We examine the model of two-dimensional quadratic gravity as a consequence of symmetry breaking within the framework of background field (BF) theory. This theory is essentially an extension of BF theory, introducing an additional polynomial term that operates on both the gauge and background fields. We analyze the theory using the Dirac and Faddeev–Jackiw procedures, determining the form of the gauge transformation, the full structure of the constraints, the counting of degrees of freedom, and the generalized Faddeev–Jackiw brackets. Additionally, we demonstrate the coincidence of the Faddeev–Jackiw and Dirac’s brackets. Finally, we provide some remarks and discuss prospects.

我们研究了背景场 (BF) 理论框架内对称性破缺的结果的二维二次引力模型。该理论本质上是 BF 理论的扩展，引入了一个同时作用于规范场和背景场的附加多项式项。我们使用 Dirac 和 Faddeev-Jackiw 过程分析该理论，确定规范变换的形式、约束的完整结构、自由度的计算以及广义的 Faddeev-Jackiw 括号。此外，我们还证明了 Faddeev-Jackiw 括号和狄拉克括号的一致性。最后，我们提出一些评论并讨论前景。