One-loop quantization of Euclidean D3-branes in holographic backgrounds

In this note we analyze the semi-classical quantization of D3-branes in three different holographic backgrounds in type IIB string theory. The first background is Euclidean AdS₅ with S¹ × S³ boundary accompanied with a twist to preserve supersymmetry. We work out the spectrum of fluctuations around the classical D3-brane configuration, compute its one-loop partition function, and match to the non-perturbative correction to the superconformal index of $\mathcal{N}$ \( \mathcal{N} \) = 4 SYM. We then study Euclidean D3-branes in the Pilch-Warner geometry dual to the IR Leigh-Strassler fixed point of $\mathcal{N}$ \( \mathcal{N} \) = 1^* with the aim to find non-perturbative corrections to its index. Finally we study Euclidean D3-branes in the non-geometric $\mathcal{N}$ \( \mathcal{N} \) = 2 J-fold background which is dual to the gauging of the 3D Gaiotto-Witten SCFT.