Projective superspace Ariunzul Davgadorj Masaryk University, Czech - - PowerPoint PPT Presentation

Motivation for using Projective superspace Review of Projective superspace Super Yang-Mills theory in Projective superspace framework Summary

Projective superspace

Ariunzul Davgadorj

Masaryk University, Czech Republic

New Frontiers in String Theory 2018 August 2, 2018 based on works with Rikard von Unge

Ariunzul Davgadorj Projective superspace 1

Motivation for using Projective superspace Review of Projective superspace Super Yang-Mills theory in Projective superspace framework Summary

References

If you are interested in Projective superspace check for example:

1 U.L, M.R; Commun.Math.Phys 128 (1990) 191 2 F.G-R, M.R, S.W, U.L, R.von U; arxiv: hep-th 9710250 3 F.G-R; arxiv: hep-th 9712128 4 A.D, R.von U; arxiv: hep-th 1706.07000 5 others... Ariunzul Davgadorj Projective superspace 2

Motivation for using Projective superspace Review of Projective superspace Super Yang-Mills theory in Projective superspace framework Summary

Outline

1

Motivation for using Projective superspace

2

Review of Projective superspace

3

Super Yang-Mills theory in Projective superspace framework Gauge connection and transformations Yang-Mills action in chiral representation Classic action from 1-loop divergent diagram Abelian limit

4

Summary

Ariunzul Davgadorj Projective superspace 3

Motivation for using Projective superspace Review of Projective superspace Super Yang-Mills theory in Projective superspace framework Summary

Motivation

Standard R4|8 superspace is not sufficient in realizing off-shell N = 2 supersymmetric action.

• A supersymmetric theory needs a given set of auxiliary fields to

be described off-shell.

• For N = 2 supersymmetry, it’s complicated to construct a

multiplet with a finite number of auxiliaries.

• Constraints to eliminate those auxiliary fields also put the

physical fields on mass-shell.

• Need some other relaxed constraints or an unconstrained

description in some other space.

• Perhaps an infinite number of auxilaries?

Ariunzul Davgadorj Projective superspace 4

Motivation for using Projective superspace Review of Projective superspace Super Yang-Mills theory in Projective superspace framework Summary

Extend the conventional N = 2 susy group by its automorphism group SU(2) over U(1) subgroup and introduce:

• Isotwistors on full C2\{0} space
• Harmonic space on sphere S2 in spinor harmonic basis
• Projective space over Riemann sphere CP1 in terms of a

holomorphic variable

Ariunzul Davgadorj Projective superspace 5

Motivation for using Projective superspace Review of Projective superspace Super Yang-Mills theory in Projective superspace framework Summary

Review of Projective superspace

N = 2 susy algebra: {Di

α, Dj β} = {Di ˙ α, Dj ˙ β} = 0 , {Di α, Dj ˙ β} = iδi j∂α ˙ β

Diα = ∂ ∂θiα + ˙ ıθ ˙

α i

∂ ∂xα ˙

α , D1 ≡ D

D2 ≡ Q Introducing a bosonic coordinate on manifold CP1: ζ to parameterize the N = 2 Grassmann coordinates. Θα = θ2α − ζθ1α , Θ ˙

α = θ1 ˙ α + ζθ2 ˙ α

Ariunzul Davgadorj Projective superspace 6

Motivation for using Projective superspace Review of Projective superspace Super Yang-Mills theory in Projective superspace framework Summary

Define projective supercovariant derivatives that annihilate projective superfields Ω(Θ, Θ). ∇α ≡ (Dα + ζQα)Ω = 0 ∇ ˙

α ≡ (Q ˙ α − ζD ˙ α)Ω = 0

Further define the second set of orthogonal derivatives. ∆α ≡ (Qα − 1 ζ Dα) ∼ ∂ ∂Θα ∆ ˙

α ≡ (D ˙ α + 1

ζ Q ˙

α) ∼

∂ ∂Θ ˙

α

Ariunzul Davgadorj Projective superspace 7

Motivation for using Projective superspace Review of Projective superspace Super Yang-Mills theory in Projective superspace framework Summary

Projective supercovariant algebra follows: {∇, ∇} = {∇, ∇} = {∆, ∆} = {∆, ∆} = {∇, ∆} = 0 {∇α, ∆ ˙

α} = −{∆α, ∇ ˙ α} = 2˙

ı∂α ˙

α

{∇α(ζ1), ∇ ˙

α(ζ2)} = ˙

ı(ζ1 − ζ2)∂α ˙

α

A conjugation of an object is defined by applying Hermitian conjugation and an antipodal map onto the Riemann sphere. f (ζ) = (−1)pζpf ∗(−1 ζ )

Ariunzul Davgadorj Projective superspace 8

Motivation for using Projective superspace Review of Projective superspace Super Yang-Mills theory in Projective superspace framework Summary

Grassmann integration is the same as differentiation. Using the ζ parameterized Grassmann coordinates the measure acting on a function of projective superfields Ω(Θ, Θ) will be constructed: ∂ ∂Θα

2

∂ ∂Θ ˙

α 2

∼ ∆2∆2 ∼ (1 ζ ∇ − 2 ζ D)2(1 ζ ∇ + 2D)2f (Ω) ∼ D2D2f (Ω)

• dζ
• d4x

d4θ

• P =

dζ ζ

• d4xD2D2

The chiral measure will be:

• dζ
• d4x

d4θ

• C =

dζ ζ

• d4xD2Q2

Ariunzul Davgadorj Projective superspace 9

Motivation for using Projective superspace Review of Projective superspace Super Yang-Mills theory in Projective superspace framework Summary

Projective superfields of particular interests for us are polar multiplets and the tropical multiplet Υ =

∞

• n=0

Υnζn Arctic Υ =

∞

• n=0

Υn(−1 ζ )n Antarctic V (ζ) =

∞

• −∞

υnζn , υ−n = (−1)nυn Tropical A free action for the polar multiplets would be:

• d4x

d4θ

• P

dζ ζ ΥΥ

Ariunzul Davgadorj Projective superspace 10

Motivation for using Projective superspace Review of Projective superspace Super Yang-Mills theory in Projective superspace framework Summary Gauge connection and transformations Yang-Mills action in chiral representation Classic action from 1-loop divergent diagram Abelian limit

Gauge transformation and connection

Under a gauge transformation polar multiplets transform: Υ − → e˙

ıΛΥ , Υ −

→ Υe−˙

ıΛ

To make the action inv under the transformation introduce a real projective bridge V that converts Λ transformation to the Λ one. eV − → e˙

ıΛeV e−˙ ıΛ

Now using eV make all fields transform in arctic Λ or antarctic Λ representations.

• ΥA ≡ Υ −

→ e ˙

ıΛ

Υ ,

• ΥA ≡ ΥeV −

→ Υe−˙

ıΛ

• r
• ΥA ≡ eV Υ −

→ e ˙

ıΛ

Υ ,

• ΥA ≡ Υ −

→ Υe−˙

ıΛ

Ariunzul Davgadorj Projective superspace 11

Motivation for using Projective superspace Review of Projective superspace Super Yang-Mills theory in Projective superspace framework Summary Gauge connection and transformations Yang-Mills action in chiral representation Classic action from 1-loop divergent diagram Abelian limit

These newly defined polar multiplets are annihilated by ∇ and ∇. Next, one can also define vector representation. Split: eV = eUeU eU − → e ˙

ıKeUe−˙ ıΛ , eU −

→ e ˙

ıΛeUe−˙ ıK

Now let the fields transform with the ζ-independent real field K.

• Υvec ≡ eUΥ −

→ e˙

ıK

Υ ,

• Υvec ≡ ΥeU −

→ Υe−˙

ıK

They are annihilated by:

α ≡ eU∇αe−U = e−U∇αeU = ∇α + Γα(ζ) ˙ α ≡ eU∇ ˙ αe−U = e−U∇ ˙ αeU = ∇ ˙ α + Γ ˙ α(ζ)

Ariunzul Davgadorj Projective superspace 12

Motivation for using Projective superspace Review of Projective superspace Super Yang-Mills theory in Projective superspace framework Summary Gauge connection and transformations Yang-Mills action in chiral representation Classic action from 1-loop divergent diagram Abelian limit

One can identify Γα(ζ) as a gauge connection. And notice that Γα(ζ) = Γ1

α + ζΓ2 α. Which means: α = Dα + ζQα

with Dα = Dα + Γ1

α

Qα = Qα + Γ2

α

D and Q are N = 2 gauge covariant derivatives. Their algebra is: {Dα, Qβ} = ˙ ıCαβW , {Dα, D ˙

α} = {Qα, Q ˙ α} = ˙

ı

α ˙ α

Projective gauge covariant derivatives satisfy: {

α(ζ1), β(ζ2)} = ˙

ıCαβ(ζ1−ζ2)W or equiv {

α, [∂ζ, β]} = ˙

ıCαζW W is a field strength in vector representation.

Ariunzul Davgadorj Projective superspace 13

Motivation for using Projective superspace Review of Projective superspace Super Yang-Mills theory in Projective superspace framework Summary Gauge connection and transformations Yang-Mills action in chiral representation Classic action from 1-loop divergent diagram Abelian limit

Find the field strength in arctic/antarctic representation. {∇α, [e−U∂ζeU, ∇β]} = ˙ ıCαβe−UWeU ≡ ˙ ıCαβW(ζ) {∇α, [eU∂ζe−U, ∇β]} = ˙ ıCαβeUWe−U ≡ ˙ ıCαβ W(ζ) From here we define the gauge covariant ζ-derivatives. Dζ = ∂ζ + Aζ = e−U∂ζeU ,

• Dζ = ∂ζ +

Aζ = eU∂ζe−U Field strengths are expressed by the ζ-connections: W = −˙ ı∇2Aζ ,

• W = −˙

ı∇2 Aζ , where e−V (∂ζeV ) = Aζ − e−V AζeV

Ariunzul Davgadorj Projective superspace 14

Motivation for using Projective superspace Review of Projective superspace Super Yang-Mills theory in Projective superspace framework Summary Gauge connection and transformations Yang-Mills action in chiral representation Classic action from 1-loop divergent diagram Abelian limit

Notice that the Aζ contains only ≥ 0 powers of ζ and Aζ only ≤ −2 powers.

1 ζ2

∞

k=0

• ζ1

ζ2

k ≡

1 ζ21

We call now (eV − 1)n = eV (ζn) − 1 := Xn which will play the essential role of the unconstrained prepotential Write the relation between the arctic and the antarctic connections in terms of X Solve the equation recursively in powers of X ∂X = Aζ − Aζ + XAζ − AζX A(1)

ζ

− A(1)

ζ

= ∂ζX , A(n+1)

ζ

− A(n+1)

ζ

= −XA(n)

ζ

+ A(n)

ζ X

Now introduce operators Π± that project into positive and negative powers of ζ.

Ariunzul Davgadorj Projective superspace 15

Motivation for using Projective superspace Review of Projective superspace Super Yang-Mills theory in Projective superspace framework Summary Gauge connection and transformations Yang-Mills action in chiral representation Classic action from 1-loop divergent diagram Abelian limit

A(1)

ζ

= Π+(∂ζX) , A(2)

ζ

= −Π+(XΠ+(∂ζX) + Π−(∂ζX)X) , A(3)

ζ

= Π+ [XΠ+(XΠ+(∂ζX)) + Π−(XΠ+(∂ζX))X +XΠ+(Π−(∂ζX)X)) + Π−(Π−(∂ζX)X)X]

• A(1)

ζ

= −Π−(∂ζX) ,

• A(2)

ζ

= Π−(XΠ+(∂ζX) + Π−(∂ζX)X) ,

• A(3)

ζ

= −Π−[XΠ+(XΠ+(∂ζX)) + Π−(XΠ+(∂ζX))X +XΠ+(Π−(∂ζX)X)) + Π−(Π−(∂ζX)X)X]

Ariunzul Davgadorj Projective superspace 16

Motivation for using Projective superspace Review of Projective superspace Super Yang-Mills theory in Projective superspace framework Summary Gauge connection and transformations Yang-Mills action in chiral representation Classic action from 1-loop divergent diagram Abelian limit

Now do the recursion to find the full form of the connections to be: Aζ =

∞

• n=1

(−1)n+1

• dζ1 . . .
• dζn

(eV − 1)1 . . . (eV − 1)n ζ21 . . . ζn,n−1 1 ζ10ζn,0

• Aζ =

∞

• n=1

(−1)n+1

• dζ1 . . .
• dζn

(eV − 1)1 . . . (eV − 1)n ζ21 . . . ζn,n−1 1 ζ01ζ0,n Aζ and Aζ transform correctly under the infinitesimal non-abelian gauge transformation: δeV = iΛeV − eV iΛ Aζ → −i∂Λ + [iΛ, Aζ] ,

• Aζ → −i∂Λ + [iΛ,

Aζ]

Ariunzul Davgadorj Projective superspace 17

Motivation for using Projective superspace Review of Projective superspace Super Yang-Mills theory in Projective superspace framework Summary Gauge connection and transformations Yang-Mills action in chiral representation Classic action from 1-loop divergent diagram Abelian limit

This can be shown by picking up the terms that has the same power in X from δA(n+1)

ζ

and δA(n)

ζ

and then using the induction we find: [δA(n+1)

ζ

+ δA(n)

ζ ](n) = [iΛ, A(n) ζ ] ,

[δ A(n+1)

ζ

+ δ A(n)

ζ ](n) = [iΛ,

A(n)

ζ ] ,

δAζ =

∞

• n=1

δA(n)

ζ

=

∞

• n=1

[δA(n)

ζ ](n−1) + ∞

• n=1

[δA(n)

ζ ](n)

= [δA(1)

ζ ](0) + ∞

• n=1

[δA(n+1)

ζ

+ δA(n)

ζ ](n)

= −i∂Λ + [iΛ, Aζ]

Ariunzul Davgadorj Projective superspace 18

Motivation for using Projective superspace Review of Projective superspace Super Yang-Mills theory in Projective superspace framework Summary Gauge connection and transformations Yang-Mills action in chiral representation Classic action from 1-loop divergent diagram Abelian limit

Action in chiral representation

N = 2 Yang-Mills theory can be constructed within the conventional R4|8 superspace in terms of covariant chiral field

• strength. However we would like to write the action in projective

space using unconstrained prepotential V . S = 1

2

• [d4θ]CTr(WW) = 1

2

• [d4θ]C

dζ

ζ Tr(WW)

Since we have found the proper expression for the connection Aζ and consequently W in terms of eV − 1, so we can write the action using this unconstrained prepotential X:

Ariunzul Davgadorj Projective superspace 19

Motivation for using Projective superspace Review of Projective superspace Super Yang-Mills theory in Projective superspace framework Summary Gauge connection and transformations Yang-Mills action in chiral representation Classic action from 1-loop divergent diagram Abelian limit

S = 1 2

• [d4θ]C
• dζ Tr(WW)

ζ = −1 2

• [d4θ]C
• dζ Tr(∇2A∇2A)

ζ = 1 4

• [d4θ]C
• dζ∇2 Tr(∇αA∇αA)

ζ = 1 4

∞

• n=2

n−1

• k=1
• [d4θ]C
• dζ∇2 Tr(∇αA(k)∇αA(n−k))

ζ After ζ manipulations and using cyclicity of Tr one can get:

S = − 1

4

∞

n=2(−1)n n m=2

• [d4θ]C
• dζ1 . . . dζnD2 Tr(DαX1...DαXm...Xn))

ζ21...ζn,n−1ζ1n (ζ1−ζm)2 ζ2

1ζ2 m

Ariunzul Davgadorj Projective superspace 20

Motivation for using Projective superspace Review of Projective superspace Super Yang-Mills theory in Projective superspace framework Summary Gauge connection and transformations Yang-Mills action in chiral representation Classic action from 1-loop divergent diagram Abelian limit

The most divergent part of the 1-loop diagram

ζ1 ζ2 ζ3 ∇4

2

∇4

3

∇4

2

∇4

1

∇4

1

∇4

3

S = ∞

n=2 (−1)n n

• d8θ dζ1...dζn

ζ21...ζ1n Tr

• eV − 1
• 1 . . .
• eV − 1
• n
• Ariunzul Davgadorj

Projective superspace 21

Motivation for using Projective superspace Review of Projective superspace Super Yang-Mills theory in Projective superspace framework Summary Gauge connection and transformations Yang-Mills action in chiral representation Classic action from 1-loop divergent diagram Abelian limit

The full measure is: d8θ = [d4θ]CD2Q2 = 1

ζ2 [d4θ]P∇2∇2.

Then using the fact that X is projective convert all the Q derivatives into D’s and ζ-manipulating, finally the ”loop action” will turn to the same expression derived from the ”chiral action”. The action is invariant under infinitesimal gauge transformation: δ(eV − 1) = ˙ ıΛ − ˙ ıΛ + ˙ ıΛ(eV − 1) − (eV − 1)˙ ıΛ Variation of the action leads to the correct e.o.m (Bianchi identity equals to zero). ∇2W = ∇2W = 0 δS =

• d8θ
• dζTr(e−V δeV Aζ)

Ariunzul Davgadorj Projective superspace 22

Motivation for using Projective superspace Review of Projective superspace Super Yang-Mills theory in Projective superspace framework Summary Gauge connection and transformations Yang-Mills action in chiral representation Classic action from 1-loop divergent diagram Abelian limit

Abelian limit

One can consistency check if this non-Abelian action produce the right action in the Abelian limit. Aζ − Aζ =

∞

• (−1)nX n∂ζX = ∂ζ ln(1 + X) = ∂ζV

SAbelian = −1 2

• [d4θ]C
• dζ1dζ2D2Q2 Tr(V1V2)

ζ12ζ21 = −1 2

• [d4θ]CTr(D2υ−1D2υ−1)

= −

• D2D2Tr(D2υ1D2υ−1 − 1

2υ0DD2Dυ0) =

• D2D2Tr(ψD2D2ψ + 1

2υDD2Dυ)

Ariunzul Davgadorj Projective superspace 23

Motivation for using Projective superspace Review of Projective superspace Super Yang-Mills theory in Projective superspace framework Summary

Summary

In projective superspace we deal with closed contour integrals. The so called ”ǫ-prescription” which describes the way out when there is a coinciding singularity problem by shifting the contours by little ǫ.

1 ζ1(1+ǫ)−ζ2(1−ǫ) ≡ 1 ζ1

∞

n=0( ζ2 ζ1 )n

Result: We have described N = 2 super Yang-Mills action in terms of unconstrained prepotential V in pure projective formalism. Next step

Find the low energy effective action. many more...

Ariunzul Davgadorj Projective superspace 24