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.

]]>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.

]]>added publication data to this item:

- Maxim Kontsevich, Graeme Segal,
*Wick rotation and the positivity of energy in quantum field theory*, The Quarterly Journal of Mathematics**72**1-2 (2021) 673–699 [arXiv:2105.10161, doi:10.1093/qmath/haab027]

added pointer to the recent writeup:

- Maxim Kontsevich, Graeme Segal,
*Wick rotation and the positivity of energy in quantum field theory*(arXiv:2105.10161)

and this recent talk:

- Graeme Segal,
*Wick Rotation and the Positivity of Energy in Quantum Field Theory*, talk at IAS Physics Group Meeting, December 2021 (video recording)

following up on talks revolving around the same idea, such as

- Graeme Segal,
*Wick rotation and the positivity of energy in quantum field theory*, talk at Institut des Hautes Études Scientifiques (IHÉS), June 2014 (video recording)

which the entry already used to be linking to.

]]>added these pointers:

On the (non-)existence of Wick rotation for quantum field theory on curved spacetimes:

MathOverflow comments:

See also:

- {#HollandsWald14} Stefan Hollands, Robert Wald, p. 7-8 in:
*Quantum fields in curved spacetime*, Physics Reports Volume 574, 16 April 2015, Pages 1-35 (arXiv:1401.2026, doi:10.1016/j.physrep.2015.02.001)

added picture of the rotation doing it’s thing

]]>added a few more words to the Idea-section, and pointed more prominently to Fulling-Ruijsenaars 87

]]>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.

]]>Does this Wick rotation story ’lift’ to some relation between stable cohomotopy spaces?

]]>added an overview-diagram to the Idea section (copied over from *Euclidean field theory*)

added a bit more text to the Idea-section at *Wick rotation* and in particular added cross-links with *Osterwalder-Schrader theorem*.