## 1 Introduction  
Optical modes tend to diffract after transmitting in a waveguide, which is detrimental to the long-haul transmission of optical signals. If the diffraction is balanced by a self-focusing effect, which is the result of the inherent nonlinearity of some media, solitons are formed and maintain their profiles and power over a relatively long distance. Nonlinearities could be local or nonlocal. In the nonlocal case, the nonlinear response depends on the light incidence area and the neighborhood, while the local nonlinear response depends only on the single incidence area. Practically, nonlocal nonlinearity is demonstrated in various materials, including nematic liquid crystals [1; 2; 3], thermal nonlinear media [4; 5; 6], and so on. Nonlocal nonlinearity strongly affects the propagation of light, providing the potential to form and stabilize many types of solitons, such as defect solitons [7], breather solitons [8], surface solitons [9], multipole solitons [10], and vortex solitons [11]. On the other hand, parity-time (PT) symmetry has drawn much attention since it was first proposed by [PERSON] and [PERSON] [12]. A PT-symmetric system is required to satisfy the PT transformation, which is defined by the parity operator \(\mathbf{P}\) as \(\mathbf{P}\psi(\mathbf{r},t)=\psi(-\mathbf{r},t)\) and the time reversal operator \(\mathbf{T}\) as \(\mathbf{T}\psi(\mathbf{r},t)=\psi^{*}(\mathbf{r},-t)\), where \(\psi(\mathbf{r},t)\) is a wave function in quantum mechanics. A PT-symmetric system is proved to have entirely real spectra even if its Hamiltonian is non-Hermitian [12; 13]. In optics, the refractive index of a two-dimensional (2D) PT-symmetric system is required to be complex and meet \(\mathrm{n}(\mathrm{x},\mathrm{y})=\mathrm{n}^{*}(-\mathrm{x},-\mathrm{y})\); that is, its real part is an even function, and the imaginary part is an odd function. From a practical point of view, the imaginary part of the refractive index is realized by inducing optical gain and loss [14; 15]. By means of a PT-symmetric refractive index, a nonlinear medium with complex potential can be fabricated to support complicated solitons, such as multipole solitons [16; 17; 18] and vortex solitons [19; 20; 21]. Much research has focused on solitons in square nonlinear optical lattices, which can be induced by interfering two pairs of ordinarily plane-polarized light [22]. In contrast to square lattices, triangular lattices are more compact and they support larger bandgaps [23]. Thus, triangular lattices are expected to exhibit new features of solitons upon propagation, such as the periodic beam oscillation inside a waveguide [24], and a certain parameter range in which solitons can exist or be stabilized [25; 26]. However, most of the previous works on solitons in triangular lattices are about local solitons. The evolution dynamics during the propagation of solitons in nonlocal triangular lattices with PT symmetry are still unexplored. In practice, the dynamics and stability of spatial optical solitons and the interaction between them can be used in optical interconnects [27; 28], beam manipulation [29], optical computing [30], all-optical switching [31; 32], and logic gates [33; 34; 35]. With appropriate improvements, solitons in nonlocal nonlinear and PT-symmetric triangular lattices may achieve the same function, thus providing a scheme for creating reconfigurable all-optical circuits. In this paper, we investigate in detail the propagation and stability of the multipole solitons, as well as the evolution of the vortex solitons in a nonlocal triangular optical lattice with PT-symmetric complex potentials. These two types of soliton have complex phases and wave fronts, which means that they will have more complex dynamics and carry more information than fundamental solitons. We find that the dipole and the six-pole solitons can be found in the first bandgap of the specific triangular lattices.