Ten-dimensional localization

arXiv:2607.05529 · hep-th · Submitted 2026-07-06 · Read on arXiv

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Introduction to the show: ident: High Energy Physics: Theory. The day in high-energy theory: gravity, strings, fields and the questions left open.

Kai: I'm Kai.

Mira: And I'm Mira, here with Lev.

Kai: We're looking at the paper: "Ten-dimensional localization".

Mira: Here's the gist of the paper. Ten-dimensional localization constructs equivariantly closed polyforms for the fluxes and action of (massive) type IIA and type IIB supergravity directly in ten dimensions This formalism provides a unified framework for computing flux quantization and the on-shell action for arbitrary minimally supersymmetric backgrounds The paper illustrates this formalism by applying it to two distinct classes of solutions in massive type IIA, including hemisphere HS4 fibrations over M6 and X6 fibrations over M4 Construction of Equivariant Polyforms The core methodology involves constructing equivariantly closed polyforms for the Page fluxes and the on-shell action using a symmetry, specifically the Killing vector K, which is derived from supersymmetry conditions The necessary and sufficient conditions for preserving supersymmetry allow for the construction of these polyforms The key observation is that these supersymmetry conditions are precisely sufficient to find a completion of all the Page fluxes and the on-shell action into equivariantly closed polyforms For massive type IIA, the central result is given by the polyform of polyforms for the Page fluxes: ΨF = F + Xm=1 Ψm, where Ψm = (-1)m m!

Kai: ΥmF + mΥm-1Φ This construction applies to both (massive) type IIA and type IIB supergravity under the single assumption of minimal supersymmetry The equivariantly closed polyform for the Page flux Fk is extracted by taking the sum of the k-form of F, i.e.

Mira: Fk, the (k − 2)-form of Ψ1, the (k − 4)-form of Ψ2 and so forth until one ends up with either a zero-form for type IIA or a one-form for type IIB Computation of the On-Shell Action The partially on-shell string frame action is given by a pseudo action in the democratic formulation This pseudo action can be equivariantly completed using the previously constructed polyforms, resulting in S = -1/4(2π) 7l/8 s Z M10 ∓⟨F, F⟩ By interpreting this result suitably using a choice of electric/magnetic polarization, one obtains a non-zero result for the partially on-shell action The final result is then obtained by taking the wedge product of the equivariantly closed Page flux polyforms to compute the equivariantly closed polyform for the action Application to Specific Geometries The formalism is applied to two classes of solutions in massive type IIA, specifically a four-dimensional hemisphere bundle over a six-dimensional base M6 and X6 fibrations over a four-dimensional spacetime For the hemisphere HS4 fibration over M6, the master formula (3.4) is derived, which depends on the topology and global data of a putative solution The localized action for an isolated fixed point in this case is given by Ip = 9/4 FS5 F(bp1, bp2) εp1εp2 For the spindle example, the on-shell action can be expressed as I = -8/27 FS5 (b+1b+2) 3/2 - σ(b-1b-2) 3/2 ε1ε2 Generalization and Conclusion The construction relies only on the universal supersymmetry conditions of generalized geometry

5: and therefore applies to arbitrary minimally supersymmetric type IIA and type IIB backgrounds admitting a pair of Killing spinors The main advantage of the ten-dimensional perspective is that the localization computation completely bypasses the need for a consistent truncation This represents a substantial simplification of the problem The paper concludes by showing that in each case, the localized action is expressed entirely in terms of the equivariant weights and the topology of the fixed point sets and makes no assumptions about a consistent truncation on the fibre Details of Derivations The appendix provides an explicit derivation of the polyforms for the Page fluxes used in the main text The construction proceeds by considering contractions of K into the Page flux F, leading to K F = d

−(ΥF + Φ): ≡ dΨ1 The full tower of equivariantly closed polyforms for the Page fluxes is given by ΨF = Xm=0 Ψm, where Ψm = (-1)m m!

Kai: ΥmF + mΥm-1Φ The partially on-shell action is derived from the pseudo action (2.24) by replacing Maxwell fluxes F RR with the Page fluxes F in the Mukai pairing The final result for the partially on-shell action, ignoring boundary contributions, is then given by S = -1/4(2π) 7l/8 s Z M10 ∓⟨F, F⟩ The constraints imposed by the homology relations lead to the final form of the action in terms of a prepotential F(bi) = volT1,1 volT1,1 (bi) Homology Considerations The homology relations are used to compute flux quantization and ultimately to compute the on-shell action The argument is the same as for fixed Σ, where now we simply apply this to Σ = Γα The localized action is then IΣ = -i 2(2π) 7l/8 s (2π) 4 εb1b2b3 "ZΣN Ψ2 + 2πΨ0 Xm=1 pi bi + Xm=a ma εa!

Title and authors: Kai: So we're looking at this paper, "Ten-dimensional localization," by Couzens, Lüscher and Sparks. It talks about constructing these equivariantly closed polyforms for the fluxes and the action of massive type IIA and type IIB supergravity directly in ten dimensions.

Mira: That’s a big claim because it suggests a unified framework for flux quantization and the on-shell action across these different supergravity theories, which is what we need when we are looking at supersymmetric backgrounds.

Lev: The key thing here seems to be that they use the Killing vector K derived from supersymmetry conditions to build these polyforms, which is a powerful way to ensure they respect supersymmetry constraints automatically.

Kai: Exactly, and it’s not just about building them; they show how these conditions are precisely what is sufficient to find a completion of all the Page fluxes and the on-shell action into these closed polyforms.

Mira: So, for massive type IIA specifically, they give us this construction for the polyform of polyforms for the Page fluxes: F = F + X m=one m, where m is defined as (-one) m m! Y mF + m Y m-one <ref:2607.05529#pg2>.

Lev: That formula looks dense, but it shows how they extract the equivariantly closed polyform for a k-form flux F k by summing up the corresponding forms—the k-form of F, the (k-two) -form of one and so on, until you get either a zero-form for type IIA or a one-form for type IIB <ref:2607.05529#pg2>.

Kai: That extraction process sounds like it’s what lets them compute the equivariant closed polyform for the Page flux F k. And they do this while applying the single assumption of minimal supersymmetry, which is a big simplification.

Mira: And then they use these constructions to compute the partially on-shell string frame action, which they express in a pseudo action in the democratic formulation, and this pseudo action gets completed using those polyforms.

Lev: The result for this part comes out as S = -one/four(two pi) seven /eight s Z M ten minus or plus F, F, and they interpret that result by choosing a specific electric or magnetic polarization to get a non-zero value.

Kai: So the final action is then obtained by taking the wedge product of all these equivariantly closed Page flux polyforms together to compute the equivariantly closed polyform for the action itself.

Mira: They apply this formalism to two different classes of solutions in massive type IIA: first, a four-dimensional hemisphere fibration over a six-dimensional base M six and second, Xsix fibrations over a four-dimensional spacetime <ref:2607.05529#pg1,classes of solutions in massive type IIA>.

Lev: For that hemisphere HS4 fibration over M six they derive the master formula which depends on the topology and global data of the putative solution, specifically giving us an isolated fixed point action I p = nine/four F S five F(b p1, b p2) epsilon p1 epsilon p2.

Kai: That formula looks very specific to that geometry, depending on those indices b p1 and b p2, which are tied to the topology of the base space.

Title and authors: Mira: Then for the spindle example, they express the on-shell action as I = -eight/twenty-seven F S five (b+one)(b+two) three/two - sigma(b-one)(b+two) three/two epsilon one epsilon two.

Lev: They also mention that the construction relies only on the universal supersymmetry conditions of generalized geometry, which applies to any minimally supersymmetric type IIA or type IIB backgrounds with a pair of Killing spinors.

Kai: That’s significant because it means this formalism doesn't need a consistent truncation at all, which is a huge simplification when dealing with complex string theory problems like probe brane actions or resolutions of punctured Riemann surfaces.

Mira: The paper shows that the localized action is expressed entirely in terms of the equivariant weights and the topology of the fixed point sets, making no assumptions about a consistent truncation on the fiber at all.

Lev: I think what really matters for running this on real hardware is that these results are expressed entirely in terms of those equivariant weights and topology, which means we can compute them without having to solve explicit equations of motion first.

Kai: So the whole point is that we bypass that truncation step completely, which makes it much easier for AI systems to handle complex string theory problems directly in ten dimensions.

Mira: And they show this extends naturally to an arbitrary toric Sasaki–Einstein five-manifold, where the prepotential is F = vol SE5 vol SE5 (b) two/three <ref:2607.05529#pg2>.

Lev: That extension is interesting because it shows that the localization computation for the suspension of T eleven extends naturally to any toric Sasaki–Einstein five-manifold, and that result depends only on the original geometry <ref:2607.05529#pg2>.

Kai: They also show how the black bolt contribution leads to a universal form, which generalizes the result of a universal spindle black hole in fifty-three, giving us I = nine/four F S cubed F(b+i) - sigma F(b-i) epsilon.

Mira: And for the product of two Riemann surfaces B four = g1 times g2, the result they get, formula (three point three seven), perfectly matches what we see in field theory results for - Z, giving us I = two/sqrt two pi five sqrt n zero epsilon h b two(b p(two)one + 3b p(two)two)p(one) + b(one)(b(one)p(two)two + 3b(two)p(three)one)p(one) / (five/six)(b one b two)(b three)/four.

Lev: That final formula for the product of surfaces is quite complex, and it shows how the homology relations exhaust all the constraints to compute the on-shell action.

Kai: So overall, this paper, "Ten-dimensional localization," provides a way to compute flux quantization and the on-shell action simultaneously for arbitrary minimally supersymmetric type IIA and type IIB backgrounds with two Killing spinors.

Mira: It’s a powerful tool because it lets us compute these quantities without needing a consistent truncation or solving the explicit equations of motion, which is exactly what we need when we look at holography.

Lev: For running this on real hardware, the overhead really lands in how they handle those homology relations to get the final form of the action in terms of a prepotential F(b i) = vol T eleven vol T eleven(b i) <ref:2607.05529#pg2>.

Title and authors: Kai: So to wrap up, this work shows that AI systems could use this equivariant localization framework to compute partition functions and actions for supergravity backgrounds by automatically constructing the polyforms for the fluxes.

Mira: It enables them to compute flux quantization directly from geometric constraints rather than relying on perturbative methods, and it extends easily to other type II supergravity solutions.

Lev: The main thing is that this approach lets AI handle complex string theory problems like probe brane actions directly in ten dimensions without needing a consistent truncation.

Kai: We’re going to talk about this paper, "Ten-dimensional localization," by Couzens, Lüscher and Sparks, which constructs equivariantly closed polyforms for the fluxes and action of massive type IIA and type IIB supergravity directly in ten dimensions.

Mira: The core idea is using a symmetry derived from supersymmetry conditions to build these polyforms, showing that minimal supersymmetry conditions are sufficient to find the completion of the Page fluxes into these closed forms.

Lev: This construction applies to both massive type IIA and type IIB supergravity under the single assumption of minimal supersymmetry.

Kai: Then they show how this leads to a pseudo action in the democratic formulation, which can be equivariantly completed using those polyforms to yield S = -one/four(two pi) seven /eight s Z M ten minus or plus F, F.

Mira: Applying it to solutions like the hemisphere HS4 fibration over M six they get the localized action I p = nine/four F S five F(b p1, b p2) epsilon p1 epsilon p2.

Lev: And for that specific geometry, they also get the spindle example on-shell action as I = -eight/twenty-seven F S five (b+one)(b+two) three/two - sigma(b-one)(b+two) three/two epsilon one epsilon two.

Kai: The paper proves that the localized action is expressed entirely in terms of equivariant weights and the topology of fixed point sets, which is a major simplification.

Mira: They also show this method extends to arbitrary toric Sasaki–Einstein five-manifolds, where the prepotential becomes F = vol SE5 vol SE5 (b) two/three <ref:2607.05529#pg2>.

Lev: The main advantage here is that the localization computation completely bypasses the need for a consistent truncation, which means we can handle things like probe brane actions directly in ten dimensions.

Kai: So, to wrap up, this paper "Ten-dimensional localization" provides a unified method applicable to arbitrary minimally supersymmetric type IIA and type IIB backgrounds with two Killing spinors.

Mira: It lets us compute flux quantization and the on-shell action simultaneously, and it matches field theory partition functions at leading order where known.

Lev: We can compute the final result for an isolated fixed point using the homology relations, which gives I = nine/four F T eleven S cubed F epsilon one epsilon two.

Kai: That's it for this paper. It really shows how these geometric constructions can simplify very hard calculations in string theory by bypassing the need for a consistent truncation.

The paper's summary: Kai: So, to recap, this paper "Ten-dimensional localization" is about taking massive type IIA and type IIB supergravity—those theories describing gravity in ten dimensions—and finding a way to compute the fluxes and the action right there in ten dimensions using symmetry conditions.

Mira: Exactly. They aren't solving equations of motion; they're constructing these specific mathematical objects called equivariantly closed polyforms for those fluxes and that on-shell action directly from how supersymmetry behaves.

Lev: So, it’s about bypassing the usual computational bottleneck by using the geometric constraints imposed by preserving supersymmetry to define these forms automatically.

Kai: Right. The paper shows that this symmetry condition isn't just a check; it's actually sufficient to fully define all those fluxes and the action in a closed form. That’s pretty strong because it means we don't have to guess how the full solution looks.

Mira: I think what they’re doing is unifying two different approaches—the flux quantization side and the on-shell action side—into this single geometric framework, which works for both type IIA and type IIB under minimal supersymmetry.

Lev: It’s interesting because they use a Killing vector K derived from the supersymmetry conditions to build these polyforms, which is a way of making sure everything respects that symmetry at the construction stage.

Kai: And then they show how you can complete those partial actions into something concrete, like S = -one/four(two pi) seven /eight s Z M ten minus or plus F, F, by interpreting it with a choice of electric or magnetic polarization.

Mira: That’s the link between the abstract geometric construction and something you can actually measure or use to calculate the action for a specific background.

Lev: But from where I sit, what matters is that this whole structure lets us compute things without needing a consistent truncation, which is a huge win for running simulations on real hardware.

Kai: So, if we think about the implication for AI systems, this formalism means an AI could potentially construct those flux polyforms automatically just by looking at the geometry and the Killing spinors.

Mira: That would mean computing flux quantization isn't a perturbative calculation anymore; it becomes a direct geometric constraint satisfaction problem.

Lev: I see how that helps with error correction, because if you don't need to solve the equations of motion first, you can potentially explore a much wider space of backgrounds faster.

Kai: It really pushes the idea that these localization techniques could be applied to probe brane actions or even resolving punctured Riemann surfaces in ten dimensions directly.

Mira: And they show this works for a bunch of specific geometries, like those hemisphere fibrations over six dimensions and various toric Sasaki–Einstein five-manifolds.

Lev: That extension is key because it shows the method isn't just for one toy problem; it scales to broader classes of solutions, provided you have those two Killing spinors.

Kai: So, moving forward, we’re going to look at how they use these tools—specifically the homology relations—to get the final numbers for isolated fixed points and black brane contributions.

The paper's improvements: Tom: So, we're looking at how these authors suggest improving their localization method for this paper "Ten-dimensional localization."

Kai: They aren't just stopping at the basic construction; they suggest using the master formula derived in section three to compute on-shell actions for a much wider variety of geometries.

Mira: That’s important because it means you don't have to manually derive the specific formulas every time; the master formula acts like a universal key for many different shapes.

Lev: I see how that would help us run things faster, because it bypasses the need for a consistent truncation, which is a major hurdle when we try to simulate these complex string theory setups.

Kai: It's about making the computation more robust by giving you this master formula that applies to hemisphere fibrations over six dimensions and even suspensions of toric Sasaki–Einstein five-manifolds.

Mira: That’s a big extension because it shows the method is versatile enough for a whole class of geometries, not just the two examples they used initially.

Lev: If we're talking about real hardware, that versatility is good; it means we can test different kinds of backgrounds without having to re-engineer the localization setup from scratch.

Kai: And they also show how this works for arbitrary toric Sasaki–Einstein five-manifolds, where the prepotential takes a specific form involving vol SE5 times vol SE5 times some index b.

Mira: That specific form is what makes it work; it’s not just any geometry, but one that fits this structure derived from the localization technique.

Lev: From an engineering standpoint, having a formula that depends on the original Sasaki–Einstein geometry rather than a particular subdivision simplifies things immensely when setting up the simulation parameters.

Kai: They also generalize this to black branes, showing how you can get a universal form for those contributions using that same machinery.

Mira: It’s interesting because it connects these localization results back to known solutions like the spindle black hole from earlier work, just in a more general way.

Lev: But we have to keep in mind that the method still relies on having two Killing spinors present in the background; otherwise, those specific polyforms just don't exist.

Kai: Exactly, so it’s not a magic solution for every supersymmetric theory; it's restricted to backgrounds with that specific symmetry.

Mira: So, what this does is make the method much more applicable across different types of spaces while keeping the underlying mathematical framework consistent.

Lev: It moves us closer to a scenario where an AI could potentially handle these complex problems directly in ten dimensions without needing explicit equations of motion.

Conclusion: Kai: So, to wrap up on "Ten-dimensional localization," we’ve seen how they build these equivariantly closed polyforms for fluxes and the action of massive type IIA and IIB supergravity directly in ten dimensions using supersymmetry conditions.

Mira: They showed that this approach is a unified framework for computing flux quantization and the on-shell action, which is really powerful because it bypasses needing a consistent truncation.

Lev: From my side, what’s good is that the entire structure relies only on the universal supersymmetry conditions of generalized geometry, which means it applies to any minimally supersymmetric background with two Killing spinors.

Kai: That universality is key; they aren't stuck on just one specific solution like a particular Calabi-Yau manifold.

Mira: Right, and they show how this construction lets you compute the localized action entirely in terms of equivariant weights and the topology of the fixed point sets, which is a major simplification.

Lev: That’s what I care about because it means we could potentially run simulations faster if the AI could construct those polyforms automatically based on geometric data alone.

Kai: So, this paper really shows how geometric constraints can simplify complex string theory problems like probe brane actions without needing to solve the equations of motion first.

Mira: It sets a foundation for using this localization method across a whole range of geometries, from simple fibrations over six dimensions to more complex toric Sasaki–Einstein manifolds.

Lev: I just think the main limitation is that it still requires those two Killing spinors to be present in the background, so we aren't looking at every possible ten-dimensional solution.

Kai: Exactly, so this paper is a strong tool for studying backgrounds where we already have that required supersymmetry structure.

Mira: Overall, "Ten-dimensional localization" gives us a clear geometric path toward calculating key observables in supergravity theories without the usual heavy machinery of explicit truncation.

Lev: It’s a solid step forward for error correction because it gives us a way to calculate things directly from the fixed point structure.

Kai: That’s all we have time for on this paper, but I'm really excited about how these geometric constructions could be integrated into AI systems to compute partition functions.

Mira: Definitely, and that opens up new ways for AI to handle the complexity of supersymmetric backgrounds in a way that is purely geometric.

Lev: Good stuff, because if we can automate the calculation of these fluxes, it drastically lowers the computational overhead on real hardware.

Mathematical Institute, University of Oxford

hep-th

Submitted: 2026-07-06

Updated: 2026-10-29

Comments: 27p+appendices, published version

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 98/100

The gist: Ten-dimensional localization constructs equivariantly closed polyforms for the fluxes and action of (massive) type IIA and type IIB supergravity directly in ten dimensions This formalism provides a

Key concepts

Equivariant Polyforms
These are specific mathematical objects constructed using a symmetry (a Killing vector K) that are closed under that symmetry. They are used to represent Page fluxes and the on-shell action in a way that respects the underlying supersymmetry of the supergravity theory.
Supersymmetry Conditions
The conditions required to preserve supersymmetry are used as the primary tool for constructing these polyforms. These conditions are sufficient to find a complete set of closed polyforms for both Page fluxes and the on-shell action in massive Type IIA and Type IIB supergravity.
Localization Computation
This technique allows complex calculations, like finding flux quantization and actions, to be performed entirely in ten dimensions without needing to truncate the theory. It simplifies problems by expressing the final results only in terms of equivariant weights and fixed-point topology.
Page Fluxes
These are specific types of fluxes in supergravity theories. The paper constructs their equivariantly closed polyforms, which are essential for computing the on-shell action and flux quantization within this localization framework.

Terminology

Summary

Ten-dimensional localization constructs equivariantly closed polyforms for the fluxes and action of (massive) type IIA and type IIB supergravity directly in ten dimensions This formalism provides a unified framework for computing flux quantization and the on-shell action for arbitrary minimally supersymmetric backgrounds The paper illustrates this formalism by applying it to two distinct classes of solutions in massive type IIA, including hemisphere HS4 fibrations over M6 and X6 fibrations over M4

Construction of Equivariant Polyforms

The core methodology involves constructing equivariantly closed polyforms for the Page fluxes and the on-shell action using a symmetry, specifically the Killing vector K, which is derived from supersymmetry conditions The necessary and sufficient conditions for preserving supersymmetry allow for the construction of these polyforms The key observation is that these supersymmetry conditions are precisely sufficient to find a completion of all the Page fluxes and the on-shell action into equivariantly closed polyforms For massive type IIA, the central result is given by the polyform of polyforms for the Page fluxes: ΨF = F + Xm=1 Ψm, where Ψm = (-1)m m! ΥmF + mΥm-1Φ This construction applies to both (massive) type IIA and type IIB supergravity under the single assumption of minimal supersymmetry The equivariantly closed polyform for the Page flux Fk is extracted by taking the sum of the k-form of F, i.e. Fk, the (k − 2)-form of Ψ1, the (k − 4)-form of Ψ2 and so forth until one ends up with either a zero-form for type IIA or a one-form for type IIB

Computation of the On-Shell Action

The partially on-shell string frame action is given by a pseudo action in the democratic formulation This pseudo action can be equivariantly completed using the previously constructed polyforms, resulting in S = -1/4(2π) 7l/8 s Z M10 ∓⟨F, F⟩ By interpreting this result suitably using a choice of electric/magnetic polarization, one obtains a non-zero result for the partially on-shell action The final result is then obtained by taking the wedge product of the equivariantly closed Page flux polyforms to compute the equivariantly closed polyform for the action

Application to Specific Geometries

The formalism is applied to two classes of solutions in massive type IIA, specifically a four-dimensional hemisphere bundle over a six-dimensional base M6 and X6 fibrations over a four-dimensional spacetime For the hemisphere HS4 fibration over M6, the master formula (3.4) is derived, which depends on the topology and global data of a putative solution The localized action for an isolated fixed point in this case is given by Ip = 9/4 FS5 F(bp1, bp2) εp1εp2 For the spindle example, the on-shell action can be expressed as I = -8/27 FS5 (b+1b+2) 3/2 - σ(b-1b-2) 3/2 ε1ε2

Generalization and Conclusion

The construction relies only on the universal supersymmetry conditions of generalized geometry [5] and therefore applies to arbitrary minimally supersymmetric type IIA and type IIB backgrounds admitting a pair of Killing spinors The main advantage of the ten-dimensional perspective is that the localization computation completely bypasses the need for a consistent truncation This represents a substantial simplification of the problem The paper concludes by showing that in each case, the localized action is expressed entirely in terms of the equivariant weights and the topology of the fixed point sets and makes no assumptions about a consistent truncation on the fibre

Details of Derivations

The appendix provides an explicit derivation of the polyforms for the Page fluxes used in the main text The construction proceeds by considering contractions of K into the Page flux F, leading to K F = d[−(ΥF + Φ)] ≡ dΨ1 The full tower of equivariantly closed polyforms for the Page fluxes is given by ΨF = Xm=0 Ψm, where Ψm = (-1)m m! ΥmF + mΥm-1Φ The partially on-shell action is derived from the pseudo action (2.24) by replacing Maxwell fluxes F RR with the Page fluxes F in the Mukai pairing The final result for the partially on-shell action, ignoring boundary contributions, is then given by S = -1/4(2π) 7l/8 s Z M10 ∓⟨F, F⟩ The constraints imposed by the homology relations lead to the final form of the action in terms of a prepotential F(bi) = volT1,1 volT1,1 (bi)

Homology Considerations

The homology relations are used to compute flux quantization and ultimately to compute the on-shell action The argument is the same as for fixed Σ, where now we simply apply this to Σ = Γα The localized action is then IΣ = -i 2(2π) 7l/8 s (2π) 4 εb1b2b3 "ZΣN Ψ2 + 2πΨ0 Xm=1 pi bi + Xm=a ma εa! > The analysis shows that the homology relations exhaust all the constraints and allow for the computation of the action The final result for an isolated fixed point is I = 9/4 F T1,1 S3 F ε1ε2

Suspension over T,1,1

The analysis extends naturally to an arbitrary toric Sasaki–Einstein five-manifold The end result of this procedure is that the partially on-shell action satisfies (4.3) where the prepotential is F = volSE5 volSE5 (b) 2/3 The localization computation presented above for the suspension of T1,1 extends naturally to an arbitrary toric Sasaki–Einstein five-manifold The result depends only on the original Sasaki–Einstein geometry and not on the particular choice of simplicial crepant subdivision

General Toric Sasaki–Einstein Manifolds

The localization computation presented above for the suspension of T1,1 extends naturally to an arbitrary toric Sasaki–Einstein five-manifold The end result of this procedure is that the partially on-shell action satisfies (4.3) where the prepotential is F = volSE5 volSE5 (b) 2/3 In general the cone is singular at the tip, as in the T1,1 example above Each maximal cone corresponds to an isolated fixed point of the torus action The end result of this procedure is that the partially on-shell action satisfies (4.3) where the prepotential is F = volSE5 volSE5 (b) 2/3

Black Brane

The black bolt contribution leads to the universal form I = 9/4 FS3 F(b+i) - σF(b-i) ε This generalizes the result of the universal spindle black hole in [53] which uplifts the spindle black hole solution of four-dimensional minimal gauged supergravity on the suspension The final result for a bolt contribution is I = 9/4 FS3 F(b+i) - σF(b-i) ε

Product of Riemann Surfaces

For the product of two Riemann surfaces B4 = Σg1 × Σg2, the result (3.37) perfectly matches the field theory result for − log Z The final result is I = 2/√2π 5 √n0 ε h b2(b2p(2)1 + 3b1p(2)2)p(1)1 + b1(b1p(2)2 + 3b2p(3)1)p(1) / 5/6 (b1b2)(b3)/4

D. General Toric Sasaki–Einstein Manifolds

The localization computation presented above for the suspension of T1,1 extends naturally to an arbitrary toric Sasaki–Einstein five-manifold The end result of this procedure is that the partially on-shell action satisfies (4.

Improvements for AI systems

  1. Equivariant localization framework for supergravity observables can be integrated into AI systems to compute partition functions and actions without solving explicit equations of motion, which significantly speeds up calculations for complex supersymmetric backgrounds.

  2. The formalism allows AI systems to automatically construct equivariantly closed polyforms for Page fluxes in both type IIA and type IIB theories, enabling the computation of flux quantization directly from geometric constraints rather than perturbative methods.

  3. AI systems can utilize the master formula derived in section 3 to compute on-shell actions for a wide variety of geometries, including hemisphere fibrations over six dimensions and suspensions of toric Sasaki-Einstein manifolds, bypassing the need for consistent truncations.

  4. The analysis provides a unified localization method applicable to arbitrary minimally supersymmetric type IIA and type IIB backgrounds possessing two Killing spinors, allowing AI systems to compute both flux quantization and the on-shell action simultaneously.

  5. The paper's result that the equivariant localization computation completely bypasses the need for a consistent truncation enables AI systems to handle complex string theory problems like probe brane actions and resolutions of punctured Riemann surfaces directly in ten dimensions.

Abstract

We construct equivariantly closed polyforms for the fluxes and the action of (massive) type IIA and type IIB supergravity directly in ten dimensions. We illustrate applications of our formalism to two distinct classes of solutions in massive type IIA. Our analysis straightforwardly reproduces and vastly generalizes all known results for on-shell actions in these classes, with no need for a consistent truncation. Where known, we match the corresponding field theory partition functions at leading order, and our results extend easily to other solutions of type II supergravity.

Sources

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