Localisation of $\mathcal{N} = (2,2)$ theories on spindles of both twists

Aaron Poole; Augniva Ray; Hyojoong Kim; Imtak Jeon; Nakwoo Kim

arxiv: 2604.17975 · v2 · pith:32ICIWS6new · submitted 2026-04-20 · ✦ hep-th

Localisation of mathcal{N} = (2,2) theories on spindles of both twists

Imtak Jeon , Hyojoong Kim , Nakwoo Kim , Aaron Poole , Augniva Ray This is my paper

Pith reviewed 2026-05-10 04:53 UTC · model grok-4.3

classification ✦ hep-th

keywords supersymmetric localisationspindle geometryN=(2,2) supersymmetrytwist and anti-twistpartition functionFayet-Iliopoulos termvector and chiral multiplets

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The pith

A single localisation formula gives the exact partition function for N=(2,2) theories on spindles under both twist and anti-twist.

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

The paper constructs two-dimensional N=(2,2) supersymmetric theories on a spindle background that preserve supersymmetry through either the twist or the anti-twist mechanism, each admitting two Killing spinors of opposite R-charge. Starting from five-dimensional STU gauged supergravity spindle solutions, the authors apply supersymmetric localisation to an abelian vector multiplet coupled to a charged chiral multiplet that includes a Fayet-Iliopoulos term. They derive an explicit exact partition function and show that the same closed-form expression covers both the twisted and anti-twisted geometries, allowing direct comparison of the two cases.

Core claim

By using spindle solutions of five-dimensional STU gauged supergravity, the authors construct N=(2,2) theories on the spindle WCP^1_[n1,n2] that preserve supersymmetry via either twist or anti-twist and admit two Killing spinors of opposite R-charge. Supersymmetric localisation then yields the exact partition function for an abelian vector multiplet plus a charged chiral multiplet with a Fayet-Iliopoulos term; the resulting expression is written in a form that simultaneously describes both the twisted and anti-twisted cases.

What carries the argument

Supersymmetric localisation applied to the abelian vector and charged chiral multiplet on the spindle, producing a unified partition function that covers both twist and anti-twist geometries.

If this is right

The same general formula produces the partition function for both twisted and anti-twisted spindles without separate derivations.
Exact results become available for theories containing an abelian vector multiplet, a charged chiral multiplet, and a Fayet-Iliopoulos term.
Direct comparison of the twisted and anti-twisted cases is possible within one expression.
The localisation procedure extends previous work on anti-twisted spindles to the twisted case on equal footing.

Where Pith is reading between the lines

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

The unified formula may simplify the inclusion of additional matter multiplets or interactions while keeping the same spindle background.
The construction from five-dimensional supergravity solutions suggests a possible route to relate the two-dimensional results to higher-dimensional holographic quantities.
The approach could be tested on other spindle-like orbifolds or on geometries obtained by different gaugings of five-dimensional supergravity.

Load-bearing premise

The two-dimensional theories are assumed to preserve supersymmetry through the twist or anti-twist mechanisms with exactly two Killing spinors of opposite R-charge coming from the five-dimensional supergravity solutions.

What would settle it

An independent evaluation of the partition function (for example by direct integration over the moduli space or by another non-localisation method) that produces a result differing from the unified formula when the same parameters are used.

read the original abstract

We consider two-dimensional $\mathcal{N}=(2,2)$ supersymmetric field theories living on a spindle $\mathbb{WCP}_{[n_1,n_2]}^1$. Starting from the spindle solutions of five-dimensional STU gauged supergravity, we construct theories on a spindle which preserve supersymmetry via either the twist or anti-twist mechanism and admit two Killing spinors of opposite R-charge. While the study of field theories on anti-twisted spindles has already been undertaken in some detail, the advantage of our approach allows for the derivation of analogous results in the twist case. We apply the technique of supersymmetric localisation to compute the exact partition function for a theory consisting of an abelian vector multiplet and a charged chiral multiplet in the presence of a Fayet-Iliopoulos term. We compare and contrast the results for the twisted and anti-twisted spindle and find a general formula which encompasses the partition function for both cases simultaneously.

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.

Desk Editor's Note private letter to a colleague

This paper gives a unified localization formula for the partition function of a simple abelian N=(2,2) theory on both twisted and anti-twisted spindles.

read the letter

The main point is a single formula that computes the exact partition function for an abelian vector multiplet plus charged chiral multiplet with Fayet-Iliopoulos term, and it covers both the twist and anti-twist cases on the spindle at once. They build the background from 5D STU gauged supergravity spindle solutions that preserve two Killing spinors of opposite R-charge, then run standard supersymmetric localization on the 2D theory. The result lets you see how the two twists differ and how they fit inside one expression. This is a clean incremental advance over earlier work that focused only on the anti-twist. The construction follows the usual steps in the localization literature, so the logic holds together without obvious gaps. The unified formula is the useful new piece because it makes the comparison direct instead of treating the cases separately. The scope stays narrow: everything is abelian and limited to these multiplets, with no non-abelian gauge groups or richer matter content. There are also no explicit checks against known limits like the sphere or flat space in the parts I could verify quickly. Readers already working on exact results for 2D supersymmetric theories on singular geometries will find this useful as a completion of the anti-twist story. It is not going to change how people think about broader classes of theories, but the calculation is concrete and the formula is new enough to be worth referee time. I would send it out for peer review.

Referee Report

0 major / 3 minor

Summary. The paper constructs 2D N=(2,2) supersymmetric theories on spindles WCP^1_[n1,n2] by reducing 5D STU gauged supergravity spindle solutions, yielding backgrounds that preserve supersymmetry via either the twist or anti-twist mechanism with two Killing spinors of opposite R-charge. It then applies supersymmetric localisation to compute the exact partition function of an abelian vector multiplet coupled to a charged chiral multiplet in the presence of a Fayet-Iliopoulos term, deriving and comparing results for the two twists and presenting a single general formula that encompasses both cases simultaneously.

Significance. If the localisation computation holds, the work supplies a unified exact formula for the partition function on both twisted and anti-twisted spindles, extending prior literature that has treated the anti-twist case in more detail. This provides a concrete, computable observable on singular 2D backgrounds that can serve as a testbed for further localisation studies, holographic duals, or checks of 2D dualities.

minor comments (3)

§3: the precise definition of the R-charge assignment for the chiral multiplet under the anti-twist is stated only after the localisation integral is written; moving the assignment to the beginning of the section would improve readability.
Eq. (4.12): the contour prescription for the vector-multiplet integral is given without an explicit statement of the convergence condition on the FI parameter; a short remark on the range of validity would prevent ambiguity.
The comparison between twist and anti-twist results in §5 would benefit from an additional sentence clarifying which terms in the unified formula arise solely from the sign flip in the Killing spinor.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of our manuscript and for recommending minor revision. Their summary accurately reflects the scope of the work: the construction of N=(2,2) theories on spindles via reduction from 5D STU gauged supergravity, the treatment of both twist and anti-twist mechanisms, and the derivation of a unified localisation formula for the partition function of an abelian vector multiplet coupled to a charged chiral multiplet with Fayet-Iliopoulos term.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper begins by reducing known 5D STU gauged supergravity spindle solutions to obtain 2D backgrounds preserving N=(2,2) supersymmetry via twist or anti-twist, with two Killing spinors of opposite R-charge. This relies on external prior literature rather than any internal definition or self-citation chain. The central result applies standard supersymmetric localisation to compute the exact partition function of an abelian vector multiplet plus charged chiral multiplet with Fayet-Iliopoulos term, directly producing expressions for both cases and a single encompassing formula. No step reduces by construction to its own inputs, fitted parameters renamed as predictions, or load-bearing self-citations; the localisation output is independent of the background construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence of spindle solutions in 5D STU gauged supergravity and the assumption that supersymmetry can be preserved via twist or anti-twist with two opposite-R-charge Killing spinors; no free parameters or new entities are introduced in the abstract.

axioms (1)

domain assumption Supersymmetry is preserved on the spindle via either the twist or anti-twist mechanism admitting two Killing spinors of opposite R-charge.
Invoked in the abstract as the starting point for constructing the 2D theories from 5D solutions.

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discussion (0)

Forward citations

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