On Flux Quantization in F-Theory
|
|
- Octavia Preston
- 5 years ago
- Views:
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.
Instantons in string theory via F-theory
Instantons in string theory via F-theory Andrés Collinucci ASC, LMU, Munich Padova, May 12, 2010 arxiv:1002.1894 in collaboration with R. Blumenhagen and B. Jurke Outline 1. Intro: From string theory to
More informationF-theory effective physics via M-theory. Thomas W. Grimm!! Max Planck Institute for Physics (Werner-Heisenberg-Institut)! Munich
F-theory effective physics via M-theory Thomas W. Grimm Max Planck Institute for Physics (Werner-Heisenberg-Institut) Munich Ahrenshoop conference, July 2014 1 Introduction In recent years there has been
More informationCalabi-Yau Fourfolds with non-trivial Three-Form Cohomology
Calabi-Yau Fourfolds with non-trivial Three-Form Cohomology Sebastian Greiner arxiv: 1512.04859, 1702.03217 (T. Grimm, SG) Max-Planck-Institut für Physik and ITP Utrecht String Pheno 2017 Sebastian Greiner
More informationHeterotic Torsional Backgrounds, from Supergravity to CFT
Heterotic Torsional Backgrounds, from Supergravity to CFT IAP, Université Pierre et Marie Curie Eurostrings@Madrid, June 2010 L.Carlevaro, D.I. and M. Petropoulos, arxiv:0812.3391 L.Carlevaro and D.I.,
More informationDavid R. Morrison. String Phenomenology 2008 University of Pennsylvania 31 May 2008
: : University of California, Santa Barbara String Phenomenology 2008 University of Pennsylvania 31 May 2008 engineering has been a very successful approach to studying string vacua, and has been essential
More informationarxiv: v2 [hep-th] 3 Jul 2015
Prepared for submission to JHEP NSF-KITP-15-068, MIT-CTP-4677 P 1 -bundle bases and the prevalence of non-higgsable structure in 4D F-theory models arxiv:1506.03204v2 [hep-th] 3 Jul 2015 James Halverson
More informationTHE MASTER SPACE OF N=1 GAUGE THEORIES
THE MASTER SPACE OF N=1 GAUGE THEORIES Alberto Zaffaroni CAQCD 2008 Butti, Forcella, Zaffaroni hepth/0611229 Forcella, Hanany, Zaffaroni hepth/0701236 Butti,Forcella,Hanany,Vegh, Zaffaroni, arxiv 0705.2771
More informationCompactifications of F-Theory on Calabi Yau Threefolds II arxiv:hep-th/ v2 31 May 1996
DUKE-TH-96-107 HUTP-96/A012 hep-th/9603161 March, 1996 (Revised 5/96) Compactifications of F-Theory on Calabi Yau Threefolds II arxiv:hep-th/9603161v2 31 May 1996 David R. Morrison Department of Mathematics,
More informationA-field and B-field from Freed-Witten anomaly
A-field and B-field from Freed-Witten anomaly Raffaele Savelli SISSA/ISAS - Trieste Based on a work with L. Bonora and F. Ferrari Ruffino (arxiv: 0810.4291) Seminal paper by Freed & Witten: hep-th/9907189
More informationF- 理論におけるフラックスコンパクト化. Hajime Otsuka( 大塚啓 ) (Waseda U.) Physics Lett. B. 774 (2017) 225 with Y. Honma (National Tsing-Hua U.) Sangyo U.
F- 理論におけるフラックスコンパクト化 Hajime Otsuka( 大塚啓 ) (Waseda U.) Physics Lett. B. 774 (2017) 225 with Y. Honma (National Tsing-Hua U.) 2018/1/29@Kyoto Sangyo U. Outline Introduction Flux compactification in type
More informationString Theory Compactifications with Background Fluxes
String Theory Compactifications with Background Fluxes Mariana Graña Service de Physique Th Journées Physique et Math ématique IHES -- Novembre 2005 Motivation One of the most important unanswered question
More informationON THE SEN LIMIT SQUARED
ON THE SEN LIMIT SQUARED JAMES FULLWOOD AND DONGXU WANG Abstract. We introduce a class of F-theory vacua which may be viewed as a specialization of the so-called E 6 fibration, and construct a weak coupling
More informationMIFPA PiTP Lectures. Katrin Becker 1. Department of Physics, Texas A&M University, College Station, TX 77843, USA. 1
MIFPA-10-34 PiTP Lectures Katrin Becker 1 Department of Physics, Texas A&M University, College Station, TX 77843, USA 1 kbecker@physics.tamu.edu Contents 1 Introduction 2 2 String duality 3 2.1 T-duality
More informationAnomalies and Remnant Symmetries in Heterotic Constructions. Christoph Lüdeling
Anomalies and Remnant Symmetries in Heterotic Constructions Christoph Lüdeling bctp and PI, University of Bonn String Pheno 12, Cambridge CL, Fabian Ruehle, Clemens Wieck PRD 85 [arxiv:1203.5789] and work
More informationString Theory. A general overview & current hot topics. Benjamin Jurke. Würzburg January 8th, 2009
String Theory A general overview & current hot topics Benjamin Jurke 4d model building Non-perturbative aspects Optional: Vafa s F-theory GUT model building Würzburg January 8th, 2009 Compactification
More informationHeterotic type IIA duality with fluxes and moduli stabilization
Heterotic type IIA duality with fluxes and moduli stabilization Andrei Micu Physikalisches Institut der Universität Bonn Based on hep-th/0608171 and hep-th/0701173 in collaboration with Jan Louis, Eran
More informationCounting black hole microstates as open string flux vacua
Counting black hole microstates as open string flux vacua Frederik Denef KITP, November 23, 2005 F. Denef and G. Moore, to appear Outline Setting and formulation of the problem Black hole microstates and
More informationGeneralized N = 1 orientifold compactifications
Generalized N = 1 orientifold compactifications Thomas W. Grimm University of Wisconsin, Madison based on: [hep-th/0602241] Iman Benmachiche, TWG [hep-th/0507153] TWG Madison, Wisconsin, November 2006
More informationWeb of threefold bases in F-theory and machine learning
and machine learning 1510.04978 & 1710.11235 with W. Taylor CTP, MIT String Data Science, Northeastern; Dec. 2th, 2017 1 / 33 Exploring a huge oriented graph 2 / 33 Nodes in the graph Physical setup: 4D
More informationMagdalena Larfors
Uppsala University, Dept. of Theoretical Physics Based on D. Chialva, U. Danielsson, N. Johansson, M.L. and M. Vonk, hep-th/0710.0620 U. Danielsson, N. Johansson and M.L., hep-th/0612222 2008-01-18 String
More informationSupersymmetric Standard Models in String Theory
Supersymmetric Standard Models in String Theory (a) Spectrum (b) Couplings (c) Moduli stabilisation Type II side [toroidal orientifolds]- brief summary- status (a),(b)&(c) Heterotic side [Calabi-Yau compactification]
More information2 Type IIA String Theory with Background Fluxes in d=2
2 Type IIA String Theory with Background Fluxes in d=2 We consider compactifications of type IIA string theory on Calabi-Yau fourfolds. Consistency of a generic compactification requires switching on a
More informationAn exploration of threefold bases in F-theory
1510.04978 & upcoming work with W. Taylor CTP, MIT String Pheno 2017; Jul. 6th, 2017 F-theory landscape program Classify distinct F-theory compactifications to 4D F-theory compactification on an elliptic
More informationD-brane instantons in Type II orientifolds
D-brane instantons in Type II orientifolds in collaboration with R. Blumenhagen, M. Cvetič, D. Lüst, R. Richter Timo Weigand Department of Physics and Astronomy, University of Pennsylvania Strings 2008
More informationNew Aspects of Heterotic Geometry and Phenomenology
New Aspects of Heterotic Geometry and Phenomenology Lara B. Anderson Harvard University Work done in collaboration with: J. Gray, A. Lukas, and E. Palti: arxiv: 1106.4804, 1202.1757 J. Gray, A. Lukas and
More informationThe geometry of Landau-Ginzburg models
Motivation Toric degeneration Hodge theory CY3s The Geometry of Landau-Ginzburg Models January 19, 2016 Motivation Toric degeneration Hodge theory CY3s Plan of talk 1. Landau-Ginzburg models and mirror
More informationWhat is F-theory? David R. Morrison. University of California, Santa Barbara
University of California, Santa Barbara Physics and Geometry of F-theory 2015 Max Plack Institute for Physics, Munich 25 February 2015 arxiv:1503.nnnnn Inspired in part by Grassi Halverson Shaneson arxiv:1306.1832
More informationAspects!of!Abelian!and!Discrete!Symmetries!! in!f<theory!compac+fica+on!
TexasA&Mworkshop,2015 AspectsofAbelianandDiscreteSymmetries inf
More informationString Phenomenology ???
String Phenomenology Andre Lukas Oxford, Theoretical Physics d=11 SUGRA IIB M IIA??? I E x E 8 8 SO(32) Outline A (very) basic introduction to string theory String theory and the real world? Recent work
More informationMapping 6D N=1 supergravities to F-theory
Mapping 6D N=1 supergravities to F-theory The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation As Published Publisher Kumar, Vijay,
More informationarxiv: v1 [hep-th] 12 Oct 2007
CHERN CLASS IDENTITIES FROM TADPOLE MATCHING IN TYPE IIB AND F-THEORY PAOLO ALUFFI AND MBOYO ESOLE arxiv:0710.2544v1 [hep-th] 12 Oct 2007 Abstract. In light of Sen s weak coupling limit of F-theory as
More informationDualities and Topological Strings
Dualities and Topological Strings Strings 2006, Beijing - RD, C. Vafa, E.Verlinde, hep-th/0602087 - work in progress w/ C. Vafa & C. Beasley, L. Hollands Robbert Dijkgraaf University of Amsterdam Topological
More informationFixing all moduli in F-theory and type II strings
Fixing all moduli in F-theory and type II strings 0504058 Per Berglund, P.M. [0501139 D. Lüst, P.M., S. Reffert, S. Stieberger] 1 - Flux compactifications are important in many constructions of string
More informationF-theory and the classification of elliptic Calabi-Yau manifolds
F-theory and the classification of elliptic Calabi-Yau manifolds FRG Workshop: Recent progress in string theory and mirror symmetry March 6-7, 2015 Washington (Wati) Taylor, MIT Based in part on arxiv:
More informationGeneralized Global Symmetries
Generalized Global Symmetries Anton Kapustin Simons Center for Geometry and Physics, Stony Brook April 9, 2015 Anton Kapustin (Simons Center for Geometry and Physics, Generalized StonyGlobal Brook) Symmetries
More informationKnots and Mirror Symmetry. Mina Aganagic UC Berkeley
Knots and Mirror Symmetry Mina Aganagic UC Berkeley 1 Quantum physics has played a central role in answering the basic question in knot theory: When are two knots distinct? 2 Witten explained in 88, that
More informationThe Toric SO(10) F-theory Landscape
The Toric SO(10) F-theory Landscape Paul-Konstantin Oehlmann DESY/Hamburg University In preperation arxiv:170x.xxxx in collaboration with: W. Buchmueller, M. Dierigl, F. Ruehle Virginia Tech,Blacksburg
More informationKnot Homology from Refined Chern-Simons Theory
Knot Homology from Refined Chern-Simons Theory Mina Aganagic UC Berkeley Based on work with Shamil Shakirov arxiv: 1105.5117 1 the knot invariant Witten explained in 88 that J(K, q) constructed by Jones
More informationG 2 manifolds and mirror symmetry
G 2 manifolds and mirror symmetry Simons Collaboration on Special Holonomy in Geometry, Analysis and Physics First Annual Meeting, New York, 9/14/2017 Andreas Braun University of Oxford based on [1602.03521]
More informationYet Another Alternative to Compactification
Okayama Institute for Quantum Physics: June 26, 2009 Yet Another Alternative to Compactification Heterotic five-branes explain why three generations in Nature arxiv: 0905.2185 [hep-th] Tetsuji KIMURA (KEK)
More informationLooking Beyond Complete Intersection Calabi-Yau Manifolds. Work in progress with Hans Jockers, Joshua M. Lapan, Maurico Romo and David R.
Looking Beyond Complete Intersection Calabi-Yau Manifolds Work in progress with Hans Jockers, Joshua M. Lapan, Maurico Romo and David R. Morrison Who and Why Def: X is Calabi-Yau (CY) if X is a Ricci-flat,
More informationMordell-Weil Torsion, Anomalies, and Phase Transitions.
Mordell-Weil Torsion, Anomalies, and Phase Transitions. Mboyo Esole, Monica Jinwoo Kang, Shing-Tung Yau, arxiv:171.0337v1 [hep-th] 6 Dec 017 Department of Mathematics, Northeastern University, Boston,
More informationString-Math 18, Sendai, June 18-22, Global Constraints on Matter Representations in F-theory. Mirjam Cvetič
String-Math 18, Sendai, June 18-22, 2018 Global Constraints on Matter Representations in F-theory Mirjam Cvetič Outline: I. F-theory Compactification: brief overview of non-abelian gauge symmetries, matter,
More informationHeterotic Flux Compactifications
Heterotic Flux Compactifications Mario Garcia-Fernandez Instituto de Ciencias Matemáticas, Madrid String Pheno 2017 Virginia Tech, 7 July 2017 Based on arxiv:1611.08926, and joint work with Rubio, Tipler,
More informationCalabi-Yau Spaces in String Theory
Habilitationsschrift Calabi-Yau Spaces in String Theory Johanna Knapp Institut fu r Theoretische Physik Technische Universita t Wien Wiedner Hauptstraße 8-0 040 Wien O sterreich Wien, September 05 Abstract
More informationComputability of non-perturbative effects in the string theory landscape
Computability of non-perturbative effects in the string theory landscape IIB/F-theory perspective Iñaki García Etxebarria Nov 5, 2010 Based on [1009.5386] with M. Cvetič and J. Halverson. Phenomenology
More informationChern numbers and Hilbert Modular Varieties
Chern numbers and Hilbert Modular Varieties Dylan Attwell-Duval Department of Mathematics and Statistics McGill University Montreal, Quebec attwellduval@math.mcgill.ca April 9, 2011 A Topological Point
More informationYet Another Alternative to Compactification by Heterotic Five-branes
The University of Tokyo, Hongo: October 26, 2009 Yet Another Alternative to Compactification by Heterotic Five-branes arxiv: 0905.285 [hep-th] Tetsuji KIMURA (KEK) Shun ya Mizoguchi (KEK, SOKENDAI) Introduction
More informationTopological reduction of supersymmetric gauge theories and S-duality
Topological reduction of supersymmetric gauge theories and S-duality Anton Kapustin California Institute of Technology Topological reduction of supersymmetric gauge theories and S-duality p. 1/2 Outline
More informationAnomalies, Gauss laws, and Page charges in M-theory. Gregory Moore. Strings 2004, Paris. Related works: Witten , , ,
Anomalies, Gauss laws, and Page charges in M-theory Gregory Moore Strings 2004, Paris Related works: Witten 9609122,9610234,9812012,9912086 Diaconescu, Moore, and Witten 00 Diaconescu, Freed, and Moore
More informationOrientiholes Γ 2 Γ 1. Frederik Denef, Mboyo Esole and Megha Padi, arxiv:
Orientiholes Γ 2 Γ 1 Γ 0 Γ 1 Γ 2 Frederik Denef, Mboyo Esole and Megha Padi, arxiv:0901.2540 Outline Motivation and basic idea Review of N = 2 black hole bound states Type IIA orientiholes Motivation and
More informationNon-associative Deformations of Geometry in Double Field Theory
Non-associative Deformations of Geometry in Double Field Theory Michael Fuchs Workshop Frontiers in String Phenomenology based on JHEP 04(2014)141 or arxiv:1312.0719 by R. Blumenhagen, MF, F. Haßler, D.
More informationString Theory in a Nutshell. Elias Kiritsis
String Theory in a Nutshell Elias Kiritsis P R I N C E T O N U N I V E R S I T Y P R E S S P R I N C E T O N A N D O X F O R D Contents Preface Abbreviations xv xvii Introduction 1 1.1 Prehistory 1 1.2
More informationE 6 Yukawa couplings in F-theory as D-brane instanton effects
E 6 Yukawa couplings in F-theory as D-brane instanton effects Iñaki García Etxebarria based on [1612.06874] with Andrés Collinucci Motivation F-theory is a beautiful and powerful framework for string model
More informationStringy Corrections, SUSY Breaking and the Stabilization of (all) Kähler moduli
Stringy Corrections, SUSY Breaking and the Stabilization of (all) Kähler moduli Per Berglund University of New Hampshire Based on arxiv: 1012:xxxx with Balasubramanian and hep-th/040854 Balasubramanian,
More informationCandidates for Inflation in Type IIB/F-theory Flux Compactifications
Candidates for Inflation in Type IIB/F-theory Flux Compactifications Irene Valenzuela IFT UAM/CSIC Madrid Geometry and Physics of F-Theory, Munich 2015 Garcia-Etxebarria,Grimm,Valenzuela [hep-th/1412.5537]
More informationSome new torsional local models for heterotic strings
Some new torsional local models for heterotic strings Teng Fei Columbia University VT Workshop October 8, 2016 Teng Fei (Columbia University) Strominger system 10/08/2016 1 / 30 Overview 1 Background and
More informationSome Issues in F-Theory Geometry
Outline Some Issues in F-Theory Geometry Arthur Hebecker (Heidelberg) based on work with A. Braun, S. Gerigk, H. Triendl 0912.1596 A. Braun, R. Ebert, R. Valandro 0907.2691 as well as two earlier papers
More informationD-branes in non-abelian gauged linear sigma models
D-branes in non-abelian gauged linear sigma models Johanna Knapp Vienna University of Technology Bonn, September 29, 2014 Outline CYs and GLSMs D-branes in GLSMs Conclusions CYs and GLSMs A Calabi-Yau
More informationPietro Fre' SISSA-Trieste. Paolo Soriani University degli Studi di Milano. From Calabi-Yau manifolds to topological field theories
From Calabi-Yau manifolds to topological field theories Pietro Fre' SISSA-Trieste Paolo Soriani University degli Studi di Milano World Scientific Singapore New Jersey London Hong Kong CONTENTS 1 AN INTRODUCTION
More informationF-theory with Quivers
University of Trieste String Phenomenology 2017 (Virginia Tech) Based on work in progress with A. Collinucci, M. Fazzi and D. Morrison Introduction In the M/F-theory geometric engineering, Abelian gauge
More informationAspects of (0,2) theories
Aspects of (0,2) theories Ilarion V. Melnikov Harvard University FRG workshop at Brandeis, March 6, 2015 1 / 22 A progress report on d=2 QFT with (0,2) supersymmetry Gross, Harvey, Martinec & Rohm, Heterotic
More informationDeformations of calibrated D-branes in flux generalized complex manifolds
Deformations of calibrated D-branes in flux generalized complex manifolds hep-th/0610044 (with Luca Martucci) Paul Koerber koerber@mppmu.mpg.de Max-Planck-Institut für Physik Föhringer Ring 6 D-80805 München
More informationF-Theory, Spinning Black Holes and Multi-string Branches
F-Theory, Spinning Black Holes and Multi-string Branches arxiv:1509.00455v3 [hep-th] 2 Dec 2015 Babak Haghighat 1,2, Sameer Murthy 3, Cumrun Vafa 1, Stefan Vandoren 4 1 Jefferson Physical Laboratory, Harvard
More informationStandard Models from Heterotic M-theory
Standard Models from Heterotic M-theory Ron Donagi 1, Burt A. Ovrut 2, Tony Pantev 1 and Daniel Waldram 34 1 Department of Mathematics, University of Pennsylvania Philadelphia, PA 19104 6395, USA 2 Department
More informationFlux Compactification of Type IIB Supergravity
Flux Compactification of Type IIB Supergravity based Klaus Behrndt, LMU Munich Based work done with: M. Cvetic and P. Gao 1) Introduction 2) Fluxes in type IIA supergravity 4) Fluxes in type IIB supergravity
More informationLinear connections on Lie groups
Linear connections on Lie groups The affine space of linear connections on a compact Lie group G contains a distinguished line segment with endpoints the connections L and R which make left (resp. right)
More informationDepartement de Physique Theorique, Universite de Geneve, Abstract
hep-th/950604 A NOTE ON THE STRING ANALOG OF N = SUPER-SYMMETRIC YANG-MILLS Cesar Gomez 1 and Esperanza Lopez 1 Departement de Physique Theorique, Universite de Geneve, Geneve 6, Switzerland Abstract A
More informationBPS states, Wall-crossing and Quivers
Iberian Strings 2012 Bilbao BPS states, Wall-crossing and Quivers IST, Lisboa Michele Cirafici M.C.& A.Sincovics & R.J. Szabo: 0803.4188, 1012.2725, 1108.3922 and M.C. to appear BPS States in String theory
More informationSolution Set 8 Worldsheet perspective on CY compactification
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics String Theory (8.821) Prof. J. McGreevy Fall 2007 Solution Set 8 Worldsheet perspective on CY compactification Due: Monday, December 18, 2007
More informationarxiv:hep-th/ v3 11 Oct 2004
UPR-1068-T, hep-th/0403061 Supersymmetric Pati-Salam Models from Intersecting D6-branes: A Road to the Standard Model Mirjam Cvetič, 1 Tianjun Li, 2 and Tao Liu 1 1 Department of Physics and Astronomy,
More informationHeterotic Standard Models
19 August 2008 Strings 08 @ CERN The High Country region of the string landscape Goal: Study string vacua which reproduce the MSSM (or close cousins thereof) at low energies String landscape is huge, but
More informationA Landscape of Field Theories
A Landscape of Field Theories Travis Maxfield Enrico Fermi Institute, University of Chicago October 30, 2015 Based on arxiv: 1511.xxxxx w/ D. Robbins and S. Sethi Summary Despite the recent proliferation
More informationNon-Geometric Calabi- Yau Backgrounds
Non-Geometric Calabi- Yau Backgrounds CH, Israel and Sarti 1710.00853 A Dabolkar and CH, 2002 Duality Symmetries Supergravities: continuous classical symmetry, broken in quantum theory, and by gauging
More informationOn the Virtual Fundamental Class
On the Virtual Fundamental Class Kai Behrend The University of British Columbia Seoul, August 14, 2014 http://www.math.ubc.ca/~behrend/talks/seoul14.pdf Overview Donaldson-Thomas theory: counting invariants
More informationAn introduction to heterotic mirror symmetry. Eric Sharpe Virginia Tech
An introduction to heterotic mirror symmetry Eric Sharpe Virginia Tech I ll begin today by reminding us all of ordinary mirror symmetry. Most basic incarnation: String theory on a Calabi-Yau X = String
More informationElements of Topological M-Theory
Elements of Topological M-Theory (with R. Dijkgraaf, S. Gukov, C. Vafa) Andrew Neitzke March 2005 Preface The topological string on a Calabi-Yau threefold X is (loosely speaking) an integrable spine of
More informationarxiv: v2 [hep-th] 23 Mar 2018
MPP-2018-20 Infinite Distances in Field Space and Massless Towers of States Thomas W. Grimm 1, Eran Palti 2, Irene Valenzuela 1 arxiv:1802.08264v2 [hep-th] 23 Mar 2018 1 Institute for Theoretical Physics
More informationCohomology jump loci of quasi-projective varieties
Cohomology jump loci of quasi-projective varieties Botong Wang joint work with Nero Budur University of Notre Dame June 27 2013 Motivation What topological spaces are homeomorphic (or homotopy equivalent)
More informationThe Geometry of F 4 -Models
The Geometry of F 4 -Models Mboyo Esole, Patrick Jefferson, Monica Jinwoo Kang arxiv:704.0825v [hep-th] 26 Apr 207 Department of Mathematics, Northeastern University 360 Huttington Avenue, Boston, MA 025,
More informationThe global gauge group structure of F-theory compactifications with U(1)s
The global gauge group structure of F-theory compactifications with U(1)s based on arxiv:1706.08521 with Mirjam Cvetič Ling Lin Department of Physics and Astronomy University of Pennsylvania String Phenomenology,
More informationGauge Threshold Corrections for Local String Models
Gauge Threshold Corrections for Local String Models Stockholm, November 16, 2009 Based on arxiv:0901.4350 (JC), 0906.3297 (JC, Palti) Local vs Global There are many different proposals to realise Standard
More informationBlack Hole Microstate Counting using Pure D-brane Systems
Black Hole Microstate Counting using Pure D-brane Systems HRI, Allahabad, India 11.19.2015 UC Davis, Davis based on JHEP10(2014)186 [arxiv:1405.0412] and upcoming paper with Abhishek Chowdhury, Richard
More information2d-4d wall-crossing and hyperholomorphic bundles
2d-4d wall-crossing and hyperholomorphic bundles Andrew Neitzke, UT Austin (work in progress with Davide Gaiotto, Greg Moore) DESY, December 2010 Preface Wall-crossing is an annoying/beautiful phenomenon
More informationHolomorphic line bundles
Chapter 2 Holomorphic line bundles In the absence of non-constant holomorphic functions X! C on a compact complex manifold, we turn to the next best thing, holomorphic sections of line bundles (i.e., rank
More informationRefined Chern-Simons Theory, Topological Strings and Knot Homology
Refined Chern-Simons Theory, Topological Strings and Knot Homology Based on work with Shamil Shakirov, and followup work with Kevin Scheaffer arxiv: 1105.5117 arxiv: 1202.4456 Chern-Simons theory played
More informationAn Algorithmic Approach to Heterotic Compactification
An Algorithmic Approach to Heterotic Compactification Lara B. Anderson Department of Physics, University of Pennsylvania, and Institute for Advanced Study, Princeton Work done in collaboration with: LBA,
More informationGeometry of the Calabi-Yau Moduli
Geometry of the Calabi-Yau Moduli Zhiqin Lu 2012 AMS Hawaii Meeting Department of Mathematics, UC Irvine, Irvine CA 92697 March 4, 2012 Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 1/51
More informationCompact T-branes. Teórica. Fernando Marchesano. Instituto de Física UAM-CSIC
Compact T-branes Fernando Marchesano Instituto de Física Teórica UAM-CIC Compact T-branes Fernando Marchesano Based on: F.M., avelli, chwieger 1707.03797 Cooking a compactification Building a 4d string
More informationElliptic Calabi-Yau fourfolds and 4D F-theory vacua
Elliptic Calabi-Yau fourfolds and 4D F-theory vacua Dave Day F-theory at 20 conference Burke Institute, Caltech February 25, 2016 Washington (Wati) Taylor, MIT Based in part on arxiv: 1201.1943, 1204.0283,
More informationThe exact quantum corrected moduli space for the universal hypermultiplet
The exact quantum corrected moduli space for the universal hypermultiplet Bengt E.W. Nilsson Chalmers University of Technology, Göteborg Talk at "Miami 2009" Fort Lauderdale, December 15-20, 2009 Talk
More informationSphere Partition Functions, Topology, the Zamolodchikov Metric
Sphere Partition Functions, Topology, the Zamolodchikov Metric, and Extremal Correlators Weizmann Institute of Science Efrat Gerchkovitz, Jaume Gomis, ZK [1405.7271] Jaume Gomis, Po-Shen Hsin, ZK, Adam
More informationRealistic D-Brane Models on Warped Throats: Fluxes, Hierarchies and Moduli Stabilization
Preprint typeset in JHEP style - HYPER VERSION DAMTP-2003-133 IFT-UAM/CSIC-03-50 FTUAM-03-28 hep-th/0312051 arxiv:hep-th/0312051v2 10 Dec 2003 Realistic D-Brane Models on Warped Throats: Fluxes, Hierarchies
More informationN = 2 supersymmetric gauge theory and Mock theta functions
N = 2 supersymmetric gauge theory and Mock theta functions Andreas Malmendier GTP Seminar (joint work with Ken Ono) November 7, 2008 q-series in differential topology Theorem (M-Ono) The following q-series
More informationRelating DFT to N=2 gauged supergravity
Relating DFT to N=2 gauged supergravity Erik Plauschinn LMU Munich Chengdu 29.07.2016 based on... This talk is based on :: Relating double field theory to the scalar potential of N=2 gauged supergravity
More informationarxiv:hep-th/ v2 28 Mar 2000
PUPT-1923 arxiv:hep-th/0003236v2 28 Mar 2000 A Note on Warped String Compactification Chang S. Chan 1, Percy L. Paul 2 and Herman Verlinde 1 1 Joseph Henry Laboratories, Princeton University, Princeton
More informationThe Geometry of the SU(2) G 2 -model.
The Geometry of the SU() G -model. Mboyo Esole and Monica Jinwoo Kang arxiv:85.4v [hep-th] 8 May 8 Department of Mathematics, Northeastern University 6 Huttington Avenue, Boston, MA 5, U.S.A. Department
More informationGeneralized complex geometry and topological sigma-models
Generalized complex geometry and topological sigma-models Anton Kapustin California Institute of Technology Generalized complex geometry and topological sigma-models p. 1/3 Outline Review of N = 2 sigma-models
More informationLecture 8: 1-loop closed string vacuum amplitude
Lecture 8: 1-loop closed string vacuum amplitude José D. Edelstein University of Santiago de Compostela STRING THEORY Santiago de Compostela, March 5, 2013 José D. Edelstein (USC) Lecture 8: 1-loop vacuum
More informationGeometry of Conformal Field Theory
Geometry of Conformal Field Theory Yoshitake HASHIMOTO (Tokyo City University) 2010/07/10 (Sat.) AKB Differential Geometry Seminar Based on a joint work with A. Tsuchiya (IPMU) Contents 0. Introduction
More information