What is the theory behind an active region? If we check data in solar monitors, there are some cases where there is no sunspot but there is still an active region.
Why is that the case?
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Active regions are observed in EUV and x-rays while sunspots are observed in white light. Thus, sunspots tend to be identified as photospheric phenomena and active regions coronal. It can be the case that an active region is due to a sunspot, but it need not be. See the link to the corona Wiki page and then look for the sub-header on Active Regions.
As an aside, the type of photons used to make an observation often correlate with altitude. For instance, EUV and x-rays are typically found exclusively in the corona and white light down near the photosphere. While both active regions and sunspots correspond to enhanced magnetic fields in spatially confined regions, they are seen at different altitudes. Generally, a sunspot will have an active region overhead due to its enhanced magnetic fields, but an active region need not have a corresponding sunspot.
What is the theory behind an active region?
The plasma in the photosphere, chromosphere, and corona are tied to the magnetic field, something called the frozen-in condition. Bulk fluid-like motions can cause magnetic fields to accumulate in spatially confined regions. If the fields get strong enough, they will actually reduce the thermal pressure of the plasma by pushing out the hotter particles, resulting in a sunspot if the region gets large enough.
Active regions can be the precursors to sunspots or never form into sunspots and just embody locally enhanced magnetic fields. Sunspots are regions where the magnetic pressure dominates over the particle pressure, pushing out hotter particles. The total pressure stays constant, i.e., the magnetic pressure plus the particle thermal pressure.
Why is that the case?
Sunspots are an extreme, if you will. They are the result of the magnetic forces dominating over the fluid forces. Compared to active regions, they are rare. Active regions are just regions of enhanced magnetic field identified as groups of loop structures seen in EUV and/or X-rays. They form for the same reasons as sunspots but are just not as intense.