"There is hope."
"Hay" is the present-tense form of the verb, which means "there is/are". "Haya" is the subjunctive form of the verb, which is a little harder to translate, because the use of the subjunctive is not the same in English. Examples: "Hay esperanza" ("There is hope"), but "Deseo que haya esperanza" ("I wish there were hope"). Subjunctive is used when there is doubt or insecurity inherent in the phrase.
Because "esperanza" is the object of the sentence, not the subject (although in English, it is the subject). If it were the subject, you would have to use the definite article; nouns can't be the subject if they don't have an article, but objects don't require them.
A noun 'can' be a subject without a definite article -- certainly with an indefinite article (un hombre) or an adjective (mi hermano).
Okay, that is helpful! I'd never heard that before--only that this type of abstract nouns generally need the article in Spanish where they don't in English.
Because ya darn don't use definite articles with hay! ;)
I'm 99% sure.