feat(SUSY/N1): add chiral scalar configuration space and coordinate CLMs#1113
feat(SUSY/N1): add chiral scalar configuration space and coordinate CLMs#1113pariandrea wants to merge 1 commit into
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Introduce the minimal label and configuration data for the N=1 chiral scalar sector: `ChiralIndex`, the configuration space `ChiralScalarValue n = Fin n → ℂ`, the derived anti-chiral view via complex conjugation, and the per-coordinate ℝ-linear projections `chiralCoordCLM` / `antiChiralCoordCLM`. Self-contained over Mathlib; entry point for the forthcoming Wirtinger calculus layers. Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
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| /-- The label data of an N=1 chiral sector: simply the number of complex | ||
| scalars. The chiral and anti-chiral indices both range over `Fin n`. -/ | ||
| abbrev ChiralIndex : Type := ℕ |
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I have two questions about this definition:
- I think
ChiralIndexmight be the wrong name here, as an element ofChiralIndexis the number of chiral fields, whilst I think a chiral index is really an element ofFin n. - what stops you generalizing this to an arbitary
Fintype, rather thanFin n?
I wonder if something like the following is better (I think discussed before):
structure Model (ChiralIndexingType : Type) [Fintype ChiralIndexingType]
(AntiChiralIndexingType : Type) [Fintype AntiChiralIndexingType] where
equiv : ChiralIndexingType ≃ AntiChiralIndexingType| /-- The chiral scalar configuration: an assignment of complex values to each | ||
| chiral label. Declared as `abbrev` so unification can apply Mathlib's | ||
| calculus lemmas about `α → ℂ` directly. -/ | ||
| abbrev ChiralScalarValue (n : ChiralIndex) := Fin n → ℂ |
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I think ChiralScalarConfiguration would be a better name. Maybe add to the doc-string something like:
The term of this type gives a value to each of the chiral scalar fields.
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I think you likely also need a type AntiChiralScalarConfiguration
| /-- The anti-chiral view of a chiral scalar configuration: pointwise complex | ||
| conjugation. Realises the physical statement that in the scalar sector, | ||
| barring is complex conjugation. -/ | ||
| def antiChiral (u : ChiralScalarValue n) : ChiralScalarValue n := |
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Should be able to lift this to a CLM.
| public import Physlib.Relativity.Tensors.TensorSpecies.Basic | ||
| public import Physlib.Relativity.Tensors.Tensorial | ||
| public import Physlib.Relativity.Tensors.UnitTensor | ||
| public import Physlib.SUSY.N1.Basic |
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We already have Physlib.Particles.SuperSymmetry, maybe we should put it there? Also if it is about Wirtinger calculus, maybe we should put this PR in Physlib.Mathematics
Motivation
This is the foundational layer of a formalization of the N=1 chiral scalar sector
in Physlib. The broader goal is to build the Wirtinger differential calculus on
complex field space — the
∂/∂φ^Iand∂/∂φ̄^Ioperators that underpinsupersymmetric and Kähler geometry — and ultimately a worked Kähler-geometry
application (the log Kähler potential on
H^nand its metric).Following review feedback on the larger branch, that work is being split into
reviewable pieces. This PR contributes only the basic data layer: the labels, the
configuration space, and the coordinate maps that everything downstream is built on.
It is fully self-contained over Mathlib, so it can be reviewed and merged
independently of the calculus that will follow.
What this adds
Physlib/SUSY/N1/Basic.lean:ChiralIndex— the label data of the chiral sector reduces to a singlen : ℕ, the number of complex scalars. Chiral and anti-chiral indices share the one labelset
Fin n; in the scalar sector the bar is the identity, so no separate pairing iscarried as data.
ChiralScalarValue n = Fin n → ℂ— the physical scalar configuration space,carrying exactly
2nreal degrees of freedom. Declared as anabbrevso Mathlib'sα → ℂcalculus lemmas apply by unification.ChiralScalarValue.antiChiral— the anti-chiral view as the derivedpointwise complex conjugate, not an independent input. Every
φ̄formula factorsthrough this, keeping the configuration space physical and the
2ncount honest.chiralCoordCLM I/antiChiralCoordCLM I— the per-coordinate projectionsz^I = u Iandz̄^I = star (u I)as continuous ℝ-linear maps. Both are continuousand real-linear, but only
chiralCoordCLMis ℂ-linear —antiChiralCoordCLMfactorsthrough conjugation and must live at the
→L[ℝ]level.