Learn Hardcore Portuguese (Portugal): Se bem que a polícia vigie a manifestação, deixa os manifestantes circular em paz. - Although the police monitor the demonstration, they let the protesters move around in peace.

Because se bem que is a conjunction that introduces a concessive clause (similar to “even though / although”), and it normally requires the subjunctive mood in Portuguese.

Vigiar (to watch/monitor) in the present indicative (fact) for a polícia (3rd person singular) is: vigia.
In the present subjunctive (used after se bem que, embora, mesmo que, etc.), it becomes: vigie.

So:

A polícia vigia a manifestação. – The police watches the protest. (statement of fact, indicative)
Se bem que a polícia vigie a manifestação, … – Even though the police watches the protest, … (subjunctive after a concessive conjunction)

Using vigia in this sentence would sound ungrammatical in standard Portuguese.

What exactly does se bem que mean, and how is it different from embora or apesar de?

Se bem que is a fixed expression meaning roughly “even though / although”. It introduces something that could be an obstacle, but in fact does not prevent what comes next.

Se bem que a polícia vigie a manifestação, deixa os manifestantes circular em paz.
→ Even though the police is monitoring the protest, it lets the protesters move around in peace.

Comparison:

Embora a polícia vigie a manifestação, …
Very similar meaning. Embora is also followed by the subjunctive and is very common.
Apesar de a polícia vigiar a manifestação, …
Here you use apesar de
- infinitive (vigiar). This is another way to say “despite the fact that…”

So:

se bem que
- subjunctive
embora
- subjunctive
apesar de
- infinitive (or apesar de que
  - subjunctive, more formal)

In everyday speech, embora is more frequent and often sounds more neutral. Se bem que is also used, sometimes with a slightly more “commenting” or reflective tone, but the difference is subtle.

Why does se bem que trigger the subjunctive, even though the speaker knows the police really is monitoring the protest?

In Portuguese, the subjunctive is not only about doubt or unreality; it is also used for:

Attitudes (concession, wish, possibility)
Subordinate clauses introduced by certain conjunctions, including concessive ones

Conjunctions like:

se bem que
embora
mesmo que
ainda que

almost always take the subjunctive, regardless of whether the fact is actually true in real life.

So even if the speaker clearly believes it:

Se bem que a polícia vigie a manifestação, …

still uses the subjunctive, because the mood is controlled by the type of clause (concessive introduced by se bem que), not by the speaker’s doubt.

How is vigie formed from vigiar? What is its full present subjunctive paradigm?

Vigie is the 3rd person singular of the present subjunctive of vigiar.

Present subjunctive of vigiar:

que eu vigie
que tu vigies
que ele / ela / você vigie
que nós vigiemos
que vós vigieis
que eles / elas / vocês vigiem

The change from g to gi before e and i keeps the soft /ʒ/ sound (like the j in jeans in some accents).

Why does a polícia take a singular verb (vigie, deixa) if it refers to many officers?

In Portuguese, a polícia is a collective noun and is grammatically feminine singular.

a polícia vigia
a polícia vigie
a polícia deixa

Even though the group consists of many people, the noun itself behaves like “the police force / the police as an institution”, which is singular.

So you use singular verb agreement:

A polícia está aqui. – The police is here.
A polícia vigia a manifestação. – The police watches the protest.
Se bem que a polícia vigie a manifestação, … – Even though the police watches the protest, …

In English, you might also say “The police are…”; Portuguese does not do that with a polícia in standard usage.

What is the difference between a polícia and o polícia / os polícias?

a polícia
- Feminine singular.
- Refers to the police as an institution or a force.
- Takes singular verbs: a polícia está, a polícia vigia, a polícia deixa.
o polícia / os polícias (European Portuguese)
- Masculine (singular/plural).
- Refers to an individual police officer or police officers.
- Behaves like an ordinary countable noun:
  - O polícia fala com os manifestantes. – The police officer talks to the protesters.
  - Os polícias vigiam a manifestação. – The police officers monitor the protest.

In your sentence, a polícia = “the police (as a body)”.

Who is the subject of deixa in this sentence, and why is there no pronoun like ela?

The subject of deixa is still a polícia.

The full structure is:

Se bem que a polícia vigie a manifestação, (a polícia) deixa os manifestantes circular em paz.

Portuguese normally drops subject pronouns (eu, tu, ele, ela, nós, vocês, eles, elas) when the subject is clear from context or from the verb form. Here:

deixa is 3rd person singular
the only logical 3rd person singular subject in the context is a polícia

So adding ela (… ela deixa os manifestantes…) would sound unusual or emphatic; the normal version simply omits it.

Why is circular in the infinitive (deixa os manifestantes circular) and not conjugated like circulam?

This is because deixar is being used as a causative / permissive verb, similar to English “let / allow”:

deixar alguém fazer algo – to let someone do something

The pattern is:

deixar (conjugated) + direct object (the person) + bare infinitive (the action)

So:

deixa os manifestantes circular em paz
= (it) lets the protesters move around in peace

If you said:

deixa os manifestantes circulam em paz

that would be ungrammatical, because deixar in this meaning does not take a finite verb; it takes an infinitive.

An alternative structure is:

deixa que os manifestantes circulem em paz
(deixar que
- subjunctive), which is also correct but syntactically different.

Could we say deixa-os circular em paz instead of deixa os manifestantes circular em paz?

Yes. In European Portuguese:

Deixa os manifestantes circular em paz.
Deixa-os circular em paz.

both are correct.

Points to note:

os manifestantes is the direct object of deixa.
You can replace it with the clitic pronoun os:
- deixa os manifestantes circular → deixa-os circular

In European Portuguese, the unstressed object pronoun usually attaches to the verb (enclisis) when the verb is in the affirmative and not preceded by certain triggers (like não, se, etc.), so deixa-os is the expected shape.

Both versions mean the same; with deixa-os, the reference to the manifestantes must already be clear from context.

What exactly does manifestação mean here, and how does it relate to manifestantes?

manifestação
- A public demonstration / protest / rally, usually political or social.
- a manifestação = “the protest / the demonstration”
manifestantes
- The people taking part in the protest: protesters / demonstrators.

So:

a polícia vigie a manifestação
→ the police is monitoring the protest event
deixa os manifestantes circular em paz
→ it lets the protesters (the participants) move around in peace

Is circular em paz an idiom, or just literal?

It is mostly literal:

circular = to move around, circulate (walk, drive, go from place to place)
em paz = in peace, peacefully

Together:

circular em paz ≈ “to move around peacefully, without being disturbed/harassed”

It’s not a fixed idiom with a special hidden meaning; it just implies the police is not interfering or bothering the protesters as they move.

Could we say Se bem que a polícia esteja a vigiar a manifestação, … instead of vigie a manifestação? What changes?

Yes, in European Portuguese you can say:

Se bem que a polícia esteja a vigiar a manifestação, deixa os manifestantes circular em paz.

Differences:

vigie a manifestação
- Simple present subjunctive.
- More neutral, focusing on the fact that the police watches/monitors.
esteja a vigiar a manifestação
- Present subjunctive of estar
  - a
    - infinitive (vigiar).
- This is the typical progressive aspect in European Portuguese.
- Emphasises the ongoing action right now: “is in the process of watching/monitoring”.

Both are grammatically fine; the choice is stylistic and aspectual.

Is se bem que common in spoken European Portuguese, or does it sound very formal?

Se bem que is used in spoken European Portuguese, but:

It can sound a bit more formal or bookish than embora, depending on the speaker and context.
In everyday speech many people would more naturally say:
- Embora a polícia vigie a manifestação, deixa os manifestantes circular em paz.

That said, se bem que is far from rare, and it also appears in conversation, often when the speaker is adding a kind of reflective or concessive comment, similar to English:

“Even though…, still …” / “Mind you, although…”