2d N = (2, 2) supersymmetry with U(1) RV in curved space

Transcription

1 2d N = (2, 2) supersymmetry with U(1) RV in curved space Stefano Cremonesi Imperial College London SUSY 2013, ICTP Trieste August 27, 2013

2 Summary Based on: C. Closset, SC, to appear. F. Benini, SC, Partition functions of N = (2, 2) gauge theories on S 2 and vortices, arxiv: [hep-th]. What we do: Define N = (2, 2) field theories on compact Riemann surfaces Classify supersymmetric backgrounds Why: Exact results by localization Type II strings on Calabi-Yau 4d N = 2 field theories

3 Plan 1 2d N = (2, 2) SUSY in flat space 2 2d N = (2, 2) SUSY with U(1) RV in curved space Killing spinor equations Supersymmetry algebra Supermultiplets Supersymmetric Lagrangians 3 Supersymmetric backgrounds 4 Outlook

4 2d N = (2, 2) SUSY in flat space

5 2d N = (2, 2) on flat R 2 [Hori: Trieste Lectures on Mirror Symmetry, 2002] Superalgebra {Q ±, Q ±} = P ±± {Q, Q ±} = 0 {Q, Q +} = Z { Q, Q +} = Z {Q, Q +} = W { Q, Q +} = W [M, Q ] = ± 1 2 Q [M, Q ] = ± 1 2 Q U(1) RA U(1) RV automorphism [R V, Q ] = Q [R V, Q ] = + Q [R A, Q ] = ±Q [R A, Q ] = Q [R V, Z ] = 2Z [R V, Z ] = 2 Z [R A, W ] = 2W [R A, W ] = 2 W [R V, W ] = [R V, W ] = 0 [R A, Z ] = [R A, Z ] = 0 Z 2 mirror automorphism: θ θ Q Q, R V R A, Z W, Z W

6 2d N = (2, 2) on flat R 2 : supermultiplets Chiral Multiplet Φ: D Φ = D +Φ = 0. Can have nontrivial R V, W, W charges. Vector Multiplet: V = Ṽ V + Φ + Φ. Twisted Chiral Multiplet Ω: D Ω = D +Ω = 0. [Gates, Hull, Roček 1984] Can have nontrivial R A, Z, Z charges. Twisted Vector Multiplet: V = V V + Ω + Ω.

7 2d N = (2, 2) with U(1) RV on flat R 2 Z = Z = 0, Axial R-symmetry not required. Flat space superalgebra {Q ±, Q ±} = P ±± {Q, Q ±} = 0 {Q, Q +} = 0 { Q, Q +} = 0 {Q, Q +} = W { Q, Q +} = W [M, Q ] = ± 1 2 Q [M, Q ] = ± 1 2 Q [R V, Q ] = Q [R V, Q ] = + Q [R V, W ] = 0 [R V, W ] = 0 We will generalize this supersymmetry algebra to curved space à la [Festuccia, Seiberg 2011].

8 2d N = (2, 2) SUSY with U(1) RV in curved space

9 Coupling to background supergravity Couple the 2d R-multiplet [Dumitrescu, Seiberg 2011] to external new minimal SUGRA multiplet, similarly to [Closset, Dumitrescu, Festuccia, Komargodski 2012] in 3d. Supercurrent R-multiplet T µν S µα j µ R V G µ Gµ New min SUGRA multiplet h µν ψ µα A µ Cµ C µ j µ R V, ε µν Gν, ε µν G ν: conserved currents for R V, W, W. A µ, C µ, C µ: conjugate gauge fields.

10 Killing spinor equations Frame: {e 1 = g 1 2 dz, e 1 = g 1 2 dz} Metric: ds 2 = g dz dz Killing spinor equations ( 1 i A 1 )ζ = 0 ( 1 i A 1 )ζ = 1 2 H ζ+ ( 1 i A 1 )ζ + = 1 2 H ζ ( 1 i A 1 )ζ + = 0 ( 1 + i A 1 ) ζ = 0 ( 1 + i A 1 ) ζ = 1 2 H ζ + ( 1 + i A 1 ) ζ + = 1 2 H ζ ( 1 + i A 1 ) ζ + = 0 where H = iε µν µ Cν, H = iε µν µc ν.

11 2d N = (2, 2) with U(1) RV in curved space Curved space supersymmetry algebra {δ ζ, δ ζ}ϕ (r,w, w) = 2i (L K + ζ + ζ (w r ) ( 2 H ζ ζ+ w r 2 H )) {δ ζ, δ η}ϕ (r,w, w) = 0 {δ ζ, δ η }ϕ (r,w, w) = 0 ϕ (r,w, w) where L K ϕ (r,w, w) = ( K µ D µ + i 2 (DµKν)Sµν) ϕ (r,w, w) is the totally covariant Lie derivative along the Killing vector K µ = ζγ µ ζ, D µϕ (r,w, w) ( µ ira µ + 1 (w C ) 2 µ wc µ) ϕ (r,w, w) the totally covariant derivative of a field ϕ (r,w, w).

12 Supersymmetry transformations Can be written for a general multiplet. E.g.: Chiral multiplet of charges (r, w, w) δφ = 2(ζ +ψ ζ ψ +) δψ = 2ζ F 2i ζ ( w r ) 2 H φ + 2 2i ζ +D 1 φ δψ + = 2ζ +F 2i ζ + ( w r 2 H ) φ + 2 2i ζ D 1 φ δf = ( 2i w r 2 ) H ζ +ψ ( 2i w r 2 ) 2 2 H + 2 2iD 1 ( ζ +ψ +) 2 2iD 1 ( ζ ψ ) ζ ψ +

13 Supersymmetric Lagrangians Kinetic Lagrangian L ΦΦ = 4D 1 φd1 φ F F + 2i ψ +D 1 ψ + 2i ψ D 1 ψ + + (w r ) ( 2 H w r 2 H ) r φφ 4 R φφ ( + i w r H) ψ+ψ i 2 ( w r 2 H ) ψ ψ + ( wh + w H) φφ+ F-term Lagrangian from superpotential W (Φ i ) of r = 2, w = w = 0 L F + F = F i i W + ψ i ψ j + i j W + F ĩ ĩ W + ψĩ + ψ j ĩ j W Similarly vector, twisted chiral ([Gomis, Lee 2012] on S 2 ), twisted vector multiplet.

14 Supersymmetric backgrounds

15 Supersymmetric backgrounds: 1 supercharge ζ R-symmetry line bundle L: m c 1 (L) = 1 2π Σ g da = n(g 1) Z. R V -charge 1 Killing spinors ζ (z, z) = λs 0 g 1+n 4 f 1 (z), ζ +(z, z) = λ s 0 g 1 n 4 f 2 (z) f 1 (z) Γ( K 1 n ), f 2 (z) Γ( K 1+n ) f 1 (z), f 2 (z): regular holomorphic sections. λ, s 0 : non-vanishing functions. U(1) RV connection A A z = n 2 ωz i z ln (λs 0), A z = n 2 ω z i z ln (λ/s 0 ) H, H z f 1 (z) + 1+n f 2 1(z) z ln g + f 1 (z) z ln s0 2 = 1 Hf 2 2(z) 1 1 n s g zf 2 (z) + 1 n f 2 2(z) z ln g f 2 (z) z ln s0 2 = 1 Hf 2 1 (z) s0 2 1+n g 2.

16 Supersymmetric backgrounds: 1 supercharge ζ deg( K 1±n ) = 1 g m. g > 1 n = +1: (ζ, ζ +) = (0, 1), A = ω, H = 0, H A-twist n = 1: (ζ, ζ +) = (1, 0), A = 1 2 ω, H = 0, H A-twist g = 1: T 2 Trivial bundles: constant f 1, f 2. g = 0: S 2 n = +1: A-twist, U(1)-equivariant A-twist n = 1: A-twist, U(1)-equivariant A-twist n = 0: f 1 (z), f 2 (z) O(1). One zero each. Can have single supercharge.

17 Supersymmetric backgrounds: more supercharges Integrability conditions: ζ 2ζ F 11 + i ζ (R 2H H) iζ H = 0 2ζ +F 11 i ζ+(r 2H H) + iζ 4 1 H = 0 Integrability conditions: ζ 2 ζ F 11 + i ζ 4 (R 2H H) i ζ + 1 H = 0 2 ζ +F 11 i ζ 4 +(R 2H H) + i ζ 1 H = 0 F = da, R Ricci scalar (= 2/L 2 on round S 2 L of radius L). 2+2 supercharges Flat T 2 F = da = 0, R = 2H H, µh = 0, µ H = 0 Round S 2 L with H H = 1/L 2 : [Benini, SC 2012; Doroud, Gomis, Le Floch, Lee 2012] and constant complexified U(1) RA rotations thereof.

18 Supersymmetric backgrounds: more supercharges 2+0 supercharges F = da = 0, R = 2H H, zh = 0, z H = supercharges: Alternatives globally defined Killing vector K = ζγ µ ζ µ: - T 2 - U(1)-isometric squashed S 2 : n = ±1: (U(1)-equivariant) A/A-twist n = 0: f 1 = 1, f 2 = iz (and conjugate), generalizing [Gomis, Lee 2012]. No isometries: A/A-twist (possibly with H or H = 0).

19 Outlook

20 Outlook We have defined 2d N = (2, 2) field theories with U(1) RV curved space. symmetry in We have classified supersymmetric backgrounds. Subclasses of these backgrounds can uplift to 3d or 4d. We use these results to study Q-cohomology: whether and how the partition function and correlators depend on various deformations. We hope that our study will lead to new exact results in superstring theory, geometry and 4d gauge theories.