On Flux Quantization in F-Theory

Transcription

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

2 Motivations

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

4 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.

5 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.

6 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.

7 Plan of the talk

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

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

10 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.

11 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.

12 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.

13 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)

14 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: 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

15 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: 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

16 The quantum theory of M2 propagating in CY 4 may suffer from a global anomaly E. Witten hep-th/ 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

17 The quantum theory of M2 propagating in CY 4 may suffer from a global anomaly E. Witten hep-th/ 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)

18 The quantum theory of M2 propagating in CY 4 may suffer from a global anomaly E. Witten hep-th/ 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)

19 The quantum theory of M2 propagating in CY 4 may suffer from a global anomaly E. Witten hep-th/ 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

20 The quantum theory of M2 propagating in CY 4 may suffer from a global anomaly E. Witten hep-th/ 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

21 The quantum theory of M2 propagating in CY 4 may suffer from a global anomaly E. Witten hep-th/ 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

22 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!

23 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!

24 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!

25 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

26 Non-abelian singularities

27 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.

28 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.

29 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

30 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 n n 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

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

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

33 By adjunction one finds: c 2 ( CY 4 ) = 11α αH + 182H 2 c 2 (CY 4 ) +(n 28)EH c 2 After computing intersection numbers with the package SAGE, we obtained: c = 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 ) 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!

34 By adjunction one finds: c 2 ( CY 4 ) = 11α αH + 182H 2 c 2 (CY 4 ) +(n 28)EH c 2 After computing intersection numbers with the package SAGE, we obtained: c = 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 ) 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?

35 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]

36 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

37 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

38 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

39 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: gcy 4 G 4 G 4 =ch 2 (F 2 )

40 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)

41 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 = exactly the total tadpole predicted by F-theory!

42 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 = exactly the total tadpole predicted by F-theory! By construction the gauge flux on the D7 stack is: F 2 = (28 n) H 0 1 expected quantization rule in terms of n

43 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 = exactly the total tadpole predicted by F-theory! By construction the gauge flux on the D7 stack is: F 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)

44 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

45 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

46 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

47 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 checked with SAGE for N 4

48 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

49 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

50 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

51 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

52 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 )

53 Conclusions

54 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.

55 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.

56 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.

57 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.

58 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.

59 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).

60 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.

61 Work in progress

62 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.

63 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/

64 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/ 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/

65 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/ 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/ 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.