Singular Monopoles and Instantons on Curved Backgrounds Sergey Cherkis (Berkeley, Stanford, TCD) C k U(n) U(n) U(n) Odense 2 November 2010
Outline: Classical Solutions & their Charges Relations between Instantons and Monopoles: Calorons and Monopoles Instantons on TNk and Singular Monopoles Singular Monopole Solutions (New) Screening of t Hooft operators Brane Configurations SQCD in three dimensions Moduli spaces of singular monopoles Some problems and details at http://www.maths.tcd.ie/~islands
Use of Monopoles and Instantons Instantons - Nonperturbative effects in quantum gauge theories - 1/2 BPS states in higher-dimensional gauge theories - D-brane physics: Branes within branes - Heterotic string theory compactifications - Quantum gauge theories with walls of impurities Monopoles - magnetic solitons in Yang-Mills-Higgs theory - 1/2 BPS states in Super-Yang-Mills theory - Brane physics: Chalmers-Hanany-Witten configurations - Electric-magnetic dual of gauge bosons - Effective infrared dynamics of super-qcd in 2+1 dimensions - Gravitational Instantons
Yang-Mills U(n) Yang-Mills theory is specified by a bundle and gauge field Hermitian one-form A = A µ dx µ Γ(adE T M 4 ) It has the field strength two-form U(n) gauge transformations And the Yang-Mills action is
This action is limited below by a topological charge (a BPS bound) with equality iff More geometrically, at infinity the gauge field approaches pure gauge The instanton number is the index of this map over the sphere From: http://hof.povray.org/
Instantons An instanton is a gauge field with finite action satisfying the self-duality equation F = F In a slightly different notation F µν = µ A ν ν A µ + [A µ, A ν ] F = F µν dx µ dx ν F µν = 1 g µνρσ g ρα g σβ F αβ For example in flat space the self-duality equation reads Since with equality iff instantons deliver minima to the Yang-Mills action. Leading contributions to the QFT path integral
One SU(2) Instanton It is convenient to think of as a space of quaternions, of SU(2) as unit quaternions, and its Lie algebra su(2) as imaginary quaternions. Quaternions Spinors quaternionic units I, J, K or satisfying I 2 =J 2 =K 2 =IJK=-1 I= J= K= Basis of self-dual forms and anti-self-dual forms
Belavin-Polyakov-Schwartz-Tyupkin solution One instanton at the origin and at a generic point B This is a five-parameter family of solutions: - four position parameters - one size parameter A large family of multi-instanton solutions is delivered by Corrigan-Fairlie- t Hooft-Wilczek ansatz and t Hooft solution. A complete construction on 8m0-3 parameter family. is found by Atiyah, Hitchin, Drinfeld, and Manin.
Monopoles Yang-Mills-Higgs Theory Gauge field Higgs field Energy of a static configuration: BPS Energy Bound Geometrically and m is the degree of this map. This bound is saturated by solutions of the Bogomolny Equation In flat three-space
Dirac BPS Monopole Monopole Solutions solves the U(1) Bogomolny equation: Wu-Yang Monopole For a three vector let it is easily embedded into U(2) and into SU(2) Φ(t) = \n 2 t ν j, A(t) =\nω j, After an SU(2) gauge transformation it can be put into a Wu-Yang form
t Hooft-Polyakov BPS Monopole From: http://www.phys.uu.nl/~thooft/ Solves SU(2) Bogomolny equation Completely smooth solution. Asymptotically approaches Wu-Yang monopole.
Instantons and Monopoles are intimately related Self-duality Eq. in D=4 Nahm Eq. in D=1 Bogomolny Eq. in D=3
Monopole and Caloron A Caloron is a finite action self-dual field on a space with a periodic direction. Name due to its role in gauge theory at finite temperature. Period of is 2π/T. A static solution of self-dulaity equation solved Bogomolny Eq. Caloron Charges Nye and Singer Finite action => At infinity caloron field is θ-independent => monopole charges (m1, m2,..., mn) Total instanton charge m0
Some Simple Geometry Flat Space R 4 Pure imaginary ā = a q = q 0 + iq 1 + jq 2 + kq 3 = ae i ψ 2 ds 2 = dqd q = 1 4 aiā = ix 1 + jx 2 + kx 3 1 x dx2 + x (dψ + ω dx) 2 Taub-NUT TN ds 2 = l + 1 x dx 2 + 1 (dψ + ω dx) 2 l + 1 x multi-taub-nut TN k Gibbons-Hawking ds 2 = l + 1 x x 1 + 1 x x 2 +... + 1 dx 2 + (dψ + (ω 1 + ω 2 +... + ω k ) dx) 2 x x k l + 1 x x 1 + 1 x x 2 +... + 1 x x k
Instantons on TNk & Singular Monopoles Taub-NUT (A0 ALF) Instanton on Taub-NUT ds 2 = Vd x 2 + (dτ + ω)2 V V = l + 1 x, dω = 3 dv τ τ +4π, if there is a gauge transformation making A and Φ τ-independent, then ˆF = ˆF Smooth connection of TN = singular monopole Kronheimer τ-invariant self-dual field on Taub-NUT = singular monopole
Singular Monopole SCh & Chris Blair t Hooft-Polyakov Solution Dirac singularities Solution Φ = k j=1 \n j 2t j ν j asingularity the monopole z T j t j Our Solution
Monopole Screening Effect Energy Density and Tr Φ 2 Profiles 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 d z 3 SCh & Brian Durcan Monopole d Singularity r x z z 3 5 Tr Φ 2 Contour lines (0.5 & 0.007) 4 3 Energy Density Contour lines 2 1 1 2 3 4 d 1
Singular Monopoles in String Theory Theory on the internal D3-branes in extreme infrared is 2+1 dimensional U(2) gauge theory: BPS charge m monopole with k Dirac singularities D5 D3... D3... U(m) gauge theory Massive Quarks masses given by D3 posiitons at infinity
In extreme infrared N=4, D=3 the gauge theory is described by the sigma-model with its moduli space of vacua as its target space. The dynamics of the effective gauge theory is the geodesic motion on this space. Quantum moduli space of vacua of N=4, D=3 U(m) super-yang- Mills with k massive quarks = classical moduli space of charge m monopoles with k Dirac singularities.
Moduli Spaces of Singular Monopoles There are a few techniques one can use to compute the moduli spaces of singular monopoles: spectral data of the scattering problem or Nahm data. We used these with Kapustin to obtain: Moduli space of one monopole with k singularities = TNk Moduli space of two monopoles with k singularities = Dk ALF space