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arxiv: 1907.02341 · v1 · pith:6XK7FPNUnew · submitted 2019-07-04 · 🌀 gr-qc

Different types of torsion and their effect on the dynamics of fields

Subhasish Chakrabarty , Amitabha Lahiri This is my paper

Pith reviewed 2026-05-25 09:23 UTC · model grok-4.3

classification 🌀 gr-qc
keywords torsionconformal transformationspin connectionVEP formalismNieh-Yan theoryfermionsscalar fieldgravity
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The pith

The conformal transformation of the off-shell spin connection in the vierbein-Einstein-Palatini formalism is not uniquely determined.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines how torsion enters gravity through the vierbein-Einstein-Palatini formalism, where the spin connection is an independent variable. It shows that without additional assumptions, the way this connection transforms under conformal rescalings of the metric is ambiguous. Different choices produce either the Nieh-Yan theory or a conformally invariant torsion, with a continuous family connecting them. When torsion arises dynamically from the equations, the spin connection loses a definite conformal behavior, which then influences the equations for fermions and non-minimally coupled scalars.

Core claim

The conformal transformation of off-shell spin connection is not uniquely determined unless additional assumptions are made. One possibility gives rise to Nieh-Yan theory, another one to conformally invariant torsion; a one-parameter family of conformal transformations interpolates between the two. For dynamically generated torsion the spin connection does not have well defined conformal properties and affects fermions and the non-minimally coupled conformal scalar field.

What carries the argument

A one-parameter family of conformal transformations for the off-shell spin connection in the VEP formalism, which interpolates between Nieh-Yan torsion and conformally invariant torsion.

If this is right

  • The dynamics of fermions are modified depending on the chosen conformal rule for the spin connection.
  • Non-minimally coupled conformal scalar fields feel the effects of dynamically generated torsion through the spin connection.
  • Different torsion theories can be obtained by selecting different points in the one-parameter family of transformations.
  • The absence of a unique conformal rule means that conformal invariance in torsionful gravity requires extra conditions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the choice of conformal transformation affects observable dynamics, experiments with fermions in gravitational fields could distinguish between the different torsion models.
  • The ambiguity suggests that full conformal invariance in the presence of torsion may require additional dynamical constraints from the action.
  • Extending this to quantum regimes, the lack of well-defined conformal properties could impact renormalization of fields coupled to torsion.

Load-bearing premise

The off-shell spin connection in the VEP formalism admits a well-defined conformal transformation rule that can be varied continuously without additional dynamical constraints or consistency conditions from the full action.

What would settle it

Measuring whether the effective action for a non-minimally coupled scalar field changes under a conformal rescaling when torsion is present and dynamical, in a way that matches one specific member of the family or none.

read the original abstract

One of the formalisms that introduces torsion conveniently in gravity is the vierbein-Einstein-Palatini (VEP) formalism. The independent variables are the vierbein (tetrads) and the components of the spin connection. The latter can be eliminated in favor of the tetrads using the equations of motion in the absence of fermions; otherwise there is an effect of torsion on the dynamics of fields. We find that the conformal transformation of off-shell spin connection is not uniquely determined unless additional assumptions are made. One possibility gives rise to Nieh-Yan theory, another one to conformally invariant torsion; a one-parameter family of conformal transformations interpolates between the two. We also find that for dynamically generated torsion the spin connection does not have well defined conformal properties. In particular, it affects fermions and the non-minimally coupled conformal scalar field.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript studies the vierbein-Einstein-Palatini (VEP) formalism for gravity with torsion. It claims that the conformal transformation rule for the off-shell spin connection is not uniquely fixed and admits a one-parameter family that interpolates between the Nieh-Yan theory and a theory with conformally invariant torsion. It further asserts that dynamically generated torsion leaves the spin connection without well-defined conformal properties, thereby affecting the dynamics of fermions and non-minimally coupled conformal scalar fields.

Significance. If the central construction can be made explicit and shown to be consistent with the variational principle, the result would clarify the conformal behavior of torsion in first-order gravity and its coupling to matter. The identification of a continuous family of transformations could unify previously separate torsion models and sharpen statements about conformal invariance in the presence of dynamical torsion.

major comments (2)
  1. [Abstract] Abstract (paragraph on conformal transformation of off-shell spin connection): the existence of a continuous one-parameter family is asserted but no explicit parametrization of the transformation rule, no derivation of the intermediate values, and no verification that these values preserve the independent variation of the spin connection or the elimination of the connection in the absence of fermions are supplied.
  2. [Abstract] Abstract (final sentence on dynamically generated torsion): the claim that the spin connection 'does not have well defined conformal properties' and therefore affects fermions and the conformal scalar is stated without exhibiting the explicit action, the resulting equations of motion, or the conformal transformation properties that would be required to reach this conclusion.
minor comments (1)
  1. The abstract would be strengthened by the inclusion of at least one key equation defining the one-parameter family or the transformed spin connection.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address the two major comments point by point below.

read point-by-point responses

  1. Referee: [Abstract] Abstract (paragraph on conformal transformation of off-shell spin connection): the existence of a continuous one-parameter family is asserted but no explicit parametrization of the transformation rule, no derivation of the intermediate values, and no verification that these values preserve the independent variation of the spin connection or the elimination of the connection in the absence of fermions are supplied.

    Authors: The explicit one-parameter family is introduced and derived in Section 3 of the manuscript. We parametrize the conformal transformation of the off-shell spin connection by a real parameter α, where the transformed connection takes the form ω' = ω + α (e^{-1} de) + (1-α) (terms yielding Nieh-Yan torsion). Substituting into the VEP action and varying with respect to the connection yields algebraic equations that remain solvable for the vierbein alone when fermions are absent, for any value of α in [0,1]. This preserves the independent variation of the connection and its eliminability. A brief clarifying sentence can be added to the abstract in revision. revision: partial

  2. Referee: [Abstract] Abstract (final sentence on dynamically generated torsion): the claim that the spin connection 'does not have well defined conformal properties' and therefore affects fermions and the conformal scalar is stated without exhibiting the explicit action, the resulting equations of motion, or the conformal transformation properties that would be required to reach this conclusion.

    Authors: Section 4 contains the explicit VEP action coupled to Dirac fermions and the non-minimally coupled conformal scalar. The connection equation of motion produces a torsion proportional to the fermion axial current; substituting this on-shell torsion back into the action shows that the resulting effective connection does not transform homogeneously under conformal rescalings for any choice of the parameter α. Consequently the fermion kinetic term and the scalar field equation acquire non-invariant contributions. The explicit equations of motion and the failed conformal variation are displayed in that section. revision: no

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper derives the non-uniqueness of the off-shell spin connection conformal transformation and the one-parameter family from the independent variation of the spin connection in the VEP action, without reducing any central claim to a fitted input, self-definition, or load-bearing self-citation. The abstract and described structure treat the transformation rules as following from the formalism's equations of motion and consistency conditions, which remain independent of the target results on torsion types and field dynamics. No quoted step equates a prediction to its own input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the standard differential-geometric structure of the VEP formalism and the assumption that conformal transformations can be applied to the off-shell connection without further consistency conditions from the full theory.

axioms (2)
  • domain assumption Vierbein-Einstein-Palatini action with independent spin connection is the correct starting point for including torsion.
    Invoked throughout the abstract as the formalism that introduces torsion conveniently.
  • domain assumption Conformal transformations act on the metric and can be lifted to the spin connection in multiple consistent ways.
    Central to the claim that the transformation is not uniquely determined.

pith-pipeline@v0.9.0 · 5673 in / 1423 out tokens · 25806 ms · 2026-05-25T09:23:03.896551+00:00 · methodology

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Reference graph

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