on vertical 4-cycles of the exceptional divisors. Possibly half-quantized 7-brane gauge fluxes along Cartan directions of the gauge group. This happens when the 7-brane stack wraps a non-spin manifold. Freed-Witten anomaly: S 2 non spin F 2 c 1(S 2 ) H 2 (S 2, Z) 2 Claim proven in the case of singularities of Kodaira type I ns 2N namely for Sp(N) gauge groups, N being the rank.

Non-abelian singularities No Weierstrass representation of c 2 CY 4 after blow-up possibly odd only on vertical 4-cycles of the exceptional divisors. Possibly half-quantized 7-brane gauge fluxes along Cartan directions of the gauge group. This happens when the 7-brane stack wraps a non-spin manifold. Freed-Witten anomaly: S 2 non spin F 2 c 1(S 2 ) H 2 (S 2, Z) 2 Claim proven in the case of singularities of Kodaira type I ns 2N namely for Sp(N) gauge groups, N being the rank. Illustrative example: N = 1 and B 3 = P 3 Force an Sp(1)=SU(2) singularity on a divisor {P n =0} P 3 Blow-up CY 4 Compute the along the cod. 2 locus c 2 X = Y = P n =0 of the resolved fourfold CY4

To use toric methods, add to the ambient fivefold a coordinate, σ, and the new equation σ = P n (x 1,...,x 4 ) The resolution of the SU(2)-singular Weierstrass model will be described by the projective weights: M 5 P 3 x 1 x 2 x 3 x 4 σ X Y Z v vσ = P n 1 1 1 1 n 8 12 0 0 n 0 0 0 0 0 2 3 1 0 0 0 0 0 0 1 1 1 0 1 0 hyperplane class H 0-section class α exceptional divisor class E Y 2 + a 1 XY Z + a 3,1 σyz 3 = vx 3 + a 2 X 2 Z 2 + a 4,1 σxz 4 + a 6,2 σ 2 Z 6 The resolved fiber on the 7-branes splits in two components: Cartan node v =0 Affine node σ =0

By adjunction one finds: c 2 ( CY 4 ) = 11α 2 + 92αH + 182H 2 c 2 (CY 4 ) +(n 28)EH c 2

By adjunction one finds: c 2 ( CY 4 ) = 11α 2 + 92αH + 182H 2 c 2 (CY 4 ) +(n 28)EH c 2 After computing intersection numbers with the package SAGE, we obtained: c 2 2 4 = 1 2 n(n 28)2 if n is odd, the second Chern class is odd!

By adjunction one finds: c 2 ( CY 4 ) = 11α 2 + 92αH + 182H 2 c 2 (CY 4 ) +(n 28)EH c 2 After computing intersection numbers with the package SAGE, we obtained: c 2 2 4 = 1 2 n(n 28)2 if n is odd, the second Chern class is odd! For n odd we must turn on a half-quantized G4 flux. Choosing G 4 = c 2 2 n M2 = χ( CY 4 ) 24 1 2 G 4 G 4 = 972 = χ(cy 4) 24 The D3 charge is conserved as it should: the blow-up transition is a process of recombination/separation of branes in type IIB!

By adjunction one finds: c 2 ( CY 4 ) = 11α 2 + 92αH + 182H 2 c 2 (CY 4 ) +(n 28)EH c 2 After computing intersection numbers with the package SAGE, we obtained: c 2 2 4 = 1 2 n(n 28)2 if n is odd, the second Chern class is odd! For n odd we must turn on a half-quantized G4 flux. Choosing G 4 = c 2 2 n M2 = χ( CY 4 ) 24 1 2 G 4 G 4 = 972 = χ(cy 4) 24 The D3 charge is conserved as it should: the blow-up transition is a process of recombination/separation of branes in type IIB! What is the D7-brane gauge flux induced by such G4 flux?

Sen s weak coupling limit Tachyon condensation of 4 D9-branes with 4 anti-d9-branes: T = 0 η η 0 ρ + ξ ψ 0 ψ τ 0 0 Pn P n 0 O7 at {ξ =0} CY 3 CY 3 =WP 4 1,1,1,1,4[8]

Sen s weak coupling limit Tachyon condensation of 4 D9-branes with 4 anti-d9-branes: T = 0 η η 0 ρ + ξ ψ 0 ψ τ 0 0 Pn P n 0 O7 at {ξ =0} CY 3 CY 3 =WP 4 1,1,1,1,4[8] Whitney umbrella D7 of class 2 x (16-n)H

Sen s weak coupling limit Tachyon condensation of 4 D9-branes with 4 anti-d9-branes: T = 0 η η 0 ρ + ξ ψ 0 ψ τ 0 0 Pn P n 0 O7 at {ξ =0} CY 3 CY 3 =WP 4 1,1,1,1,4[8] Whitney umbrella D7 of class 2 x (16-n)H Sp(1) stack of class 2 x nh spin for n even non-spin for n odd

Sen s weak coupling limit Tachyon condensation of 4 D9-branes with 4 anti-d9-branes: T = 0 η η 0 ρ + ξ ψ 0 ψ τ 0 0 Pn P n 0 O7 at {ξ =0} CY 3 CY 3 =WP 4 1,1,1,1,4[8] Whitney umbrella D7 of class 2 x (16-n)H Sp(1) stack of class 2 x nh The total tadpole-cancelling D7-brane wraps the manifold: spin for n even non-spin for n odd det T = P 2 n(η 2 + ξ 2 (ρτ ψ 2 )) = 0 class 32H

Sen s weak coupling limit Tachyon condensation of 4 D9-branes with 4 anti-d9-branes: T = 0 η η 0 ρ + ξ ψ 0 ψ τ 0 0 Pn P n 0 O7 at {ξ =0} CY 3 CY 3 =WP 4 1,1,1,1,4[8] Whitney umbrella D7 of class 2 x (16-n)H Sp(1) stack of class 2 x nh The total tadpole-cancelling D7-brane wraps the manifold: spin for n even non-spin for n odd det T = P 2 n(η 2 + ξ 2 (ρτ ψ 2 )) = 0 class 32H The right configuration of D9/anti-D9-branes is found by imposing: Generic shape for the singular D7. Gauge flux F2 on the Sp(1) stack such that: 1 2 Collinucci, Denef, Esole arxiv:0805.1573 gcy 4 G 4 G 4 =ch 2 (F 2 )

The result is: D9 1 D9 2 D9 3 D9 4 O((n 14)H) O( 2H) O ((14 n)h) O ( 14H) n<12 D9 1 D9 2 D9 3 D9 4 O((14 n)h) O(2H) O ((n 14)H) O (14H)

The result is: D9 1 D9 2 D9 3 D9 4 O((n 14)H) O( 2H) O ((14 n)h) O ( 14H) n<12 D9 1 D9 2 D9 3 D9 4 O((14 n)h) O(2H) O ((n 14)H) O (14H) The standard K-theory formula leads to the following induced D3 charge (as computed in the covering space): Q D3 = 1944 = 2 972 exactly the total tadpole predicted by F-theory!

The result is: D9 1 D9 2 D9 3 D9 4 O((n 14)H) O( 2H) O ((14 n)h) O ( 14H) n<12 D9 1 D9 2 D9 3 D9 4 O((14 n)h) O(2H) O ((n 14)H) O (14H) The standard K-theory formula leads to the following induced D3 charge (as computed in the covering space): Q D3 = 1944 = 2 972 exactly the total tadpole predicted by F-theory! By construction the gauge flux on the D7 stack is: F 2 = 1 1 0 2 (28 n) H 0 1 expected quantization rule in terms of n

The result is: D9 1 D9 2 D9 3 D9 4 O((n 14)H) O( 2H) O ((14 n)h) O ( 14H) n<12 D9 1 D9 2 D9 3 D9 4 O((14 n)h) O(2H) O ((n 14)H) O (14H) The standard K-theory formula leads to the following induced D3 charge (as computed in the covering space): Q D3 = 1944 = 2 972 exactly the total tadpole predicted by F-theory! By construction the gauge flux on the D7 stack is: F 2 = 1 1 0 2 (28 n) H 0 1 expected quantization rule in terms of n The opposite contributions of D7 and image-d7 are manifest. The flux is along the Cartan direction of the adjoint of SU(2) the gauge group is broken to U(1)

Generalization to Sp(N) Sen s limit of F-theory on CY 4 P 3 with singularity of type along X = Y = P n =0 obtained by tachyon condensation of I ns 2N

Generalization to Sp(N) Sen s limit of F-theory on CY 4 P 3 with singularity of type along X = Y = P n =0 obtained by tachyon condensation of I ns 2N 2N+2 D9 and 2N+2 anti-d9 endowed with the gauge bundle O((14 nn)h) O(2H) N [O((in 14)H) O((14 (i 1)n)H)] i=1

Generalization to Sp(N) Sen s limit of F-theory on CY 4 P 3 with singularity of type along X = Y = P n =0 obtained by tachyon condensation of I ns 2N 2N+2 D9 and 2N+2 anti-d9 endowed with the gauge bundle O((14 nn)h) O(2H) F 2 = 1 2 H N [O((in 14)H) O((14 (i 1)n)H)] i=1 Gauge flux on the stack of N D7 + N image-d7 branes: N i=1 (28 n(2i 1)) C 2i 1 (C 2i 1 ) jk = δ ij δ ik δ i+n,j δ i+n,k The shifted quantization arises along all the Cartan directions. Such flux breaks Sp(N) to U(1) N

Generalization to Sp(N) Sen s limit of F-theory on CY 4 P 3 with singularity of type along X = Y = P n =0 obtained by tachyon condensation of I ns 2N 2N+2 D9 and 2N+2 anti-d9 endowed with the gauge bundle O((14 nn)h) O(2H) F 2 = 1 2 H N [O((in 14)H) O((14 (i 1)n)H)] i=1 Gauge flux on the stack of N D7 + N image-d7 branes: N i=1 (28 n(2i 1)) C 2i 1 (C 2i 1 ) jk = δ ij δ ik δ i+n,j δ i+n,k The shifted quantization arises along all the Cartan directions. Such flux breaks Sp(N) to U(1) N Geometric tadpole: Tadpole grav = 1944 nn 2 It agrees with the F-theory expectation 2 χ( CY 4 ) 24 (28 nn) 2 + n 2 N 2 1 3 checked with SAGE for N 4

Generalization to any base B3 For an Sp(N) singularity on a smooth divisor S 2 B 3 (π : CY 3 B 3 ) n D3 = 1 29 π c 1 (B 3 ) 3 + c 2 (CY 3 ) π c 1 (B 3 ) physical D3 charge, 4 CY 3 independent of N

Generalization to any base B3 For an Sp(N) singularity on a smooth divisor S 2 B 3 (π : CY 3 B 3 ) n D3 = 1 29 π c 1 (B 3 ) 3 + c 2 (CY 3 ) π c 1 (B 3 ) physical D3 charge, 4 CY 3 independent of N Gauge-induced tadpole: n gauge D3 = N 8 CY 3 π PD B3 S 2 (7 c 1 (B 3 ) NPD B3 S 2 ) 2 + N2 1 3 (PD B3 S 2 ) 2

Generalization to any base B3 For an Sp(N) singularity on a smooth divisor S 2 B 3 (π : CY 3 B 3 ) n D3 = 1 29 π c 1 (B 3 ) 3 + c 2 (CY 3 ) π c 1 (B 3 ) physical D3 charge, 4 CY 3 independent of N Gauge-induced tadpole: n gauge D3 = N 8 CY 3 π PD B3 S 2 (7 c 1 (B 3 ) NPD B3 S 2 ) 2 + N2 1 3 Gauge invariant flux on the D7 stack: F 2 = B 2 + F 2 = 1 2 N i=1 (PD B3 S 2 ) 2 [7 c 1 (B 3 ) (2i 1) PD B3 S 2 ] S2 C 2i 1

Generalization to any base B3 For an Sp(N) singularity on a smooth divisor S 2 B 3 (π : CY 3 B 3 ) n D3 = 1 29 π c 1 (B 3 ) 3 + c 2 (CY 3 ) π c 1 (B 3 ) physical D3 charge, 4 CY 3 independent of N Gauge-induced tadpole: n gauge D3 = N 8 CY 3 π PD B3 S 2 (7 c 1 (B 3 ) NPD B3 S 2 ) 2 + N2 1 3 Gauge invariant flux on the D7 stack: F 2 = B 2 + F 2 = 1 2 N i=1 (PD B3 S 2 ) 2 [7 c 1 (B 3 ) (2i 1) PD B3 S 2 ] S2 C 2i 1 Subtlety: When the class of O7 in CY3 is odd, a B-field must be turned on! B 2 = p 2 c 1(B 3 ) p =0 B3 spin p =1 B 3 non spin This is again due to Freed-Witten anomaly

Generalization to any base B3 For an Sp(N) singularity on a smooth divisor S 2 B 3 (π : CY 3 B 3 ) n D3 = 1 29 π c 1 (B 3 ) 3 + c 2 (CY 3 ) π c 1 (B 3 ) physical D3 charge, 4 CY 3 independent of N Gauge-induced tadpole: n gauge D3 = N 8 CY 3 π PD B3 S 2 (7 c 1 (B 3 ) NPD B3 S 2 ) 2 + N2 1 3 Gauge invariant flux on the D7 stack: F 2 = B 2 + F 2 = 1 2 N i=1 (PD B3 S 2 ) 2 [7 c 1 (B 3 ) (2i 1) PD B3 S 2 ] S2 C 2i 1 Subtlety: When the class of O7 in CY3 is odd, a B-field must be turned on! B 2 = p 2 c 1(B 3 ) p =0 B3 spin p =1 B 3 non spin The quantization rule of F 2 is determined by w 2 (S 2 ) and one has: This is again due to Freed-Witten anomaly w 2 (S 2 )=w 2 (B 3 S2 )+w 2 (N B3 S 2 )

Conclusions

Conclusions Using M/F-theory duality the gauge flux quantization rule for F- theory 7-branes is deduced from the one of the M-theory G4 flux.

Conclusions Using M/F-theory duality the gauge flux quantization rule for F- theory 7-branes is deduced from the one of the M-theory G4 flux. In the absence of non-abelian singularities for the F-theory CY fourfold no shifted quantizations arise.

Conclusions Using M/F-theory duality the gauge flux quantization rule for F- theory 7-branes is deduced from the one of the M-theory G4 flux. In the absence of non-abelian singularities for the F-theory CY fourfold no shifted quantizations arise. In the case of F-theory fourfolds with Sp(N)-type singularities, G4 is half-quantized when the D7-stack is non-spin.

Conclusions Using M/F-theory duality the gauge flux quantization rule for F- theory 7-branes is deduced from the one of the M-theory G4 flux. In the absence of non-abelian singularities for the F-theory CY fourfold no shifted quantizations arise. In the case of F-theory fourfolds with Sp(N)-type singularities, G4 is half-quantized when the D7-stack is non-spin. A gauge flux à la Freed-Witten is induced at weak coupling on the D7-stack, which breaks Sp(N) to U(1) N.

Conclusions Using M/F-theory duality the gauge flux quantization rule for F- theory 7-branes is deduced from the one of the M-theory G4 flux. In the absence of non-abelian singularities for the F-theory CY fourfold no shifted quantizations arise. In the case of F-theory fourfolds with Sp(N)-type singularities, G4 is half-quantized when the D7-stack is non-spin. A gauge flux à la Freed-Witten is induced at weak coupling on the D7-stack, which breaks Sp(N) to U(1) N. The gauge-induced tadpole agrees with the one predicted by F- theory for G 4 = c 2 /2. The geometric tadpoles also match.

Conclusions Using M/F-theory duality the gauge flux quantization rule for F- theory 7-branes is deduced from the one of the M-theory G4 flux. In the absence of non-abelian singularities for the F-theory CY fourfold no shifted quantizations arise. In the case of F-theory fourfolds with Sp(N)-type singularities, G4 is half-quantized when the D7-stack is non-spin. A gauge flux à la Freed-Witten is induced at weak coupling on the D7-stack, which breaks Sp(N) to U(1) N. The gauge-induced tadpole agrees with the one predicted by F- theory for G 4 = c 2 /2. The geometric tadpoles also match. The total D3 charge is conserved in the transition from the smooth configuration (kinematical constraint).

Conclusions Using M/F-theory duality the gauge flux quantization rule for F- theory 7-branes is deduced from the one of the M-theory G4 flux. In the absence of non-abelian singularities for the F-theory CY fourfold no shifted quantizations arise. In the case of F-theory fourfolds with Sp(N)-type singularities, G4 is half-quantized when the D7-stack is non-spin. A gauge flux à la Freed-Witten is induced at weak coupling on the D7-stack, which breaks Sp(N) to U(1) N. The gauge-induced tadpole agrees with the one predicted by F- theory for G 4 = c 2 /2. The geometric tadpoles also match. The total D3 charge is conserved in the transition from the smooth configuration (kinematical constraint). A half-quantized B-field must be turned on for O7-planes of odd degree to compensate for the lack of bulk spin structure.

Work in progress

Work in progress Address the cases of the other non-abelian singularities, especially the unitary ones. Construct explicit models and try to find a general pattern for the flux quantization. A hint can be obtained by iterating Fulton s formula for the Chern classes of blown-up manifolds.

Work in progress Address the cases of the other non-abelian singularities, especially the unitary ones. Construct explicit models and try to find a general pattern for the flux quantization. A hint can be obtained by iterating Fulton s formula for the Chern classes of blown-up manifolds. Understand the role in F-theory of Kapustin s improvement of FW anomaly for stacks of D-branes. A. Kapustin hep-th/9909089

Work in progress Address the cases of the other non-abelian singularities, especially the unitary ones. Construct explicit models and try to find a general pattern for the flux quantization. A hint can be obtained by iterating Fulton s formula for the Chern classes of blown-up manifolds. Understand the role in F-theory of Kapustin s improvement of FW anomaly for stacks of D-branes. A. Kapustin hep-th/9909089 Find the F-theory counterpart of the restriction to the topological type of G4 imposed by the FW-anomaly of M5. E. Witten hep-th/9912086

Work in progress Address the cases of the other non-abelian singularities, especially the unitary ones. Construct explicit models and try to find a general pattern for the flux quantization. A hint can be obtained by iterating Fulton s formula for the Chern classes of blown-up manifolds. Understand the role in F-theory of Kapustin s improvement of FW anomaly for stacks of D-branes. A. Kapustin hep-th/9909089 Find the F-theory counterpart of the restriction to the topological type of G4 imposed by the FW-anomaly of M5. E. Witten hep-th/9912086 This opens the way to a generalization of FW anomaly to general bound states of (p,q)7-branes, giving rise to general gauge group enhancing.