Semiclassical Framed BPS States

Transcription

1 Semiclassical Framed BPS States Andy Royston Rutgers University String-Math, Bonn, July 18, 2012 Based on work with Dieter van den Bleeken and Greg Moore

2 Motivation framed BPS state 1 : a BPS state in N = 2 SYM with line operator defects why study them? a simple conceptual approach to KS wall-crossing formula compute Wilson- t Hooft operator vev s exactly why study them semiclassically? 1. define quantities of interest in terms of differentio-geometric structures use mathematical tools to obtain interesting physics results physical mathematics 2. translate recent physically motivated conjectures on N = 2 spectrum into interesting new conjectures about Dirac operators on hyperkähler manifolds mathematical physics 1 Gaiotto, Moore and Neitzke (2010)

3 Outline Motivation Background N = 2 semiclassical methodology, line operators Application and Examples: physical mathematics Protected Spin Characters and their Positivity Conjectures Examples: mathematical physics

4 N = 2 SYM on R 1,3, gauge group G, no hypers UV description (A, ϕ, ψ A ) Ω 1 (R 1,3, g) Ω 0 (R 1,3, g C ) Ω 0 (S + (R 1,3 ), g C ) (1, 1, 2) of SU(2) R { S = 1 2g 2 0 F 2 + Dϕ [ϕ, ϕ] 2 + } IR description à la Seiberg Witten Coulomb branch B[u i ]; e.m. charge lattice Γ, γ 1, γ 2 Z U(1) rnk g abelian N = 2 v.m. s (a J, A J ) a D,I F a I, Z γ=(γm,γ e) = a D,I γ I m + a I γ e,i, M BPS γ = Z γ SW solution: ( a I (u), a D,I (u) ) F(a)

5 Semiclassical regime and UV IR connection Identifying low energy quantities let g = t α Φ g α; basis {H αi ; E α }, H αi simple co-roots A = A I H αi +, ϕ = a I H αi +, Γ = Γ m Γ e = Λ cr Λ wt with charges γ I m = 1 2π γ e,i = 1 2π F I S 2 S 2 (Im τ IJ ) F J Semiclassical regime F = F cl + F 1-lp + F np a D,I = a cl D,I + Z γ = Z cl γ + s.c. regime: choice of (g 0, Λ 0 ) and R s.c. B, s.t. Zγ cl dominates Zγ 1-lp, Zγ np

6 Semiclassical vanilla BPS states I recipe for studying BPS spectrum in s.c. regime: 1. construct moduli space of classical BPS field configurations 2. approximate dynamics via motion on moduli space 3. quantize associated c.c. d.o.f. s N = 4 SQM 4. s.c. BPS states = BPS states in the SQM

7 Semiclassical vanilla BPS states II Step 1: monopole moduli space H cl Re(ζ 1 Z cl γ )

8 Semiclassical vanilla BPS states II Step 1: monopole moduli space H cl = Re(ζ 1 Z cl γ ) when with iϕ ζ(x + iy ), A 0 = Y : 0A i = 0X = 0Y = 0, 3F = D (3) X, D 2 (3)Y + [X, [X, Y ]] = 0, (p) (s) (p) primary, Bogomolny BPS eqn. (s) secondary BPS eqn, F i0 = D i Y. maximal bound: ζ = Arg( Z cl γ )

9 Semiclassical vanilla BPS states II Step 1: monopole moduli space H cl = Re(ζ 1 Z cl γ ) when with iϕ ζ(x + iy ), A 0 = Y : 0A i = 0X = 0Y = 0, 3F = D (3) X, D 2 (3)Y + [X, [X, Y ]] = 0, (p) (s) (p) primary, Bogomolny BPS eqn. moduli (s) secondary BPS eqn, F i0 = D i Y. unique solution, given Y maximal bound: ζ = Arg( Zγ cl ) M(γ m; X ) = { (A i, X ) (p), X X F 1 γ r mvol 2 S 2 admits T ad SO(3) SU(2) R action: }/ G R 3 R M Z G ad T ad global g.t. s; hyperkähler isometries SO(3) isometry inherited from spatial rotations SU(2) R action on T M: generators = cmplx structures J a (A)b

10 Semiclassical vanilla BPS states III Step 2: motion on moduli space low energy approx: give time dep. to collective coords z a (t), λ a (t)... Step 3: Quantize ˆλ a = γ a, ˆp z a = id a... Step 4: s.c. BPS states: /D M G Ψ BPS = 0 G a tri-holomorphic vector field on M; originates from Y γe Hu;(γ BPS ker m,γ e) L 2 /DM(γm,X ) G(Y ) t so(3) su(2) R action lifts to kernel; t so(3) acts by Lie derivative su(2) R generators = J ˆ (A) = 1 J 8 (A)ab[γ a, γ b ]

11 Line operators of type ζ, L ζ defect at x = 0 R 3, preserving so(3) su(2) R bosonic symmetry SUSY s R A α = ζ 1/2 Q A α + ζ 1/2 (σ 0 ) α β Q βa, where ζ = 1 a modification of the theory in the UV; H H L Line operator charges, L ζ (P, Q) (P, Q) (Λ wt (G) Λ wt (G))/W L ζ (0, Q): Wilson operator in irrep R(Q) L ζ (P, 0): t Hooft operator, defined by boundary conditions F P 2 vol S 2 + reg, iζ 1 ϕ = P 2r + reg, r = x x (b)

12 (Semiclassical) Framed BPS states a state H L, preserving R A α s saturates bound E Re(ζ 1 Z) space of such states: H BPS L;u Semiclassical description: = γ ΓL H BPS L;u;γ classical BPS equations same as before, with iζ 1 ϕ = X + iy { }/ M(P; γ m, X ) = (A, X ) (p), (b), X X F 1 G r γ 2 m vol S 2 admits T ad SO(3) SU(2) R action c.c. expansion of low energy action, quantization proceeds identically framed BPS states: γe HL;u;(γ BPS ker m,γ e) L 2 /DM(P;γm;X ) G(Y )

13 Application: A vanishing theorem Vanilla case: There is a special locus on the Coulomb branch where we know the spectrum exactly. On this locus if γ m not a simple co-root, implying that dim M > 0, then the kernel vanishes. Argument: special locus {Y = 0} R s.c. Y = 0 G = 0, so /D M G /D M /D M Ψ = 0 ( /D M ) 2 Ψ = 0 M h.k. ==== Ψ = 0 0 = Ψ Ψ = M M Ψ 2 Ψ = 0 Ψ = 0 Similarly in the framed case when dim M > 0

14 Example 2 G = SU(3), γ m = (1, 1) relative moduli space is Taub-NUT ds 2 M = µ ds2 TN (l) and G M = µ 1 l 2 P ψ where µ = 2π x 1x 2 g 2 0 x 1+x 2, l = π g 2 0 µ, with x I = α I, X and y I = α I, Y ( ) P = π y1 g 2 0 x 1 y2 x 2 zero modes of /D G Ψ = (λ A, 0) T, with λ A = ϕ 1 o A + ϕ 2 ι A ϕ 1 = r j+ 1 2 r+l e (P j 1 2 )r/l D j j,m (ψ, θ, φ)e iψ/2, ϕ 2 = 0, (P > 1 2 ) j {0, 1 2, 1,..., [ P 1 2 ]}, and m { j, j + 1,..., j} 2 Pope (1978); Lee, Weinberg and Yi (1997); Gauntlett, Kim, Park and Yi (1999)

15 Example G = SU(3), γ m = (1, 1) Checks walls consistent with Im ( Z γ1 Z γ2 ) = 0 in s.c. regime, where γ 1 = (1, 0; n 1, 0), γ 2 = (0, 1; 0, n 2 ) and we identify j = n 1 n 2 for j large, and near wall (0 < P j 1 2 1), wavefunction is sharply peaked at r = 2R bnd, where R bnd consistent 3 with Denef s bound state radius 4 : R bnd = 1 2 γ 1, γ 2 Z γ 1 + Z γ2 Im ( Z γ1 Zγ2 ) 3 Up to a pesky factor of 2 we haven t yet tracked down. 4 Denef (2002), Denef and Moore (2007)

16 Protected Spin Characters 5 vanilla protected spin character Wigner long rep: ρ hh ρ hh h, E > Z γ (u) generically short rep: ρ hh h, E = Z γ (u) ρ hh = (0, 1 ) ( 1, 0) half-hypermultiplet, and 2 2 h an arbitrary so(3) su(2) R rep. protected spin character counts rigid BPS states: Ω(γ, u; y) := Tr H BPS u,γ ( y)2i3 y 2J3, where Hu,γ BPS = ρ hh H u,γ BPS framed protected spin character long rep: ρ hh h L, E > Re(ζ 1 Z γ (u)) generically short rep: h L, E = Re(ζ 1 Z γ (u)) Ω(L ζ, γ, u; y) := Tr H BPS L,u,γ ( y)2i3 y 2J3 5 GMN 2010

17 Positivity Conjectures presence of su(2) R essential for rigidity of Ω, Ω interesting question: What su(2) R reps appear in Ω, Ω? surprising empirical answer: only the trivial one note: Ω(y) = Tr H BPS y 2J3 Ω(1) = dim H BPS Positivity Conjectures: PC1: short reps in H BPS u,, H BPS L,u always have I 3 = 0 PC2: I 3 eigenvalues all (half-)integral note: still Ω(y = 1) = dim H BPS

18 Positivity Conjectures presence of su(2) R essential for rigidity of Ω, Ω interesting question: What su(2) R reps appear in Ω, Ω? surprising empirical answer: only the trivial one note: Ω(y) = Tr H BPS y 2J3 Ω(1) = dim H BPS Positivity Conjectures: PC1: short reps in H u, BPS, HL,u BPS always have I 3 = 0 no exotics PC2: I 3 eigenvalues all (half-)integral (half-)integral R-spin note: still Ω(y = 1) = dim H BPS

19 PC s and the Kernel of /D G I SU(2) R as the commutant of holonomy recall SU(2) R acts on T M with cmplx structures as generators lifts to action on S D (M) via J (A) = 1 8 J (A)ab[ˆλ a, ˆλ b ] Hol( M ) USp(2N) SO(4N) preserves cmplx structures USp(2N) SU(2) f R f Spin(4N) SO(4N) features: 1. f canonically defined from T X M H N, USp(2N) U(N, H) 2. f (1, 1) = ω, volume form in Spin(4N) 3. f (ρ Dirac) = N+1 k=1 R k S k, where S k the k-dim SU(2) rep

20 PC s and the Kernel of /D G II Consequences for the Dirac operator Spin(4N) USp(2N) SU(2) R ker /D G N+1 k=1 n kr k S k PC1 n k = 0, k > 1; ker /D G is irrep of USp(2N) PC2 ker /D G is chiral our previous example consistent with PC1

21 Conclusions We reviewed the semiclassical construction of the space of one-particle BPS states in N = 2 theories as the kernel of certain twisted Dirac operators on the moduli space of classical (singular) monopole solutions. We described the action of so(3) su(2) R on the kernels. We translated the Positivity Conjectures of GMN into statements about the kernels. In particular, the no exotics and (half-)integral R-spins only conjectures imply that the kernels are chiral.

22 Conclusions We reviewed the semiclassical construction of the space of one-particle BPS states in N = 2 theories as the kernel of certain twisted Dirac operators on the moduli space of classical (singular) monopole solutions. We described the action of so(3) su(2) R on the kernels. We translated the Positivity Conjectures of GMN into statements about the kernels. In particular, the no exotics and (half-)integral R-spins only conjectures imply that the kernels are chiral. Thanks!