The N = 2 Gauss-Bonnet invariant in and out of superspace
|
|
- Lily Mason
- 5 years ago
- Views:
Transcription
1 The N = 2 Gauss-Bonnet invariant in and out of superspace Daniel Butter NIKHEF College Station April 25, 2013 Based on work with B. de Wit, S. Kuzenko, and I. Lodato Daniel Butter (NIKHEF) Super GB 1 / 29
2 Motivation The Gauss-Bonnet is a very nice higher derivative invariant: L GB = 1 4 εmnpq R mn ab R pq cd ε abcd = C abcd C abcd 2R ab R ab R2. Although it is topological by itself, it often appears multiplied by a scalar function in specific applications (e.g. anomalies, 5D to 4D reductions, etc.). Its supersymmetric version will appear in corresponding situations. The manifestly supersymmetric 4D N = 1 GB is well-known. [Townsend and van Nieuwenhuizen; Ferrara and Villasante; Buchbinder and Kuzenko] This reflects the completeness of our understanding of 4D N = 1 higher derivative terms. We should better understand higher derivative terms in 4D N = 2! Daniel Butter (NIKHEF) Super GB 2 / 29
3 Outline 1 Superspace philosophy 2 The N = 1 Gauss-Bonnet in superspace and an N = 2 mystery 3 Superconformal tensor calculus in superspace 4 The Gauss-Bonnet invariant in and out of N = 2 superspace Daniel Butter (NIKHEF) Super GB 3 / 29
4 Outline 1 Superspace philosophy 2 The N = 1 Gauss-Bonnet in superspace and an N = 2 mystery 3 Superconformal tensor calculus in superspace 4 The Gauss-Bonnet invariant in and out of N = 2 superspace Daniel Butter (NIKHEF) Super GB 4 / 29
5 Quick review: N = 1 supersymmetry in spinor notation In a Weyl-basis, the γ-matrices are ( ) 1 0 γ 5 =, γ m = 0 1 ( 0 ) (σ m ) α α ( σ m ) αα 0 A Dirac fermion Ψ and its conjugate Ψ look like ( ) χα ( ) Ψ = ψ α, Ψ = ψ α χ α N = 1 supersymmetry in four dimensions: {Q α, Q α } = 2i (σ m ) α α m. Daniel Butter (NIKHEF) Super GB 5 / 29
6 Superspace makes supersymmetry manifest Superspace: add new Grassmann coordinates θ α and Interpret the supersymmetry generator Q α D α as Chiral multiplet is short D α = θ α + i(σm ) α α θ α m D α Φ = 0 = Φ = φ + θ α ψ α + θ 2 F + (x-derivative terms) The free massless chiral multiplet action is given by ( d 4 x d 2 θ d 2 θ ΦΦ = d 4 i x φ φ 2 ψ α ( σ m ) αα m ψ α + F F ) θ α If we want to gauge a U(1) symmetry, the standard lore is to add an explicit vector multiplet prepotential V, do Wess-Zumino gauge, etc. Daniel Butter (NIKHEF) Super GB 6 / 29
7 How to avoid prepotentials and Wess-Zumino gauges Encode gauge connection in superspace: A = dx m A m A = dz M A M. Build covariant derivative D A = (D α, D α, D a ) in superspace. This is supersymmetric minimal substitution. Keep the same action ΦΦ but replace the flat chiral constraint with a covariant chiral constraint D α Φ = 0 D α Φ = 0. We define the components of Φ not by a θ expansion but by φ Φ θ=0, ψ α = D α Φ θ=0, F = 1 4 Dα D α Φ θ=0. The supergravity story is essentially analogous. 1 Lift all connections to superconnections. 2 Replace flat derivatives with covariant derivatives. Daniel Butter (NIKHEF) Super GB 7 / 29
8 How to avoid prepotentials and Wess-Zumino gauges Encode gauge connection in superspace: A = dx m A m A = dz M A M. Build covariant derivative D A = (D α, D α, D a ) in superspace. This is supersymmetric minimal substitution. Keep the same action ΦΦ but replace the flat chiral constraint with a covariant chiral constraint D α Φ = 0 D α Φ = 0. We define the components of Φ not by a θ expansion but by φ Φ θ=0, ψ α = D α Φ θ=0, F = 1 4 Dα D α Φ θ=0. The supergravity story is essentially analogous. 1 Lift all connections to superconnections. 2 Replace flat derivatives with covariant derivatives. Daniel Butter (NIKHEF) Super GB 7 / 29
9 From superspace to tensor calculus and back again For every covariant field e.g. ψ α, there is some superfield, e.g. D α Φ with ψ α = D α Φ θ=0 Supersymmetry on the component field corresponds to covariant spinor derivative on the superfield: δ Q ψ α = ξ β Q β ψ α + ξ β Q βψ α δ ξ D α Φ θ=0 = ξ β D β D α Φ θ=0 + ξ β D βd α Φ θ=0 Given covariant superfields, one can evaluate all their components and derive supersymmetry transformations. The converse is also possible: one can often lift component results to superspace expressions. Properties/symmetries of the tensor calculus reflect those of the superspace. Poincaré tensor calculus Poincaré superspace Superconformal tensor calculus (Super)conformal superspace Daniel Butter (NIKHEF) Super GB 8 / 29
10 Outline 1 Superspace philosophy 2 The N = 1 Gauss-Bonnet in superspace and an N = 2 mystery 3 Superconformal tensor calculus in superspace 4 The Gauss-Bonnet invariant in and out of N = 2 superspace Daniel Butter (NIKHEF) Super GB 9 / 29
11 N = 1 Poincaré superspace Old minimal Poincaré supergravity involves the field content e a m, ψ α m, V }{{} m, }{{} M massive vector complex scalar Superspace geometry involves curvatures which are built out of the superfields W αβγ, G a, R. These contain (respectively) C abcd, R ab and R. General integrals in superspace look like S D = d 4 x e [L] D = d 4 x d 2 θ d 2 θ E L, L is unconstrained S F = d 4 x e [L c ] F = d 4 x d 2 θ E L c, D α L c = 0 A caveat: any D-term can be written as an F -term. Let s consider only honest F -terms. Daniel Butter (NIKHEF) Super GB 10 / 29
12 N = 1 Poincaré superspace Old minimal Poincaré supergravity involves the field content e a m, ψ α m, V }{{} m, }{{} M massive vector complex scalar Superspace geometry involves curvatures which are built out of the superfields W αβγ, G a, R. These contain (respectively) C abcd, R ab and R. General integrals in superspace look like S D = d 4 x e [L] D = d 4 x d 2 θ d 2 θ E L, L is unconstrained S F = d 4 x e [L c ] F = d 4 x d 2 θ E L c, D α L c = 0 A caveat: any D-term can be written as an F -term. Let s consider only honest F -terms. Daniel Butter (NIKHEF) Super GB 10 / 29
13 N = 1 Poincaré superspace Old minimal Poincaré supergravity involves the field content e a m, ψ α m, V }{{} m, }{{} M massive vector complex scalar Superspace geometry involves curvatures which are built out of the superfields W αβγ, G a, R. These contain (respectively) C abcd, R ab and R. General integrals in superspace look like S D = d 4 x e [L] D = d 4 x d 2 θ d 2 θ E L, L is unconstrained S F = d 4 x e [L c ] F = d 4 x d 2 θ E L c, D α L c = 0 A caveat: any D-term can be written as an F -term. Let s consider only honest F -terms. Daniel Butter (NIKHEF) Super GB 10 / 29
14 N = 1 actions and higher derivatives The Einstein-Hilbert Lagrangian is L EH = 3 2 d 4 x d 2 θ d 2 1 θ E 2 M P 2 d 4 x e R In N = 1, the only purely chiral (non-singular) invariants are L c = c 1 W αβγ W αβγ + c 2 3 (W αβγ W αβγ ) 2 + [L c ] F c 1 C abcd C abcd + c 2 3 (C ) 2 (Dψ) 2 + The other (non-singular) terms generically are D-term invariants with L = c 3 R R + c 4 G a G a + c 5 D α D α R + c 6 4 (D α G a ) 4 + c 7 3 (W αβγ ) 2 G 2 + [L D ] c 3 R 2 + c 4 R ab R ab + c 5 R + c 6 4 (R ab ) 4 + c 7 3 (C ) 2 (Dψ) 2 + Daniel Butter (NIKHEF) Super GB 11 / 29
15 N = 1 actions and higher derivatives The Einstein-Hilbert Lagrangian is L EH = 3 2 d 4 x d 2 θ d 2 1 θ E 2 M P 2 d 4 x e R In N = 1, the only purely chiral (non-singular) invariants are L c = c 1 W αβγ W αβγ + c 2 3 (W αβγ W αβγ ) 2 + [L c ] F c 1 C abcd C abcd + c 2 3 (C ) 2 (Dψ) 2 + The other (non-singular) terms generically are D-term invariants with L = c 3 R R + c 4 G a G a + c 5 D α D α R + c 6 4 (D α G a ) 4 + c 7 3 (W αβγ ) 2 G 2 + [L D ] c 3 R 2 + c 4 R ab R ab + c 5 R + c 6 4 (R ab ) 4 + c 7 3 (C ) 2 (Dψ) 2 + Daniel Butter (NIKHEF) Super GB 11 / 29
16 N = 1 actions and higher derivatives The Einstein-Hilbert Lagrangian is L EH = 3 2 d 4 x d 2 θ d 2 1 θ E 2 M P 2 d 4 x e R In N = 1, the only purely chiral (non-singular) invariants are L c = c 1 W αβγ W αβγ + c 2 3 (W αβγ W αβγ ) 2 + [L c ] F c 1 C abcd C abcd + c 2 3 (C ) 2 (Dψ) 2 + The other (non-singular) terms generically are D-term invariants with L = c 3 R R + c 4 G a G a + c 5 D α D α R + c 6 4 (D α G a ) 4 + c 7 3 (W αβγ ) 2 G 2 + [L D ] c 3 R 2 + c 4 R ab R ab + c 5 R + c 6 4 (R ab ) 4 + c 7 3 (C ) 2 (Dψ) 2 + Daniel Butter (NIKHEF) Super GB 11 / 29
17 N = 1 actions and higher derivatives The Einstein-Hilbert Lagrangian is L EH = 3 2 d 4 x d 2 θ d 2 1 θ E 2 M P 2 d 4 x e R In N = 1, the only purely chiral (non-singular) invariants are L c = c 1 W αβγ W αβγ + c 2 3 (W αβγ W αβγ ) 2 + [L c ] F c 1 C abcd C abcd + c 2 3 (C ) 2 (Dψ) 2 + The other (non-singular) terms generically are D-term invariants with L = c 3 R R + c 4 G a G a + c 5 D α D α R + c 6 4 (D α G a ) 4 + c 7 3 (W αβγ ) 2 G 2 + [L D ] c 3 R 2 + c 4 R ab R ab + c 5 R + c 6 4 (R ab ) 4 + c 7 3 (C ) 2 (Dψ) 2 + The Gauss-Bonnet invariant is a certain combination of the highlighted terms. Moreover, the full supersymmetric result (with all the fermions) is topological. Daniel Butter (NIKHEF) Super GB 11 / 29
18 The mystery in N = 2 For simplicity, let s stick with the minimal SU(2) multiplet: e m a, ψ mα i, V m i j, W m, = gauge connections T ab ij, χ i α, D, Y ij, V m = covariant fields Superspace geometry includes curvatures W αβ }{{}, S ij, G a, Y αβ chiral Weyl superfield The minimal multiplet action is the F-term integral of a constant S minimal = 3 ( 2 M P 2 d 4 x d 4 θ E = 2 d 4 x e 1 ) 2 R + 3D + The obvious higher derivative chiral invariant is L c = W αβ W αβ [L c ] F (C ) 2 D-term invariants lead to dimension 6 and higher: L = 1 ( ) 2 S ij S ij + Y αβ Y αβ +, [L] D 1 ( ) 2 R 3 + R R + Daniel Butter (NIKHEF) Super GB 12 / 29
19 The mystery in N = 2 For simplicity, let s stick with the minimal SU(2) multiplet: e m a, ψ mα i, V m i j, W m, = gauge connections T ab ij, χ i α, D, Y ij, V m = covariant fields Superspace geometry includes curvatures W αβ }{{}, S ij, G a, Y αβ chiral Weyl superfield The minimal multiplet action is the F-term integral of a constant S minimal = 3 ( 2 M P 2 d 4 x d 4 θ E = 2 d 4 x e 1 ) 2 R + 3D + The obvious higher derivative chiral invariant is L c = W αβ W αβ [L c ] F (C ) 2 D-term invariants lead to dimension 6 and higher: L = 1 ( ) 2 S ij S ij + Y αβ Y αβ +, [L] D 1 ( ) 2 R 3 + R R + So how do we construct the N = 2 Gauss-Bonnet in superspace? Daniel Butter (NIKHEF) Super GB 12 / 29
20 The mystery in N = 2 For simplicity, let s stick with the minimal SU(2) multiplet: e m a, ψ mα i, V m i j, W m, = gauge connections T ab ij, χ i α, D, Y ij, V m = covariant fields Superspace geometry includes curvatures W αβ }{{}, S ij, G a, Y αβ chiral Weyl superfield The minimal multiplet action is the F-term integral of a constant S minimal = 3 ( 2 M P 2 d 4 x d 4 θ E = 2 d 4 x e 1 ) 2 R + 3D + The obvious higher derivative chiral invariant is L c = W αβ W αβ [L c ] F (C ) 2 D-term invariants lead to dimension 6 and higher: L = 1 ( ) 2 S ij S ij + Y αβ Y αβ +, [L] D 1 ( ) 2 R 3 + R R + So how do we construct the N = 2 Gauss-Bonnet in superspace? Daniel Butter (NIKHEF) Super GB 12 / 29
21 The mystery in N = 2 For simplicity, let s stick with the minimal SU(2) multiplet: e m a, ψ mα i, V m i j, W m, = gauge connections T ab ij, χ i α, D, Y ij, V m = covariant fields Superspace geometry includes curvatures W αβ }{{}, S ij, G a, Y αβ chiral Weyl superfield The minimal multiplet action is the F-term integral of a constant S minimal = 3 ( 2 M P 2 d 4 x d 4 θ E = 2 d 4 x e 1 ) 2 R + 3D + The obvious higher derivative chiral invariant is L c = W αβ W αβ [L c ] F (C ) 2 D-term invariants lead to dimension 6 and higher: L = 1 ( ) 2 S ij S ij + Y αβ Y αβ +, [L] D 1 ( ) 2 R 3 + R R + So how do we construct the N = 2 Gauss-Bonnet in superspace? Daniel Butter (NIKHEF) Super GB 12 / 29
22 The mystery in N = 2 For simplicity, let s stick with the minimal SU(2) multiplet: e m a, ψ mα i, V m i j, W m, = gauge connections T ab ij, χ i α, D, Y ij, V m = covariant fields Superspace geometry includes curvatures W αβ }{{}, S ij, G a, Y αβ chiral Weyl superfield The minimal multiplet action is the F-term integral of a constant S minimal = 3 ( 2 M P 2 d 4 x d 4 θ E = 2 d 4 x e 1 ) 2 R + 3D + The obvious higher derivative chiral invariant is L c = W αβ W αβ [L c ] F (C ) 2 D-term invariants lead to dimension 6 and higher: L = 1 ( ) 2 S ij S ij + Y αβ Y αβ +, [L] D 1 ( ) 2 R 3 + R R + So how do we construct the N = 2 Gauss-Bonnet in superspace? Daniel Butter (NIKHEF) Super GB 12 / 29
23 Outline 1 Superspace philosophy 2 The N = 1 Gauss-Bonnet in superspace and an N = 2 mystery 3 Superconformal tensor calculus in superspace 4 The Gauss-Bonnet invariant in and out of N = 2 superspace Daniel Butter (NIKHEF) Super GB 13 / 29
24 Review: Conformal gravity Let s briefly review conformal gravity since the superconformal case is quite similar. Introduce covariant derivative so that c = a a. e m a a = m 1 2 ω m ab M ab b m D f m a K a We may consistently impose curvature constraints to determine ω m ab and f m a. Only independent fields are e m a and b m. To recover Poincaré gravity, use K-transformation to fix b m = 0. Fix dilatations by gauging some compensator field to a constant. Daniel Butter (NIKHEF) Super GB 14 / 29
25 Review: Conformal gravity Let s briefly review conformal gravity since the superconformal case is quite similar. Introduce covariant derivative so that c = a a. e m a a = m 1 2 ω m ab M ab b m D f m a K a We may consistently impose curvature constraints to determine ω m ab and f m a. Only independent fields are e m a and b m. To recover Poincaré gravity, use K-transformation to fix b m = 0. Fix dilatations by gauging some compensator field to a constant. Daniel Butter (NIKHEF) Super GB 14 / 29
26 N = 2 superconformal vs. Poincaré tensor calculus For N = 2 conformal supergravity, we have the fundamental connections e m a, ψ m αi, b m, A m, V m i j Constraints = composite ω m ab, f m a and φ m α i Choosing a certain compensator, we can reduce to a Poincaré tensor calculus real scalar multiplet ( ) U(2) Poincaré tensor calculus U(2) superspace [Howe 81] chiral multiplet (16+16) SU(2) Poincaré tensor calculus SU(2) superspace [Grimm 80] vector multiplet (8+8) minimal SU(2) Poincaré tensor calculus reduced SU(2) superspace Poincaré supergravity multiplets still can describe superconformal theories. Then the additional compensator fields drop out of the action. But this can be complicated. Daniel Butter (NIKHEF) Super GB 15 / 29
27 N = 2 superconformal vs. Poincaré tensor calculus For N = 2 conformal supergravity, we have the fundamental connections e m a, ψ m αi, b m, A m, V m i j Constraints = composite ω m ab, f m a and φ m α i Choosing a certain compensator, we can reduce to a Poincaré tensor calculus real scalar multiplet ( ) U(2) Poincaré tensor calculus U(2) superspace [Howe 81] chiral multiplet (16+16) SU(2) Poincaré tensor calculus SU(2) superspace [Grimm 80] vector multiplet (8+8) minimal SU(2) Poincaré tensor calculus reduced SU(2) superspace Poincaré supergravity multiplets still can describe superconformal theories. Then the additional compensator fields drop out of the action. But this can be complicated. Daniel Butter (NIKHEF) Super GB 15 / 29
28 N = 2 superconformal vs. Poincaré tensor calculus For N = 2 conformal supergravity, we have the fundamental connections e m a, ψ m αi, b m, A m, V m i j Constraints = composite ω m ab, f m a and φ m α i Choosing a certain compensator, we can reduce to a Poincaré tensor calculus real scalar multiplet ( ) U(2) Poincaré tensor calculus U(2) superspace [Howe 81] chiral multiplet (16+16) SU(2) Poincaré tensor calculus SU(2) superspace [Grimm 80] vector multiplet (8+8) minimal SU(2) Poincaré tensor calculus reduced SU(2) superspace Poincaré supergravity multiplets still can describe superconformal theories. Then the additional compensator fields drop out of the action. But this can be complicated. Daniel Butter (NIKHEF) Super GB 15 / 29
29 N = 2 superconformal vs. Poincaré tensor calculus For N = 2 conformal supergravity, we have the fundamental connections e m a, ψ m αi, b m, A m, V m i j Constraints = composite ω m ab, f m a and φ m α i Choosing a certain compensator, we can reduce to a Poincaré tensor calculus real scalar multiplet ( ) U(2) Poincaré tensor calculus U(2) superspace [Howe 81] chiral multiplet (16+16) SU(2) Poincaré tensor calculus SU(2) superspace [Grimm 80] vector multiplet (8+8) minimal SU(2) Poincaré tensor calculus reduced SU(2) superspace Poincaré supergravity multiplets still can describe superconformal theories. Then the additional compensator fields drop out of the action. But this can be complicated. Daniel Butter (NIKHEF) Super GB 15 / 29
30 N = 2 superconformal vs. Poincaré tensor calculus For N = 2 conformal supergravity, we have the fundamental connections e m a, ψ m αi, b m, A m, V m i j Constraints = composite ω m ab, f m a and φ m α i Choosing a certain compensator, we can reduce to a Poincaré tensor calculus real scalar multiplet ( ) U(2) Poincaré tensor calculus U(2) superspace [Howe 81] chiral multiplet (16+16) SU(2) Poincaré tensor calculus SU(2) superspace [Grimm 80] vector multiplet (8+8) minimal SU(2) Poincaré tensor calculus reduced SU(2) superspace Poincaré supergravity multiplets still can describe superconformal theories. Then the additional compensator fields drop out of the action. But this can be complicated. Daniel Butter (NIKHEF) Super GB 15 / 29
31 N = 2 superconformal vs. Poincaré tensor calculus For N = 2 conformal supergravity, we have the fundamental connections e m a, ψ m αi, b m, A m, V m i j Constraints = composite ω m ab, f m a and φ m α i Choosing a certain compensator, we can reduce to a Poincaré tensor calculus real scalar multiplet ( ) U(2) Poincaré tensor calculus U(2) superspace [Howe 81] chiral multiplet (16+16) SU(2) Poincaré tensor calculus SU(2) superspace [Grimm 80] vector multiplet (8+8) minimal SU(2) Poincaré tensor calculus reduced SU(2) superspace Poincaré supergravity multiplets still can describe superconformal theories. Then the additional compensator fields drop out of the action. But this can be complicated. Daniel Butter (NIKHEF) Super GB 15 / 29
32 N = 2 conformal superspace For a superconformal model, use conformal superspace (no compensator!) [DB 11] Introduce covariant derivative A = ( αi, αi, a ) with connections in the full superconformal algebra. Single curvature superfield W αβ contains all of the superconformal curvatures and the matter fields T ab ij, χ i α, D All constraints and curvatures of N = 2 superconformal tensor calculus are reproduced. This was worked out long ago at the linearized level in superspace. [Bergshoeff, de Roo, de Wit 81] Daniel Butter (NIKHEF) Super GB 16 / 29
33 Outline 1 Superspace philosophy 2 The N = 1 Gauss-Bonnet in superspace and an N = 2 mystery 3 Superconformal tensor calculus in superspace 4 The Gauss-Bonnet invariant in and out of N = 2 superspace Daniel Butter (NIKHEF) Super GB 17 / 29
34 Gauss-Bonnet invariant in conformal gravity c c ln φ is K-invariant for any choice of weight w for φ and equals ln φ + D a( 2 ) 3 RD a ln φ 2R ab D b ln φ + w 6 R w 2 Rab R ab + w 6 R2. We can get very close to the usual Gauss-Bonnet in conformal gravity: L GB = C abcd C abcd + 4 w c c ln φ = L GB R + 4 ) (D w Da a ln φ RD a ln φ 2R ab D b ln φ Daniel Butter (NIKHEF) Super GB 18 / 29
35 Gauss-Bonnet invariant in conformal gravity c c ln φ is K-invariant for any choice of weight w for φ and equals ln φ + D a( 2 ) 3 RD a ln φ 2R ab D b ln φ + w 6 R w 2 Rab R ab + w 6 R2. We can get very close to the usual Gauss-Bonnet in conformal gravity: L GB = C abcd C abcd + 4 w c c ln φ = L GB R + 4 ) (D w Da D a ln φ RD a ln φ 2R ab D b ln φ This combination is known to physicists as the Riegert operator. [Riegert 84; Paneitz 08] Daniel Butter (NIKHEF) Super GB 18 / 29
36 Gauss-Bonnet invariant in conformal gravity c c ln φ is K-invariant for any choice of weight w for φ and equals ln φ + D a( 2 ) 3 RD a ln φ 2R ab D b ln φ + w 6 R w 2 Rab R ab + w 6 R2. We can get very close to the usual Gauss-Bonnet in conformal gravity: L GB = C abcd C abcd + 4 w c c ln φ = L GB R + 4 ) (D w Da a ln φ RD a ln φ 2R ab D b ln φ This combination is known to physicists as the Riegert operator. [Riegert 84; Paneitz 08] This allows you to see that under a finite Weyl transformation ( g L GB + 2 ) 3 R = g (L GB + 2 ) 3 R 4 gd a( D a σ + 2 ) 3 RD aσ 2R ab D b σ Daniel Butter (NIKHEF) Super GB 18 / 29
37 Supersymmetrizing the Gauss-Bonnet invariant Let us take the point of view that we wish to supersymmetrize L GB = C abcd C abcd + 4 w c c ln φ This can be done using superconformal methods. The first term should supersymmetrize easily to the superconformal Weyl-squared invariant. The second term can be supersymmetrized once we decide on the multiplet for φ. For both N = 1, 2, the natural choice is to take φ to be complex and the lowest component of a chiral multiplet. For both N = 1, 2 the natural quantity will be complex: L Γ = 1 8 (C ) w c c ln φ + Daniel Butter (NIKHEF) Super GB 19 / 29
38 N = 1 conformal superspace construction In N = 1 conformal superspace, only a single superspace curvature W αβγ. [DB 09] The only option is an F -term action with L c = W αβγ W αβγ [W αβγ W αβγ ] F = 1 16 C abcdc abcd 1 16 C abcd C abcd + additional terms. In flat space, it is easy to check that 1 64 we covariantize? Take chiral superfield Φ of weight w and introduce S(ln Φ) = ln Φ. d 2 θ D 2 D 2 D2 ln Φ = ln φ. Can One can check that S is chiral and also S-invariant. This means we can use it to build chiral actions. Daniel Butter (NIKHEF) Super GB 20 / 29
39 N = 1 conformal superspace construction In N = 1 conformal superspace, only a single superspace curvature W αβγ. [DB 09] The only option is an F -term action with L c = W αβγ W αβγ [W αβγ W αβγ ] F = 1 16 C abcdc abcd 1 16 C abcd C abcd + additional terms. In flat space, it is easy to check that 1 64 we covariantize? Take chiral superfield Φ of weight w and introduce S(ln Φ) = ln Φ. d 2 θ D 2 D 2 D2 ln Φ = ln φ. Can One can check that S is chiral and also S-invariant. This means we can use it to build chiral actions. Daniel Butter (NIKHEF) Super GB 20 / 29
40 N = 1 conformal superspace construction In N = 1 conformal superspace, only a single superspace curvature W αβγ. [DB 09] The only option is an F -term action with L c = W αβγ W αβγ [W αβγ W αβγ ] F = 1 16 C abcdc abcd 1 16 C abcd C abcd + additional terms. In flat space, it is easy to check that 1 64 we covariantize? Take chiral superfield Φ of weight w and introduce S(ln Φ) = ln Φ. d 2 θ D 2 D 2 D2 ln Φ = ln φ. Can One can check that S is chiral and also S-invariant. This means we can use it to build chiral actions. Daniel Butter (NIKHEF) Super GB 20 / 29
41 N = 1 conformal superspace construction We choose the chiral F -term Lagrangian is Γ := W αβγ W αβγ + 1 S(ln Φ), 4w [Γ] F = 1 8 (C ) w c c ln φ + additional terms Rewriting in N = 1 Poincaré superspace... S(ln Φ) = 1 4 ( D ( 1 2 8R) 16 D2 D2 ln Φ + D α (G α α D α ln Φ) + 4wG a G a + 8wR R 1 ) 2 wd2 R We can see that we recover the topological N = 1 Gauss-Bonnet combination d 4 x d 2 θ E Γ = d 4 x d 2 θ E W αβγ W αβγ + d 4 x d 2 θ d 2 θ E (2R R + G a G a 1 ) 8 D2 R Daniel Butter (NIKHEF) Super GB 21 / 29
42 N = 1 conformal superspace construction We choose the chiral F -term Lagrangian is Γ := W αβγ W αβγ + 1 S(ln Φ), 4w [Γ] F = 1 8 (C ) w c c ln φ + additional terms Rewriting in N = 1 Poincaré superspace... S(ln Φ) = 1 4 ( D ( 1 2 8R) 16 D2 D2 ln Φ + D α (G α α D α ln Φ) + 4wG a G a + 8wR R 1 ) 2 wd2 R We can see that we recover the topological N = 1 Gauss-Bonnet combination d 4 x d 2 θ E Γ = d 4 x d 2 θ E W αβγ W αβγ + d 4 x d 2 θ d 2 θ E (2R R + G a G a 1 ) 8 D2 R Daniel Butter (NIKHEF) Super GB 21 / 29
43 N = 2 conformal superspace construction Again: we want to supersymmetrize L Γ = 1 8 (C ) w c c ln φ The first term is easy. The second resembles the N = 2 kinetic multiplet. [de Wit, Katmadas, van Zalk 11] In flat N = 2 superspace, the second term is generated by d 4 θ D 4 ln Φ = D 4 D4 ln Φ θ=0 = ln φ In N = 2 conformal superspace, we can simply covariantize: T := 4 ln Φ. We call this the nonlinear kinetic multiplet. For this to be a valid chiral Lagrangian, we must check that it is chiral and S-invariant, αi T = S α i T = S i αt = 0. Then T is a proper conformally primary chiral multiplet, and we can write d 4 x d 4 θ E T = d 4 x e c c ln φ + Daniel Butter (NIKHEF) Super GB 22 / 29
44 N = 2 conformal superspace construction Again: we want to supersymmetrize L Γ = 1 8 (C ) w c c ln φ The first term is easy. The second resembles the N = 2 kinetic multiplet. [de Wit, Katmadas, van Zalk 11] In flat N = 2 superspace, the second term is generated by d 4 θ D 4 ln Φ = D 4 D4 ln Φ θ=0 = ln φ In N = 2 conformal superspace, we can simply covariantize: T := 4 ln Φ. We call this the nonlinear kinetic multiplet. For this to be a valid chiral Lagrangian, we must check that it is chiral and S-invariant, αi T = S α i T = S i αt = 0. Then T is a proper conformally primary chiral multiplet, and we can write d 4 x d 4 θ E T = d 4 x e c c ln φ + But how do we actually evaluate the last equation? Daniel Butter (NIKHEF) Super GB 22 / 29
45 Some details of chiral multiplets in superspace In N = 2 superspace, we need to know how to convert F -terms to component actions. ( ) d 4 x d 4 θ E L c = d 4 x e 4 L c + θ=0 The exact formula coincides with the usual chiral invariant density of N = 2 superconformal tensor calculus: [L c ] F C ε ij ψµi γ µ Λ j 1 8 ψ µi T ab jk γ ab γ µ Ψ l ε ij ε kl 1 16 A(T ab ijε ij ) ψ µi γ µν ψ νj B kl ε ik ε jl + ε ij ψµi ψ νj (F µν 1 2 A T µν kl ε kl ) 1 2 εij ε kl e 1 ε µνρσ ψµi ψ νj ( ψ ρk γ σ Ψ l + ψ ρk ψ σj A) where A,, C are the components of the chiral multiplet L c. In our case, we need to know the chiral components corresponding to T := 4 ln Φ. Daniel Butter (NIKHEF) Super GB 23 / 29
46 Components of the nonlinear kinetic multiplet The components of the ln Φ multiplet ˆĀ := ln Φ θ=0, ˆ Ψ αi := αi ln Φ θ=0,, ˆ C : 4 ln Φ The components of the nonlinear kinetic multiplet T := 4 ln Φ A T = T θ=0, Ψ αi T = αi T θ=0,, C T 4 T θ=0 A straightforward (increasingly tedious) calculation, using αi ln Φ = 0, gives A T ˆ C,, C T 4 4 ln Φ θ=0 = c c ln φ + Daniel Butter (NIKHEF) Super GB 24 / 29
47 After the dust settles... We have a new invariant [T] F 4( c + 3 D) c log φ 1 2 D a( T ab ij ij T ) cb D c log φ ( + D a ε ij D a T bcij ˆF +bc + 4 ε ij T ab ij D c + ˆF cb T bc ij T ac ij D b log φ ) + ( 6 D b D 8iD a ) R(A) ab D b log φ wr(v) + i ab j R(V) + abj i + 8w R(D) + ab abr(d)+ wd c( D a T ab ij ) T cb ij wda T ab ij D c T cb ij +, The Weyl squared invariant is L W 2 C abcd C abcd C abcd Cabcd 4R(A) ab R(A) ab + R(V) abi jr(v) abj i + 6D 2 T ac ij D a D b T bc ij T ab ij T ab kl T cd ijt cd kl Putting them together in the right combination we find (up to an explicit total derivative) C abcd C abcd 2R ab R ab R2 C abcd Cabcd + 2R(A) ab R(A)ab R(V) ab i j R(V) abj i Daniel Butter (NIKHEF) Super GB 25 / 29
48 So we have a new invariant, and it turns out to match the combination we needed to find from dimensional reduction from 5D... But what about the mystery of the minimal multiplet? Daniel Butter (NIKHEF) Super GB 26 / 29
49 A new chiral invariant in the minimal multiplet! Let s try to understand the nonlinear kinetic multiplet by writing it in SU(2) superspace: 4 ln Φ = ln Φ + wt 0 T 0 = 1 12 D ij Sij S ij Sij + 1 2Ȳ α βȳ α β. The operator is the SU(2) superspace chiral operator, generalizing D 4 of flat superspace. Under a full superspace integral, one can show that d 4 x d 4 θ E ln Φ = d 4 x d 4 θ d 4 θ E ln Φ = 0 so the dependence on Φ lies only within a total derivative. The remaining combination T 0 must be chiral (this can be checked explicitly), which is remarkable since none of its individual pieces is chiral! The minimal multiplet has an additional chiral invariant! Daniel Butter (NIKHEF) Super GB 27 / 29
50 Conclusions / Open questions (1/2) We have constructed a new chiral invariant based on N = 2 conformal supergravity coupled to a chiral multiplet. If the chiral multiplet is taken to be a vector multiplet and gauged to unity, it gives a new chiral invariant in the SU(2) minimal multiplet. It corresponds to certain actions which arise from reduction from 5D. The chiral multiplet can also be considered composite. Then in addition to D-term invariants d 4 x d 4 θ d 4 θ E H(X, X), X I H I = 0 we have intrinsic chiral invariants that look like d 4 x d 4 θ E Φ (X)T(ln Φ( X)), X I Φ I = 0, X J Φ J = w Φ In flat space, the second class corresponds to the first with the choice H = Φ ln Φ, but not in the curved case. In fact, a term of this form was exactly seen from dimensional reduction from 5D. Daniel Butter (NIKHEF) Super GB 28 / 29
51 Open questions (2/2) But there are several things we don t know. Does the new invariant class contribute to black hole entropy? but hopefully soon... The new invariant is peculiar in that it is (almost) independent of the compensator. Can we construct generic R 2 and (R ab ) 2 terms by introducing the second (tensor?) comensator? In principle, this should be so. Daniel Butter (NIKHEF) Super GB 29 / 29
A SUPERSPACE ODYSSEY. Bernard de Wit. Adventures in Superspace McGill University, Montreal April 19-20, Nikhef Amsterdam. Utrecht University
T I U A SUPERSPACE ODYSSEY Bernard de Wit Nikhef Amsterdam Adventures in Superspace McGill University, Montreal April 19-20, 2013 Utrecht University S A R U L N O L I S Æ S O I L T S T I Marc and I met
More informationRecent Progress on Curvature Squared Supergravities in Five and Six Dimensions
Recent Progress on Curvature Squared Supergravities in Five and Six Dimensions Mehmet Ozkan in collaboration with Yi Pang (Texas A&M University) hep-th/1301.6622 April 24, 2013 Mehmet Ozkan () April 24,
More informationSymmetries of curved superspace
School of Physics, University of Western Australia Second ANZAMP Annual Meeting Mooloolaba, November 27 29, 2013 Based on: SMK, arxiv:1212.6179 Background and motivation Exact results (partition functions,
More informationarxiv: v1 [hep-th] 24 Jul 2013
Nikhef-2013-024 ITP-UU-13/17 New higher-derivative invariants in N=2 supergravity and the Gauss-Bonnet term arxiv:1307.6546v1 [hep-th] 24 Jul 2013 Daniel Butter a, Bernard de Wit a,b, Sergei M. Kuzenko
More informationON ULTRAVIOLET STRUCTURE OF 6D SUPERSYMMETRIC GAUGE THEORIES. Ft. Lauderdale, December 18, 2015 PLAN
ON ULTRAVIOLET STRUCTURE OF 6D SUPERSYMMETRIC GAUGE THEORIES Ft. Lauderdale, December 18, 2015 PLAN Philosophical introduction Technical interlude Something completely different (if I have time) 0-0 PHILOSOPHICAL
More informationA Higher Derivative Extension of the Salam-Sezgin Model from Superconformal Methods
A Higher Derivative Extension of the Salam-Sezgin Model from Superconformal Methods Frederik Coomans KU Leuven Workshop on Conformal Field Theories Beyond Two Dimensions 16/03/2012, Texas A&M Based on
More informationKatrin Becker, Texas A&M University. Strings 2016, YMSC,Tsinghua University
Katrin Becker, Texas A&M University Strings 2016, YMSC,Tsinghua University ± Overview Overview ± II. What is the manifestly supersymmetric complete space-time action for an arbitrary string theory or M-theory
More informationNew superspace techniques for conformal supergravity in three dimensions
New superspace techniques for conformal supergravity in three dimensions Daniel Butter, Sergei Kuzenko, Joseph Novak School of Physics, The University of Western Australia, ARC DECRA fellow email: gabriele.tartaglino-mazzucchelli@uwa.edu.au
More informationAspects of SUSY Breaking
Aspects of SUSY Breaking Zohar Komargodski Institute for Advanced Study, Princeton ZK and Nathan Seiberg : arxiv:0907.2441 Aspects of SUSY Breaking p. 1/? Motivations Supersymmetry is important for particle
More informationOff-shell conformal supergravity in 3D
School of Physics, The University of Western Australia, ARC DECRA fellow ANZAMP meeting, Mooloolaba, 28 November 2013 Based on: Kuzenko & GTM, JHEP 1303, 113 (2013), 1212.6852 Butter, Kuzenko, Novak &
More informationAnomalies, Conformal Manifolds, and Spheres
Anomalies, Conformal Manifolds, and Spheres Nathan Seiberg Institute for Advanced Study Jaume Gomis, Po-Shen Hsin, Zohar Komargodski, Adam Schwimmer, NS, Stefan Theisen, arxiv:1509.08511 CFT Sphere partition
More informationA Short Note on D=3 N=1 Supergravity
A Short Note on D=3 N=1 Supergravity Sunny Guha December 13, 015 1 Why 3-dimensional gravity? Three-dimensional field theories have a number of unique features, the massless states do not carry helicity,
More informationLecture 9: RR-sector and D-branes
Lecture 9: RR-sector and D-branes José D. Edelstein University of Santiago de Compostela STRING THEORY Santiago de Compostela, March 6, 2013 José D. Edelstein (USC) Lecture 9: RR-sector and D-branes 6-mar-2013
More informationarxiv:hep-th/ v1 21 May 1996
ITP-SB-96-24 BRX-TH-395 USITP-96-07 hep-th/xxyyzzz arxiv:hep-th/960549v 2 May 996 Effective Kähler Potentials M.T. Grisaru Physics Department Brandeis University Waltham, MA 02254, USA M. Roče and R. von
More informationLecture 7: N = 2 supersymmetric gauge theory
Lecture 7: N = 2 supersymmetric gauge theory José D. Edelstein University of Santiago de Compostela SUPERSYMMETRY Santiago de Compostela, November 22, 2012 José D. Edelstein (USC) Lecture 7: N = 2 supersymmetric
More information1 4-dimensional Weyl spinor
4-dimensional Weyl spinor Left moving ψ α, α =, Right moving ψ α, α =, They are related by the complex conjugation. The indices are raised or lowerd by the ϵ tensor as ψ α := ϵ αβ ψ β, α := ϵ α β β. (.)
More informationTHE 4D/5D CONNECTION BLACK HOLES and HIGHER-DERIVATIVE COUPLINGS
T I U THE 4D/5D CONNECTION BLACK HOLES and HIGHER-DERIVATIVE COUPLINGS Mathematics and Applications of Branes in String and M-Theory Bernard de Wit Newton Institute, Cambridge Nikhef Amsterdam 14 March
More informationTwistor Strings, Gauge Theory and Gravity. Abou Zeid, Hull and Mason hep-th/
Twistor Strings, Gauge Theory and Gravity Abou Zeid, Hull and Mason hep-th/0606272 Amplitudes for YM, Gravity have elegant twistor space structure: Twistor Geometry Amplitudes for YM, Gravity have elegant
More informationSemper FI? Supercurrents, R symmetries, and the Status of Fayet Iliopoulos Terms in Supergravity. Keith R. Dienes
Semper FI? Supercurrents, R symmetries, and the Status of Fayet Iliopoulos Terms in Supergravity Keith R. Dienes National Science Foundation University of Maryland University of Arizona Work done in collaboration
More informationarxiv: v2 [hep-th] 3 Feb 2014
Nikhef-2014-001 ITP-UU-14/01 Non-renormalization theorems and N=2 supersymmetric backgrounds arxiv:1401.6591v2 [hep-th] 3 Feb 2014 Daniel Butter a, Bernard de Wit a,b and Ivano Lodato a a Nikhef, Science
More informationTalk at the International Workshop RAQIS 12. Angers, France September 2012
Talk at the International Workshop RAQIS 12 Angers, France 10-14 September 2012 Group-Theoretical Classification of BPS and Possibly Protected States in D=4 Conformal Supersymmetry V.K. Dobrev Nucl. Phys.
More informationTopics in nonlinear self-dual supersymmetric theories. Shane A. McCarthy
Topics in nonlinear self-dual supersymmetric theories Shane A. McCarthy This thesis is presented for the degree of Doctor of Philosophy of The University of Western Australia School of Physics. December,
More informationSymmetries, Groups Theory and Lie Algebras in Physics
Symmetries, Groups Theory and Lie Algebras in Physics M.M. Sheikh-Jabbari Symmetries have been the cornerstone of modern physics in the last century. Symmetries are used to classify solutions to physical
More informationTowards a manifestly diffeomorphism invariant Exact Renormalization Group
Towards a manifestly diffeomorphism invariant Exact Renormalization Group Anthony W. H. Preston University of Southampton Supervised by Prof. Tim R. Morris Talk prepared for UK QFT-V, University of Nottingham,
More informationN = 2 supergravity in d = 4, 5, 6 and its matter couplings
KUL-TF-XX/XXX hep-th/yymmnnn N = 2 supergravity in d = 4, 5, 6 and its matter couplings Antoine Van Proeyen, 1 Instituut voor theoretische fysica Universiteit Leuven, B-3001 Leuven, Belgium Abstract An
More information15.2. Supersymmetric Gauge Theories
348 15. LINEAR SIGMA MODELS 15.. Supersymmetric Gauge Theories We will now consider supersymmetric linear sigma models. First we must introduce the supersymmetric version of gauge field, gauge transformation,
More informationA Supergravity Dual for 4d SCFT s Universal Sector
SUPERFIELDS European Research Council Perugia June 25th, 2010 Adv. Grant no. 226455 A Supergravity Dual for 4d SCFT s Universal Sector Gianguido Dall Agata D. Cassani, G.D., A. Faedo, arxiv:1003.4283 +
More informationManifestly diffeomorphism invariant classical Exact Renormalization Group
Manifestly diffeomorphism invariant classical Exact Renormalization Group Anthony W. H. Preston University of Southampton Supervised by Prof. Tim R. Morris Talk prepared for Asymptotic Safety seminar,
More informationNew Model of massive spin-2 particle
New Model of massive spin-2 particle Based on Phys.Rev. D90 (2014) 043006, Y.O, S. Akagi, S. Nojiri Phys.Rev. D90 (2014) 123013, S. Akagi, Y.O, S. Nojiri Yuichi Ohara QG lab. Nagoya univ. Introduction
More informationSupercurrents. Nathan Seiberg IAS
Supercurrents Nathan Seiberg IAS 2011 Zohar Komargodski and NS arxiv:0904.1159, arxiv:1002.2228 Tom Banks and NS arxiv:1011.5120 Thomas T. Dumitrescu and NS arxiv:1106.0031 Summary The supersymmetry algebra
More informationSome applications of light-cone superspace
Some applications of light-cone superspace Stefano Kovacs (Trinity College Dublin & Dublin Institute for Advanced Studies) Strings and Strong Interactions LNF, 19/09/2008 N =4 supersymmetric Yang Mills
More informationarxiv:hep-th/ v1 26 Aug 1998
THU-98/30 hep-th/9808160 Rigid N =2 superconformal hypermultiplets 1 arxiv:hep-th/9808160v1 26 Aug 1998 Bernard de Wit a, Bas Kleijn a and Stefan Vandoren b a Institute for Theoretical Physics, Utrecht
More informationQuantum Fields in Curved Spacetime
Quantum Fields in Curved Spacetime Lecture 3 Finn Larsen Michigan Center for Theoretical Physics Yerevan, August 22, 2016. Recap AdS 3 is an instructive application of quantum fields in curved space. The
More informationarxiv: v2 [hep-th] 19 Feb 2013
January, 2013 Duality rotations in supersymmetric nonlinear electrodynamics revisited Sergei M. Kuzenko arxiv:1301.5194v2 [hep-th] 19 Feb 2013 School of Physics M013, The University of Western Australia
More informationHIGHER SPIN PROBLEM IN FIELD THEORY
HIGHER SPIN PROBLEM IN FIELD THEORY I.L. Buchbinder Tomsk I.L. Buchbinder (Tomsk) HIGHER SPIN PROBLEM IN FIELD THEORY Wroclaw, April, 2011 1 / 27 Aims Brief non-expert non-technical review of some old
More informationContact interactions in string theory and a reformulation of QED
Contact interactions in string theory and a reformulation of QED James Edwards QFT Seminar November 2014 Based on arxiv:1409.4948 [hep-th] and arxiv:1410.3288 [hep-th] Outline Introduction Worldline formalism
More informationGRANGIAN QUANTIZATION OF THE HETEROTIC STRING IN THE BOSONIC FORMULAT
September, 1987 IASSNS-HEP-87/draft GRANGIAN QUANTIZATION OF THE HETEROTIC STRING IN THE BOSONIC FORMULAT J. M. F. LABASTIDA and M. PERNICI The Institute for Advanced Study Princeton, NJ 08540, USA ABSTRACT
More informationRigid SUSY in Curved Superspace
Rigid SUSY in Curved Superspace Nathan Seiberg IAS Festuccia and NS 1105.0689 Thank: Jafferis, Komargodski, Rocek, Shih Theme of recent developments: Rigid supersymmetric field theories in nontrivial spacetimes
More informationFirst Year Seminar. Dario Rosa Milano, Thursday, September 27th, 2012
dario.rosa@mib.infn.it Dipartimento di Fisica, Università degli studi di Milano Bicocca Milano, Thursday, September 27th, 2012 1 Holomorphic Chern-Simons theory (HCS) Strategy of solution and results 2
More informationNon-Abelian tensor multiplet in four dimensions
PASCOS 2012 18th nternational Symposium on Particles Strings and Cosmology OP Publishing Non-Abelian tensor multiplet in four dimensions Hitoshi Nishino and Subhash Rajpoot, Department of Physics and Astronomy,
More informationChern-Simons Theory and Its Applications. The 10 th Summer Institute for Theoretical Physics Ki-Myeong Lee
Chern-Simons Theory and Its Applications The 10 th Summer Institute for Theoretical Physics Ki-Myeong Lee Maxwell Theory Maxwell Theory: Gauge Transformation and Invariance Gauss Law Charge Degrees of
More informationLorentz-covariant spectrum of single-particle states and their field theory Physics 230A, Spring 2007, Hitoshi Murayama
Lorentz-covariant spectrum of single-particle states and their field theory Physics 30A, Spring 007, Hitoshi Murayama 1 Poincaré Symmetry In order to understand the number of degrees of freedom we need
More informationExact Results in D=2 Supersymmetric Gauge Theories And Applications
Exact Results in D=2 Supersymmetric Gauge Theories And Applications Jaume Gomis Miami 2012 Conference arxiv:1206.2606 with Doroud, Le Floch and Lee arxiv:1210.6022 with Lee N = (2, 2) supersymmetry on
More informationAn extended standard model and its Higgs geometry from the matrix model
An extended standard model and its Higgs geometry from the matrix model Jochen Zahn Universität Wien based on arxiv:1401.2020 joint work with Harold Steinacker Bayrischzell, May 2014 Motivation The IKKT
More informationExact solutions in supergravity
Exact solutions in supergravity James T. Liu 25 July 2005 Lecture 1: Introduction and overview of supergravity Lecture 2: Conditions for unbroken supersymmetry Lecture 3: BPS black holes and branes Lecture
More informationarxiv:hep-th/ v1 6 Mar 2007
hep-th/0703056 Nonlinear Realizations in Tensorial Superspaces and Higher Spins arxiv:hep-th/0703056v1 6 Mar 2007 Evgeny Ivanov Bogoliubov Laboratory of Theoretical Physics, JINR, 141980, Dubna, Moscow
More informationCP n supersymmetric mechanics in the U(n) background gauge fields
Preliminaries: and free CP n mechanics CP n supersymmetric mechanics in the U(n) background gauge fields Sergey Krivonos Joint Institute for Nuclear Research Advances of Quantum Field Theory, Dubna, 2011
More informationThe Definitions of Special Geometry 1
KUL-TF-96/11 hep-th/9606073 The Definitions of Special Geometry 1 Ben Craps, Frederik Roose, Walter Troost 2 andantoinevanproeyen 3 Instituut voor theoretische fysica Universiteit Leuven, B-3001 Leuven,
More informationarxiv: v3 [hep-th] 28 Apr 2014
UUITP-20/13 LMU-ASC 87/13 December, 2013 Three-dimensional N = 2 supergravity theories: From superspace to components arxiv:1312.4267v3 [hep-th] 28 Apr 2014 Sergei M. Kuzenko a, Ulf Lindström b, Martin
More informationConnecting the ambitwistor and the sectorized heterotic strings
Connecting the ambitwistor and the sectorized heterotic strings Renann Lipinski Jusinskas February 11th - 2018 Discussion Meeting on String Field Theory and String Phenomenology - HRI, Allahabad, India
More informationIntroduction to defects in Landau-Ginzburg models
14.02.2013 Overview Landau Ginzburg model: 2 dimensional theory with N = (2, 2) supersymmetry Basic ingredient: Superpotential W (x i ), W C[x i ] Bulk theory: Described by the ring C[x i ]/ i W. Chiral
More informationAs usual, these notes are intended for use by class participants only, and are not for circulation. Week 6: Lectures 11, 12
As usual, these notes are intended for use by class participants only, and are not for circulation Week 6: Lectures, The Dirac equation and algebra March 5, 0 The Lagrange density for the Dirac equation
More informationAmplitudes & Wilson Loops at weak & strong coupling
Amplitudes & Wilson Loops at weak & strong coupling David Skinner - Perimeter Institute Caltech - 29 th March 2012 N=4 SYM, 35 years a!er Twistor space is CP 3, described by co-ords It carries a natural
More informationYangian Symmetry of Planar N = 4 SYM
Yangian Symmetry of Planar N = 4 SYM ITP, Niklas Beisert New formulations for scattering amplitudes Ludwig Maximilians Universität, München 9 September 2016 work with J. Plefka, D. Müller, C. Vergu (1509.05403);
More information2-Group Global Symmetry
2-Group Global Symmetry Clay Córdova School of Natural Sciences Institute for Advanced Study April 14, 2018 References Based on Exploring 2-Group Global Symmetry in collaboration with Dumitrescu and Intriligator
More informationKähler representations for twisted supergravity. Laurent Baulieu LPTHE. Université Pierre et Marie Curie, Paris, France. Puri, January 6th, 2011
Kähler representations for twisted supergravity Laurent Baulieu LPTHE Université Pierre et Marie Curie, Paris, France Puri, January 6th, 2011-1 - Introduction The notion of twisting a supersymmetry algebra
More informationSpinning strings and QED
Spinning strings and QED James Edwards Oxford Particles and Fields Seminar January 2015 Based on arxiv:1409.4948 [hep-th] and arxiv:1410.3288 [hep-th] Outline Introduction Various relationships between
More informationQuantum Field Theory III
Quantum Field Theory III Prof. Erick Weinberg April 5, 011 1 Lecture 6 Let s write down the superfield (without worrying about factors of i or Φ = A(y + θψ(y + θθf (y = A(x + θσ θ A + θθ θ θ A + θψ + θθ(
More informationIntroduction to Modern Quantum Field Theory
Department of Mathematics University of Texas at Arlington Arlington, TX USA Febuary, 2016 Recall Einstein s famous equation, E 2 = (Mc 2 ) 2 + (c p) 2, where c is the speed of light, M is the classical
More informationTopological DBI actions and nonlinear instantons
8 November 00 Physics Letters B 50 00) 70 7 www.elsevier.com/locate/npe Topological DBI actions and nonlinear instantons A. Imaanpur Department of Physics, School of Sciences, Tarbiat Modares University,
More informationSupergravity in Quantum Mechanics
Supergravity in Quantum Mechanics hep-th/0408179 Peter van Nieuwenhuizen C.N. Yang Institute for Theoretical Physics Stony Brook University Erice Lectures, June 2017 Vienna Lectures, Jan/Feb 2017 Aim of
More informationCP n supersymmetric mechanics in the U(n) background gauge fields
CP n supersymmetric mechanics in the U(n) background gauge fields Sergey Krivonos Joint Institute for Nuclear Research Recent Advances in Quantum Field and String Theory, Tbilisi, September 26-30, 2011
More informationOn the curious spectrum of duality-invariant higher-derivative gravitational field theories
On the curious spectrum of duality-invariant higher-derivative gravitational field theories VIII Workshop on String Field Theory and Related Aspects ICTP-SAIFR 31 May 2016 Barton Zwiebach, MIT Introduction
More informationarxiv:hep-th/ v2 14 Oct 1997
T-duality and HKT manifolds arxiv:hep-th/9709048v2 14 Oct 1997 A. Opfermann Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge, CB3 9EW, UK February
More information2d N = (2, 2) supersymmetry with U(1) RV in curved space
2d N = (2, 2) supersymmetry with U(1) RV in curved space Stefano Cremonesi Imperial College London SUSY 2013, ICTP Trieste August 27, 2013 Summary Based on: C. Closset, SC, to appear. F. Benini, SC, Partition
More informationN=1 Global Supersymmetry in D=4
Susy algebra equivalently at quantum level Susy algebra In Weyl basis In this form it is obvious the U(1) R symmetry Susy algebra We choose a Majorana representation for which all spinors are real. In
More informationarxiv:hep-th/ v1 28 Jan 1999
N=1, D=10 TENSIONLESS SUPERBRANES II. 1 arxiv:hep-th/9901153v1 28 Jan 1999 P. Bozhilov 2 Bogoliubov Laboratory of Theoretical Physics, JINR, 141980 Dubna, Russia We consider a model for tensionless (null)
More informationSmall Black Strings/Holes
Small Black Strings/Holes Based on M. A., F. Ardalan, H. Ebrahim and S. Mukhopadhyay, arxiv:0712.4070, 1 Our aim is to study the symmetry of the near horizon geometry of the extremal black holes in N =
More informationColliding scalar pulses in the Einstein-Gauss-Bonnet gravity
Colliding scalar pulses in the Einstein-Gauss-Bonnet gravity Hisaaki Shinkai 1, and Takashi Torii 2, 1 Department of Information Systems, Osaka Institute of Technology, Hirakata City, Osaka 573-0196, Japan
More informationExercises Symmetries in Particle Physics
Exercises Symmetries in Particle Physics 1. A particle is moving in an external field. Which components of the momentum p and the angular momentum L are conserved? a) Field of an infinite homogeneous plane.
More informationThe rudiments of Supersymmetry 1/ 53. Supersymmetry I. A. B. Lahanas. University of Athens Nuclear and Particle Physics Section Athens - Greece
The rudiments of Supersymmetry 1/ 53 Supersymmetry I A. B. Lahanas University of Athens Nuclear and Particle Physics Section Athens - Greece The rudiments of Supersymmetry 2/ 53 Outline 1 Introduction
More informationSpinor Representation of Conformal Group and Gravitational Model
Spinor Representation of Conformal Group and Gravitational Model Kohzo Nishida ) arxiv:1702.04194v1 [physics.gen-ph] 22 Jan 2017 Department of Physics, Kyoto Sangyo University, Kyoto 603-8555, Japan Abstract
More informationStability in Maximal Supergravity
Stability in Maximal Supergravity S. Bielleman, s171136, RuG Supervisor: Dr. D. Roest August 5, 014 Abstract In this thesis, we look for a bound on the lightest scalar mass in maximal supergravity. The
More informationGeneralized N = 1 orientifold compactifications
Generalized N = 1 orientifold compactifications Thomas W. Grimm University of Wisconsin, Madison based on: [hep-th/0602241] Iman Benmachiche, TWG [hep-th/0507153] TWG Madison, Wisconsin, November 2006
More informationSUSY Breaking in Gauge Theories
SUSY Breaking in Gauge Theories Joshua Berger With the Witten index constraint on SUSY breaking having been introduced in last week s Journal club, we proceed to explicitly determine the constraints on
More informationSupersymmetric Gauge Theories, Matrix Models and Geometric Transitions
Supersymmetric Gauge Theories, Matrix Models and Geometric Transitions Frank FERRARI Université Libre de Bruxelles and International Solvay Institutes XVth Oporto meeting on Geometry, Topology and Physics:
More informationCorrelation functions of conserved currents in N =2 superconformal theory
Class. Quantum Grav. 17 (2000) 665 696. Printed in the UK PII: S0264-9381(00)08504-X Correlation functions of conserved currents in N =2 superconformal theory Sergei M Kuzenko and Stefan Theisen Sektion
More informationBMS current algebra and central extension
Recent Developments in General Relativity The Hebrew University of Jerusalem, -3 May 07 BMS current algebra and central extension Glenn Barnich Physique théorique et mathématique Université libre de Bruxelles
More informationIs there any torsion in your future?
August 22, 2011 NBA Summer Institute Is there any torsion in your future? Dmitri Diakonov Petersburg Nuclear Physics Institute DD, Alexander Tumanov and Alexey Vladimirov, arxiv:1104.2432 and in preparation
More informationComponent versus superspace approaches to D = 4, N = 1 conformal supergravity
Prog. Theor. Exp. Phys. 2016, 073B07 36 pages DOI: 10.1093/ptep/ptw090 Component versus superspace approaches to D = 4, N = 1 conformal supergravity Taichiro Kugo 1, Ryo Yokokura 2,, and Koichi Yoshioka
More informationInteracting non-bps black holes
Interacting non-bps black holes Guillaume Bossard CPhT, Ecole Polytechnique Istanbul, August 2011 Outline Time-like Kaluza Klein reduction From solvable algebras to solvable systems Two-centre interacting
More informationNONINTEGER FLUXES, DOLBEAULT COMPLEXES, AND SUPERSYMMETRIC QUANTUM MECHANICS. based on [ ] and [ ] Hannover, August 1, 2011
NONINTEGER FLUXES, DOLBEAULT COMPLEXES, AND SUPERSYMMETRIC QUANTUM MECHANICS based on [1104.3986] and [1105.3935] Hannover, August 1, 2011 WHAT IS WRONG WITH NONINTEGER FLUX? Quantization of Dirac monopole
More informationBounds on 4D Conformal and Superconformal Field Theories
Bounds on 4D Conformal and Superconformal Field Theories David Poland Harvard University November 30, 2010 (with David Simmons-Duffin [arxiv:1009.2087]) Motivation Near-conformal dynamics could play a
More informationThe Divergence Myth in Gauss-Bonnet Gravity. William O. Straub Pasadena, California November 11, 2016
The Divergence Myth in Gauss-Bonnet Gravity William O. Straub Pasadena, California 91104 November 11, 2016 Abstract In Riemannian geometry there is a unique combination of the Riemann-Christoffel curvature
More informationQuantum Field Theory I Examination questions will be composed from those below and from questions in the textbook and previous exams
Quantum Field Theory I Examination questions will be composed from those below and from questions in the textbook and previous exams III. Quantization of constrained systems and Maxwell s theory 1. The
More informationRunning at Non-relativistic Speed
Running at Non-relativistic Speed Eric Bergshoeff Groningen University Symmetries in Particles and Strings A Conference to celebrate the 70th birthday of Quim Gomis Barcelona, September 4 2015 Why always
More informationarxiv:hep-th/ v3 15 Nov 1996
Institut für Theoretische Physik Universität Hannover Institut für Theoretische Physik Universität Hannover Institut für Theoretische Physik Hannover DESY 96 165 ITP UH 15/96 hep-th/9608131 revised version
More informationInstantons in supersymmetric gauge theories. Tobias Hansen Talk at tuesday s Werkstatt Seminar January 10, 2012
Instantons in supersymmetric gauge theories Tobias Hansen Talk at tuesday s Werkstatt Seminar January 10, 01 References [1] N. Dorey, T. J. Hollowood, V. V. Khoze and M. P. Mattis, The Calculus of many
More informationLecture 25 Superconformal Field Theory
Lecture 25 Superconformal Field Theory Superconformal Field Theory Outline: The c-theorem. The significance of R-symmetry. The structure of anomalies in SCFT. Computation of R-charges: A-maximization These
More informationarxiv: v2 [hep-th] 15 Dec 2016
Nikhef-2016-026 June, 2016 arxiv:1606.02921v2 [hep-th] 15 Dec 2016 Invariants for minimal conformal supergravity in six dimensions Daniel Butter a, Sergei M. Kuzenko b, Joseph Novak c and Stefan Theisen
More informationTHE RENORMALIZATION GROUP AND WEYL-INVARIANCE
THE RENORMALIZATION GROUP AND WEYL-INVARIANCE Asymptotic Safety Seminar Series 2012 [ Based on arxiv:1210.3284 ] GIULIO D ODORICO with A. CODELLO, C. PAGANI and R. PERCACCI 1 Outline Weyl-invariance &
More informationChern-Simons Theories and AdS/CFT
Chern-Simons Theories and AdS/CFT Igor Klebanov PCTS and Department of Physics Talk at the AdS/CMT Mini-program KITP, July 2009 Introduction Recent progress has led to realization that coincident membranes
More informationGravitational Waves and New Perspectives for Quantum Gravity
Gravitational Waves and New Perspectives for Quantum Gravity Ilya L. Shapiro Universidade Federal de Juiz de Fora, Minas Gerais, Brazil Supported by: CAPES, CNPq, FAPEMIG, ICTP Challenges for New Physics
More informationReφ = 1 2. h ff λ. = λ f
I. THE FINE-TUNING PROBLEM A. Quadratic divergence We illustrate the problem of the quadratic divergence in the Higgs sector of the SM through an explicit calculation. The example studied is that of the
More information8.821 String Theory Fall 2008
MIT OpenCourseWare http://ocw.mit.edu 8.81 String Theory Fall 008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 8.81 F008 Lecture 1: Boundary of AdS;
More informationStress-energy tensor is the most important object in a field theory and have been studied
Chapter 1 Introduction Stress-energy tensor is the most important object in a field theory and have been studied extensively [1-6]. In particular, the finiteness of stress-energy tensor has received great
More informationarxiv: v2 [hep-th] 15 Apr 2009
CERN-PH-TH/2009-040 Globally and locally supersymmetric effective theories for light fields arxiv:0904.0370v2 [hep-th] 15 Apr 2009 Leonardo Brizi a, Marta Gómez-Reino b and Claudio A. Scrucca a a Institut
More informationLAGRANGIANS FOR CHIRAL BOSONS AND THE HETEROTIC STRING. J. M. F. LABASTIDA and M. PERNICI The Institute for Advanced Study Princeton, NJ 08540, USA
October, 1987 IASSNS-HEP-87/57 LAGRANGIANS FOR CHIRAL BOSONS AND THE HETEROTIC STRING J. M. F. LABASTIDA and M. PERNICI The Institute for Advanced Study Princeton, NJ 08540, USA ABSTRACT We study the symmetries
More informationThe Dirac Propagator From Pseudoclassical Mechanics
CALT-68-1485 DOE RESEARCH AND DEVELOPMENT REPORT The Dirac Propagator From Pseudoclassical Mechanics Theodore J. Allen California Institute of Technology, Pasadena, CA 9115 Abstract In this note it is
More information