Can a jewish male wear sleeveless shirt on a hot day? Is there a direct prohibition?

Shalom, Thank you for your question. There are several levels of what is the correct clothing for a man. Firstly, there is the correct clothing for prayer. Then there are guidelines for Torah scholars. Lastly, there are clothing for wearing in public as opposed to what one may wear in private. I assume your question relates to this last category – may a man wear a sleeveless shirt in public? This will depend very much on what is generally acceptable in your location. For example, if one works as a life guard (at a men’s only pool, of course), then wearing a sleeveless shirt is the norm, and not a problem. If one is a lawyer in a downtown office, this would not do. So, if wearing such shirts is widespread by community you are part of, in such a place it would be acceptable. (I can imagine it being acceptable for men to wear such shirts in their gardens, or at a picnic etc). If though, such clothing is only worn by other segments of society, or at other places and times, then one should not wear such items. (See Shaarim Metzuim BeHalacha, 2, footnote 9). In general, although one is not obligated by a clear law, one should strive in their clothing to two aspirations. Firstly, to dress in a modest fashion. This includes (but does not end with) covering one’s body. Secondly, to represent a good image of a Jew. This means dressing in a way that “advertises” Judaism in a good and proud light. For both these reasons, a sleeveless shirt for a man is (in most situations) not a good idea. In places were wearing it would be an embarrassment, or viewed as immodest it is forbidden. Blessings.