Stability data, irregular connections and tropical curves

Transcription

1 Stability data, irregular connections and tropical curves Mario Garcia-Fernandez Instituto de iencias Matemáticas (Madrid) Barcelona Mathematical Days 7 November 2014 Joint work with S. Filippini and J. Stoppa, arxiv: MGF (IMAT) BMD Barcelona, 7 November 1 / 14

2 Hitchin systems, stability data and irregular connections Hitchin systems Spaces of stability Irregular connections A) Hitchin integrable systems (Higgs bundles) B) Spaces of stability conditions (à la Bridgeland) ) Irregular connections on P 1 (after P. Boalch) MGF (IMAT) BMD Barcelona, 7 November 2 / 14

3 Hitchin systems Riemann surface, genus g(). SL(2, )-Higgs bundle: (E, φ), E holomorphic vector bundle, Λ 2 E = O, φ: E E K holomorphic, tr φ = 0. Theorem (Hitchin 86): the moduli space M Dol of (E, φ) s is a completely integrable, algebraic, Hamiltonian system. Hitchin map: M Dol B = H 0 (K 2 ): [(E, φ)] det φ, with fibres abelian varieties. Observation 1 (Diaconescu-Donagi-Dijkgraaf-Hofman-Pantev 05): B parameterises a family of non-compact alabi-yau threefolds. Take V := K 1/2 K 1/2 and consider p : S 2 V (rank 3) with the natural map det: S 2 V Hom(V, V ) K 2 : u det u. Then, X b = {det = p b} S 2 V, b H 0 (K 2 ) Locally v 1 v 2 y 2 = det φ (affine conic fibration over ). MGF (IMAT) BMD Barcelona, 7 November 3 / 14

4 Hitchin systems Riemann surface, genus g(). SL(2, )-Higgs bundle: (E, φ), E holomorphic vector bundle, Λ 2 E = O, φ: E E K holomorphic, tr φ = 0. Theorem (Hitchin 86): the moduli space M Dol of (E, φ) s is a completely integrable, algebraic, Hamiltonian system. Hitchin map: M Dol B = H 0 (K 2 ): [(E, φ)] det φ, with fibres abelian varieties. Observation 1 (Diaconescu-Donagi-Dijkgraaf-Hofman-Pantev 05): B parameterises a family of non-compact alabi-yau threefolds. Take V := K 1/2 K 1/2 and consider p : S 2 V (rank 3) with the natural map det: S 2 V Hom(V, V ) K 2 : u det u. Then, X b = {det = p b} S 2 V, b H 0 (K 2 ) Locally v 1 v 2 y 2 = det φ (affine conic fibration over ). MGF (IMAT) BMD Barcelona, 7 November 3 / 14

5 Hitchin systems Riemann surface, genus g(). SL(2, )-Higgs bundle: (E, φ), E holomorphic vector bundle, Λ 2 E = O, φ: E E K holomorphic, tr φ = 0. Theorem (Hitchin 86): the moduli space M Dol of (E, φ) s is a completely integrable, algebraic, Hamiltonian system. Hitchin map: M Dol B = H 0 (K 2 ): [(E, φ)] det φ, with fibres abelian varieties. Observation 1 (Diaconescu-Donagi-Dijkgraaf-Hofman-Pantev 05): B parameterises a family of non-compact alabi-yau threefolds. Take V := K 1/2 K 1/2 and consider p : S 2 V (rank 3) with the natural map det: S 2 V Hom(V, V ) K 2 : u det u. Then, X b = {det = p b} S 2 V, b H 0 (K 2 ) Locally v 1 v 2 y 2 = det φ (affine conic fibration over ). MGF (IMAT) BMD Barcelona, 7 November 3 / 14

6 Hitchin systems Riemann surface, genus g(). SL(2, )-Higgs bundle: (E, φ), E holomorphic vector bundle, Λ 2 E = O, φ: E E K holomorphic, tr φ = 0. Theorem (Hitchin 86): the moduli space M Dol of (E, φ) s is a completely integrable, algebraic, Hamiltonian system. Hitchin map: M Dol B = H 0 (K 2 ): [(E, φ)] det φ, with fibres abelian varieties. Observation 1 (Diaconescu-Donagi-Dijkgraaf-Hofman-Pantev 05): B parameterises a family of non-compact alabi-yau threefolds. Take V := K 1/2 K 1/2 and consider p : S 2 V (rank 3) with the natural map det: S 2 V Hom(V, V ) K 2 : u det u. Then, X b = {det = p b} S 2 V, b H 0 (K 2 ) Locally v 1 v 2 y 2 = det φ (affine conic fibration over ). MGF (IMAT) BMD Barcelona, 7 November 3 / 14

7 Stability conditions from quadratic differentials X b = {det = p b} S 2 V, b H 0 (K 2 ) Locally v 1 v 2 y 2 = det φ (affine conic fibration over ). Observation 2 (Smith 13): X b can be endowed with a symplectic structure, which only depends on g(): B parameterises alabi-yau structures in a fixed symplectic manifold Y g(). Observation 3 (Thomas 01): the holomorphic volume form on a alabi-yau manifold determines a stability condition for Lagrangians. Aside: Thomas conjectures that L is stable iff it contains a special Lagrangian on its Hamiltonian isotopy class. onclusion: The base B of the Hitchin system (actually varying ), should be a space of stability conditions for Lagrangians in Y g(). MGF (IMAT) BMD Barcelona, 7 November 4 / 14

8 Stability conditions from quadratic differentials X b = {det = p b} S 2 V, b H 0 (K 2 ) Locally v 1 v 2 y 2 = det φ (affine conic fibration over ). Observation 2 (Smith 13): X b can be endowed with a symplectic structure, which only depends on g(): B parameterises alabi-yau structures in a fixed symplectic manifold Y g(). Observation 3 (Thomas 01): the holomorphic volume form on a alabi-yau manifold determines a stability condition for Lagrangians. Aside: Thomas conjectures that L is stable iff it contains a special Lagrangian on its Hamiltonian isotopy class. onclusion: The base B of the Hitchin system (actually varying ), should be a space of stability conditions for Lagrangians in Y g(). MGF (IMAT) BMD Barcelona, 7 November 4 / 14

9 Stability conditions from quadratic differentials X b = {det = p b} S 2 V, b H 0 (K 2 ) Locally v 1 v 2 y 2 = det φ (affine conic fibration over ). Observation 2 (Smith 13): X b can be endowed with a symplectic structure, which only depends on g(): B parameterises alabi-yau structures in a fixed symplectic manifold Y g(). Observation 3 (Thomas 01): the holomorphic volume form on a alabi-yau manifold determines a stability condition for Lagrangians. Aside: Thomas conjectures that L is stable iff it contains a special Lagrangian on its Hamiltonian isotopy class. onclusion: The base B of the Hitchin system (actually varying ), should be a space of stability conditions for Lagrangians in Y g(). MGF (IMAT) BMD Barcelona, 7 November 4 / 14

10 Stability conditions... and quivers Theorem (Bridgeland-Smith 13): for suitable irregular Hitchin systems there exists a triangulated category D with combinatorial description (quivers with potential), such that the base H 0 (K 2 (D)) (varying ) is a connected component of Stab(D). Aside: (A, Z) Stab(D) is given by an abelian subcategory A and an homomorphism Z : K(A) Z such that Z(K >0 (A)) H, plus conditions. MGF (IMAT) BMD Barcelona, 7 November 5 / 14

11 Stability conditions... and quivers Theorem (Bridgeland-Smith 13): for suitable irregular Hitchin systems there exists a triangulated category D with combinatorial description (quivers with potential), such that the base H 0 (K 2 (D)) (varying ) is a connected component of Stab(D). Aside: (A, Z) Stab(D) is given by an abelian subcategory A and an homomorphism Z : K(A) Z such that Z(K >0 (A)) H, plus conditions. Example: = P 1, φ with singularity of order 4 at, nilpotent leading part at the pole. Generically det φ H 0 (K 2 (7 )), with three simple zeroes. Let R be the path algebra of. Then, A = Mod fin (R) and D = D b (Mod fin ( R)) ( R completed (Ginzburg dg) algebra). MGF (IMAT) BMD Barcelona, 7 November 6 / 14

12 What is all this good for? 1) Geometric description of Stab(D) for amenable (combinatorial) D. 2) ombinatorial description of Fukaya. 3) (onjectural) description of hyperkähler geometry of M Dol! Theorem (Hitchin 86): the moduli space M Dol has a compatible hyperkähler metric g. onjecture (Gaiotto Moore Neitzke 09): the category D recovers the hyperkähler metric g. Physical Theorem (BPS spectrum of susy Gauge Theory Donaldson-Thomas invariants for D.) MGF (IMAT) BMD Barcelona, 7 November 7 / 14

13 What is all this good for? 1) Geometric description of Stab(D) for amenable (combinatorial) D. 2) ombinatorial description of Fukaya. 3) (onjectural) description of hyperkähler geometry of M Dol! Theorem (Hitchin 86): the moduli space M Dol has a compatible hyperkähler metric g. onjecture (Gaiotto Moore Neitzke 09): the category D recovers the hyperkähler metric g. Physical Theorem (BPS spectrum of susy Gauge Theory Donaldson-Thomas invariants for D.) MGF (IMAT) BMD Barcelona, 7 November 7 / 14

14 What is all this good for? 1) Geometric description of Stab(D) for amenable (combinatorial) D. 2) ombinatorial description of Fukaya. 3) (onjectural) description of hyperkähler geometry of M Dol! Theorem (Hitchin 86): the moduli space M Dol has a compatible hyperkähler metric g. onjecture (Gaiotto Moore Neitzke 09): the category D recovers the hyperkähler metric g. Physical Theorem (BPS spectrum of susy Gauge Theory Donaldson-Thomas invariants for D.) MGF (IMAT) BMD Barcelona, 7 November 7 / 14

15 What is all this good for? 1) Geometric description of Stab(D) for amenable (combinatorial) D. 2) ombinatorial description of Fukaya. 3) (onjectural) description of hyperkähler geometry of M Dol! Theorem (Hitchin 86): the moduli space M Dol has a compatible hyperkähler metric g. onjecture (Gaiotto Moore Neitzke 09): the category D recovers the hyperkähler metric g. Physical Theorem (BPS spectrum of susy Gauge Theory Donaldson-Thomas invariants for D.) MGF (IMAT) BMD Barcelona, 7 November 7 / 14

16 What is all this good for? 1) Geometric description of Stab(D) for amenable (combinatorial) D. 2) ombinatorial description of Fukaya. 3) (onjectural) description of hyperkähler geometry of M Dol! Theorem (Hitchin 86): the moduli space M Dol has a compatible hyperkähler metric g. onjecture (Gaiotto Moore Neitzke 09): the category D recovers the hyperkähler metric g. Physical Theorem (BPS spectrum of susy Gauge Theory Donaldson-Thomas invariants for D.) MGF (IMAT) BMD Barcelona, 7 November 7 / 14

17 What is all this good for? 1) Geometric description of Stab(D) for amenable (combinatorial) D. 2) ombinatorial description of Fukaya. 3) (onjectural) description of hyperkähler geometry of M Dol! Theorem (Hitchin 86): the moduli space M Dol has a compatible hyperkähler metric g. onjecture (Gaiotto Moore Neitzke 09): the category D recovers the hyperkähler metric g. Physical Theorem (BPS spectrum of susy Gauge Theory Donaldson-Thomas invariants for D.) MGF (IMAT) BMD Barcelona, 7 November 7 / 14

18 Gaiotto Moore Neitzke coordinates Observation 4: the twistor family of holomorphic symplectic structures recovers g. ϖ(ζ) = i 2ζ (ω J + iω K ) + ω I iζ 2 (ω J iω K ), ζ P 1 IDEA (GMN 09): find family X γ (, ζ): M of holomorphic Darboux coordinates for ϖ(ζ) solving an integral equation for each x M Dol, over b B: X γ (x, ζ) = Xγ sf (x, ζ) exp γ K(A b ) Ω(γ, b) γ, γ dζ ζ + ζ R <0Z b (γ ) ζ ζ ζ log(1 X γ (x, ζ )) i) (A b, Z b ) Stab(D) ii) Ω(γ, b) Q (if semistable = stable, χ(m s γ )), iii) K(A b ) = H 1 (M Dol b, Z) pairing γ, γ, angular coordinates θ γ iv) X sf γ (x, ζ) = exp(ζ 1 Z b (γ) + iθ γ + ζz b (γ)) MGF (IMAT) BMD Barcelona, 7 November 8 / 14

19 Gaiotto Moore Neitzke coordinates Observation 4: the twistor family of holomorphic symplectic structures recovers g. ϖ(ζ) = i 2ζ (ω J + iω K ) + ω I iζ 2 (ω J iω K ), ζ P 1 IDEA (GMN 09): find family X γ (, ζ): M of holomorphic Darboux coordinates for ϖ(ζ) solving an integral equation for each x M Dol, over b B: X γ (x, ζ) = Xγ sf (x, ζ) exp γ K(A b ) Ω(γ, b) γ, γ dζ ζ + ζ R <0Z b (γ ) ζ ζ ζ log(1 X γ (x, ζ )) i) (A b, Z b ) Stab(D) ii) Ω(γ, b) Q (if semistable = stable, χ(m s γ )), iii) K(A b ) = H 1 (M Dol b, Z) pairing γ, γ, angular coordinates θ γ iv) X sf γ (x, ζ) = exp(ζ 1 Z b (γ) + iθ γ + ζz b (γ)) MGF (IMAT) BMD Barcelona, 7 November 8 / 14

20 Gaiotto Moore Neitzke coordinates Observation 4: the twistor family of holomorphic symplectic structures recovers g. ϖ(ζ) = i 2ζ (ω J + iω K ) + ω I iζ 2 (ω J iω K ), ζ P 1 IDEA (GMN 09): find family X γ (, ζ): M of holomorphic Darboux coordinates for ϖ(ζ) solving an integral equation for each x M Dol, over b B: X γ (x, ζ) = Xγ sf (x, ζ) exp γ K(A b ) Ω(γ, b) γ, γ dζ ζ + ζ R <0Z b (γ ) ζ ζ ζ log(1 X γ (x, ζ )) i) (A b, Z b ) Stab(D) ii) Ω(γ, b) Q (if semistable = stable, χ(m s γ )), iii) K(A b ) = H 1 (M Dol b, Z) pairing γ, γ, angular coordinates θ γ iv) X sf γ (x, ζ) = exp(ζ 1 Z b (γ) + iθ γ + ζz b (γ)) MGF (IMAT) BMD Barcelona, 7 November 8 / 14

21 Gaiotto Moore Neitzke coordinates Observation 4: the twistor family of holomorphic symplectic structures recovers g. ϖ(ζ) = i 2ζ (ω J + iω K ) + ω I iζ 2 (ω J iω K ), ζ P 1 IDEA (GMN 09): find family X γ (, ζ): M of holomorphic Darboux coordinates for ϖ(ζ) solving an integral equation for each x M Dol, over b B: X γ (x, ζ) = Xγ sf (x, ζ) exp γ K(A b ) Ω(γ, b) γ, γ dζ ζ + ζ R <0Z b (γ ) ζ ζ ζ log(1 X γ (x, ζ )) i) (A b, Z b ) Stab(D) ii) Ω(γ, b) Q (if semistable = stable, χ(m s γ )), iii) K(A b ) = H 1 (M Dol b, Z) pairing γ, γ, angular coordinates θ γ iv) X sf γ (x, ζ) = exp(ζ 1 Z b (γ) + iθ γ + ζz b (γ)) MGF (IMAT) BMD Barcelona, 7 November 8 / 14

22 Gaiotto Moore Neitzke coordinates Observation 4: the twistor family of holomorphic symplectic structures recovers g. ϖ(ζ) = i 2ζ (ω J + iω K ) + ω I iζ 2 (ω J iω K ), ζ P 1 IDEA (GMN 09): find family X γ (, ζ): M of holomorphic Darboux coordinates for ϖ(ζ) solving an integral equation for each x M Dol, over b B: X γ (x, ζ) = Xγ sf (x, ζ) exp γ K(A b ) Ω(γ, b) γ, γ dζ ζ + ζ R <0Z b (γ ) ζ ζ ζ log(1 X γ (x, ζ )) i) (A b, Z b ) Stab(D) ii) Ω(γ, b) Q (if semistable = stable, χ(m s γ )), iii) K(A b ) = H 1 (M Dol b, Z) pairing γ, γ, angular coordinates θ γ iv) X sf γ (x, ζ) = exp(ζ 1 Z b (γ) + iθ γ + ζz b (γ)) MGF (IMAT) BMD Barcelona, 7 November 8 / 14

23 Gaiotto Moore Neitzke coordinates IDEA (GMN 09): find family X γ (, ζ): M of holomorphic Darboux coordinates for ϖ(ζ) solving an integral equation for each x M Dol, over b B: X γ (x, ζ) = Xγ sf (x, ζ) exp γ K(A b ) Important Remarks: Ω(γ, b) γ, γ dζ ζ + ζ R <0Z b (γ ) ζ ζ ζ log(1 X γ (x, ζ )) A) Xγ sf determine g sf HyperKähler, away from singular fibres, B) X γ instanton correct Xγ sf ( Gross-Siebert), ) X γ discontinuous along R <0 Z b (γ ) with Ω(γ, b) 0. D) Kontsevich-Soibelman wall-crossing formula ϖ(ζ) = i,j d log X γ i (ζ) d log X γj (ζ) smooth E) Ω(γ, b) = Ω( γ, b) (symmetric), by Bridgeland axioms. MGF (IMAT) BMD Barcelona, 7 November 9 / 14

24 Gaiotto Moore Neitzke coordinates IDEA (GMN 09): find family X γ (, ζ): M of holomorphic Darboux coordinates for ϖ(ζ) solving an integral equation for each x M Dol, over b B: X γ (x, ζ) = Xγ sf (x, ζ) exp γ K(A b ) Important Remarks: Ω(γ, b) γ, γ dζ ζ + ζ R <0Z b (γ ) ζ ζ ζ log(1 X γ (x, ζ )) A) Xγ sf determine g sf HyperKähler, away from singular fibres, B) X γ instanton correct Xγ sf ( Gross-Siebert), ) X γ discontinuous along R <0 Z b (γ ) with Ω(γ, b) 0. D) Kontsevich-Soibelman wall-crossing formula ϖ(ζ) = i,j d log X γ i (ζ) d log X γj (ζ) smooth E) Ω(γ, b) = Ω( γ, b) (symmetric), by Bridgeland axioms. MGF (IMAT) BMD Barcelona, 7 November 9 / 14

25 Gaiotto Moore Neitzke coordinates IDEA (GMN 09): find family X γ (, ζ): M of holomorphic Darboux coordinates for ϖ(ζ) solving an integral equation for each x M Dol, over b B: X γ (x, ζ) = Xγ sf (x, ζ) exp γ K(A b ) Important Remarks: Ω(γ, b) γ, γ dζ ζ + ζ R <0Z b (γ ) ζ ζ ζ log(1 X γ (x, ζ )) A) Xγ sf determine g sf HyperKähler, away from singular fibres, B) X γ instanton correct Xγ sf ( Gross-Siebert), ) X γ discontinuous along R <0 Z b (γ ) with Ω(γ, b) 0. D) Kontsevich-Soibelman wall-crossing formula ϖ(ζ) = i,j d log X γ i (ζ) d log X γj (ζ) smooth E) Ω(γ, b) = Ω( γ, b) (symmetric), by Bridgeland axioms. MGF (IMAT) BMD Barcelona, 7 November 9 / 14

26 Gaiotto Moore Neitzke coordinates IDEA (GMN 09): find family X γ (, ζ): M of holomorphic Darboux coordinates for ϖ(ζ) solving an integral equation for each x M Dol, over b B: X γ (x, ζ) = Xγ sf (x, ζ) exp γ K(A b ) Important Remarks: Ω(γ, b) γ, γ dζ ζ + ζ R <0Z b (γ ) ζ ζ ζ log(1 X γ (x, ζ )) A) Xγ sf determine g sf HyperKähler, away from singular fibres, B) X γ instanton correct Xγ sf ( Gross-Siebert), ) X γ discontinuous along R <0 Z b (γ ) with Ω(γ, b) 0. D) Kontsevich-Soibelman wall-crossing formula ϖ(ζ) = i,j d log X γ i (ζ) d log X γj (ζ) smooth E) Ω(γ, b) = Ω( γ, b) (symmetric), by Bridgeland axioms. MGF (IMAT) BMD Barcelona, 7 November 9 / 14

27 Gaiotto Moore Neitzke coordinates IDEA (GMN 09): find family X γ (, ζ): M of holomorphic Darboux coordinates for ϖ(ζ) solving an integral equation for each x M Dol, over b B: X γ (x, ζ) = Xγ sf (x, ζ) exp γ K(A b ) Important Remarks: Ω(γ, b) γ, γ dζ ζ + ζ R <0Z b (γ ) ζ ζ ζ log(1 X γ (x, ζ )) A) Xγ sf determine g sf HyperKähler, away from singular fibres, B) X γ instanton correct Xγ sf ( Gross-Siebert), ) X γ discontinuous along R <0 Z b (γ ) with Ω(γ, b) 0. D) Kontsevich-Soibelman wall-crossing formula ϖ(ζ) = i,j d log X γ i (ζ) d log X γj (ζ) smooth E) Ω(γ, b) = Ω( γ, b) (symmetric), by Bridgeland axioms. MGF (IMAT) BMD Barcelona, 7 November 9 / 14

28 A very rough approximation of GMN coordinates (Γ,, ) a symplectic lattice and V Γ a cone. onsider g V the corresponding commutative associative algebra over e γ e γ = e γ+γ, γ, γ V Fix Z Hom(Γ, Z). onsider maps X : Aut(ĝ V ) (completion), as unknowns for the integral equation: X (ζ)(e γ ) = X sf (ζ)(e γ ) exp Ω(γ, Z) γ, γ γ Γ R <0Z(γ ) dζ ζ ζ + ζ ζ ζ log (1 X (ζ )(e γ ) Remarks: A) X sf (ζ)(e γ ) = exp Aut(ĝV )(ζ 1 Z + ζz)(e γ ) = exp(ζ 1 Z(γ) + ζz(γ))e γ B) exp, log formal series, ) Ω(γ, Z) Q, γ Γ collection of numbers. Analogy: X (ζ) complexified diffeomorphism of fiber M Dol b. MGF (IMAT) BMD Barcelona, 7 November 10 / 14

29 A very rough approximation of GMN coordinates (Γ,, ) a symplectic lattice and V Γ a cone. onsider g V the corresponding commutative associative algebra over e γ e γ = e γ+γ, γ, γ V Fix Z Hom(Γ, Z). onsider maps X : Aut(ĝ V ) (completion), as unknowns for the integral equation: X (ζ)(e γ ) = X sf (ζ)(e γ ) exp Ω(γ, Z) γ, γ γ Γ R <0Z(γ ) dζ ζ ζ + ζ ζ ζ log (1 X (ζ )(e γ ) Remarks: A) X sf (ζ)(e γ ) = exp Aut(ĝV )(ζ 1 Z + ζz)(e γ ) = exp(ζ 1 Z(γ) + ζz(γ))e γ B) exp, log formal series, ) Ω(γ, Z) Q, γ Γ collection of numbers. Analogy: X (ζ) complexified diffeomorphism of fiber M Dol b. MGF (IMAT) BMD Barcelona, 7 November 10 / 14

30 Irregular connections on P 1 X (ζ)(e γ ) = X sf (ζ)(e γ ) exp Ω(γ, Z) γ, γ γ Γ R <0Z(γ ) dζ ζ ζ + ζ ζ ζ log (1 X (ζ )(e γ ) ASSUMPTION: Ω positive (Ω(γ, Z) = 0 if γ / V ). Observation 5 (GMN,, Filippini, Stoppa): The integral equation has a solution X : Aut(ĝ V ) by iteration on X sf (Z-dependent) which defines a connection (Z) on P 1 with order 2 poles at {0, } ( ζ X )X 1 (ζ) = ( ζ 2 A 1 (Z) + ζ 1 A 0 (Z) + A 1 (Z) ) dz Ω 1 (Der(ĝ V )) A) Well defined limit g(z) = lim ζ 0 X (X sf ) 1 (ζ) Aut(ĝ V ) (frame) B) Ω(γ, Z) Q satisfy KS wall-crossing formulae iff (Z) is isomonodromic. MGF (IMAT) BMD Barcelona, 7 November 11 / 14

31 Irregular connections on P 1 X (ζ)(e γ ) = X sf (ζ)(e γ ) exp Ω(γ, Z) γ, γ γ Γ R <0Z(γ ) dζ ζ ζ + ζ ζ ζ log (1 X (ζ )(e γ ) ASSUMPTION: Ω positive (Ω(γ, Z) = 0 if γ / V ). Observation 5 (GMN,, Filippini, Stoppa): The integral equation has a solution X : Aut(ĝ V ) by iteration on X sf (Z-dependent) which defines a connection (Z) on P 1 with order 2 poles at {0, } ( ζ X )X 1 (ζ) = ( ζ 2 A 1 (Z) + ζ 1 A 0 (Z) + A 1 (Z) ) dz Ω 1 (Der(ĝ V )) A) Well defined limit g(z) = lim ζ 0 X (X sf ) 1 (ζ) Aut(ĝ V ) (frame) B) Ω(γ, Z) Q satisfy KS wall-crossing formulae iff (Z) is isomonodromic. MGF (IMAT) BMD Barcelona, 7 November 11 / 14

32 Irregular connections on P 1 X (ζ)(e γ ) = X sf (ζ)(e γ ) exp Ω(γ, Z) γ, γ γ Γ R <0Z(γ ) dζ ζ ζ + ζ ζ ζ log (1 X (ζ )(e γ ) ASSUMPTION: Ω positive (Ω(γ, Z) = 0 if γ / V ). Observation 5 (GMN,, Filippini, Stoppa): The integral equation has a solution X : Aut(ĝ V ) by iteration on X sf (Z-dependent) which defines a connection (Z) on P 1 with order 2 poles at {0, } ( ζ X )X 1 (ζ) = ( ζ 2 A 1 (Z) + ζ 1 A 0 (Z) + A 1 (Z) ) dz Ω 1 (Der(ĝ V )) A) Well defined limit g(z) = lim ζ 0 X (X sf ) 1 (ζ) Aut(ĝ V ) (frame) B) Ω(γ, Z) Q satisfy KS wall-crossing formulae iff (Z) is isomonodromic. MGF (IMAT) BMD Barcelona, 7 November 11 / 14

33 onformal vs large limit g(z)(e γ ) = e γ exp γ, W T GT 0 (Z) T GT 0 dζ (Z) = R <0Z(γ T ) ζ exp(ζ 1 Z(γT ) + ζ Z(γ T ))e γt G T (ζ, Z) T dζ ζ + ζ G T (ζ, Z) = ζ ζ ζ exp(ζ 1 Z(γT ) + ζ Z(γ T ))e γt G T (ζ, Z) T R <0Z(γ T ) W T = Γ-valued weight, associated to Γ-decorated rooted tree T. Theorem (, Filippini, Stoppa): onsider R > 0 a real constant After change of variable t = R 1 ζ there exists a well-defined limit ( Z lim R 0 g(rz) (RZ)g(RZ) 1 = d t 2 + f (Z) ) dt, t where f (Z) is Joyce s holomorphic generating function for DT invariants. As R, the instanton contribution G (t, RZ) as Z crosses a wall in Hom(Γ, Z) encodes counts of tropical curves immersed in R 2. MGF (IMAT) BMD Barcelona, 7 November 12 / 14

34 onformal vs large limit g(z)(e γ ) = e γ exp γ, W T GT 0 (Z) T GT 0 dζ (Z) = R <0Z(γ T ) ζ exp(ζ 1 Z(γT ) + ζ Z(γ T ))e γt G T (ζ, Z) T dζ ζ + ζ G T (ζ, Z) = ζ ζ ζ exp(ζ 1 Z(γT ) + ζ Z(γ T ))e γt G T (ζ, Z) T R <0Z(γ T ) W T = Γ-valued weight, associated to Γ-decorated rooted tree T. Theorem (, Filippini, Stoppa): onsider R > 0 a real constant After change of variable t = R 1 ζ there exists a well-defined limit ( Z lim R 0 g(rz) (RZ)g(RZ) 1 = d t 2 + f (Z) ) dt, t where f (Z) is Joyce s holomorphic generating function for DT invariants. As R, the instanton contribution G (t, RZ) as Z crosses a wall in Hom(Γ, Z) encodes counts of tropical curves immersed in R 2. MGF (IMAT) BMD Barcelona, 7 November 12 / 14

35 Perspectives: Symmetric case, Link with geometric setup, onformal limit as coordinates in submanifold of opers (Gaiotto), MGF (IMAT) BMD Barcelona, 7 November 13 / 14

36 GRÀIES! MGF (IMAT) BMD Barcelona, 7 November 14 / 14