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added a bit more text to the Idea-section at Wick rotation and in particular added cross-links with Osterwalder-Schrader theorem.
added an overview-diagram to the Idea section (copied over from Euclidean field theory)
Does this Wick rotation story ’lift’ to some relation between stable cohomotopy spaces?
Right, so by the discussion at stable splitting of mapping spaces we have that the real cohomology of the space of spatially compactly supported cocycles in degree-4 cohomotopy on Minkowski spacetime (“Lorentzian instantons”) is equivalently correlators of Euclidean field theory on $R^3 \times S^1$, hence Is Wick-rotated n-point functions of relativistic field theory on Minkowski spacetime at finite temperature.
added a few more words to the Idea-section, and pointed more prominently to Fulling-Ruijsenaars 87
added these pointers:
On the (non-)existence of Wick rotation for quantum field theory on curved spacetimes:
MathOverflow comments:
See also:
added pointer to the recent writeup:
and this recent talk:
following up on talks revolving around the same idea, such as
which the entry already used to be linking to.
added publication data to this item:
Is it known to some degree of rigour what happens to a BPS state under Wick rotation? In general, there are different allowed forms of supersymmetries depending on the signature of spacetime. In Section 4 of 2208.02267, it is briefly claimed that if one Wick rotates a 4d theory on a $2+2$ signature (where the corresponding Spin group exhibits the exceptional isomorphism $\text{Spin}(2,2)\cong SL(2,\mathbb{R})\times SL(2,\mathbb{R})$) to a $3+1$ signature, one does not have a supersymmetric theory anymore, yet there are supposed to be states protected against radiative corrections. I’m wondering if this can be used to obtain additional states not manifestly protected in string theory by looking at BPS states on, for example, the analogues in other signatures.
Why do you say “yet” in “yet there are supposed to be states protected against radiative corrections”?
It sounds like you are tacitly assuming that radiative corrections is something only seen after Wick rotation?
Concerning your main question:
I don’t know (how BPS states appear after Wick rotation).
Generally, Wick rotation of non-trivial fields (notably spinors) is a subtle busines; I’d be surprised if the community that typically cares about BPS states has been careful enough to sort out their Wick rotation.
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