A sigh from a Jewish person can repair what is broken in the world. Rabbi Nachman of Breslov taught this not as poetry but as metaphysics. The sigh, the deep exhalation of grief or longing, is an extension of the ruach (רוח), the breath, and the breath is the very substance from which the world was made.
"By the breath of His mouth, their entire hosts were created" (Psalms 33:6). The renewal of the world will also come through breath: "You will send Your spirit and they will be created; You renew the face of the earth" (Psalms 104:30). And within each human being, the breath is the life-force itself: "He breathed into his nostrils the breath of life" (Genesis 2:7).
When something is lacking, what is really missing is the ruach, the life-force of that thing. A sigh is the extension of breath, corresponding to erekh apayim (ארך אפים), divine patience, literally "extended breath." By sighing over what is lacking, you draw fresh life-force to fill the void. The lack is restored.
But where does this fresh life-force come from? From the tzaddik (a righteous person) of the generation, who is bound to the Torah, where the essential ruach-of-life resides. "And the spirit of God hovers over the waters' surface" (Genesis 1:2), and the "waters" are the Torah (Tikkunei Zohar 36). The tzaddik, bound to Torah, becomes the source from which others draw vitality.
This is the meaning of King David's harp, which had five strings corresponding to the Five Books of the Torah. A northern wind, the ruach tzefonit (רוח צפונית), would blow upon it and the harp would play by itself (Berakhot 3b). The hidden spirit in the human heart, the ruach ha-tzafun (רוח הצפון), corresponds to this wind. When the Torah scholar sighs and extends their breath, they draw melody from the same source that played David's harp at midnight.