Instantons, monopoles, Skyrmions and hyperbolic spacewinyard/talks/YTF2013.pdf · 2017. 7. 27. ·...

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Instantons, monopoles, Skyrmions and hyperbolic space Thomas Winyard Durham University YTF 2013 June 20, 2015 Thomas Winyard Instantons, monopoles, Skyrmions and hyperbolic space

Transcript of Instantons, monopoles, Skyrmions and hyperbolic spacewinyard/talks/YTF2013.pdf · 2017. 7. 27. ·...

Page 1: Instantons, monopoles, Skyrmions and hyperbolic spacewinyard/talks/YTF2013.pdf · 2017. 7. 27. · instantons which have the data: ( x) = L M x 0 1 N (8) where L is a row of N quaternions

Instantons, monopoles, Skyrmions and hyperbolicspace

Thomas Winyard

Durham UniversityYTF 2013

June 20, 2015

Thomas Winyard Instantons, monopoles, Skyrmions and hyperbolic space

Page 2: Instantons, monopoles, Skyrmions and hyperbolic spacewinyard/talks/YTF2013.pdf · 2017. 7. 27. · instantons which have the data: ( x) = L M x 0 1 N (8) where L is a row of N quaternions

Thomas Winyard Instantons, monopoles, Skyrmions and hyperbolic space

Page 3: Instantons, monopoles, Skyrmions and hyperbolic spacewinyard/talks/YTF2013.pdf · 2017. 7. 27. · instantons which have the data: ( x) = L M x 0 1 N (8) where L is a row of N quaternions

Pure SU(N) Yang-Mills theory in 4-dim

S = −1

8

∫d4xTr (FµνF

µν) (1)

We want a finite action, hence must fix some BC’s

|x | → ∞ ⇒ Aµ = −∂µg∞ (g∞)−1 (2)

∂R4 = S3∞ → SU (2) , π3 (SU (2)) = Z (3)

The integer N ∈ Z is the topological charge of the theory.

Thomas Winyard Instantons, monopoles, Skyrmions and hyperbolic space

Page 4: Instantons, monopoles, Skyrmions and hyperbolic spacewinyard/talks/YTF2013.pdf · 2017. 7. 27. · instantons which have the data: ( x) = L M x 0 1 N (8) where L is a row of N quaternions

We can split the action into the following form:

3S = − 1

16

∫ {Tr ((Fµν ∓? Fµν) (Fµν ∓? Fµν))± 2Tr

(F ?µνFµν

)}d4x

(4)The bogomolny bound is:

S ≥ π2 |N| (5)

N = − 1

8π2

∫Tr(F ?µνFµν

)d4x (6)

Thomas Winyard Instantons, monopoles, Skyrmions and hyperbolic space

Page 5: Instantons, monopoles, Skyrmions and hyperbolic spacewinyard/talks/YTF2013.pdf · 2017. 7. 27. · instantons which have the data: ( x) = L M x 0 1 N (8) where L is a row of N quaternions

This bound is attained only when the fields are either self-dual oranti-self-dual.

Fµν = ± ? Fµν (7)

hence Yang-Mills instantons are BPS, E = 2π2 |N|

There is an 8N parameter family of solutions:

Each instanton has position(4) + SU (2) orientation (3) + scale (1)

Thomas Winyard Instantons, monopoles, Skyrmions and hyperbolic space

Page 6: Instantons, monopoles, Skyrmions and hyperbolic spacewinyard/talks/YTF2013.pdf · 2017. 7. 27. · instantons which have the data: ( x) = L M x 0 1 N (8) where L is a row of N quaternions

We can write instantons in terms of algebraic equations ofquaternionic matricies. We are only really interested in SU(2)instantons which have the data:

∆ (x) =

(LM

)− x

(0

1N

)(8)

where L is a row of N quaternions and M a pure quaternionic NxNmatrix.To construct our instanton from the ADHM data, we must find aunit norm column vector ψ (x) satisfying

ψ†∆ (x) = 0 (9)

giving a gauge field of the form

Aµ = ψ†∂µψ (10)

Thomas Winyard Instantons, monopoles, Skyrmions and hyperbolic space

Page 7: Instantons, monopoles, Skyrmions and hyperbolic spacewinyard/talks/YTF2013.pdf · 2017. 7. 27. · instantons which have the data: ( x) = L M x 0 1 N (8) where L is a row of N quaternions

Derricks theorem implies that pure Yang-Mills had no topologicalsoliton solutions in three spatial dimensions. However if the fieldsare coupled to Higgs scalar fields, then stable monopole solutionsare possible.

We consider an SU(2) Yang-Mills theory with gauge potential Aµand the adjoint Higgs field Φ.

L =

∫ {1

8Tr (FµνF

µν)− 1

4Tr (DµΦDµΦ)− λ

4

(1− |Φ|2

)2}(11)

Note this is a dimensional reduction of the Instanton lagrangian.The SYDM equation becomes

?F = DΦ (12)

Thomas Winyard Instantons, monopoles, Skyrmions and hyperbolic space

Page 8: Instantons, monopoles, Skyrmions and hyperbolic spacewinyard/talks/YTF2013.pdf · 2017. 7. 27. · instantons which have the data: ( x) = L M x 0 1 N (8) where L is a row of N quaternions

We can constuct N-monopole solutions using the Nahm equations:

dTi

ds=

1

2iεijk [Tj ,Tk ] (13)

where T1,T2 and T3 are NxN Hermitian matrix functions in therange S ∈ [−1, 1].There is only one known family of monopoles called theErcolani-Sinha solution.Can also be described as specral curves (Rieman surface encodingthe monopole data).

Thomas Winyard Instantons, monopoles, Skyrmions and hyperbolic space

Page 9: Instantons, monopoles, Skyrmions and hyperbolic spacewinyard/talks/YTF2013.pdf · 2017. 7. 27. · instantons which have the data: ( x) = L M x 0 1 N (8) where L is a row of N quaternions

Some numerical solutions have been found of monopoles byimposing strict symmetries on the data and numerically performingthe Nahm transform.

B = 3 (Td) tetrahedral, B=4 (Oh) cubic, B=5 (D2d) dodecahedral

Thomas Winyard Instantons, monopoles, Skyrmions and hyperbolic space

Page 10: Instantons, monopoles, Skyrmions and hyperbolic spacewinyard/talks/YTF2013.pdf · 2017. 7. 27. · instantons which have the data: ( x) = L M x 0 1 N (8) where L is a row of N quaternions

Hyperbolic monopoles on a general 3-manifold with metric gijtakes the form

1

2

√gεijkg

jlgkmFlm = DiΦ (14)

Atiyah showed that circle invariant instantons are equivalent tohyperbolic monopoles.Consider R4 with x3 + ix4 = re iχ:

ds2 = dx21 + dx22 + dr2 + r2dχ2 (15)

We then remove the plane r = 0, then this metric is conformal to:

ds2 =1

r2(dx21 + dx22 + dr2

)+ dχ2 (16)

which is the metric for H3XS1 and the removed plane is thehyperbolic boundary.Hence there is a conformal equivalence between

(R4 − R2

)/S1

and hyperbolic 3-space H3.

Thomas Winyard Instantons, monopoles, Skyrmions and hyperbolic space

Page 11: Instantons, monopoles, Skyrmions and hyperbolic spacewinyard/talks/YTF2013.pdf · 2017. 7. 27. · instantons which have the data: ( x) = L M x 0 1 N (8) where L is a row of N quaternions

A circle invariant instanton we can reduce along the circledirection.As SDYM are conformally invariant this will give a hyperbolicmonopole.Can construct hyperbolic monopoles from ADHM data if it is circleinvariant.

Thomas Winyard Instantons, monopoles, Skyrmions and hyperbolic space

Page 12: Instantons, monopoles, Skyrmions and hyperbolic spacewinyard/talks/YTF2013.pdf · 2017. 7. 27. · instantons which have the data: ( x) = L M x 0 1 N (8) where L is a row of N quaternions

Some solutions constructed from circle invariant ADHM data(Manton, Sutcliffe - Platonic hyperbolic monopoles)

Thomas Winyard Instantons, monopoles, Skyrmions and hyperbolic space

Page 13: Instantons, monopoles, Skyrmions and hyperbolic spacewinyard/talks/YTF2013.pdf · 2017. 7. 27. · instantons which have the data: ( x) = L M x 0 1 N (8) where L is a row of N quaternions

The Skyrme model is a non-linear theory of pions that has solitonsolutions that can be interpreted as baryons. While not containingquarks it is considered a low energy effective field theory of QCD.

L =

∫ {−1

2Tr (RµR

µ) +1

16Tr ([Rµ,Rν ] [Rµ,Rν ])

}d3x (17)

where Rµ = (∂µU)U† is an su(2) valued current of the scalarU (t, x).

Thomas Winyard Instantons, monopoles, Skyrmions and hyperbolic space

Page 14: Instantons, monopoles, Skyrmions and hyperbolic spacewinyard/talks/YTF2013.pdf · 2017. 7. 27. · instantons which have the data: ( x) = L M x 0 1 N (8) where L is a row of N quaternions

We impose the natural boundary conditions U (x)→ 12 as|x | → ∞. U : S3 → S3

The topological charge or Baryon number is given by:

B = − 1

24π2

∫εijkTr (RiRjRk) d3x ∈ Z = π3 (SU (2)) (18)

Faddeev-Bogomolny bound

E ≥ 12π2 |B| (19)

not attainable for B 6= 0 (Skyrmions are not BPS)

Thomas Winyard Instantons, monopoles, Skyrmions and hyperbolic space

Page 15: Instantons, monopoles, Skyrmions and hyperbolic spacewinyard/talks/YTF2013.pdf · 2017. 7. 27. · instantons which have the data: ( x) = L M x 0 1 N (8) where L is a row of N quaternions

For B = 1 we have spherical symmetry and so can apply a hedhogansatz

U (x) = exp (if (r) x̂ · τ) (20)

where f (r) is a profile function with f (0) = π and f (∞) = 0.

Thomas Winyard Instantons, monopoles, Skyrmions and hyperbolic space

Page 16: Instantons, monopoles, Skyrmions and hyperbolic spacewinyard/talks/YTF2013.pdf · 2017. 7. 27. · instantons which have the data: ( x) = L M x 0 1 N (8) where L is a row of N quaternions

Here are some numerical minimal energy density surfaces:

Thomas Winyard Instantons, monopoles, Skyrmions and hyperbolic space

Page 17: Instantons, monopoles, Skyrmions and hyperbolic spacewinyard/talks/YTF2013.pdf · 2017. 7. 27. · instantons which have the data: ( x) = L M x 0 1 N (8) where L is a row of N quaternions

We can form a good approximation of Skyrmions by computing theholonomy of SU(2) instantons in R4 along lines parallel to thex4-axis.

U (x) = Pexp(∫ ∞∞

A4 (x , x4) dx4

)(21)

where P represents path ordering.

This produces a Skyrme field with baryon number equal to theinstanton charge, B = N. A simple example is given by the chargeone case.

Thomas Winyard Instantons, monopoles, Skyrmions and hyperbolic space

Page 18: Instantons, monopoles, Skyrmions and hyperbolic spacewinyard/talks/YTF2013.pdf · 2017. 7. 27. · instantons which have the data: ( x) = L M x 0 1 N (8) where L is a row of N quaternions

The charge one instanton is given by the ’t Hooft ansatz

Aµ =i

2σµν∂ν logη (22)

η = 1 +λ2

|x |2(23)

The holonomy of this instanton gives a Skyrme field that matchesthe hedgehog anstatz with a profile function of the form:

f (r) = π

(1− r√

λ2 + r2

)(24)

Thomas Winyard Instantons, monopoles, Skyrmions and hyperbolic space

Page 19: Instantons, monopoles, Skyrmions and hyperbolic spacewinyard/talks/YTF2013.pdf · 2017. 7. 27. · instantons which have the data: ( x) = L M x 0 1 N (8) where L is a row of N quaternions

We can compare the numerical and approximate profile functions:

this gives an energy of E = 1.243 only 1% above that of the truesolution.

Thomas Winyard Instantons, monopoles, Skyrmions and hyperbolic space

Page 20: Instantons, monopoles, Skyrmions and hyperbolic spacewinyard/talks/YTF2013.pdf · 2017. 7. 27. · instantons which have the data: ( x) = L M x 0 1 N (8) where L is a row of N quaternions

We can still obtain Skyrmions from instantons in hyperbolic spaceby computing instanton holonomy along the circles parametrizedby :

U (ρ, θ, φ) = Pexp(∫ 2π

0Aχdχ

)(25)

The circles whose holonomy we use to generate hyperbolic Skyrmefields have no points in common, hence the holonomy is onlydefined up to conjugation unless we fix a gauge on the whole ofhyperbolic space.

Thomas Winyard Instantons, monopoles, Skyrmions and hyperbolic space

Page 21: Instantons, monopoles, Skyrmions and hyperbolic spacewinyard/talks/YTF2013.pdf · 2017. 7. 27. · instantons which have the data: ( x) = L M x 0 1 N (8) where L is a row of N quaternions

Hyperbolic monopoles are constructed directly by using circleinvariant instantons.

Hyperbolic Skyrmions can be approximated by taking the holonomyaround a circle (which for circle invariant instantons is trivial).

So can we approximate hyperbolic monopoles and hyperbolicSkyrmions?

Thomas Winyard Instantons, monopoles, Skyrmions and hyperbolic space

Page 22: Instantons, monopoles, Skyrmions and hyperbolic spacewinyard/talks/YTF2013.pdf · 2017. 7. 27. · instantons which have the data: ( x) = L M x 0 1 N (8) where L is a row of N quaternions

What is the best gauge to approximate hyperbolic skyrmionsfrom instanton holonomies.

Can we model hyperbolic skyrmions numerically, both staticsand dynamics.

Can we model hyperbolic monopoles numerically.

How closely are hyperbolic monopoles and hyperbolicskyrmions related.

Thomas Winyard Instantons, monopoles, Skyrmions and hyperbolic space

Page 23: Instantons, monopoles, Skyrmions and hyperbolic spacewinyard/talks/YTF2013.pdf · 2017. 7. 27. · instantons which have the data: ( x) = L M x 0 1 N (8) where L is a row of N quaternions

Numerics for hyperbolic monopoles is hard.

What about the moduli space approximation?

But in hyperbolic space the field on the boundary tells you theposition of the soliton, so each ’snapshot’ changes the field on theboundary

Moduli space approximation blows up similarly to the CP1 modelas it costs an infinite amount of energy to move between eachsolution.

Can we model hyperbolic skyrmions to tell us about hyperbolicmonopoles?

Thomas Winyard Instantons, monopoles, Skyrmions and hyperbolic space

Page 24: Instantons, monopoles, Skyrmions and hyperbolic spacewinyard/talks/YTF2013.pdf · 2017. 7. 27. · instantons which have the data: ( x) = L M x 0 1 N (8) where L is a row of N quaternions

Conclusion:

Hyperbolic monopoles are constructed directly by using circleinvariant instantons.

Hyperbolic Skyrmions can be approximated by taking theholonomy around a circle.

Can we understand this link better?

Can approximate hyperbolic skyrmions and constructhyperbolic monopoles using the same ADHM data.

Can we use this to model difficult problems in the othermodel?

Thomas Winyard Instantons, monopoles, Skyrmions and hyperbolic space