I'd totally die in the heat with 〜じゃう

Grammar Explanation: Doing completely or ending up doing with 〜てしまう and 〜ちゃう

On a summer day, Hori and her friends in class can’t ignore the heat.

由紀(ゆき):: 「外(そと)の暑(あつ)さってなんなの…？　クーラーないと死(し)んじゃうって…っ」; “What's with this heat outside...? I'd totally die without the AC...”; Literal: “What is this outside heat...? Without an air conditioner, I would end up dying...”; 「この日差(ひざ)しのなか体育(たいいく)とか絶対(ぜったい)無理(むり)だわ…」; “There's no way I can do PE in this blazing sun...”

Key Points

1死(し)んじゃう = “I’ll end up dying”

死(し)んじゃう is the casual spoken contraction of 死(し)んでしまう, from 死(し)ぬ + 〜でしまう

2Why じゃう follows ん

For verbs whose て-form ends in で, 〜でしまう contracts to 〜じゃう in casual speech.
死(し)ぬ becomes 死(し)んで, then 死(し)んでしまう becomes 死(し)んじゃう.

3クーラーないと sets the condition

クーラー(が)ないと means “if there is no AC” or “without the AC”; the subject marker が is omitted in casual speech.

この (This): この means “this” or “these” and comes directly before a noun. It points to something near the speaker or something the speaker is referring to in the conversation.
体育 (Physical education): 体育(たいいく) means physical education, exercise taught to develop fitness and physical skills.
絶対 (Absolute): 絶対(ぜったい) means absolute or unconditional. As an adverb, it means definitely or under no circumstances, depending on the sentence.
無理 (Impossible): 無理(むり) means impossible, unreasonable, or too demanding. In conversation, it can mean “that’s impossible” or warn someone not to push too hard.