Magnetic Monopoles and N = 2 super Yang Mills

Magnetic Monopoles and N = 2 super Yang Mills Andy Royston Texas A&M University MCA Montréal, July 25, 2017 1404.5616, 1404.7158, 1512.08923, 1512.08924 with G. Moore and D. Van den Bleeken; work in progress with D. Brennan and G. Moore lecture notes available at http://ckottke.ncf.edu/senworkshop/notes

Main Idea Recent developments in 4D, N = 2 SYM? wall crossing, defects, no-exotics,... Geometry of monopole moduli space Fredholm properties of Dirac-like operators L 2 cohomology,...

Main Idea Recent developments in 4D, N = 2 SYM Seiberg Witten BPS states semiclassical? wall crossing, defects, no-exotics,... Geometry of monopole moduli space Fredholm properties of Dirac-like operators L 2 cohomology,...

Outline Moduli spaces of (singular) monopoles Consequences of N = 2 susy and quantization The SW sc map and its predictions Asymptotic analysis

(Singular) monopoles YMH on R 1,3 : S ymh = 1 [(F g0 2 A, F A ) + (d A X, d A X)] R 1,3 simple compact Lie group G, algebra g Magnetic Monopoles: F A = 3 d A X on R 3 s.t. (bc ) : X = X γ m 2r + F A = γ m 2 sin θdθdφ + as r, where X t, a Cartan subalgebra, with basis {H I } γ m Λ cr t

(Singular) monopoles YMH on R 1,3 : S ymh = 1 [(F g0 2 A, F A ) + (d A X, d A X)] R 1,3 simple compact Lie group G, algebra g t Hooft defects: (bc 0 ): (P Hom(U(1), T )) X = P 2r + F A = P 2 sin θdθdφ + as r 0 t Hooft ( 78); Kapustin ( 05) Singular magnetic monopoles: F A = 3 d A X on R 3 s.t. (bc 0 ) and (bc ) hold Kronheimer ( 85); Cherkis, Kapustin ( 97)

Moduli space of (singular) monopoles M(P; γ m, X ) = space of such solutions / local g.t. s Pauly ( 98); Cherkis, Kapustin ( 97, 98); Kapustin, Witten ( 06); MRV ( 14); Foscolo ( 14)

Moduli space of (singular) monopoles M(γ m, X ) = space of such solutions / local g.t. s Weinberg ( 78); Taubes ( 83); Donaldson ( 84) Atiyah, Hitchin ( 88),...

Moduli space of (singular) monopoles M(P; γ m, X ) = space of such solutions / local g.t. s Hyperkähler Pauly ( 98); Cherkis, Kapustin ( 97, 98); Kapustin, Witten ( 06); MRV ( 14); Foscolo ( 14)

Moduli space of (singular) monopoles M(P; γ m, X ) = space of such solutions / local g.t. s Hyperkähler Pauly ( 98); Cherkis, Kapustin ( 97, 98); Kapustin, Witten ( 06); MRV ( 14); Foscolo ( 14) Dimension: 4 (# of fundamental constituents) Isometries: M : M : R 3 so(3) rot t so(3) rot t

Examples Vanilla Examples {1, 0,..., 0}, M 0 = {pt}, {nm} I = {2, 0,..., 0}, M 0 = Atiyah Hitchin m fold, {1, 1, 0,..., 0}, M 0 = Taub-NUT Defect examples SO(3) gauge theory: P = p 2 H, γ m = ñ m H, s.t. p, ñ m Z ñ m = 0, M = {pt}, ñ m = 1, M = Taub-NUT/Z p, ñ m = 2, p = 1, M = Dancer m fold

Embedding into N = 2 SYM S = S[A, ϕ, ψ a ] = 1 g 2 0 R 1,3 [(F A, F A ) + (d A ϕ, d A ϕ) + ] ϕ a complex Higgs field, ψ a, a = 1, 2, an SU(2) R doublet of Weyl fermions

Embedding into N = 2 SYM S = S[A, ϕ, ψ a ] = 1 g 2 0 R 1,3 [(F A, F A ) + (d A ϕ, d A ϕ) + ] ϕ a complex Higgs field, ψ a, a = 1, 2, an SU(2) R doublet of Weyl fermions Supersymmetry: δ ξ S = 0 δ ξ ϕ = 2iɛ ab ξ a ψ b, δ ξ ψ a = = Noether charges Q a

Embedding into N = 2 SYM S = S[A, ϕ, ψ a ] = 1 g 2 0 ϕ a complex Higgs field, R 1,3 [(F A, F A ) + (d A ϕ, d A ϕ) + ] ψ a, a = 1, 2, an SU(2) R doublet of Weyl fermions Supersymmetry: δ ξ S = 0 δ ξ ϕ = 2iɛ ab ξ a ψ b, δ ξ ψ a = = Noether charges Q a Algebra: {Q a, Q b } + δ a bh, {Q a, Q b } + ɛ ab Z H is the Hamiltonian Z = 2 (if g 0 2 S 2 A F A, ϕ) = 4πi (γ g m, ϕ ) (γ 0 2 e, ϕ ), central charge

Embedding into N = 2 SYM BPS bound: Set R ζ 1/2 Q + ζ 1/2 Q, ( ζ = 1) Then 0 {R, R } + = H + Re(ζ 1 Z) = H Re(ζ 1 Z)

Embedding into N = 2 SYM BPS bound: Set R ζ 1/2 Q + ζ 1/2 Q, ( ζ = 1) Then 0 {R, R } + = H + Re(ζ 1 Z) = H Re(ζ 1 Z) Saturated on fields A, ϕ =: ζ(y + ix), satisfying (F A ) ij = ɛ ijk (d A X) k, ( 3 d A 3 d A + ad(x) 2 )Y = 0, (F A ) i0 = (d A Y ) i What about ζ? no defects ( vanilla BPS f.c. s): maximize the bound: ζ = ζ van := Z/ Z

Quantum N = 2 SYM Generalities H (no defects) The Hilbert space of states H Lζ (defects) is a representation space for the (super)symmetry algebra. BPS states The subspace of BPS states H BPS H (vanilla BPS states) H BPS L ζ H Lζ (framed BPS states) consists of short representations on which ˆR is represented by 0. for a more detailed description...

Quantum N = 2 SYM A tale of two approaches A: Seiberg Witten B: semiclassical

Seiberg Witten approach (vanilla) Γ u B u u e.-values of ϕ

Seiberg Witten approach (vanilla) H BPS u B u u e.-values of ϕ (vanilla) BPS states: H BPS u = γ Γ u H BPS u,γ = ( com (H0 ) BPS ) u,γ γ Γ u

Seiberg Witten approach (vanilla) H BPS u B u u e.-values of ϕ (vanilla) BPS states: H BPS u = γ Γ u H BPS u,γ = ( com (H0 ) BPS ) u,γ γ Γ u (H 0) BPS u,γ mass: an so(3) rot su(2) R rep. space Z γ(u)

Seiberg Witten approach (vanilla) s 12 H BPS u,γ B u u e.-values of ϕ (vanilla) BPS states: H BPS u = γ Γ u H BPS u,γ = ( com (H0 ) BPS ) u,γ γ Γ u (H 0) BPS u,γ mass: an so(3) rot su(2) R rep. space Z γ(u) wall-crossing: (H 0) BPS u,γ 1 +γ 2 0 as u crosses W (γ 1, γ 2) B s 12 : γ 1 γ 2

Seiberg Witten approach (framed) Core-halo picture for framed BPS states Gaiotto, Moore, Neitzke ( 10) There is an analogous story for the space of framed BPS states, H BPS L ζ,u = γ H BPS L ζ,u,γ in terms of the core-halo picture. Framed BPS states also undergo wall-crossing.

Semiclassical approach Collective coordinate ansatz, quantization Manton ( 84) A(t, x) = A mono ( x; z m (t)) + g 0 δa(t, x) z m (t), m = 1,..., dim M, collective coordinates δa to be integrated out susy QM where states are { spinors on M (0, )-forms Gauntlett ( 93); Gauntlett, Harvey ( 94); Gauntlett, N. Kim, Park, Yi ( 99); Gauntlett, C. Kim, Lee, Yi ( 00); MRV ( 15)

Semiclassical approach Semiclassical framed BPS states H scbps P,X,Y,γ m,γ e := ker (γe) ( L D / Y ) 2 M(P;γ m,x )

Semiclassical approach Semiclassical framed BPS states H scbps P,X,Y,γ m,γ e := ker (γe) ( L D / Y ) 2 M(P;γ m,x )... or L 2 coho. of ig(y) (0,1)

Semiclassical approach Semiclassical framed BPS states H scbps P,X,Y,γ m,γ e := ker (γe) ( L D / Y ) 2 M(P;γ m,x )... or L 2 coho. of ig(y) (0,1) Semiclassical vanilla BPS states (H 0 ) scbps X,Y,γ m,γ e in terms of L 2 kernel on M 0

The sc SW Map Conjecture: H scbps P,X,Y,γ m,γ e = H BPS L ζ (P),u,γ

The sc SW Map Conjecture: H scbps P,X,Y,γ m,γ e = H BPS L ζ (P),u,γ provided: X = Im(ζ 1 a(u)) Y = Im(ζ 1 a D (u)) γ m γ e = γ { a(u), ad (u) are (dual) SW special coords. }

The sc SW Map Conjecture: H scbps P,X,Y,γ m,γ e = H BPS L ζ (P),u,γ provided: X = Im(ζ 1 a(u)) Y = Im(ζ 1 a D (u)) γ m γ e = γ { a(u), ad (u) are (dual) SW special coords. } consistent with X = X + and Y = 4π g 2 0 Y +, and predicts quantum corrections

Prediction 1: Generalized Sen from no-exotics a priori, (H 0 ) BPS u,γ and H BPS L,u,γ can be arbitrary su(2) R reps

Prediction 1: Generalized Sen from no-exotics a priori, (H 0 ) BPS u,γ... but always trivial! and H BPS L,u,γ can be arbitrary su(2) R reps

Prediction 1: Generalized Sen from no-exotics a priori, (H 0 ) BPS u,γ... but always trivial! and H BPS L,u,γ can be arbitrary su(2) R reps = no-exotics: This is always true on B Gaiotto, Moore, Neitzke ( 10); Chuang, Diaconescu, Manschot, Moore, Soibelman ( 13); Del Zotto, Sen ( 14); Cordova, Dumitrescu

Prediction 1: Generalized Sen from no-exotics a priori, (H 0 ) BPS u,γ... but always trivial! and H BPS L,u,γ can be arbitrary su(2) R reps = no-exotics: This is always true on B Semiclassical su(2) R use (0, )-forms: λ = q λ(0,q) ω := ω 3 cmplx. coords. ω ± := ω 1 ± iω 2 hol.-sympl. form Î 3 λ (0,q) = 1 2 (q N)λ(0,q), Î + λ = ω + λ Î λ = ι ω λ (dim M = 4N)

Prediction 2: Fredholmness and WCF for Dirac Conjecture: Dirac operators in the family /D Y0 M 0(γ m,x ) for (X, Y 0 ) W + t γ m fail to be Fredholm only if there are charges γ 1,2 = γ m,1,2 γ e,1,2 such that γ m,1 + γ m,2 = γ m, γ 1, γ 2 (γ e,2, γ m,1 ) (γ e,1, γ m,2 ) 0, ker L 2( /D Y0 M 0(γ m,1,x )) 0 and ker L 2( /D Y0 M 0(γ m,2,x )) 0, and (Y 0, γ m,1 ) + (γ e,1 γ m,1 (γ e,x ) (γ m,x ), X ) = 0. At these real co-dimension one walls in (X, Y 0 )-space, the kernel jumps in a way dictated by the (Kontsevich Soibelman) WCF.

Framed ex: G = SO(3), p = 1 2, γ h = 1 n e 200 1.0-2 -1 c -1-2 Y c -1-1 100 c 0 SW sc map 0.5 0 0.0 c 0 c 0-100 3 c 2-200 -200-100 0 100 200 2 c 1 W (γ h ) 1 B -0.5 c 2 3 2-1.0 0.0 0.2 0.4 0.6 0.8 1.0 c 1 scw (γ h ) 1 X

Framed ex: G = SO(3), p = 1 2, γ h = 1 n e 200 1.0-2 -1 c -1-2 Y c -1-1 100 c 0 SW sc map 0.5 0 0.0 c 0 c 0-100 3 c 2-200 -200-100 0 100 200 2 c 1 W (γ h ) 1 B -0.5 c 2 3 2-1.0 0.0 0.2 0.4 0.6 0.8 1.0 c 1 scw (γ h ) 1 X M = Taub NUT

Framed ex: G = SO(3), p = 1 2, γ h = 1 n e 200 1.0-2 -1 c -1-2 Y c -1-1 100 c 0 SW sc map 0.5 0 0.0 c 0 c 0-100 3 c 2-200 -200-100 0 100 200 2 c 1 W (γ h ) 1 B M = Taub NUT /D Y -0.5 c 2 3 2-1.0 0.0 0.2 0.4 0.6 0.8 1.0 c 1 scw (γ h ) Pope ( 77); Gauntlett, N. Kim, Park, Yi ( 99); Jante, Schroers ( 13); MRV ( 15) 1 X

Asymptotic Analysis Goal: Prove Fredholm and wall-crossing properties by utilizing techniques from the compactification on manifolds-with-corners approach. Kottke, Singer ( 15) Fritszch, Kottke, Melrose, Singer Current work with Brennan and Moore: asymptotic analysis of /D Y near boundary face corresponding to a generic 2-partition: γ m = γ m,1 + γ m,2 ds 2 (M 0 ) = ds 2 (M 1 0 ) + ds2 (M 2 0 ) + ds2 (TN) + O(ε 2 ) can reproduce location of walls, primitive WCF

Conclusions Summary compared semiclassical and Seiberg Witten descriptions of (framed) BPS states SW sc map allows one to translate no-exotics and wall-crossing into precise and nontrivial predictions about the geometry of M Outlook rigorous verification of these predictions requires L 2 control over forms, spinors on M 0, M technology that is currently being developed Fritszch, Kottke, Melrose, Singer; Bielawski

Seiberg Witten approach (framed) Γ L,u B u susy t Hooft defects: L ζ (P)

Seiberg Witten approach (framed) H BPS L,u B u susy t Hooft defects: L ζ (P) framed BPS states: H BPS u = γ H BPS L,u,γ Gaiotto, Moore, Neitzke ( 10)

Seiberg Witten approach (framed) H BPS L,u B u susy t Hooft defects: L ζ (P) framed BPS states: H BPS u = γ H BPS L,u,γ an so(3)rot su(2) R rep. space mass: Re [ ζ 1 Z γ(u) ] H BPS L,u,γ Gaiotto, Moore, Neitzke ( 10)

Seiberg Witten approach (framed) s h H BPS L,u,γ B u susy t Hooft defects: L ζ (P) s h : framed BPS states: H BPS u = γ H BPS L,u,γ γ h γ h γ h γ h H BPS L,u,γ an so(3)rot su(2) R rep. space mass: Re [ ζ 1 Z γ(u) ] framed wall-crossing at W (γ h ) via core-halo picture γ h γ h γ h γ c γ h γ h γ h

Seiberg Witten approach (framed) s h H BPS L,u,γ B u γ h γ h susy t Hooft defects: L ζ (P) γ h s h : γ h framed BPS states: H BPS u H BPS L,u,γ = γ H BPS L,u,γ an so(3)rot su(2) R rep. space mass: Re [ ζ 1 Z γ(u) ] γ h γ c γ h framed wall-crossing at W (γ h ) via core-halo picture γ h γ h γ h γ h

Embedding into N = 2 SYM S = 1 g 2 0 R 1,3 { (F A, F A ) + (d A ϕ, d A ϕ) 1 ([ϕ, ϕ], [ϕ, ϕ])+ 4 i [ 2( ψ a, /D + (A)ψ a ) + ɛ ab (ψ a, [ψ b, ϕ]) + ɛ ab ( ψ a, [ ψ b, ϕ]) ] 1 } ϕ a complex Higgs field, ψ a, a = 1, 2, an SU(2) R doublet of Weyl fermions Supersymmetry: δ ξ S = 0 δ ξ ϕ = 2iɛ ab ξ a ψ b, δ ξ ψ a = = Noether charges Q a Algebra: {Q a, Q b } + δ a bh, {Q a, Q b } + ɛ ab Z H is the Hamiltonian Z = 2 (if g 0 2 S 2 A F A, ϕ) = 4πi (γ g m, ϕ ) (γ 0 2 e, ϕ ), central charge

Framed ex: G = SO(3), p = 1 2, n m = 1 200 1.0-2 -1 c -1-2 Y c -1-1 100 c 0 SW sc map 0.5 0 0.0 c 0 c 0-100 3 c 2-200 -200-100 0 100 200 2 c 1 W (γ h ) 1 B -0.5 c 2 3 2-1.0 0.0 0.2 0.4 0.6 0.8 1.0 c 1 scw (γ h ) 1 X

Framed ex: G = SO(3), p = 1 2, n m = 1 1.0-2 4-1 -2 Y c -1-1 c 0 c -1 0.5 2 SW sc map 0 0.0 c 0-2 3 c 0-0.5-4 c 2 c 1 1 2-4 -2 0 2 4 W (γ h ) B c 2 3 2-1.0 0.0 0.2 0.4 0.6 0.8 1.0 c 1 scw (γ h ) 1 X

Framed ex: G = SO(3), p = 1 2, n m = 1 200 1.0-2 -1 c -1-2 Y c -1-1 100 c 0 SW sc map 0.5 0 0.0 c 0 c 0-100 3 c 2-200 -200-100 0 100 200 2 c 1 W (γ h ) 1 B -0.5 c 2 3 2-1.0 0.0 0.2 0.4 0.6 0.8 1.0 c 1 scw (γ h ) 1 X M = Taub NUT

Framed ex: G = SO(3), p = 1 2, n m = 1 200 1.0-2 -1 c -1-2 Y c -1-1 100 c 0 SW sc map 0.5 0 0.0 c 0 c 0-100 3 c 2-200 -200-100 0 100 200 2 c 1 W (γ h ) 1 B M = Taub NUT /D Y -0.5 c 2 3 2-1.0 0.0 0.2 0.4 0.6 0.8 1.0 c 1 scw (γ h ) Pope ( 77); Gauntlett, N. Kim, Park, Yi ( 99); Jante, Schroers ( 13); MRV ( 15) 1 X