Dirac-Born-Infeld Actions and the Dilaton in N=2 Supersymmetry

Dirac-Born-Infeld actions and the dilaton in N=2 Supersymmetry

Nicola Ambrosetti

Institute for Theoretical Physics Albert Einstein Center for fundamental physics University of Bern

SUSY 2010, Bonn

Based on: NA, Antoniadis, Derendinger, Tziveloglou, “Nonlinear Supersymmetry, Brane-bulk Interactions and Super-Higgs without Gravity” (arXiv:0911.5212) and “The Hypermultiplet with Heisenberg Isometry in N=2 Global and Local Supersymmetry” (arXiv:1005.0323)

N. Ambrosetti (Bern U.) DBI and dilaton in N=2 SUSY 2010, Bonn 1 / 11 General setting

Type II string compactifications on Calabi-Yau to D = 4 ⇒ N = 2 supersymmetry in the bulk D-branes break half of supersymmetry which is then nonlinearly realised on their worldvolume Gauge field on D-brane described by Dirac-Born-Infeld (DBI) action Complete effective action includes a topological term with coupling to RR fields Dilaton belongs to Universal Hypermultiplet, which has Heisenberg symmetry preserved in string perturbation theory Shift symmetries of the Universal Hypermultiplet allow dualisation of one or two scalars into two-forms ⇒ single-tensor or double-tensor descriptions

N. Ambrosetti (Bern U.) DBI and dilaton in N=2 SUSY 2010, Bonn 2 / 11 Description

Global N = 2 supersymmetry realised on N = 1 superfields Hypermultiplet dualised into single-tensor to allow off-shell supersymmetric description DBI and nonlinear supersymmetry introduced by constrained superfields Lagrangian given by supersymmetrisation of b ∧ F interaction with one linear and one nonlinear supersymmetry Nonlinear N = 2 vector multiplet coupled to full single-tensor: no orientifold truncation Bosonic fields are: In abelian vector: Aµ (Fµν) and the auxiliary d2. In single-tensor: bµν , C (dilaton),Φ (complex RR scalar), 4-form Cµνρσ (see later).

N. Ambrosetti (Bern U.) DBI and dilaton in N=2 SUSY 2010, Bonn 3 / 11 Dilaton-Brane coupling and DBI

After integration of auxiliary field d2 the bosonic Lagrangian is √ Re φ h q C 2 q i LDBI ,bos. = 4κ 1 − 1 + 2(Re φ)2 − det(ηµν + 2 2κFµν) κ 1 1 + µνρσ Im φ F F − b F + C 4 µν ρσ 4 µν ρσ 24κ µνρσ κ (mass−2) is the nonlinear scale. We recovered the DBI action with field dependent coefficient in front of the square root generated by integration of auxiliary field P F Full topological term ∼ k Ck ∧ e Invariance under non-linearly realised 2nd SUSY

C arises as the MP → ∞ limit of the dilaton for a quaternionic-K¨ahlermanifold with Heisenberg symmetry (in string perturbation theory) [AADT, 1005.0323]

N. Ambrosetti (Bern U.) DBI and dilaton in N=2 SUSY 2010, Bonn 4 / 11 Analysis of the vacuum: super-Higgs without gravity There is a scalar potential "s # Re φ C 2 V (C, Re φ) = 1 + − 1 4κ 2(Re φ)2

Im φ is a ﬂat direction. SUSY vacuum at hCi = 0, for any value of Re φ. Properties of SUSY vacuum: φ is massless 2 1 C is massive mC = 4κhRe φi , same mass as the vector Partial breaking of supersymmetry N = 2 to N = 1 Gaugino transforms as a goldstino, but is not massless The mixing term χλ gave a mass to the gaugino, but 2nd SUSY preserved since its variation is canceled by the variation of the 4-form.

N. Ambrosetti (Bern U.) DBI and dilaton in N=2 SUSY 2010, Bonn 5 / 11 Realisation of N = 2 on N = 1 superﬁelds

Off-shell realisation with two real N = 1 superfields V1, V2 (16B + 16F ): √ ∗ i ∗ δ V1 = −√ (ηD − ηD)V2 , δ V2 = 2i(ηD − ηD)V1 2 Works for two off-shell N = 2 multiplets (not for hypermultiplets):

1 Impose supersymmetric constraint on V1 and V2: V1 = L linear( DDL = 0), V2 = Φ + Φ, Φ chiral. ⇒ Single-tensor multiplet (L, Φ), 8B + 8F oﬀ-shell. Bosons: real C, complexΦ, bµν with δbµν = 2∂[µΛν], auxiliary fΦ.

2 Impose gauge invariance on V1 and V2 Use the single-tensor (Λ`, Λc ) as gauge parameters:

δgauge V1 = Λ` , δgauge V2 = Λc + Λc

⇒ N = 2 abelian vector multiplet, 8B + 8F oﬀ-shell.

N. Ambrosetti (Bern U.) DBI and dilaton in N=2 SUSY 2010, Bonn 6 / 11 The constrained vector multiplet Gauge invariant vector multiplet: 1 1 X = DDV , W = − DDD V . 2 1 α 4 α 2 Can be embedded into usual N = 2 chiral superﬁeld √ α 1 W = X + 2iθe Wα − θeθeDDX 4 Nonlinear supersymmetry can be obtained imposing a quadratic constraint

1 2 1 1 1 W − θeθe = W2 − θeθeW = 0 ⇐⇒ WW − X DDX = X 2κ κ 2 κ

Need to deform the 2nd SUSY transformation of Wα Gaugino transforms as a goldstino. Solution of the constraint X (WW ) contains the Born-Infeld

N. Ambrosetti (Bern U.) DBI and dilaton in N=2 SUSY 2010, Bonn 7 / 11 Single-tensor version II

The N = 1 superfields( L, Φ) supersymmetrise Hµνρ = 3∂[µBνρ] with 8B + 8F off-shell. To supersymmetrise bµν one needs 16B + 16F with gauge invariance: α α˙ (Y , χα, Φ) with L = D χα − Dα˙ χ and Y a chiral superfield containing the four-form Cµνρσ We can construct a chiral N = 2 superfield √ α i 1 Y = Y + 2θe χα − θeθe Φ + DDY 2 4 Bosonic fields: N = 1 superfield Field Gauge invariance Number of fields χα bµν δbµν = 2 ∂[µΛν] 6B − 3B = 3B C 1B Φ Φ 2B fΦ 2B (auxiliary) Y Cµνρσ δ Cµνρσ = 4 ∂[µΛνρσ] 1B − 1B = 0B

N. Ambrosetti (Bern U.) DBI and dilaton in N=2 SUSY 2010, Bonn 8 / 11 Supersymmetric coupling

D-brane gauge ﬁeld is described by constrained vector multiplet 1 2 Wnl = W − 2κ θeθe, Wnl = 0 Dilaton multiplet is described by single-tensor Y N = 2 Supersymmetric coupling with one linear and one nonlinear supersymmetry is given by Z 2 2 L = d θd θeiYWnl + c.c. + single-tensor kinetic terms

which is the supersymmetrisation of b ∧ F (Chern-Simons).

N. Ambrosetti (Bern U.) DBI and dilaton in N=2 SUSY 2010, Bonn 9 / 11 Conclusions

Global N = 2 supersymmetry and Heisenberg symmetry uniquely determine the form of the theory Generalized the derivation of the DBI from constrained superfields to the coupling to the dilaton (single-tensor) with one linear and one nonlinear supersymmetry Nonlinear N = 2 vector multiplet coupled to full single-tensor: no orientifold truncation needed New super-Higgs without gravity Low energy effective theory description of string theory compactified on rigid Calabi-Yau including perturbative corrections

N. Ambrosetti (Bern U.) DBI and dilaton in N=2 SUSY 2010, Bonn 10 / 11 Web of supersymmetric dualities

Double-tensor (L, L0) 6 ?

Single-tensor ST-ST duality Single-tensor E-M duality Magnetic dual St¨uckelberg Chern-Simons - Single-tensor - gauging (L, Φ) (L, Φ) (L0, Φ0) 6 ? Hypermultiplet (Φ, Φ0)

N. Ambrosetti (Bern U.) DBI and dilaton in N=2 SUSY 2010, Bonn 11 / 11