L2 kernels of Dirac-type operators on monopolemoduli spaces

Andy Royston

Texas A&M University

String-Math, TSIMF, China, Dec 31, 2015

Based on 1512.08923, 1512.08924 with Greg Moore and Dieter Van den Bleeken

Goal

The past ten years have seen tremendous developments inN = 2, d = 4 QFT:wall crossing, defects and framed BPS states, “no exotics”, relations tothe (2, 0) theory, localization computations, . . .

Our goal is to determinewhat these developments imply for the differential geometry associatedwith the semiclassical description—i.e. for Dirac/Dolbeault-typeoperators on the hyperkähler moduli spaces of (singular) monopoles

Spoiler:All nontrivial L2-cohomology of ∂ − iG∧ is concentrated in the middledegreei.e. (0,N) forms when the dimension of the moduli space is 4N.

Outline

• The moduli space of (singular) monopoles

• Consequences of N = 2 supersymmetry

• Mathematical Predictions

• Vanilla example

• Framed example

Moduli space of (singular) monopoles (I)

• Yang–Mills–Higgs theory on R1,3 with compact simple gauge group G

1g2

0

∫R1,3

(F , ?F ) + (DX , ?DX ) ,

( , ) a Killing form on g, g∗ s.t. (αlong, αlong) = 2; g0 the YM coupling

• Magnetic monopoles: solutions to F = ?3DX on R3, such that

X = X∞ −γm

2r + · · · , F =12γm sin θdθdφ+ · · · ,

as r →∞ where• X∞ ∈ g regular =⇒ defines CSA t ⊂ g with

simple roots αI ∈ t∗ ,simple co-roots HI ∈ t .

• γm ∈ Λcr

Moduli space of (singular) monopoles (II)

Def’n:M(γm,X∞) is the set of such solutions, identified bygauge transformations that approach 1 as r →∞

• (M, g , Jr ) a smooth hyperkähler manifold1:g , Jr induced from energy functional and hk structure of R4, onwhich A = Aidx i + Xdx4 satisfies F = ?4F and is x4-transl’n-inv.

• dimension is2 dimR(M) = 4∑

I nIm, where γm =

∑I nI

mHI

• Lie algebra of Killing vector fields R3 ⊕ so(3)⊕ t corresponding totranslations, rotations, and asymptotically nontrivial gauge transfo’s.

Def’n: G : t→ isomH(M) is a Lie algebra homomorphism s.t.• G(H) gives directional derivative at A, d

ds A = −DεH , where• εH is x4-transl’n-inv., solves D2εH = 0, and has εH → H as r →∞.

1Atiyah, Hitchin (1988)2Weinberg (1978); Taubes (1983); Donaldson (1984)

Moduli space of (singular) monopoles (III)

• The universal cover is metrically a product:

M(γm,X∞) = R3cm × RX∞ ×M0(γm,X∞) ,

R3cm generated by the translation Killing vectors; RX∞ generated by

G(X∞);M0 is the strongly centered moduli space3

• M = M/D, where the group of Deck transformations, D ∼= Z, isgenerated by an isometry φ of M.

• Fact: A subgroup LZ ⊆ Z is generated by action of gauge transfo’s:

∃h ∈ Λmw s.t. exp(2πG(h)) = φL ,

where L = gcdIpInIm with pI = 2

(αI ,αI )∈ 1, 2, 3.

3Atiyah, Hitchin (1988); Hitchin, Manton, Murray (1995)

Moduli space of (singular) monopoles (IV)Include an ’t Hooft defect at origin4:

X = − P2r + O(r−1/2) , F =

12P sin θdθdφ+ O(r−3/2) , (?)

as r → 0, where P ∈ Λmw.

Def’n: M(P, γm,X∞) is the set of solutions to Bogomolny,with same b.c.’s as r →∞, additionally satisfying (?), andidentified by gauge transformations.5

• Hyperkähler manifold w/ possible singularities on co-dim 4 loci,associated with monopole bubbling

• The relative magnetic charge is γm := γm − P−,where P− ∈ [P] with 〈αI ,P−〉 ≤ 0.

• dimR(M) = 4∑

I nIm, where γm =

∑I nI

mHI .• Lie algebra of Killing vector fields so(3)⊕ t.

4’t Hooft (1978); Kapustin (2005)5Kronheimer (1985); Pauly (1998); Kapustin, Witten (2006); MRV (2014)

Consequences of N = 2 susy (I)Quantum Theory — Seiberg–Witten Description

1 Space of vacua u ∈ B; local system of e.m. charges γ ∈ Γ→ B.

• (Vanilla) BPS states: HBPSu = ⊕γ∈ΓuHBPS

u,γ ; mass = |Zγ(u)|

• Factor out center-of-mass d.o.f.’s: HBPSu,γ = ρhh ⊗ (HBPS

0 )u,γ

• (HBPS0 )u,γ is an so(3)rot ⊕ su(2)R representation

• Protected spin characters Ω(u, γ; y) obey wall crossing formulae

2 ’t Hooft defects → supersymmetric ’t Hooft defects, Lζ([P]);preserve 4 out of 8 supercharges, labeled by a phase ζ

• Framed BPS states6 are BPS states in the theory in the presence ofdefects

• HBPSL,u = ⊕γ∈ΓL,u H

BPSL,u,γ ; mass = −Re[ζ−1Zγ(u)]

• framed PSC’s Ω(L, u, γ; y) obey wall crossing formulae; core/halo picturenear the walls

6Gaiotto, Moore, Neitzke (2010)

Consequences of N = 2 susy (II)

Classical modifications• a second Higgs, Y , and a potential energy

∫R3 ([X ,Y ], ?3[X ,Y ])

⇒ Y → Y∞ ∈ t as r →∞

• classical BPS dyons with7 γe = γe(γm,X∞,Y∞) :M(M)→ t∗

Semiclassical construction of (framed) BPS states• Collective coordinate (c.c.) ansatz:

A = Amono(~x ; zm(t)) + a(x) ,

zm, m = 1, . . . , 4N = dimM are local coordinates onM (M);a is a quantum fluctuation field with zero-modes excluded.

• integrate out a → get effective theory of collective coordinates

7Lee, Yi (1998); Bak et.al. (1999); Tong (1999)

Consequences of N = 2 susy (III)

Semiclassical Framed BPS states• fermions carry 4N real c.c.’s as well

• quantization of c.c.’s8 leads to supersymmetric QM in which statesare represented by Dirac spinors onM (or (0, ∗)-forms onM)

• supercharge op. represented by Dirac-like operator

/DG(Y∞)

M(P,γm,X∞):= /DM − i /G(Y∞) ,

• electric charge op. represented by spinorial Lie derivative,γe = i

∑I αI(£K I ), where K I generate 2π-periodic triholomorphic

isometries s.t. eigenvalues γe ∈ Λrt.

• Define:H scBPS

P,X∞,Y∞,γm,γe:= kerγe

L2

(/DG(Y∞)

M(P,γm,X∞)

).

8Gauntlett (1993); Gauntlett, Harvey (1994); Gauntlett, Kim, Park, Yi (1999)

Consequences of N = 2 susy (IV)Semiclassical vanilla BPS statesDecompose /DG and γe with respect to factorization of M:

/DG(Y∞)

M(γm,X∞)Ψγe = 0

γeΨγe = γeΨγe

⇒ Ψγe = s e−iqcmχ ⊗Ψγe,00 , where

• s is a constant spinor on R4,

• χ is a coordinate on RX∞ normalized such that φ = exp(2π∂χ)⊗ φ0with φ0 an isometry ofM0,

• γe = qcmγ∗m + γe,0 with qcm := 〈γe,X∞〉

(γm,X∞) ,

• and /DG0M0

Ψγe,00 = 0, where

/DG0(Y⊥∞)

M0(γm,X∞) := /DM0 − i /G0(Y⊥∞) .

Here G0(H) is metric projection of G(H) toM0 and (γm,Y⊥∞) = 0.

Consequences of N = 2 susy (V)

Semiclassical vanilla BPS states cont’d• γe ∈ Λrt ⇒ φL(Ψγe ) = Ψγe , but Ψγe is not necessarily invariantunder φ without further conditions

• there exists a map γe 7→ kγe ∈ Z/LZ such that

φ(Ψγe ) = Ψγe ⇐⇒ φ0(Ψγe,00 ) = e i(c(γe,0)+kγe/L)Ψ

γe,00 .

• Then

(HscBPS0 )X∞,Y⊥∞,γm,γe :=

(kerγe,0

L2

(/DG0(Y⊥∞)

M0(γm,X∞)

))(kγe )

.

Consequences of N = 2 susy (VI)

Conjecture: For any u ∈ Bwc ∈ B, we have

H scBPSP,X∞,Y∞,γm,γe

∼= HBPSLζ([P]),u,γ & (HscBPS

0 )X∞,Y⊥∞,γm,γe∼= (HBPS

0 )u,γ ,

provided we identify

X∞ = Im[ζ−1a(u)

], Y∞ = Im

[ζ−1aD(u)

],

where the weak coupling duality frame u 7→ a(u) is fixed by requiring〈αI ,X∞〉 > 0 ∀I, and in the vanilla case Zγ(u) = −ζ|Zγ(u)|.Furthermore, in this frame, ΓL,u ∼= (Λcr + P)⊕ Λwt and we identify

γ = γm ⊕ γe .

The weak coupling regime Bwc ⊂ B of the Coulomb branch consists ofthose u such that the series expansion of aD(a) around ∞ converges.

Mathematical Predictions (I)

No exotics9: HBPSL,u,γ , (HBPS

0 )u,γ transform trivially under SU(2)R

Semiclassical realization of SU(2)R• su(2)R acts via complex structures 1

2Jr

• homomorphism SU(2)R × Sp(N)hol → Spin(4N), under which(−1, 1) 7→ the Clifford volume form⇒ No exotics implies the kernel is chiral

• Set ωr = g(·, Jr (·)). (M,ω3) is a Kähler manifold withω± = ω1 ± iω2, ω+ ∈ Λ(0,2); su(2)R action on Λ(0,∗) is

I 3|Λ(0,q) =12 (q − N)1 , I + = iω+ ∧ , I− = −iιω− .

No exotics prediction:All L2 Dolbeault cohomology of ∂ − iG(Y∞)(0,1)∧ is concentrated in themiddle degree and primitive with respect to I±.

9GMN (2010); Chuang et.al. (2014); Del Zotto, Sen (2014); Cordova, Dumitrescu

Mathematical Predictions (II)

SC identification of BPS spaces ⇒ PSC’s Ω(y),Ω(y) are indexcharacters of Dirac operators (at fixed electric charge) w.r.t. so(3)

isometry. Hence by no exotics they are characters of ker /DG+ .

Vanilla Fredholm prediction:Consider the family of Dirac operators /DG0(Y⊥∞)

M0(γm,X∞). Generically, theseoperators are Fredholm. However the kernel will jump at co-dimensionone walls defined by

(γi,m,Y⊥∞) + 〈(γi,e)0,X∞〉 = 0 , i = 1, 2 ,

where γ1,2 = γ1,2,m ⊕ γ1,2,e must satisfy1. γm = γ1,m + γ2,m,2. ⟪γ1, γ2⟫ := 〈γ1,e, γ2,m〉 − 〈γ2,e, γ1,m〉 6= 0, and3. (HscBPS

0 )i=1,2 6= 0.

At such walls, the characters of ker(/DG0M0

)+ jump according to theKontsevich–Soibelman wall crossing formula.10

10Kontsevich, Soibelman (2008)

Mathematical Predictions (III)

Framed Fredholm prediction:Consider the family of Dirac operators /DG(Y∞)

M(P,γm,X∞). Generically, these

operators are Fredholm. However the kernel will jump at co-dimensionone walls defined by

(γh,m,Y∞) + 〈γh,e,X∞〉 = 0 ,

where the “halo charge” γh = γh,m ⊕ γh,e ∈ Λcr ⊕ Λrt must be occupied:(HscBPS

0 )γh 6= 0.

At such walls, the characters of ker(/DGM)+ jump in accord with the

wall crossing formula for framed BPS states.11

11Gaiotto, Moore, Neitzke (2010)

Vanilla Example (I)

• Consider case γm = H1 + H2 for rank two simple g:12

ds2M = Md~xcm · d~xcm + M−1dχ2 + ds2

M0, with

ds2M0

= µ ds2TN(`)(~r ,ψ)

where dsTN(`)(~r ,ψ) is the (single-centered) Taub–NUT metric viewed asa circle fibration over R3 \ 0 3 ~r , with fiber coordinate ψ ∼ ψ+ 2π,and asymptotic radius 2`.

• The paramaters are

M = m1 + m2 , µ =m1m2

m1 + m2, ` =

p2µ ,

with mi = (Hi ,X∞) and p = 1, 2, 3 for g = su(3), so(5), g2 respectively.

• The quotient by D ∼= Z imposes (χ,ψ) ∼(χ+ 2π,ψ+ 2πm2

pM

).

• G0(Y⊥∞) = 1p (H1,Y⊥∞)∂ψ

12Lee, Weinberg, Yi (1996); Gauntlett, Kim, Park, Yi (1999)

Vanilla Example (II)

0.0 0.2 0.4 0.6 0.8 1.01.0

0.5

0.0

0.5

1.0

x1

y 1Ne, 0 1

Ne, 0 2

Ne, 0 3

Ne, 0 1

Ne, 0 2

Ne, 0 3

Structure of the kernel13 with x1 := m1/M and y1 := p(H1,Y⊥∞)/M• trivial in middle chamber containing locus y1 = 0• dyon tower γe = nα1 + (pn − Ne,0)α2, n ∈ Z, of angular momentummultiplets of spin j = 1

2 (|Ne,0| − 1) created when Ne,0-wall crossed13Pope (1978)

Vanilla Example (III)

5 10 20 50 100r

0.2

0.4

0.6

0.8

1.0Pr

C 50C 48

C 46C 44

C 42 C 40

Radial probability densities for states with |Ne,0| = 78, as we approachthe MSW at |C | = 1

2 |Ne,0| = 39, where C = y12x1(1−x1) .

• Compute P ′(rext) = 0, and find

rext = rDenef :=⟪γ1, γ2⟫

2 Im [ζ−1Zγ1 (u)],

using map of parameters.14

14Denef (2002)

Framed Example (I)

Consider g = su(2) with defectP = p

2 Hα, |p| ∈ N

Let Wn denote the MSW in theweak coupling regime for haloparticles with γn = Hα ⊕ nα

Then using the map u 7→ X∞,Y∞,we have Wn: y + nx = 0, wherey = (Hα,Y∞), x = 〈α,X∞〉

0.0 0.2 0.4 0.6 0.8 1.01.0

0.5

0.0

0.5

1.0

x

y

W1

W2

W1

W2W

3

c0

c1

c1c2

WΑ

WΑ

Framed Example (II)

• Consider the generating function Fn(p; y) :=∑γ Ωn(p, γ; y)Xγ , where

XγXγ′ = y⟪γ,γ⟫Xγ+γ′

• The possible charges are γ = (nm − |p|2 )Hα ⊕ neα, with nm ∈ Z≥0 andne ∈ Z, so

Fn(p; y) = X−|p|/21

∑nm,ne

Ωn(p, nm, ne; y)yne(2nm−|p|)X nm1 X ne

2 .

• Ω(p, nm, ne; y) are characters of ker(/DGM)+, where dimM = 4nm.

• We have the closed form solution:

Fn(p; y) =[X−1/2

1 X−n/22 (Un(fn)− X−1/2

2 Un−1(fn))]|p|

, (n ≥ 0) ,

with fn =12

[X 1/2

2 + X−1/22 (1 + y2n+3X1X n+1

2 )],

where X1 := XHα , X2 = Xα, and Un(cos θ) = sin((n+1)θ)sin θ .

Framed Example (III)Semiclassical checks:

• nm = 0: A unique state with ne = 0 in cn,∀n; the only state in c0

X Consistent with sc analysis onM = pt..

• nm = 1: Spin j = 12 (|ne| − 1) multiplet enters spectrum after

crossing Wn=|p|ne (in direction away from c0)X Consistent with sc analysis onM = TN/Z|p|

Many more predictions!• nm = 2: M is the 8-dimensional Dancer manifold15 for |p| = 1.We give the detailed spectrum for any |p|;

• The spectrum is finite in every chamber. In cn there is a uniquestate of maximal nm given by nm = |p|n, carrying electric chargene = |p|n2

15Dancer (1992)

Future Directions

• Physically, it is clear that the Dirac operators fail to be Fredholmwhen the gap between the boundstate and the continuum closes. Itshould be possible to derive the condition for the walls from anasymptotic analysis16 ofM orM0.

• We are not aware of an L2 index theorem for these types ofoperators, but a localization computation of the vev of the linedefect wrapping S1 in the theory on R3 × S1 could lead to one.

• The semiclassical construction can be extended to more generalN = 2 theories in the presence of general Wilson–’t Hooft defects.17

Thank you!

16as in Stern, Yi (2000)17Brennan, Moore (work in progress)