[Paper Review] Non-anticommutative N=(1,1) Euclidean Superspace

Евгений Алексеевич Иванов, Olaf Lechtenfeld+1 more|ArXiv.org|Feb 7, 2004

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TL;DR

This paper introduces nilpotent deformations of four-dimensional N=(1,1) Euclidean superspace by deforming the anticommutation relations of half the fermionic coordinates via a bi-differential Poisson operator, preserving chirality and harmonic analyticity while breaking N=(1,1) supersymmetry to N=(1,0) or N=(1,1/2). The key contribution is the construction of off-shell, gauge-invariant superfield actions for deformed N=2 Maxwell and hypermultiplet theories in harmonic superspace using star products generated by nilpotent operators with P³=0.

ABSTRACT

We study deformations of four-dimensional N=(1,1)Euclidean superspace induced by non-anticommuting fermionic coordinates. We essentially use the harmonic superspace approach and consider nilpotent bi-differential Poisson operators only, which generalizes the recently studied chiral deformation of N=(1/2,1/2) superspace. We present non-anticommutative Euclidean analogs of N=2 Maxwell and hypermultiplet off-shell actions. The talk is based on the paper hep-th/0308012.

Motivation & Objective

• To generalize chiral nilpotent deformations of N=(1/2,1/2) superspace to full N=(1,1) Euclidean supersymmetry.
• To preserve harmonic analyticity and chirality in the deformed theory, crucial for N=2 superfield formulations.
• To construct off-shell, gauge-invariant superfield actions for deformed N=2 Maxwell and hypermultiplet theories in harmonic superspace.
• To analyze the breaking of N=(1,1) supersymmetry to N=(1,0) or N=(1,1/2) under nilpotent deformations.
• To clarify the role of pseudoconjugation in Euclidean N=(1,1) superspace and its compatibility with the deformation structure.

Proposed method

• Utilizes the harmonic superspace formalism to extend chiral nilpotent deformations to N=(1,1) Euclidean superspace.
• Applies a nilpotent bi-differential Poisson operator P with P³=0, generating a star product A⋆B = A e^P B.
• Imposes non(anti)commutativity only on half the fermionic coordinates, specifically θα⋆θβ = θαθβ + ½Cαβ, while keeping other relations undeformed.
• Constructs deformed superfield actions using star-commutators in place of ordinary commutators in the harmonic superfield formalism.
• Ensures gauge invariance and chirality by building actions holomorphic in the chiral superfield strength W, such as S_W ∼ ∫ d⁴x d⁴θ (W² + a W⋆W⋆W).
• Uses the harmonic derivative D⁺⁺ and the connection V⁺⁺ to define deformed gauge theories with infinite-vertex actions involving star products.

Experimental results

Research questions

• RQ1 How can nilpotent deformations of N=(1,1) Euclidean superspace be constructed while preserving harmonic analyticity and chirality?
• RQ2 What is the structure of the deformed N=2 Maxwell and hypermultiplet actions in harmonic superspace under such deformations?
• RQ3 How does the pseudoconjugation operation in Euclidean N=(1,1) superspace affect the reality conditions and consistency of the deformed theory?
• RQ4 To what extent is N=(1,1) supersymmetry preserved or broken in the deformed theory, and what are the residual symmetries?
• RQ5 Can gauge-invariant, off-shell actions be constructed for deformed N=2 super Yang-Mills theories using the star product formalism?

Key findings

• The deformed N=2 Maxwell theory is constructed using a gauge-invariant action S_W ∼ ∫ d⁴x d⁴θ (W² + a W⋆W⋆W), which remains invariant under star-commutator gauge transformations.
• The nilpotent deformation preserves chirality and harmonic analyticity, ensuring the consistency of the harmonic superspace formalism under non-anticommutativity.
• The deformation breaks N=(1,1) supersymmetry to N=(1,0) or N=(1,1/2), depending on the rank of the deformation matrix Cαβ.
• The star product is generated by a nilpotent operator P with P³=0, ensuring that the deformed theory remains local and free of nonlocal divergences.
• The component form of the deformed action remains finite and mild, with corrections appearing only as higher-derivative terms in the Feynman rules.
• The theory admits a consistent formulation in chiral coordinates, with the superfield strength W and W̄ independent and covariantized via star-commutators.