I was going to create a Witch build, but I've decided to go with Sorcerer, because it fits her backstory perfectly. However, there is a problem: I need to dump her Charisma. This is for roleplay reasons. I imagine this character as incredibly socially awkward, and a high Charisma is incompatible with that. So I came up with the solution of having her cast with Wisdom instead. Essentially, her spells work through force of will, not personality.
Does switching casting stats effect the game balance of a Sorcerer in any way, shape, or form?
There's actually an Archtype that does just that. It trades the Bloodline Arcane (if I remember correctly) for Wisdom Casting. It's exclusive to the Celestial Bloodline, but I can't think of any reason it necessarily should be.
Should I have to give away anything for Wisdom casting?
Yes it does change things and affect game balance but only on the theoretical level. In practice as long as everyone at the table is good with it it's not a terrible thing.
If you really wanted to you could use the cross-blooded archetype to get the Empyreal bloodline and whatever the normal blood line for the witch is and have everything Kosher with RAW, but that's only if someone insists upon it.
As has been mentioned up-thread, Empyreal is the Wildblooded archetype for the Celestial bloodline. You give up the Celestial bloodline's Arcana (gives summoned creatures DR/evil) in return for Wisdom-based spellcasting and a +2 bonus on Heal and Knowledge (Religion) checks. You also give up the Celestial bloodline's wings (L 9) and instead get the ability channel energy like a cleric a few levels lower (also received at L 9). See Ultimate Magic, or the PF SRD.
Just to add the link to bellona's comment, the Empyreal Wildblooded Archetype can be found here
Empyreal
Your heavenly power derives from insight rather than force of personality.Associated Bloodline: Celestial.
Bloodline Arcana: Unlike most sorcerers whose innate magic is powered by force of personality, you use pure willpower to master and fuel your magic. You use your Wisdom, rather than your Charisma, to determine all class features and effects relating to your sorcerer class, such as bonus spells per day, maximum spell level you can cast, and the save DCs of your spells. You gain a +2 bonus on all Heal and Knowledge (religion) checks.
Bloodline Powers: Your quasi-divine nature gives you strange powers.
Sacred Cistern (Su): At 9th level, your bloodline makes you a natural receptacle of divine energy. You can channel energy once per day as a cleric of your sorcerer level – 4. This bloodline power replaces wings of heaven.
You arent giving up the bloodline arcana, but wisdom casting is PART of your bloodline arcana.
Thank you, Kolokotroni, for the proper link and most of the clarification! :)
But ... the introduction at the start of the section on Wildblooded archetypes states that such a Wildblooded sorcerer should "use the normal bloodline's class skills, bonus spells, and bonus feats, and the mutated bloodline's bloodline arcana. Use the normal bloodline's bloodline powers, except when the mutated bloodline replaces one of those powers." (UM, p. 70)
Which I read as meaning that the Wildblooded version does _not_ use the original bloodline's arcana, but uses instead the mutated one. (Otherwise the normal bloodline's arcana would be listed with the normal bloodline's class skills, bonus spells, and bonus feats.)
The issue is, I want the Stormborn bloodline, not a Crossblooded bloodline or Celestial. I brought up the question of whether letting any Sorcerer cast with Wisdom would be unbalancing to designer James Jacobs, and he says it shouldn't really effect game balance at all, so I guess I got my answer.