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Triality in Little String Theories Stefan Hohenegger GGI Workshop: - PowerPoint PPT Presentation

May 10, 2023 •221 likes •2.08k views

Triality in Little String Theories Stefan Hohenegger GGI Workshop: String Theory from a Worldsheet Perspective Galileo Galilei Institute, 29 Apr. 2019 based on work in collaboration with: Brice Bastian, Amer Iqbal and Soo-Jong Rey hep-th

  1. Parallel M5-branes M-branes (M2- and M5) are extended in objects in 11 -dimensional M-theory They can be arranged in a fashion to preserve (some amount of) supersymmetry: brane webs String-like objects arise at the intersection of M5- and M2-branes stretched M2-branes M-string provide description of (almost) tensionless strings in 6. dim. relevant for SCFT N = (2 , 0) many dual realisations allowing to explicitly compute quantities (e.g. partition function)

  2. Parallel M5-branes M-branes (M2- and M5) are extended in objects in 11 -dimensional M-theory They can be arranged in a fashion to preserve (some amount of) supersymmetry: brane webs String-like objects arise at the intersection of M5- and M2-branes stretched M2-branes M-string provide description of (almost) tensionless strings in 6. dim. relevant for SCFT N = (2 , 0) many dual realisations allowing to explicitly compute quantities (e.g. partition function) notably: F-theory compactification on toric, non-compact Calabi-Yau threefolds [Morrison, Vafa 1996] [Heckman, Morrison, Vafa 2013] [Del Zotto, Heckman, Tomasiello, Vafa 2014] [Heckman 2014] [Haghighat, Klemm, Lockhart, Vafa 2014] [Heckman, Morrison, Rudelius, Vafa 2015] [SH, Iqbal, Rey 2015] [Bhardwaj, Del Zotto, Heckman, Morrison, Rudelius, Vafa 2016]

  3. Parallel M5-branes M-branes (M2- and M5) are extended in objects in 11 -dimensional M-theory They can be arranged in a fashion to preserve (some amount of) supersymmetry: brane webs String-like objects arise at the intersection of M5- and M2-branes stretched M2-branes M-string provide description of (almost) tensionless strings in 6. dim. relevant for SCFT N = (2 , 0) many dual realisations allowing to explicitly compute quantities (e.g. partition function) notably: F-theory compactification on toric, non-compact Calabi-Yau threefolds depending on the details of the brane configuration, a large class of different Little Strings (or their duals) can be realised and studied very explicitly

  4. Parallel M5-branes M-branes (M2- and M5) are extended in objects in 11 -dimensional M-theory They can be arranged in a fashion to preserve (some amount of) supersymmetry: brane webs String-like objects arise at the intersection of M5- and M2-branes stretched M2-branes M-string provide description of (almost) tensionless strings in 6. dim. relevant for SCFT N = (2 , 0) many dual realisations allowing to explicitly compute quantities (e.g. partition function) notably: F-theory compactification on toric, non-compact Calabi-Yau threefolds depending on the details of the brane configuration, a large class of different Little Strings (or their duals) can be realised and studied very explicitly low energy limit associated with non-abelian supersymmetric field theories (mass deformed theories upon compactification to 4 dimensions) N = 2 ∗

  5. Parallel M5-branes M-branes (M2- and M5) are extended in objects in 11 -dimensional M-theory They can be arranged in a fashion to preserve (some amount of) supersymmetry: brane webs String-like objects arise at the intersection of M5- and M2-branes stretched M2-branes M-string provide description of (almost) tensionless strings in 6. dim. relevant for SCFT N = (2 , 0) many dual realisations allowing to explicitly compute quantities (e.g. partition function) notably: F-theory compactification on toric, non-compact Calabi-Yau threefolds depending on the details of the brane configuration, a large class of different Little Strings (or their duals) can be realised and studied very explicitly low energy limit associated with non-abelian supersymmetric field theories (mass deformed theories upon compactification to 4 dimensions) N = 2 ∗ Class of theories exhibits interesting (and non-expected) dualities!

  6. Parallel M5-branes M-branes (M2- and M5) are extended in objects in 11 -dimensional M-theory They can be arranged in a fashion to preserve (some amount of) supersymmetry: brane webs String-like objects arise at the intersection of M5- and M2-branes stretched M2-branes M-string provide description of (almost) tensionless strings in 6. dim. relevant for SCFT N = (2 , 0) many dual realisations allowing to explicitly compute quantities (e.g. partition function) notably: F-theory compactification on toric, non-compact Calabi-Yau threefolds depending on the details of the brane configuration, a large class of different Little Strings (or their duals) can be realised and studied very explicitly low energy limit associated with non-abelian supersymmetric field theories (mass deformed theories upon compactification to 4 dimensions) N = 2 ∗ Class of theories exhibits interesting (and non-expected) dualities! in this talk: triality

  7. Little Strings 6-dimensional systems: - gravity is decoupled - have an intrinsic string scale - obtained from type II string theory through the decoupling limit while g st → 0 ` st = fixed Little String Theories with 16 supercharges (A-series) N = (2 , 0) IIb LST of type with supersymmetry A N − 1 -) decoupling limit of N M5-branes with transverse space S 1 × R 4

  8. Little Strings 6-dimensional systems: - gravity is decoupled - have an intrinsic string scale - obtained from type II string theory through the decoupling limit while g st → 0 ` st = fixed Little String Theories with 16 supercharges (A-series) N = (2 , 0) IIb LST of type with supersymmetry A N − 1 -) decoupling limit of N M5-branes with transverse space S 1 × R 4 R 4 -) decoupling limit of a stack of N NS5-branes in type IIA with transverse space -) type IIB string theory on orbifold background A N − 1

  9. Little Strings 6-dimensional systems: - gravity is decoupled - have an intrinsic string scale - obtained from type II string theory through the decoupling limit while g st → 0 ` st = fixed Little String Theories with 16 supercharges (A-series) N = (2 , 0) IIb LST of type with supersymmetry A N − 1 -) decoupling limit of N M5-branes with transverse space S 1 × R 4 R 4 -) decoupling limit of a stack of N NS5-branes in type IIA with transverse space -) type IIB string theory on orbifold background A N − 1 A N − 1 N = (1 , 1) IIa LST of type with supersymmetry R 4 -) decoupling limit of a stack of N NS5-branes in type IIB with transverse space A N − 1 -) type IIA string theory on orbifold background

  10. Little Strings 6-dimensional systems: - gravity is decoupled - have an intrinsic string scale - obtained from type II string theory through the decoupling limit while g st → 0 ` st = fixed Little String Theories with 16 supercharges (A-series) N = (2 , 0) IIb LST of type with supersymmetry A N − 1 -) decoupling limit of N M5-branes with transverse space S 1 × R 4 related by T-duality R 4 -) decoupling limit of a stack of N NS5-branes in type IIA with transverse space -) type IIB string theory on orbifold background A N − 1 A N − 1 N = (1 , 1) IIa LST of type with supersymmetry R 4 -) decoupling limit of a stack of N NS5-branes in type IIB with transverse space A N − 1 -) type IIA string theory on orbifold background

  11. Little Strings 6-dimensional systems: - gravity is decoupled - have an intrinsic string scale - obtained from type II string theory through the decoupling limit while g st → 0 ` st = fixed Little String Theories with 16 supercharges (A-series) N = (2 , 0) IIb LST of type with supersymmetry A N − 1 -) decoupling limit of N M5-branes with transverse space S 1 × R 4 related by T-duality R 4 -) decoupling limit of a stack of N NS5-branes in type IIA with transverse space -) type IIB string theory on orbifold background A N − 1 A N − 1 N = (1 , 1) IIa LST of type with supersymmetry R 4 -) decoupling limit of a stack of N NS5-branes in type IIB with transverse space A N − 1 -) type IIA string theory on orbifold background BPS states from the point of view of M5-branes correspond to M2-branes ending on them

  12. Little Strings 6-dimensional systems: - gravity is decoupled - have an intrinsic string scale - obtained from type II string theory through the decoupling limit while g st → 0 ` st = fixed Little String Theories with 8 supercharges: particular class obtained as A M − 1 N = (1 , 0) orbifold of IIa LST of type with supersymmetry Z N -) decoupling limit of M M5-branes with transverse space S 1 × ALE A N − 1 related by T-duality R 4 / Z N -) decoupling limit of a stack of N NS5-branes in type IIB with transverse space orbifold of IIb LST of type with supersymmetry A N − 1 N = (1 , 0) Z M S 1 × ALE A M − 1 -) decoupling limit of N M5-branes with transverse space R 4 / Z M -) decoupling limit of a stack of N NS5-branes in type IIA with transverse space

  13. Little Strings 6-dimensional systems: - gravity is decoupled - have an intrinsic string scale - obtained from type II string theory through the decoupling limit while g st → 0 ` st = fixed Little String Theories with 8 supercharges: particular class obtained as A M − 1 N = (1 , 0) orbifold of IIa LST of type with supersymmetry Z N -) decoupling limit of M M5-branes with transverse space S 1 × ALE A N − 1 related by T-duality R 4 / Z N -) decoupling limit of a stack of N NS5-branes in type IIB with transverse space orbifold of IIb LST of type with supersymmetry A N − 1 N = (1 , 0) Z M S 1 × ALE A M − 1 -) decoupling limit of N M5-branes with transverse space R 4 / Z M -) decoupling limit of a stack of N NS5-branes in type IIA with transverse space Explicit computation of BPS partition function using various methods [Haghighat, Iqbal, Kozçaz, Lockhart, Vafa 2013] [Haghighat, Kozçaz, Lockhart, Vafa 2013] [SH, Iqbal 2013] [SH, Iqbal, Rey 2015]

  14. Little Strings 6-dimensional systems: - gravity is decoupled - have an intrinsic string scale - obtained from type II string theory through the decoupling limit while g st → 0 ` st = fixed Little String Theories with 8 supercharges: particular class obtained as A M − 1 N = (1 , 0) orbifold of IIa LST of type with supersymmetry Z N -) decoupling limit of M M5-branes with transverse space S 1 × ALE A N − 1 related by T-duality R 4 / Z N -) decoupling limit of a stack of N NS5-branes in type IIB with transverse space orbifold of IIb LST of type with supersymmetry A N − 1 N = (1 , 0) Z M S 1 × ALE A M − 1 -) decoupling limit of N M5-branes with transverse space R 4 / Z M -) decoupling limit of a stack of N NS5-branes in type IIA with transverse space Explicit computation of BPS partition function using various methods [Haghighat, Iqbal, Kozçaz, Lockhart, Vafa 2013] [Haghighat, Kozçaz, Lockhart, Vafa 2013] in this talk: [SH, Iqbal 2013] further dualities [SH, Iqbal, Rey 2015]

  15. Brane Configurations The most general configuration of branes in M-theory in 11 dimensions looks like 0 1 2 3 4 5 6 7 8 9 10 M5-branes • • • • • • M2-branes • • • | {z } | {z } ALE AM − 1 ∼ R 4 / Z M R 4 ||

  16. Brane Configurations The most general configuration of branes in M-theory in 11 dimensions looks like 0 1 2 3 4 5 6 7 8 9 10 M5-branes • • • • • • M2-branes • • • | {z } | {z } ALE AM − 1 ∼ R 4 / Z M R 4 || non-compact case: R M5-branes distributed along non-comp. (6)-direction with M2-branes stretched between them x 6 . . . t f N − 1 t f 1

  17. Brane Configurations The most general configuration of branes in M-theory in 11 dimensions looks like 0 1 2 3 4 5 6 7 8 9 10 M5-branes • • • • • • M2-branes • • • | {z } | {z } ALE AM − 1 ∼ R 4 / Z M R 4 || compact case: S 1 non-compact case: R ρ M5-branes arranged on a circle R 6 = 2 π i M5-branes distributed along non-comp. (6)-direction × . . . × x 6 × with M2-branes stretched between them . . . t f 2 × × × t f N t f 1 x 6 . . . t f N − 1 t f 1

  18. Brane Configurations The most general configuration of branes in M-theory in 11 dimensions looks like 0 1 2 3 4 5 6 7 8 9 10 M5-branes • • • • • • M2-branes • • • | {z } | {z } ALE AM − 1 ∼ R 4 / Z M R 4 || compact case: S 1 non-compact case: R ρ M5-branes arranged on a circle R 6 = 2 π i M5-branes distributed along non-comp. (6)-direction × . . . × x 6 × with M2-branes stretched between them . . . t f 2 × × × t f N t f 1 necessary for little-string interpretation tensionful string going around S 1 x 6 × . . . t f N − 1 t f 1 limit where all M5-branes form a single stack

  19. Brane Configurations The most general configuration of branes in M-theory in 11 dimensions looks like (0) (1) 2 3 4 5 6 7 8 9 10 M5-branes • • • • • • M2-branes • • • | {z } | {z } T 2 ∼ S 1 × S 1 | {z } ALE AM − 1 ∼ R 4 / Z M R 4 || τ Compactification: T 2 ∼ S 1 × S 1 Compactify (0,1) to with radii and R 0 R 1 =: 2 π i

  20. Brane Configurations The most general configuration of branes in M-theory in 11 dimensions looks like (0) (1) 2 3 4 5 6 7 8 9 10 M5-branes • • • • • • M2-branes • • • � � � � � � ✏ 1 � � � � � � ✏ 2 τ Compactification: T 2 ∼ S 1 × S 1 Compactify (0,1) to with radii and R 0 R 1 =: 2 π i Deformations: there are two types of deformations with respect to the compactified (0,1)-directions introducing complex coordinates and ( w 1 , w 2 ) = ( x 7 + ix 8 , x 9 + ix 10 ) ( z 1 , z 2 ) = ( x 2 + ix 3 , x 4 + ix 5 ) (0)-direct.: U (1) ✏ 1 × U (1) ✏ 2 : ( z 1 , z 2 ) → ( e 2 ⇡ i ✏ 1 z 1 , e 2 ⇡ i ✏ 2 z 2 ) ( w 1 , w 2 ) → ( e − i ⇡ ( ✏ 1 + ✏ 2 ) w 1 , e − i ⇡ ( ✏ 1 + ✏ 2 ) w 2 ) and (1)-direct.: U (1) m : ( w 1 , w 2 ) → ( e 2 π im w 1 , e − 2 π im w 2 )

  21. Brane Configurations The most general configuration of branes in M-theory in 11 dimensions looks like (0) (1) 2 3 4 5 6 7 8 9 10 M5-branes • • • • • • M2-branes • • • � � � � � � ✏ 1 � � � � � � ✏ 2 τ Compactification: T 2 ∼ S 1 × S 1 Compactify (0,1) to with radii and R 0 R 1 =: 2 π i Deformations: there are two types of deformations with respect to the compactified (0,1)-directions introducing complex coordinates and ( w 1 , w 2 ) = ( x 7 + ix 8 , x 9 + ix 10 ) ( z 1 , z 2 ) = ( x 2 + ix 3 , x 4 + ix 5 ) (0)-direct.: U (1) ✏ 1 × U (1) ✏ 2 : ( z 1 , z 2 ) → ( e 2 ⇡ i ✏ 1 z 1 , e 2 ⇡ i ✏ 2 z 2 ) ( w 1 , w 2 ) → ( e − i ⇡ ( ✏ 1 + ✏ 2 ) w 1 , e − i ⇡ ( ✏ 1 + ✏ 2 ) w 2 ) and (1)-direct.: U (1) m : ( w 1 , w 2 ) → ( e 2 π im w 1 , e − 2 π im w 2 ) gauge theory: Omega-background mass-deformation [Nekrasov 2012]

  22. Dual Setups to Brane Configurations For vanishing mass deformation ( ) the M-brane configuration is dual to D5-NS5-branes in IIB m = 0

  23. Dual Setups to Brane Configurations For vanishing mass deformation ( ) the M-brane configuration is dual to D5-NS5-branes in IIB m = 0 0 1 2 3 4 5 6 7 8 9 D5 branes – • • • • • • NS5 branes – • • • • • • | {z } | {z } | {z } gauge theory ( p,q ) − plane transverse R 3

  24. Dual Setups to Brane Configurations For vanishing mass deformation ( ) the M-brane configuration is dual to D5-NS5-branes in IIB m = 0 0 1 2 3 4 5 6 7 8 9 D5 branes – • • • • • • NS5 branes – • • • • • • | {z } | {z } | {z } gauge theory ( p,q ) − plane transverse R 3 = 1 = 2 = 3 = M . . . | | 1 1 . . . | | 2 2 . . . | | 3 3 . . . . . . . . . . . . | | N N . . . = 1 = 2 = 3 = M

  25. Dual Setups to Brane Configurations For vanishing mass deformation ( ) the M-brane configuration is dual to D5-NS5-branes in IIB m = 0 0 1 2 3 4 5 6 7 8 9 D5 branes – • • • • • • NS5 branes – • • • • • • | {z } | {z } | {z } gauge theory ( p,q ) − plane transverse R 3 M D5-branes = 1 = 2 = 3 = M . . . | | 1 1 . . . | | NS5-branes N 2 2 . . . | | 3 3 . . . . . . . . . . . . | | N N . . . = 1 = 2 = 3 = M

  26. Dual Setups to Brane Configurations For vanishing mass deformation ( ) the M-brane configuration is dual to D5-NS5-branes in IIB m = 0 0 1 2 3 4 5 6 7 8 9 D5 branes – • • • • • • NS5 branes – • • • • • • | {z } | {z } | {z } gauge theory ( p,q ) − plane transverse R 3 = 1 = 2 = 3 = M . . . Deformation: | | 1 1 . . . | | ⇒ = 2 2 (1,1) brane . . . | | 3 3 . . . . . . . . . . . . | | N N . . . = 1 = 2 = 3 = M

  27. Dual Setups to Brane Configurations For vanishing mass deformation ( ) the M-brane configuration is dual to D5-NS5-branes in IIB m = 0 0 1 2 3 4 5 6 7 8 9 D5 branes – • • • • • • NS5 branes – • • • • • • | {z } | {z } | {z } gauge theory ( p,q ) − plane transverse R 3 = 1 = 2 = 3 = M . . . Deformation: | | 1 1 . . . | | ⇒ = 2 2 (1,1) brane . . . | | 3 3 uplift the deformed type II configuration to M-th. . . . . . . . . . . . . on an elliptically fibered Calabi-Yau threefold X N,M [Leung, Vafa 1997] | | N N . . . = 1 = 2 = 3 = M

  28. Dual Setups to Brane Configurations For vanishing mass deformation ( ) the M-brane configuration is dual to D5-NS5-branes in IIB m = 0 0 1 2 3 4 5 6 7 8 9 D5 branes – • • • • • • NS5 branes – • • • • • • | {z } | {z } | {z } gauge theory ( p,q ) − plane transverse R 3 = 1 = 2 = 3 = M . . . Deformation: | | 1 1 . . . | | ⇒ = 2 2 (1,1) brane . . . | | 3 3 uplift the deformed type II configuration to M-th. . . . . . . . . . . . . on an elliptically fibered Calabi-Yau threefold X N,M [Leung, Vafa 1997] | | N N topic diagram of same as deformed brane web X N,M . . . = 1 = 2 = 3 = M

  29. Dual Construction of LSTs: Toric Calabi-Yau 3folds Specific, 2-parameter series of toric, double elliptically fibered Calabi-Yau threefolds X N,M

  30. Dual Construction of LSTs: Toric Calabi-Yau 3folds Specific, 2-parameter series of toric, double elliptically fibered Calabi-Yau threefolds X N,M Toric Web Diagram: – N = web on a torus ( N, M ) M – 2 · · · – 1 · · · = M · · · = 2 · · · · · · M legs = 1 = · · · 2 – N – 2 = 1 – 1 legs N

  31. Dual Construction of LSTs: Toric Calabi-Yau 3folds Specific, 2-parameter series of toric, double elliptically fibered Calabi-Yau threefolds X N,M Toric Web Diagram: – N = web on a torus ( N, M ) M – 2 double elliptic fibration structure · · · – 1 with parameters ( ρ , τ ) · · · = τ M · · · = 2 · · · · · · = 1 = · · · 2 – N – 2 = 1 – 1 ρ

  32. Dual Construction of LSTs: Toric Calabi-Yau 3folds Specific, 2-parameter series of toric, double elliptically fibered Calabi-Yau threefolds X N,M Toric Web Diagram: – N = web on a torus ( N, M ) M – 2 double elliptic fibration structure · · · – 1 with parameters ( ρ , τ ) · · · different parameters representing 3 NM = τ M the area of various curves of the CY3 · · · C = Z 2 · · · d = ω C · · · = 1 = · · · 2 – N – 2 = 1 – 1 ρ

  33. Dual Construction of LSTs: Toric Calabi-Yau 3folds Specific, 2-parameter series of toric, double elliptically fibered Calabi-Yau threefolds X N,M Toric Web Diagram: – N = web on a torus ( N, M ) M – 2 double elliptic fibration structure · · · – 1 with parameters ( ρ , τ ) · · · different parameters representing 3 NM = τ M the area of various curves of the CY3 · · · C = Z 2 · · · d = ω C · · · Kähler form = 1 = · · · 2 – N – 2 = 1 – 1 ρ

  34. Dual Construction of LSTs: Toric Calabi-Yau 3folds Specific, 2-parameter series of toric, double elliptically fibered Calabi-Yau threefolds X N,M Toric Web Diagram: – N h ( M − 1) N +2 v N h MN = web on a torus ( N, M ) M – 2 h ( M − 1) N +1 v 2 m MN h MN − 1 double elliptic fibration structure · · · m ( M − 1) N +1 – 1 v 1 v MN with parameters ( ρ , τ ) m ( M − 1) N +2 · · · different parameters representing v ( M − 1) N +2 3 NM h MN = τ M the area of various curves of the CY3 C · · · v 3 N v ( M − 1) N +1 h 2 N = Z 2 · · · v 2 N +2 d = ω m 2 N h N +2 h 2 N − 1 C · · · Kähler form v 2 N +1 v 2 N h N +1 h N -) horizontal lines h 1 ,...,NM m N +2 NM = 1 v N +2 -) vertical lines v 1 ,...,NM NM h 2 N h 2 h N − 1 m N m N +1 = · · · -) diagonal lines m 1 ,...,NM 2 NM v N +1 v N h 1 m 2 – N v 2 h N m 1 – 2 = 1 v 1 – 1 ρ

  35. Dual Construction of LSTs: Toric Calabi-Yau 3folds Specific, 2-parameter series of toric, double elliptically fibered Calabi-Yau threefolds X N,M Toric Web Diagram: – N h ( M − 1) N +2 v N h MN = web on a torus ( N, M ) M – 2 h ( M − 1) N +1 v 2 m MN h MN − 1 double elliptic fibration structure · · · m ( M − 1) N +1 – 1 v 1 v MN with parameters ( ρ , τ ) m ( M − 1) N +2 · · · different parameters representing v ( M − 1) N +2 3 NM h MN = τ M the area of various curves of the CY3 C · · · v 3 N v ( M − 1) N +1 h 2 N = Z 2 · · · v 2 N +2 d = ω m 2 N h N +2 h 2 N − 1 C · · · Kähler form v 2 N +1 v 2 N h N +1 h N -) horizontal lines h 1 ,...,NM m N +2 NM = 1 v N +2 -) vertical lines v 1 ,...,NM NM h 2 N h 2 h N − 1 m N m N +1 = · · · -) diagonal lines m 1 ,...,NM 2 NM v N +1 v N h 1 m 2 – N only independent NM + 2 v 2 h N m 1 – 2 parameters due to consistency = 1 v 1 conditions – 1 ρ

  36. Dual Construction of LSTs: Toric Calabi-Yau 3folds Specific, 2-parameter series of toric, double elliptically fibered Calabi-Yau threefolds X N,M Toric Web Diagram: – N h ( M − 1) N +2 v N h MN = web on a torus ( N, M ) M – 2 h ( M − 1) N +1 v 2 m MN h MN − 1 double elliptic fibration structure · · · m ( M − 1) N +1 – 1 v 1 v MN with parameters ( ρ , τ ) m ( M − 1) N +2 · · · different parameters representing v ( M − 1) N +2 3 NM h MN = τ M the area of various curves of the CY3 C · · · v 3 N v ( M − 1) N +1 h 2 N = Z 2 · · · v 2 N +2 d = ω m 2 N h N +2 h 2 N − 1 C · · · Kähler form v 2 N +1 v 2 N h N +1 h N -) horizontal lines h 1 ,...,NM m N +2 NM = 1 v N +2 -) vertical lines v 1 ,...,NM NM h 2 N h 2 h N − 1 m N m N +1 = · · · -) diagonal lines m 1 ,...,NM 2 NM v N +1 v N h 1 m 2 – N only independent NM + 2 v 2 h N m 1 – 2 parameters due to consistency = 1 v 1 conditions – 1 ρ

  37. Dual Construction of LSTs: Toric Calabi-Yau 3folds Specific, 2-parameter series of toric, double elliptically fibered Calabi-Yau threefolds X N,M Toric Web Diagram: h 0 web on a torus ( N, M ) double elliptic fibration structure m 0 v 0 with parameters ( ρ , τ ) different parameters representing 3 NM v the area of various curves of the CY3 C m Z d = ω h C Kähler form -) horizontal lines h 1 ,...,NM NM h + m = h 0 + m 0 -) vertical lines v 1 ,...,NM NM v + m 0 = m + v 0 -) diagonal lines m 1 ,...,NM NM only independent different possible choices for NM + 2 parameters due to consistency set of independent parameters conditions

  38. BPS States and Topological String Free Energy: Counts number of BPS configurations, i.e. M2-branes wrapping holomorphic curves on the CY3 . X N,M

  39. BPS States and Topological String Free Energy: Counts number of BPS configurations, i.e. M2-branes wrapping holomorphic curves on the CY3 . Captured by topological free energy of X N,M F N,M = ln Z N,M X N,M [Haghighat, Iqbal, Kozçaz, Lockhart, Vafa 2013] [Haghighat, Kozcaz, Lockhart, Vafa 2013] [SH, Iqbal 2013]

  40. BPS States and Topological String Free Energy: Counts number of BPS configurations, i.e. M2-branes wrapping holomorphic curves on the CY3 . Captured by topological free energy of X N,M F N,M = ln Z N,M X N,M [Haghighat, Iqbal, Kozçaz, Lockhart, Vafa 2013] [Haghighat, Kozcaz, Lockhart, Vafa 2013] [SH, Iqbal 2013] Compute the topological string partition function using the refined topological vertex Z N,M [Aganagic, Klemm, Marino, Vafa 2003] [Iqbal, Kozçaz, Vafa 2007]

  41. BPS States and Topological String Free Energy: Counts number of BPS configurations, i.e. M2-branes wrapping holomorphic curves on the CY3 . Captured by topological free energy of X N,M F N,M = ln Z N,M X N,M [Haghighat, Iqbal, Kozçaz, Lockhart, Vafa 2013] [Haghighat, Kozcaz, Lockhart, Vafa 2013] [SH, Iqbal 2013] Compute the topological string partition function using the refined topological vertex Z N,M – N h ( M − 1) N +2 v N h MN = M – 2 h ( M − 1) N +1 v 2 m MN h MN − 1 · · · m ( M − 1) N +1 – 1 v 1 v MN m ( M − 1) N +2 · · · v ( M − 1) N +2 h MN = M · · · v 3 N v ( M − 1) N +1 h 2 N = 2 · · · v 2 N +2 m 2 N h N +2 h 2 N − 1 · · · v 2 N +1 v 2 N h N +1 h N m N +2 = 1 v N +2 h N − 1 h 2 N h 2 m N m N +1 = · · · 2 v N +1 v N h 1 m 2 – N v 2 h N m 1 – 2 = 1 v 1 – 1

  42. BPS States and Topological String Free Energy: Counts number of BPS configurations, i.e. M2-branes wrapping holomorphic curves on the CY3 . Captured by topological free energy of X N,M F N,M = ln Z N,M X N,M [Haghighat, Iqbal, Kozçaz, Lockhart, Vafa 2013] [Haghighat, Kozcaz, Lockhart, Vafa 2013] [SH, Iqbal 2013] Compute the topological string partition function using the refined topological vertex Z N,M – N h ( M − 1) N +2 assign trivalent vertex to each intersection -) v N h MN = M – 2 h ( M − 1) N +1 v 2 m MN h MN − 1 · · · m ( M − 1) N +1 – 1 v 1 v MN m ( M − 1) N +2 · · · v ( M − 1) N +2 h MN = M · · · v 3 N v ( M − 1) N +1 h 2 N = 2 · · · v 2 N +2 m 2 N h N +2 h 2 N − 1 · · · v 2 N +1 v 2 N ν h N +1 h N m N +2 = µ 1 v N +2 • h N − 1 h 2 N h 2 m N m N +1 = · · · 2 v N +1 v N h 1 m 2 – N λ v 2 h N m 1 – 2 = 1 v 1 – 1

  43. BPS States and Topological String Free Energy: Counts number of BPS configurations, i.e. M2-branes wrapping holomorphic curves on the CY3 . Captured by topological free energy of X N,M F N,M = ln Z N,M X N,M [Haghighat, Iqbal, Kozçaz, Lockhart, Vafa 2013] [Haghighat, Kozcaz, Lockhart, Vafa 2013] [SH, Iqbal 2013] Compute the topological string partition function using the refined topological vertex Z N,M – N h ( M − 1) N +2 assign trivalent vertex to each intersection -) v N h MN = ⌘ | η | + | λ | − | µ | || µ || 2 t − || µ t || 2 || ν || 2 M – 2 ⇣ q h ( M − 1) N +1 v 2 ˜ X 2 C λ µ ν = q Z ν ( t, q ) m MN q h MN − 1 2 2 2 t · · · m ( M − 1) N +1 – 1 η v 1 v MN m ( M − 1) N +2 × s λ t / η ( t − ρ q − ν ) s µ/ η ( q − ρ t − ν t ) · · · v ( M − 1) N +2 h MN = j − i +1 q ν i − j ⌘ − 1 M ⇣ 1 − t ν t ˜ Y · · · v 3 N Z ν ( t, q ) = , v ( M − 1) N +1 h 2 N = 2 ( i,j ) ∈ ν · · · v 2 N +2 m 2 N h N +2 h 2 N − 1 · · · v 2 N +1 v 2 N ν h N +1 h N m N +2 = µ 1 v N +2 • h N − 1 h 2 N h 2 m N m N +1 = · · · 2 v N +1 v N h 1 m 2 – N λ v 2 h N m 1 – 2 = 1 v 1 – 1

  44. BPS States and Topological String Free Energy: Counts number of BPS configurations, i.e. M2-branes wrapping holomorphic curves on the CY3 . Captured by topological free energy of X N,M F N,M = ln Z N,M X N,M [Haghighat, Iqbal, Kozçaz, Lockhart, Vafa 2013] [Haghighat, Kozcaz, Lockhart, Vafa 2013] [SH, Iqbal 2013] Compute the topological string partition function using the refined topological vertex Z N,M – N h ( M − 1) N +2 assign trivalent vertex to each intersection -) v N Notation: h MN = q = e 2 ⇡ i ✏ 1 t = e − 2 ⇡ i ✏ 2 ⌘ | η | + | λ | − | µ | || µ || 2 t − || µ t || 2 || ν || 2 and M – 2 ⇣ q h ( M − 1) N +1 v 2 ˜ X 2 C λ µ ν = q Z ν ( t, q ) m MN q h MN − 1 2 2 2 t µ , ν , λ · · · integer partitions m ( M − 1) N +1 – 1 η v 1 v MN ` m ( M − 1) N +2 × s λ t / η ( t − ρ q − ν ) s µ/ η ( q − ρ t − ν t ) X | µ | = µ i · · · v ( M − 1) N +2 h MN ` = i =1 j − i +1 q ν i − j ⌘ − 1 M ⇣ 1 − t ν t ˜ Y · · · µ i v 3 N Z ν ( t, q ) = , v ( M − 1) N +1 ` h 2 N X = || µ || 2 = µ 2 2 ( i,j ) ∈ ν i · · · v 2 N +2 m 2 N h N +2 h 2 N − 1 i =1 · · · s µ/ η skew Schur function v 2 N +1 v 2 N ν h N +1 h N m N +2 = µ 1 v N +2 • h N − 1 h 2 N h 2 m N m N +1 = · · · 2 v N +1 v N h 1 m 2 – N λ v 2 h N m 1 – 2 = 1 v 1 – 1

  45. BPS States and Topological String Free Energy: Counts number of BPS configurations, i.e. M2-branes wrapping holomorphic curves on the CY3 . Captured by topological free energy of X N,M F N,M = ln Z N,M X N,M [Haghighat, Iqbal, Kozçaz, Lockhart, Vafa 2013] [Haghighat, Kozcaz, Lockhart, Vafa 2013] [SH, Iqbal 2013] Compute the topological string partition function using the refined topological vertex Z N,M – N h ( M − 1) N +2 assign trivalent vertex to each intersection -) v N h MN = ⌘ | η | + | λ | − | µ | || µ || 2 t − || µ t || 2 || ν || 2 M – 2 ⇣ q h ( M − 1) N +1 v 2 ˜ X 2 C λ µ ν = q Z ν ( t, q ) m MN q h MN − 1 2 2 2 t · · · m ( M − 1) N +1 – 1 η v 1 v MN m ( M − 1) N +2 × s λ t / η ( t − ρ q − ν ) s µ/ η ( q − ρ t − ν t ) · · · v ( M − 1) N +2 h MN = j − i +1 q ν i − j ⌘ − 1 M ⇣ 1 − t ν t ˜ Y · · · v 3 N Z ν ( t, q ) = , v ( M − 1) N +1 h 2 N = 2 ( i,j ) ∈ ν · · · v 2 N +2 m 2 N h N +2 h 2 N − 1 -) glue vertices according to web diagram · · · v 2 N +1 v 2 N h N +1 h N m N +2 = 1 λ 2 v N +2 h N − 1 h 2 N h 2 m N m N +1 = · · · µ 2 2 • v N +1 v N h 1 m 2 m – N ν v 2 µ 1 h N m 1 • – 2 = 1 v 1 λ 1 – 1

  46. BPS States and Topological String Free Energy: Counts number of BPS configurations, i.e. M2-branes wrapping holomorphic curves on the CY3 . Captured by topological free energy of X N,M F N,M = ln Z N,M X N,M [Haghighat, Iqbal, Kozçaz, Lockhart, Vafa 2013] [Haghighat, Kozcaz, Lockhart, Vafa 2013] [SH, Iqbal 2013] Compute the topological string partition function using the refined topological vertex Z N,M – N h ( M − 1) N +2 assign trivalent vertex to each intersection -) v N h MN = ⌘ | η | + | λ | − | µ | || µ || 2 t − || µ t || 2 || ν || 2 M – 2 ⇣ q h ( M − 1) N +1 v 2 ˜ X 2 C λ µ ν = q Z ν ( t, q ) m MN q h MN − 1 2 2 2 t · · · m ( M − 1) N +1 – 1 η v 1 v MN m ( M − 1) N +2 × s λ t / η ( t − ρ q − ν ) s µ/ η ( q − ρ t − ν t ) · · · v ( M − 1) N +2 h MN = j − i +1 q ν i − j ⌘ − 1 M ⇣ 1 − t ν t ˜ Y · · · v 3 N Z ν ( t, q ) = , v ( M − 1) N +1 h 2 N = 2 ( i,j ) ∈ ν · · · v 2 N +2 m 2 N h N +2 h 2 N − 1 -) glue vertices according to web diagram · · · v 2 N +1 v 2 N h N +1 h N m N +2 = X ( − e 2 π im ) | ν | C µ 1 λ 1 ν C µ t 1 λ 2 v N +2 2 λ t 2 ν t h N − 1 h 2 N h 2 m N m N +1 = ν · · · µ 2 2 • v N +1 v N h 1 m 2 m – N ν v 2 µ 1 h N m 1 • – 2 = 1 v 1 λ 1 – 1

  47. BPS States and Topological String Free Energy: Counts number of BPS configurations, i.e. M2-branes wrapping holomorphic curves on the CY3 . Captured by topological free energy of X N,M F N,M = ln Z N,M X N,M [Haghighat, Iqbal, Kozçaz, Lockhart, Vafa 2013] [Haghighat, Kozcaz, Lockhart, Vafa 2013] [SH, Iqbal 2013] Compute the topological string partition function using the refined topological vertex Z N,M – N h ( M − 1) N +2 assign trivalent vertex to each intersection -) v N h MN = ⌘ | η | + | λ | − | µ | || µ || 2 t − || µ t || 2 || ν || 2 M – 2 ⇣ q h ( M − 1) N +1 v 2 ˜ X 2 C λ µ ν = q Z ν ( t, q ) m MN q h MN − 1 2 2 2 t · · · m ( M − 1) N +1 – 1 η v 1 v MN m ( M − 1) N +2 × s λ t / η ( t − ρ q − ν ) s µ/ η ( q − ρ t − ν t ) · · · v ( M − 1) N +2 h MN = j − i +1 q ν i − j ⌘ − 1 M ⇣ 1 − t ν t ˜ Y · · · v 3 N Z ν ( t, q ) = , v ( M − 1) N +1 h 2 N = 2 ( i,j ) ∈ ν · · · v 2 N +2 m 2 N h N +2 h 2 N − 1 -) glue vertices according to web diagram · · · v 2 N +1 v 2 N h N +1 h N m N +2 = X ( − e 2 π im ) | ν | C µ 1 λ 1 ν C µ t 1 v N +2 2 λ t 2 ν t h N − 1 h 2 N h 2 m N m N +1 = ν · · · 2 v N +1 v N -) choose preferred direction h 1 m 2 – N v 2 h N m 1 – 2 = 1 v 1 – 1

  48. BPS States and Topological String Free Energy: Counts number of BPS configurations, i.e. M2-branes wrapping holomorphic curves on the CY3 . Captured by topological free energy of X N,M F N,M = ln Z N,M X N,M [Haghighat, Iqbal, Kozçaz, Lockhart, Vafa 2013] [Haghighat, Kozcaz, Lockhart, Vafa 2013] [SH, Iqbal 2013] Compute the topological string partition function using the refined topological vertex Z N,M – N h ( M − 1) N +2 assign trivalent vertex to each intersection -) v N h MN = ⌘ | η | + | λ | − | µ | || µ || 2 t − || µ t || 2 || ν || 2 M – 2 ⇣ q h ( M − 1) N +1 v 2 ˜ X 2 C λ µ ν = q Z ν ( t, q ) m MN q h MN − 1 2 2 2 t · · · m ( M − 1) N +1 – 1 η v 1 v MN m ( M − 1) N +2 × s λ t / η ( t − ρ q − ν ) s µ/ η ( q − ρ t − ν t ) · · · v ( M − 1) N +2 h MN = j − i +1 q ν i − j ⌘ − 1 M ⇣ 1 − t ν t ˜ Y · · · v 3 N Z ν ( t, q ) = , v ( M − 1) N +1 h 2 N = 2 ( i,j ) ∈ ν · · · v 2 N +2 m 2 N h N +2 h 2 N − 1 -) glue vertices according to web diagram · · · v 2 N +1 v 2 N h N +1 h N m N +2 = X ( − e 2 π im ) | ν | C µ 1 λ 1 ν C µ t 1 v N +2 2 λ t 2 ν t h N − 1 h 2 N h 2 m N m N +1 = ν · · · 2 v N +1 v N -) choose preferred direction h 1 m 2 – N v 2 must be common to all vertices of diagram h N m 1 – 2 = 1 v 1 – 1

  49. preferred direction

  50. 3 different choices for the : preferred direction

  51. – N 3 different choices for the : preferred direction = M – 2 1) horizontal: · · · – 1 · · · = M · · · = 2 · · · · · · = 1 = · · · 2 – N – 2 = 1 – 1

  52. – N 3 different choices for the : preferred direction = M – 2 1) horizontal: decompose diagram into vertical strips · · · – 1 · · · = = M · · · v 1 α 1 = 2 · · · m 1 β t 1 · · · v 2 = 1 · · · = · · · 2 – N vM − 1 α M − 1 – 2 = mM − 1 1 β t M − 1 – 1 vM α M mM β t M v 1 =

  53. – N 3 different choices for the : preferred direction = M – 2 1) horizontal: decompose diagram into vertical strips · · · – 1 W α 1 ... α M building block: β 1 ... β M ( { v } , { m } ) · · · = = M · · · v 1 α 1 = 2 · · · m 1 β t 1 · · · v 2 = 1 · · · = · · · 2 – N vM − 1 α M − 1 – 2 = mM − 1 1 β t M − 1 – 1 vM α M mM β t M v 1 =

  54. – N 3 different choices for the : preferred direction = M – 2 1) horizontal: decompose diagram into vertical strips · · · – 1 W α 1 ... α M building block: β 1 ... β M ( { v } , { m } ) · · · 2) vertical: = M · · · = 2 · · · · · · = 1 = · · · 2 – N – 2 = 1 – 1

  55. – N 3 different choices for the : preferred direction = M – 2 1) horizontal: decompose diagram into vertical strips · · · – 1 W α 1 ... α M building block: β 1 ... β M ( { v } , { m } ) · · · 2) vertical: decompose diagram into horizontal strips = M · · · = 2 · · · · · · α N = 1 α 2 = = · · · h 1 2 α 1 mN h 3 hN – N · · · m 2 h 2 – 2 = β t 1 N m 1 h 1 = – 1 β t 2 β t 1

  56. – N 3 different choices for the : preferred direction = M – 2 1) horizontal: decompose diagram into vertical strips · · · – 1 W α 1 ... α M building block: β 1 ... β M ( { v } , { m } ) · · · 2) vertical: decompose diagram into horizontal strips = M · · · W α 1 ... α N building block: β 1 ... β N ( { h } , { m } ) = 2 · · · · · · α N = 1 α 2 = = · · · h 1 2 α 1 mN h 3 hN – N · · · m 2 h 2 – 2 = β t 1 N m 1 h 1 = – 1 β t 2 β t 1

  57. – N 3 different choices for the : preferred direction = M – 2 1) horizontal: decompose diagram into vertical strips · · · – 1 W α 1 ... α M building block: β 1 ... β M ( { v } , { m } ) · · · 2) vertical: decompose diagram into horizontal strips = M · · · W α 1 ... α N building block: β 1 ... β N ( { h } , { m } ) = 2 · · · 3) diagonal: · · · = 1 = · · · 2 – N – 2 = 1 – 1

  58. … – N 3 different choices for the : preferred direction = M – 2 1) horizontal: decompose diagram into vertical strips · · · – 1 W α 1 ... α M building block: β 1 ... β M ( { v } , { m } ) … · · · 2) vertical: decompose diagram into horizontal strips = M · · · W α 1 ... α N building block: β 1 ... β N ( { h } , { m } ) = 2 · · · 3) diagonal: decompose diagram into diagonal strips · · · = 1 = · · · 2 – N α 1 – 2 … = h 1 1 = α 2 – 1 v 1 h 2 α NM v 2 h NM k β t h 3 k 1 · · · v NM k β t h 1 = 2 β t NM where k = gcd( N, M ) k

  59. 3 different choices for the : preferred direction b a 1 S α 1 1) horizontal: decompose diagram into vertical strips a b a 2 α 2 W α 1 ... α M building block: β 1 ... β M ( { v } , { m } ) α 3 β t 2) vertical: decompose diagram into horizontal strips b b 1 1 · · · W α 1 ... α N building block: β 1 ... β N ( { h } , { m } ) β t b b 2 2 · · · b a L 3) diagonal: decompose diagram into diagonal strips α L · · · α 1 ... α NM building block: ( { h } , { v } ) W k β 1 ... β NM β t k L − 1 generic form of the building block a β t L J α i β j ( b Q i,i − j ; q, t ) J β j α i (( b Y b Q i,i − j ) − 1 Q ρ ; q, t ) b L L β 1 ... β L = W L ( ∅ ) · ˆ W α 1 ... α L Z · p p q/t ; q, t ) J β j β i ( ˙ J α i α j ( Q i,i − j Q i,j − i t/q ; q, t ) i,j =1

  60. 3 different choices for the : preferred direction b a 1 S α 1 1) horizontal: decompose diagram into vertical strips a b a 2 α 2 W α 1 ... α M building block: β 1 ... β M ( { v } , { m } ) α 3 β t 2) vertical: decompose diagram into horizontal strips b b 1 1 · · · W α 1 ... α N building block: β 1 ... β N ( { h } , { m } ) β t b b 2 2 · · · b a L 3) diagonal: decompose diagram into diagonal strips α L · · · α 1 ... α NM building block: ( { h } , { v } ) W k β 1 ... β NM β t k L − 1 generic form of the building block a β t L J α i β j ( b Q i,i − j ; q, t ) J β j α i (( b Y b Q i,i − j ) − 1 Q ρ ; q, t ) b L L β 1 ... β L = W L ( ∅ ) · ˆ W α 1 ... α L Z · p p q/t ; q, t ) J β j β i ( ˙ J α i α j ( Q i,i − j Q i,j − i t/q ; q, t ) i,j =1 with (1 − b q r − 1 2 t s − 1 2 )(1 − b ρ q s − 1 2 t r − 1 L Q − 1 Y Y Q i,j Q k − 1 2 ) ∞ i,j Q k ρ W L ( ∅ ) = , q r t s − 1 )(1 − ˙ (1 − Q i,j Q k − 1 Q i,j Q k − 1 q s − 1 t r ) ρ ρ i,j =1 k,r,s =1 ⇣ j − i +1 q ν i − j ⌘ − 1 Y L Y || α t k || 2 || α k || 2 1 − t ν t ˆ Z α k ( q, t ) ˜ ˜ ˜ Z = t q Z α t k ( t, q ) , Z ν ( t, q ) = 2 2 i =1 ( i,j ) ∈ ν ∞ Y J µ ν ( Q k − 1 J µ ν ( x ; t, q ) = x ; t, q ) , ρ k =1 ⇣ ⌘ ⇣ ⌘ 1 − x t ν t j − i + 1 2 q µ i − j + 1 1 − x t − µ t j + i − 1 2 q − ν i + j − 1 Y Y J µ ν ( x ; t, q ) = 2 × 2 ( i,j ) ∈ µ ( i,j ) ∈ ν

  61. 3 different choices for the : preferred direction b a 1 S α 1 1) horizontal: decompose diagram into vertical strips a b a 2 α 2 W α 1 ... α M building block: β 1 ... β M ( { v } , { m } ) α 3 β t 2) vertical: decompose diagram into horizontal strips b b 1 1 · · · W α 1 ... α N building block: β 1 ... β N ( { h } , { m } ) β t b b 2 2 · · · a L b 3) diagonal: decompose diagram into diagonal strips α L · · · α 1 ... α NM building block: ( { h } , { v } ) W k β 1 ... β NM β t k L − 1 generic form of the building block a β t L J α i β j ( b Q i,i − j ; q, t ) J β j α i (( b Y b Q i,i − j ) − 1 Q ρ ; q, t ) b L L β 1 ... β L = W L ( ∅ ) · ˆ W α 1 ... α L Z · p p q/t ; q, t ) J β j β i ( ˙ J α i α j ( Q i,i − j Q i,j − i t/q ; q, t ) i,j =1 with Notation: j − 1 i Y Y b (1 − b q r − 1 2 t s − 1 2 )(1 − b ρ q s − 1 2 t r − 1 L Q − 1 ( Q a r Q − 1 Y Y Q i,j Q k − 1 2 ) ∞ i,j Q k Q i,j = Q S b r ) Q a i − k , ρ W L ( ∅ ) = , r =1 k =1 q r t s − 1 )(1 − ˙ (1 − Q i,j Q k − 1 Q i,j Q k − 1 q s − 1 t r ) ⇢ 1 ρ ρ i,j =1 k,r,s =1 if j = L Q j Q i,j = ⇣ j − i +1 q ν i − j ⌘ − 1 if j 6 = L k =1 Q a i − k Y L Y || α t k || 2 || α k || 2 1 − t ν t ˆ Z α k ( q, t ) ˜ ˜ ˜ Z = t q Z α t k ( t, q ) , Z ν ( t, q ) = 2 2 j Y ˙ Q i,j = Q b i + k i =1 ( i,j ) ∈ ν ∞ k =1 Y J µ ν ( Q k − 1 J µ ν ( x ; t, q ) = x ; t, q ) , Q S = e − S and ρ k =1 Q a i = e − b a i ⇣ ⌘ ⇣ ⌘ 1 − x t ν t j − i + 1 2 q µ i − j + 1 1 − x t − µ t j + i − 1 2 q − ν i + j − 1 Y Y J µ ν ( x ; t, q ) = Q b i = e − b 2 × 2 b i ( i,j ) ∈ µ ( i,j ) ∈ ν

  62. 3 different choices for the : preferred direction b a 1 S α 1 1) horizontal: decompose diagram into vertical strips a b a 2 α 2 W α 1 ... α M building block: β 1 ... β M ( { v } , { m } ) α 3 β t 2) vertical: decompose diagram into horizontal strips b b 1 1 · · · W α 1 ... α N building block: β 1 ... β N ( { h } , { m } ) β t b b 2 2 · · · b a L 3) diagonal: decompose diagram into diagonal strips α L · · · α 1 ... α NM building block: ( { h } , { v } ) W k β 1 ... β NM β t k L − 1 generic form of the building block a β t L J α i β j ( b Q i,i − j ; q, t ) J β j α i (( b Y b Q i,i − j ) − 1 Q ρ ; q, t ) b L L β 1 ... β L = W L ( ∅ ) · ˆ W α 1 ... α L Z · p p q/t ; q, t ) J β j β i ( ˙ J α i α j ( Q i,i − j Q i,j − i t/q ; q, t ) i,j =1

  63. 3 different choices for the : preferred direction b a 1 S α 1 1) horizontal: decompose diagram into vertical strips a b a 2 α 2 W α 1 ... α M building block: β 1 ... β M ( { v } , { m } ) α 3 β t 2) vertical: decompose diagram into horizontal strips b b 1 1 · · · W α 1 ... α N building block: β 1 ... β N ( { h } , { m } ) β t b b 2 2 · · · b a L 3) diagonal: decompose diagram into diagonal strips α L · · · α 1 ... α NM building block: ( { h } , { v } ) W k β 1 ... β NM β t k L − 1 generic form of the building block a β t L J α i β j ( b Q i,i − j ; q, t ) J β j α i (( b Y b Q i,i − j ) − 1 Q ρ ; q, t ) b L L β 1 ... β L = W L ( ∅ ) · ˆ W α 1 ... α L Z · p p q/t ; q, t ) J β j β i ( ˙ J α i α j ( Q i,i − j Q i,j − i t/q ; q, t ) i,j =1 suitable for all three expansions upon identifying: horizontal vertical diagonal b a i v i +1 + m i h i + m i v i + h i +1 b b i v i + m i h i + m i − 1 h i + v i S v 1 m N h 1 NM L M N k

  64. Newton Polygons Alternative view on the three gauge theories: Newton polygons as dual of web diagrams

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Newton Polygons Alternative view on the three gauge theories: Newton polygons as dual of web diagrams Example: ( N, M ) = (3 , 2) · · · − 3 10 11 12 10 11 12 h 4 7 8 9 7 8 9 = 12 2 − 2 4 5 6 4 5 6 m 6 1 2 3 1 2 3 h 6 V 9 10 11 12 10 11 12 11 − 1 v 6 7 8 9 7 8 9 m 5 h 1 h 2 = I VI 4 5 6 4 5 6 8 10 6 1 1 2 3 1 2 3 v 2 dual of web · · · · · · m 4 m 3 h 3 10 11 12 10 11 12 = II diagram 7 3 7 8 9 7 8 9 2 5 v 4 v 3 4 5 6 4 5 6 m 2 h 5 − 3 1 2 3 1 2 3 IV III 2 4 10 11 12 10 11 12 v 5 m 1 − 2 7 8 9 7 8 9 = 1 1 4 5 6 4 5 6 v 1 1 2 3 1 2 3 − 1 · · ·

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sha1_base64="zZ95vW9D7qf3on/wH2iY5VcXQ7w=">AB8XicbVBNS8NAEJ34WetX1aOXxSK0UEpaBfUgFL14USoYW0hD2Ww37dLNJuxuhBL6M7x4UPHqv/Hmv3Hb5qCtDwYe780wM8+POVPatr+tpeWV1bX13EZ+c2t7Z7ewt/+okQS6pCIR7LtY0U5E9TRTHPajiXFoc9pyx9eT/zWE5WKReJBj2LqhbgvWMAI1kZyS3eV2/Jl6aRSL3cLRbtqT4EWS0jRcjQ7Ba+Or2IJCEVmnCslFuzY+2lWGpGOB3nO4miMSZD3KeuoQKHVHnp9OQxOjZKDwWRNCU0mq/J1IcKjUKfdMZYj1Q895E/M9zEx2ceykTcaKpILNFQcKRjtDkf9RjkhLNR4ZgIpm5FZEBlphok1LehFCbf3mROPXqRdW+Py02rI0cnAIR1CGpxBA26gCQ4QiOAZXuHN0taL9W59zFqXrGzmAP7A+vwBdTGO8g=</latexit> Newton Polygons Alternative view on the three gauge theories: Newton polygons as dual of web diagrams Example: ( N, M ) = (3 , 2) · · · − 3 10 11 12 10 11 12 h 4 7 8 9 7 8 9 = 12 2 − 2 4 5 6 4 5 6 m 6 1 2 3 1 2 3 h 6 V 9 10 11 12 10 11 12 11 − 1 v 6 7 8 9 7 8 9 m 5 h 1 h 2 = I VI 4 5 6 4 5 6 8 10 6 1 1 2 3 1 2 3 v 2 dual of web · · · · · · m 4 m 3 h 3 10 11 12 10 11 12 = II diagram 7 3 7 8 9 7 8 9 2 5 v 4 v 3 4 5 6 4 5 6 m 2 h 5 − 3 1 2 3 1 2 3 IV III 2 4 10 11 12 10 11 12 v 5 m 1 − 2 7 8 9 7 8 9 = 1 1 4 5 6 4 5 6 v 1 1 2 3 1 2 3 − 1 · · · -) decomposition into two horizontal strips W α 1 α 2 α 3 β 1 β 2 β 3

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Newton Polygons Alternative view on the three gauge theories: Newton polygons as dual of web diagrams Example: ( N, M ) = (3 , 2) · · · − 3 10 11 12 10 11 12 h 4 7 8 9 7 8 9 = 12 2 − 2 4 5 6 4 5 6 m 6 1 2 3 1 2 3 h 6 V 9 10 11 12 10 11 12 11 − 1 v 6 7 8 9 7 8 9 m 5 h 1 h 2 = I VI 4 5 6 4 5 6 8 10 6 1 1 2 3 1 2 3 v 2 dual of web · · · · · · m 4 m 3 h 3 10 11 12 10 11 12 = II diagram 7 3 7 8 9 7 8 9 2 5 v 4 v 3 4 5 6 4 5 6 m 2 h 5 − 3 1 2 3 1 2 3 IV III 2 4 10 11 12 10 11 12 v 5 m 1 − 2 7 8 9 7 8 9 = 1 1 4 5 6 4 5 6 v 1 1 2 3 1 2 3 − 1 · · · -) decomposition into two horizontal strips W α 1 α 2 α 3 β 1 β 2 β 3 -) decomposition into three vertical strips W α 1 α 2 β 1 β 2

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Newton Polygons Alternative view on the three gauge theories: Newton polygons as dual of web diagrams Example: ( N, M ) = (3 , 2) · · · − 3 10 11 12 10 11 12 h 4 7 8 9 7 8 9 = 12 2 − 2 4 5 6 4 5 6 m 6 1 2 3 1 2 3 h 6 V 9 10 11 12 10 11 12 11 − 1 v 6 7 8 9 7 8 9 m 5 h 1 h 2 = I VI 4 5 6 4 5 6 8 10 6 1 1 2 3 1 2 3 v 2 dual of web · · · · · · m 4 m 3 h 3 10 11 12 10 11 12 = II diagram 7 3 7 8 9 7 8 9 2 5 v 4 v 3 4 5 6 4 5 6 m 2 h 5 − 3 1 2 3 1 2 3 IV III 2 4 10 11 12 10 11 12 v 5 m 1 − 2 7 8 9 7 8 9 = 1 1 4 5 6 4 5 6 v 1 1 2 3 1 2 3 − 1 · · · -) decomposition into two horizontal strips W α 1 α 2 α 3 β 1 β 2 β 3 -) decomposition into three vertical strips W α 1 α 2 β 1 β 2 -) for diagonal decomposition: choose different fundamental domain single strip W α 1 α 2 α 3 α 4 α 5 α 6 β 1 β 2 β 3 β 4 β 5 β 6

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Newton Polygons Alternative view on the three gauge theories: Newton polygons as dual of web diagrams Example: ( N, M ) = (3 , 2) · · · − 3 10 11 12 10 11 12 h 4 7 8 9 7 8 9 = 12 2 − 2 4 5 6 4 5 6 m 6 1 2 3 1 2 3 h 6 V 9 10 11 12 10 11 12 11 − 1 v 6 7 8 9 7 8 9 m 5 h 1 h 2 = I VI 4 5 6 4 5 6 8 10 6 1 1 2 3 1 2 3 v 2 dual of web · · · · · · m 4 m 3 h 3 10 11 12 10 11 12 = II diagram 7 3 7 8 9 7 8 9 2 5 v 4 v 3 4 5 6 4 5 6 m 2 h 5 − 3 1 2 3 1 2 3 IV III 2 4 10 11 12 10 11 12 v 5 m 1 − 2 7 8 9 7 8 9 4 = 1 1 4 5 6 4 5 6 h 1 m 1 v 1 a 1 2 3 1 2 3 − 1 5 v 1 h 2 m 5 m 4 · · · 6 v 2 1 presentation of the web diagram associated h 3 m 3 m 2 with alternative fundamental domain 1 v 3 2 h 4 m 4 m 6 2 v 4 3 h 5 m 2 m 1 3 v 5 4 h 6 m 6 m 5 v 6 5 h 1 m 3 a 6

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Newton Polygons Alternative view on the three gauge theories: Newton polygons as dual of web diagrams Example: ( N, M ) = (3 , 2) · · · − 3 10 11 12 10 11 12 h 4 7 8 9 7 8 9 = 12 2 − 2 4 5 6 4 5 6 m 6 1 2 3 1 2 3 h 6 V 9 10 11 12 10 11 12 11 − 1 v 6 7 8 9 7 8 9 m 5 h 1 h 2 = I VI 4 5 6 4 5 6 8 10 6 1 1 2 3 1 2 3 v 2 dual of web · · · · · · m 4 m 3 h 3 10 11 12 10 11 12 = II diagram 7 3 7 8 9 7 8 9 2 5 v 4 v 3 4 5 6 4 5 6 m 2 h 5 − 3 1 2 3 1 2 3 IV III 2 4 10 11 12 10 11 12 v 5 m 1 − 2 7 8 9 7 8 9 4 = 1 1 4 5 6 4 5 6 h 1 m 1 v 1 a 1 2 3 1 2 3 − 1 5 v 1 h 2 m 5 m 4 · · · 6 v 2 1 presentation of the web diagram associated h 3 m 3 m 2 with alternative fundamental domain 1 v 3 2 h 4 m 4 m 6 2 v 4 3 h 5 m 2 m 1 -) all fundamental domains equivalent 3 v 5 4 h 6 m 6 -) lead to same partition function m 5 v 6 5 h 1 m 3 a 6

  71. Topological Partition Function The full partition function is obtained by gluing together the building blocks W α 1 ... α M β 1 ... β M M,N N α 1 j ··· α M ⇣ j | ⌘ e − u ij | α i X Y Y j Z N,M = W α 1 j +1 ··· α M j +1 i =1 ,j =1 j =1 α

  72. Topological Partition Function The full partition function is obtained by gluing together the building blocks W α 1 ... α M β 1 ... β M M,N N α 1 j ··· α M ⇣ j | ⌘ e − u ij | α i X Y Y j Z N,M = W α 1 j +1 ··· α M j +1 i =1 ,j =1 j =1 α parameters used to glue the strips together

  73. Topological Partition Function The full partition function is obtained by gluing together the building blocks W α 1 ... α M β 1 ... β M M,N N α 1 j ··· α M ⇣ j | ⌘ e − u ij | α i X Y Y j Z N,M = W α 1 j +1 ··· α M j +1 i =1 ,j =1 j =1 α parameters used to glue the strips together Different choices of preferred direction afford different (but equivalent) expansions: e − ~ k · h Z ~ k ( { v } , { m } ) = Z ( N,M ) X Z N,M ( { h } , { v } , { m } , ✏ 1 , 2 ) = Z p ( { v } , { m } ) hor ~ k e − ~ k · v Z ~ k ( { h } , { m } ) = Z ( N,M ) X = Z p ( { h } , { m } ) vert ~ k e − ~ k · m Z ~ k ( { h } , { v } ) = Z ( N,M ) X = Z p ( { h } , { v } ) diag ~ k

  74. Topological Partition Function The full partition function is obtained by gluing together the building blocks W α 1 ... α M β 1 ... β M M,N N α 1 j ··· α M ⇣ j | ⌘ e − u ij | α i X Y Y j Z N,M = W α 1 j +1 ··· α M j +1 i =1 ,j =1 j =1 α parameters used to glue the strips together Different choices of preferred direction afford different (but equivalent) expansions: e − ~ k · h Z ~ k ( { v } , { m } ) = Z ( N,M ) X Z N,M ( { h } , { v } , { m } , ✏ 1 , 2 ) = Z p ( { v } , { m } ) hor ~ k e − ~ k · v Z ~ k ( { h } , { m } ) = Z ( N,M ) X = Z p ( { h } , { m } ) vert ~ k e − ~ k · m Z ~ k ( { h } , { v } ) = Z ( N,M ) X = Z p ( { h } , { v } ) diag ~ k common normalisation factor (perturbative partition function)

  75. Topological Partition Function The full partition function is obtained by gluing together the building blocks W α 1 ... α M β 1 ... β M M,N N α 1 j ··· α M ⇣ j | ⌘ e − u ij | α i X Y Y j Z N,M = W α 1 j +1 ··· α M j +1 i =1 ,j =1 j =1 α parameters used to glue the strips together Different choices of preferred direction afford different (but equivalent) expansions: e − ~ k · h Z ~ k ( { v } , { m } ) = Z ( N,M ) X Z N,M ( { h } , { v } , { m } , ✏ 1 , 2 ) = Z p ( { v } , { m } ) hor ~ k e − ~ k · v Z ~ k ( { h } , { m } ) = Z ( N,M ) X = Z p ( { h } , { m } ) vert ~ k e − ~ k · m Z ~ k ( { h } , { v } ) = Z ( N,M ) X = Z p ( { h } , { v } ) diag ~ k common normalisation factor (perturbative partition function) Compare different series expansions with instanton partition functions of quiver gauge theories.

  76. Topological Partition Function The full partition function is obtained by gluing together the building blocks W α 1 ... α M β 1 ... β M M,N N α 1 j ··· α M ⇣ j | ⌘ e − u ij | α i X Y Y j Z N,M = W α 1 j +1 ··· α M j +1 i =1 ,j =1 j =1 α parameters used to glue the strips together Different choices of preferred direction afford different (but equivalent) expansions: e − ~ k · h Z ~ k ( { v } , { m } ) = Z ( N,M ) X Z N,M ( { h } , { v } , { m } , ✏ 1 , 2 ) = Z p ( { v } , { m } ) hor ~ k e − ~ k · v Z ~ k ( { h } , { m } ) = Z ( N,M ) X = Z p ( { h } , { m } ) vert ~ k e − ~ k · m Z ~ k ( { h } , { v } ) = Z ( N,M ) X = Z p ( { h } , { v } ) diag ~ k common normalisation factor (perturbative partition function) Compare different series expansions with instanton partition functions of quiver gauge theories. Need to choose independent Kähler parameters of X N,M

  77. Bases of independent Kähler parameters For each of the expansion we can choose a suitable set of independent Kähler parameters: NM + 2

  78. Bases of independent Kähler parameters For each of the expansion we can choose a suitable set of independent Kähler parameters: NM + 2 Example: ( N, M ) = (3 , 2) − 3 h 6 = 2 − 2 m 6 h 5 − 1 v 6 h 3 m 5 h 4 = 1 v 5 m 3 m 4 h 2 = 2 v 3 v 4 − 3 m 2 h 1 v 2 − 2 m 1 = 1 v 1 − 1

  79. Bases of independent Kähler parameters For each of the expansion we can choose a suitable set of independent Kähler parameters: NM + 2 Example: ( N, M ) = (3 , 2) 1) horizontal: ( ρ , b b 1 , b b 2 ; b c 1 , b c 2 , b c 3 ; τ , E ) ρ − 3 h 6 = 2 − 2 m 6 h 5 − 1 v 6 h 3 m 5 h 4 = 1 b c 3 v 5 m 3 E = m 1 + m 2 + m 3 m 4 h 2 = τ b 2 c 2 v 3 c 1 b v 4 − 3 m 2 h 1 b b 2 v 2 − 2 m 1 = 1 b b 1 v 1 − 1

  80. Bases of independent Kähler parameters For each of the expansion we can choose a suitable set of independent Kähler parameters: NM + 2 Example: ( N, M ) = (3 , 2) 1) horizontal: ( ρ , b b 1 , b b 2 ; b c 1 , b c 2 , b c 3 ; τ , E ) series expansion: ρ − b b 1 − b b 2 − → ∞ b b 1 − → ∞ 1 2 3 b b 2 − v 1 v 2 v 3 → ∞ f d b m 4 m 6 m 5 f b d v 4 v 5 v 6 c e a m 2 m 1 m 3 a c e v 1 v 2 v 3 1 2 3

  81. Bases of independent Kähler parameters For each of the expansion we can choose a suitable set of independent Kähler parameters: NM + 2 Example: ( N, M ) = (3 , 2) 1) horizontal: ( ρ , b b 1 , b b 2 ; b c 1 , b c 2 , b c 3 ; τ , E ) series expansion: ρ − b b 1 − b b 2 − → ∞ b b 1 − → ∞ 1 2 3 b b 2 − v 1 v 2 v 3 → ∞ gauge theory: U (2) × U (2) × U (2) f d b m 4 m 6 m 5 f b d v 4 v 5 v 6 c e a m 2 m 1 m 3 a c e v 1 v 2 v 3 1 2 3

  82. Bases of independent Kähler parameters For each of the expansion we can choose a suitable set of independent Kähler parameters: NM + 2 Example: ( N, M ) = (3 , 2) 1) horizontal: ( ρ , b b 1 , b b 2 ; b c 1 , b c 2 , b c 3 ; τ , E ) series expansion: ρ − b b 1 − b b 2 − → ∞ ρ b − 3 b 1 − → ∞ h 6 b b 2 − = → ∞ 2 − 2 gauge theory: U (2) × U (2) × U (2) m 6 h 5 − 1 c 1 ; b b 1 , b b 2 , b b 3 , b v 6 2) vertical: ( τ , b b 4 ; ρ , D ) b h 3 m 5 b 4 h 4 = 1 v 5 m 3 m 4 h 2 = τ 2 b b 3 v 3 b c 1 v 4 − 3 m 2 h 1 b b 2 v 2 − 2 m 1 = 1 b b 1 v 1 − 1 D = m 1 + m 4

  83. Bases of independent Kähler parameters For each of the expansion we can choose a suitable set of independent Kähler parameters: NM + 2 Example: ( N, M ) = (3 , 2) 1) horizontal: ( ρ , b b 1 , b b 2 ; b c 1 , b c 2 , b c 3 ; τ , E ) series expansion: ρ − b b 1 − b b 2 − 3 → ∞ b b 1 − → ∞ b h 6 b 2 b 2 − → ∞ m 6 gauge theory: U (2) × U (2) × U (2) h 5 1 c 1 ; b b 1 , b b 2 , b b 3 , b 2) vertical: ( τ , b b 4 ; ρ , D ) m 5 6 h 4 6 series expansion: τ − b c 1 − → ∞ m 4 5 h 3 b a c 2 − → ∞ b h 6 5 m 3 4 h 2 4 m 2 h 1 3 m 1 h 3 2 a 1

  84. Bases of independent Kähler parameters For each of the expansion we can choose a suitable set of independent Kähler parameters: NM + 2 Example: ( N, M ) = (3 , 2) 1) horizontal: ( ρ , b b 1 , b b 2 ; b c 1 , b c 2 , b c 3 ; τ , E ) series expansion: ρ − b b 1 − b b 2 − 3 → ∞ b b 1 − → ∞ b h 6 b 2 b 2 − → ∞ m 6 gauge theory: U (2) × U (2) × U (2) h 5 1 c 1 ; b b 1 , b b 2 , b b 3 , b 2) vertical: ( τ , b b 4 ; ρ , D ) m 5 6 h 4 6 series expansion: τ − b c 1 − → ∞ m 4 5 h 3 b a c 2 − → ∞ b h 6 5 m 3 gauge theory: U (3) × U (3) 4 h 2 4 m 2 h 1 3 m 1 h 3 2 a 1

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