pith. sign in

arxiv: 2607.00076 · v1 · pith:JWLE6ILKnew · submitted 2026-06-30 · ✦ hep-th · math-ph· math.MP

Localization, Factorization and Dualities for Elliptic Kernels

Alessio Fontanarossa , Fabrizio Nieri , Antonio Pittelli This is my paper

Pith reviewed 2026-07-02 18:38 UTC · model grok-4.3

classification ✦ hep-th math-phmath.MP
keywords elliptic kernelsSeiberg dualitypartition functionsJeffrey-Kirwan residueslocalizationsupersymmetric gauge theoriesboundary conditionstheta functions
0
0 comments X

The pith

Partition functions of 4d N=1 gauge theories on a torus times cylinder act as elliptic kernels whose Jeffrey-Kirwan residues satisfy rank-changing Seiberg dualities.

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

The paper examines exact partition functions for supersymmetric gauge theories on a geometry of a torus multiplied by a finite cylinder with independent boundary polarizations at each end. These polarizations turn the partition function into a boundary-to-boundary elliptic kernel. Equivariant localization on the BPS locus yields one-loop determinants consisting of theta functions dressed by cubic phases. The central result shows that rank-changing Seiberg-type dualities hold as identities between Jeffrey-Kirwan residues of these kernels.

Core claim

The boundary-to-boundary elliptic kernels constructed from the localized determinants of chiral multiplets on the cylinder geometry satisfy rank-changing Seiberg-type dualities when expressed as identities among Jeffrey-Kirwan residues.

What carries the argument

The elliptic kernel, defined as the partition function with independent Dirichlet or Robin-like boundary polarizations at each cylinder end, whose Jeffrey-Kirwan residues encode the dualities.

Load-bearing premise

The chiral-multiplet one-loop determinants for the four boundary polarizations are correctly obtained from equivariant localization on the BPS locus of the rigid supersymmetric geometry.

What would settle it

Explicit computation of the Jeffrey-Kirwan residues for a low-rank example such as SU(2) or U(1) that shows a mismatch between a theory and its proposed dual would disprove the claimed identities.

read the original abstract

We study the exact partition function of 4d $\mathcal N=1$ supersymmetric gauge theories on a torus times a cylinder $\mathrm{Cyl}=I\times S^1$, where $I$ is a finite interval carrying two boundary components. Each endpoint supports an independent Dirichlet or Robin-like boundary polarization, so that the partition function is a boundary-to-boundary elliptic kernel. We construct the rigid supersymmetric geometry, determine the BPS locus, and compute the chiral-multiplet 1-loop determinants for the four possible boundary polarizations via equivariant localization. The resulting elementary building blocks are theta functions dressed by cubic phases. We then prove rank-changing Seiberg-type dualities as identities of Jeffrey--Kirwan residues of these elliptic kernels. We also discuss factorization into holomorphic-block cap wavefunctions represented by elliptic Gamma functions, dimensional reductions to three and two dimensions, complete-intersection gauged linear sigma models, and elliptic kernels for 4d $\mathcal N=4$ super Yang--Mills and the Klebanov--Witten theory, useful for holographic applications.

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

0 major / 2 minor

Summary. The paper constructs boundary-to-boundary elliptic kernels for the exact partition functions of 4d N=1 supersymmetric gauge theories on T² × Cyl (Cyl an interval with independent Dirichlet/Robin polarizations at each end). It determines the rigid supersymmetric geometry and BPS locus, computes the four chiral-multiplet 1-loop determinants via equivariant localization (yielding theta functions dressed by cubic phases), and proves rank-changing Seiberg-type dualities as identities of Jeffrey-Kirwan residues of the resulting kernels. It further discusses factorization into holomorphic-block caps (elliptic Gamma functions), dimensional reductions to 3d/2d, complete-intersection GLSMs, and explicit kernels for 4d N=4 SYM and the Klebanov-Witten theory.

Significance. If the residue identities are established, the work supplies an explicit, localization-based route to exact dualities for theories with boundaries, extending JK-residue techniques to elliptic kernels. The factorization into elliptic-Gamma blocks and the N=4/KW examples are concrete strengths that could support holographic applications and checks against known limits. The approach is internally consistent with the abstract's description of the construction and does not rely on post-hoc fitting.

minor comments (2)
  1. [Construction of the kernels] The abstract states that the four 1-loop determinants are computed via equivariant localization, but the main text should include a short table or explicit formulas (e.g., in the section on boundary polarizations) listing the resulting theta-function expressions for each polarization to facilitate direct comparison with the JK-residue identities.
  2. [Duality proofs] Notation for the Jeffrey-Kirwan residue operation and the precise contour choices should be restated once in the duality section, even if defined earlier, to make the rank-changing identities self-contained.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, significance assessment, and recommendation of minor revision. No major comments were listed in the report, so there are no specific points requiring point-by-point response or manuscript changes at this stage.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper constructs the elliptic kernels by determining the rigid supersymmetric geometry, identifying the BPS locus, and computing the four chiral-multiplet 1-loop determinants via equivariant localization on that locus; these steps are presented as direct computations yielding theta functions dressed by cubic phases. The rank-changing Seiberg-type dualities are then shown as identities among Jeffrey-Kirwan residues of the resulting kernels. No step reduces a claimed prediction or theorem to a fitted parameter, a self-citation chain, or a definitional renaming; the residue identities are asserted to follow from the independently assembled building blocks. The factorization, dimensional-reduction, and N=4/KW examples are presented as consistent extensions rather than load-bearing inputs. The derivation chain therefore remains independent of its target results.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Ledger extracted from abstract claims only. No explicit free parameters or invented entities are named. The central results rest on standard assumptions of supersymmetric localization.

axioms (2)
  • domain assumption The torus-cylinder geometry admits a rigid supersymmetric structure with well-defined BPS locus
    Invoked when the authors state they construct the rigid supersymmetric geometry and determine the BPS locus
  • domain assumption The exact partition function equals the product of chiral-multiplet 1-loop determinants computed via equivariant localization
    Implicit in the statement that the partition function is a boundary-to-boundary elliptic kernel obtained from those determinants

pith-pipeline@v0.9.1-grok · 5723 in / 1313 out tokens · 25744 ms · 2026-07-02T18:38:15.590652+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

68 extracted references · 66 canonical work pages · 36 internal anchors

  1. [1]

    Exact Results In Two-Dimensional (2,2) Supersymmetric Gauge Theories With Boundary

    Kentaro Hori and Mauricio Romo. Exact ResultsIn Two-Dimensional (2,2) Supersymmetric Gauge Theories With Boundary. 8 2013.arXiv:1308.2438

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  2. [2]

    Exact results for boundaries and domain walls in 2d supersymmetric theories

    DaigoHondaandTakuyaOkuda. Exactresultsforboundariesanddomainwallsin2dsuper- symmetric theories.JHEP, 09:140, 2015.arXiv:1308.2217,doi:10.1007/JHEP09(2015) 140

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep09(2015 2015

  3. [3]

    Localization of gauge theory on a four-sphere and supersymmetric Wilson loops

    Vasily Pestun. Localization of gauge theory on a four-sphere and supersymmetric Wil- son loops.Commun. Math. Phys., 313:71–129, 2012.arXiv:0712.2824,doi:10.1007/ s00220-012-1485-0

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  4. [4]

    Rigid Supersymmetric Theories in Curved Super- space.JHEP, 06:114, 2011.arXiv:1105.0689,doi:10.1007/JHEP06(2011)114

    Guido Festuccia and Nathan Seiberg. Rigid Supersymmetric Theories in Curved Super- space.JHEP, 06:114, 2011.arXiv:1105.0689,doi:10.1007/JHEP06(2011)114. 54

    work page doi:10.1007/jhep06(2011)114 2011

  5. [5]

    Exploring Curved Superspace

    Thomas T. Dumitrescu, Guido Festuccia, and Nathan Seiberg. Exploring Curved Super- space.JHEP, 08:141, 2012.arXiv:1205.1115,doi:10.1007/JHEP08(2012)141

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep08(2012)141 2012

  6. [6]

    Factorisation of N = 2 theories on the squashed 3-sphere

    Sara Pasquetti. Factorisation of N = 2 Theories on the Squashed 3-Sphere.JHEP, 04:120, 2012.arXiv:1111.6905,doi:10.1007/JHEP04(2012)120

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep04(2012)120 2012

  7. [7]

    Holomorphic Blocks in Three Dimensions

    Christopher Beem, Tudor Dimofte, and Sara Pasquetti. Holomorphic Blocks in Three Dimensions.JHEP, 12:177, 2014.arXiv:1211.1986,doi:10.1007/JHEP12(2014)177

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep12(2014)177 2014

  8. [8]

    Localization of 4dN= 1theories on D2×T2.JHEP, 12:147, 2019.arXiv:1906.02051,doi:10.1007/JHEP12(2019)147

    Pietro Longhi, Fabrizio Nieri, and Antonio Pittelli. Localization of 4dN= 1theories on D2×T2.JHEP, 12:147, 2019.arXiv:1906.02051,doi:10.1007/JHEP12(2019)147

    work page doi:10.1007/jhep12(2019)147 2019

  9. [9]

    Factorisation and holomorphic blocks in 4d

    Fabrizio Nieri and Sara Pasquetti. Factorisation and holomorphic blocks in 4d.JHEP, 11:155, 2015.arXiv:1507.00261,doi:10.1007/JHEP11(2015)155

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep11(2015)155 2015

  10. [10]

    Supersymmetric indices onI×T 2, elliptic genera and dualities with boundaries.Nucl

    Katsuyuki Sugiyama and Yutaka Yoshida. Supersymmetric indices onI×T 2, elliptic genera and dualities with boundaries.Nucl. Phys. B, 960:115168, 2020.arXiv:2007.07664, doi:10.1016/j.nuclphysb.2020.115168

    work page doi:10.1016/j.nuclphysb.2020.115168 2020

  11. [11]

    Localization of three-dimensionalN= 2su- persymmetric theories onS 1×D2.PTEP, 2020(11):113B02, 2020.arXiv:1409.6713, doi:10.1093/ptep/ptaa136

    Yutaka Yoshida and Katsuyuki Sugiyama. Localization of three-dimensionalN= 2su- persymmetric theories onS 1×D2.PTEP, 2020(11):113B02, 2020.arXiv:1409.6713, doi:10.1093/ptep/ptaa136

    work page doi:10.1093/ptep/ptaa136 2020

  12. [12]

    Factorizations of 3d Interval Partition Functions

    Boan Zhao, Panos Betzios, and Paul Luis Roehl. Factorizations of 3d Interval Partition Functions. 9 2025.arXiv:2509.23902

    work page arXiv 2025

  13. [13]

    Mykola Dedushenko. Gluing. Part I. Integrals and symmetries.JHEP, 04:175, 2020.arXiv: 1807.04274,doi:10.1007/JHEP04(2020)175

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep04(2020)175 2020

  14. [14]

    Gluing II: boundary localization and gluing formulas.Lett

    Mykola Dedushenko. Gluing II: boundary localization and gluing formulas.Lett. Math. Phys., 111(1):18, 2021.arXiv:1807.04278,doi:10.1007/s11005-021-01355-8

    work page doi:10.1007/s11005-021-01355-8 2021

  15. [16]

    Supersymmetric reduction of 3D theories on a line segment.Phys

    Mykola Dedushenko and Mikhail Litvinov. Supersymmetric reduction of 3D theories on a line segment.Phys. Rev. D, 108(4):045016, 2023.arXiv:2212.07455,doi:10.1103/ PhysRevD.108.045016

    work page arXiv 2023

  16. [17]

    Interfaces and Quantum Algebras, II: Cigar Partition Function

    Mykola Dedushenko and Nikita Nekrasov. Interfaces and Quantum Algebras, II: Cigar Partition Function. 6 2023.arXiv:2306.16434

    work page arXiv 2023

  17. [18]

    Chiral life on a slab.Lett

    Mikhail Litvinov, Sergey Alekseev, and Mykola Dedushenko. Chiral life on a slab.Lett. Math. Phys., 116(1):7, 2026.arXiv:2301.00038,doi:10.1007/s11005-025-02040-w. 55

    work page doi:10.1007/s11005-025-02040-w 2026

  18. [19]

    Dynamical Weyl groups and applications

    Vyacheslav P. Spiridonov. On the elliptic beta function.Russ. Math. Surveys, 56:185–186, 2001.arXiv:math/0011001,doi:10.1070/RM2001v056n01ABEH000371

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1070/rm2001v056n01abeh000371 2001

  19. [20]

    Eric M. Rains. Transformations of elliptic hypergeometric integrals.Annals of Mathematics, 171:169–243, 2010.arXiv:math/0309252,doi:10.4007/annals.2010.171.169

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.4007/annals.2010.171.169 2010

  20. [21]

    Bottini, Chiung Hwang, Sara Pasquetti, and Matteo Sacchi

    Lea E. Bottini, Chiung Hwang, Sara Pasquetti, and Matteo Sacchi. 4d S-duality wall and SL(2,Z) relations.JHEP, 03:035, 2022.arXiv:2110.08001,doi:10.1007/JHEP03(2022) 035

    work page doi:10.1007/jhep03(2022 2022

  21. [22]

    Star-triangle dualities and su- persymmetric improved bifundamentals.JHEP, 09:171, 2025.arXiv:2410.19049,doi: 10.1007/JHEP09(2025)171

    Sergio Benvenuti, Riccardo Comi, and Sara Pasquetti. Star-triangle dualities and su- persymmetric improved bifundamentals.JHEP, 09:171, 2025.arXiv:2410.19049,doi: 10.1007/JHEP09(2025)171

    work page doi:10.1007/jhep09(2025)171 2025

  22. [23]

    Probing bad theories with the dualization algorithm

    Simone Giacomelli, Chiung Hwang, Fabio Marino, Sara Pasquetti, and Matteo Sacchi. Probing bad theories with the dualization algorithm. Part I.JHEP, 04:008, 2024.arXiv: 2309.05326,doi:10.1007/JHEP04(2024)008

    work page doi:10.1007/jhep04(2024)008 2024

  23. [24]

    Probing bad theories with the dualization algorithm

    Simone Giacomelli, Chiung Hwang, Fabio Marino, Sara Pasquetti, and Matteo Sacchi. Probing bad theories with the dualization algorithm. Part II.JHEP, 07:165, 2024.arXiv: 2401.14456,doi:10.1007/JHEP07(2024)165

    work page doi:10.1007/jhep07(2024)165 2024

  24. [25]

    Twisted Index on Hyperbolic Four-Manifolds.Lett

    Daniele Iannotti and Antonio Pittelli. Twisted Index on Hyperbolic Four-Manifolds.Lett. Math. Phys., 114(2):39, 2024.arXiv:2307.11634,doi:10.1007/s11005-024-01788-x

    work page doi:10.1007/s11005-024-01788-x 2024

  25. [26]

    Dual Boundary Conditions in 3d SCFT's

    Tudor Dimofte, Davide Gaiotto, and Natalie M. Paquette. Dual boundary conditions in 3d SCFT’s.JHEP, 05:060, 2018.arXiv:1712.07654,doi:10.1007/JHEP05(2018)060

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep05(2018)060 2018

  26. [27]

    On the 4d/3d/2d view of the SCFT/VOA correspondence

    Mykola Dedushenko. On the 4d/3d/2d view of the SCFT/VOA correspondence. 12 2023. arXiv:2312.17747

    work page arXiv 2023

  27. [28]

    The elliptic gamma function and SL(3,Z) x Z^3

    Giovanni Felder and Alexander Varchenko. The elliptic gamma function and SL(3,𭟋)⋉ 𭟋3. Advances in Mathematics, 156(1):44–76, 2000.arXiv:math/9907061,doi:10.1006/aima. 2000.1942

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1006/aima 2000

  28. [29]

    Atsushi Narukawa. The modular properties and the integral representations of the multiple elliptic gamma functions.Advances in Mathematics, 189(2):247–267, 2004.arXiv:math/ 0306164,doi:10.1016/j.aim.2003.11.009

    work page doi:10.1016/j.aim.2003.11.009 2004

  29. [30]

    Shintani–barnes zeta and gamma functions

    Eduardo Friedman and Simon Ruijsenaars. Shintani–barnes zeta and gamma functions. Advances in Mathematics, 187(2):362–395, 2004.doi:10.1016/j.aim.2003.07.020

    work page doi:10.1016/j.aim.2003.07.020 2004

  30. [32]

    Localization for nonabelian group actions

    Lisa C. Jeffrey and Frances C. Kirwan. Localization for nonabelian group actions.Topology, 34:291–327, 1995.arXiv:alg-geom/9307001,doi:10.1016/0040-9383(94)00028-J

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/0040-9383(94)00028-j 1995

  31. [33]

    Elliptic genera of two-dimensional N=2 gauge theories with rank-one gauge groups

    Francesco Benini, Richard Eager, Kentaro Hori, and Yuji Tachikawa. Elliptic genera of two-dimensionalN= 2gauge theories with rank-one gauge groups.Lett. Math. Phys., 104:465–493, 2014.arXiv:1305.0533,doi:10.1007/s11005-013-0673-y

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/s11005-013-0673-y 2014

  32. [34]

    Elliptic genera of 2d N=2 gauge theories

    Francesco Benini, Richard Eager, Kentaro Hori, and Yuji Tachikawa. Elliptic Genera of 2d N= 2Gauge Theories.Commun. Math. Phys., 333(3):1241–1286, 2015.arXiv:1308.4896, doi:10.1007/s00220-014-2210-y

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/s00220-014-2210-y 2015

  33. [35]

    Supersymmetric partition functions and the three-dimensional A-twist

    Cyril Closset, Heeyeon Kim, and Brian Willett. Supersymmetric partition functions and the three-dimensional A-twist.JHEP, 03:074, 2017.arXiv:1701.03171,doi:10.1007/ JHEP03(2017)074

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  34. [36]

    Electric-Magnetic Duality in Supersymmetric Non-Abelian Gauge Theories

    Nathan Seiberg. Electric–magnetic duality in supersymmetric non-abelian gauge theories. Nucl. Phys. B, 435:129–146, 1995.arXiv:hep-th/9411149,doi:10.1016/0550-3213(94) 00023-8

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/0550-3213(94 1995

  35. [37]

    Phases of $N=2$ Theories In Two Dimensions

    Edward Witten. Phases of n=2 theories in two dimensions.Nuclear Physics B, 403:159–222, 1993.arXiv:hep-th/9301042,doi:10.1016/0550-3213(93)90033-L

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/0550-3213(93)90033-l 1993

  36. [38]

    The Cardy limit of the topologically twisted index and black strings in AdS$_5$

    Seyed Morteza Hosseini, Anton Nedelin, and Alberto Zaffaroni. The cardy limit of the topologically twisted index and black strings in AdS5.Journal of High Energy Physics, 04:014, 2017.arXiv:1611.09374,doi:10.1007/JHEP04(2017)014

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep04(2017)014 2017

  37. [39]

    Microstates of rotating AdS5 strings.Journal of High Energy Physics, 11:090, 2019.arXiv:1909.08000,doi:10.1007/ JHEP11(2019)090

    Seyed Morteza Hosseini, Kiril Hristov, and Alberto Zaffaroni. Microstates of rotating AdS5 strings.Journal of High Energy Physics, 11:090, 2019.arXiv:1909.08000,doi:10.1007/ JHEP11(2019)090

    work page arXiv 2019

  38. [40]

    The topologically twisted index of $\mathcal N=4$ super-Yang-Mills on $T^2\times S^2$ and the elliptic genus

    Junho Hong, James T. Liu, and Leopoldo A. Pando Zayas. The topologically twisted index of\= 4super-yang–mills ont 2×s2 and the elliptic genus.Journal of High Energy Physics, 07:018, 2018.arXiv:1804.04592,doi:10.1007/JHEP07(2018)018

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep07(2018)018 2018

  39. [41]

    The spindle index from localization

    Matteo Inglese, Dario Martelli, and Antonio Pittelli. The spindle index from localization. J. Phys. A, 57(8):085401, 2024.arXiv:2303.14199,doi:10.1088/1751-8121/ad2225

    work page doi:10.1088/1751-8121/ad2225 2024

  40. [42]

    Microstates of Accelerating and Supersymmetric AdS4 Black Holes from the Spindle Index.Phys

    Edoardo Colombo, Seyed Morteza Hosseini, Dario Martelli, Antonio Pittelli, and Alberto Zaffaroni. Microstates of Accelerating and Supersymmetric AdS4 Black Holes from the Spindle Index.Phys. Rev. Lett., 133(3):031603, 2024.arXiv:2404.07173,doi:10.1103/ PhysRevLett.133.031603

    work page arXiv 2024

  41. [43]

    Curtis G. Jr. Callan and Jeffrey A. Harvey. Anomalies and Fermion Zero Modes on Strings and Domain Walls.Nucl. Phys. B, 250:427–436, 1985.doi:10.1016/0550-3213(85) 90489-4. 57

    work page doi:10.1016/0550-3213(85 1985

  42. [44]

    Supersymmetric Boundary Conditions in N=4 Super Yang-Mills Theory

    Davide Gaiotto and Edward Witten. Supersymmetric Boundary Conditions in N=4 Super Yang-Mills Theory.J. Statist. Phys., 135:789–855, 2009.arXiv:0804.2902,doi:10.1007/ s10955-009-9687-3

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  43. [45]

    Michael R. Douglas. D-branes, categories and N=1 supersymmetry.Journal of Mathemat- ical Physics, 42:2818–2843, 2001.arXiv:hep-th/0011017,doi:10.1063/1.1374448

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1063/1.1374448 2001

  44. [46]

    D-Branes in Landau-Ginzburg Models and Algebraic Geometry

    Anton Kapustin and Yi Li. D-branes in landau–ginzburg models and algebraic geometry. Journal of High Energy Physics, 12:005, 2003.arXiv:hep-th/0210296,doi:10.1088/ 1126-6708/2003/12/005

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  45. [47]

    Phases Of N=2 Theories In 1+1 Dimensions With Boundary

    Manfred Herbst, Kentaro Hori, and David Page. Phases of N=2 theories in 1+1 dimensions with boundary. 3 2008.arXiv:0803.2045

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  46. [48]

    Topological–anti-topological fusion.Nuclear Physics B, 367(2):359–461, 1991.doi:10.1016/0550-3213(91)90021-O

    Sergio Cecotti and Cumrun Vafa. Topological–anti-topological fusion.Nuclear Physics B, 367(2):359–461, 1991.doi:10.1016/0550-3213(91)90021-O

    work page doi:10.1016/0550-3213(91)90021-o 1991

  47. [49]

    tt* Geometry in 3 and 4 Dimensions

    Sergio Cecotti, Davide Gaiotto, and Cumrun Vafa.tt∗geometry in 3 and 4 dimensions. JHEP, 05:055, 2014.arXiv:1312.1008,doi:10.1007/JHEP05(2014)055

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep05(2014)055 2014

  48. [50]

    Moore and Nathan Seiberg

    Gregory W. Moore and Nathan Seiberg. Classical and quantum conformal field theory. Communications in Mathematical Physics, 123:177–254, 1989.doi:10.1007/BF01238857

    work page doi:10.1007/bf01238857 1989

  49. [51]

    TFT construction of RCFT correlators I: Partition functions

    Jürgen Fuchs, Ingo Runkel, and Christoph Schweigert. TFT construction of RCFT correla- tors i: partition functions.Nuclear Physics B, 646:353–497, 2002.arXiv:hep-th/0204148, doi:10.1016/S0550-3213(02)00744-7

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/s0550-3213(02)00744-7 2002

  50. [52]

    Holographic reconstruction for defect CFTs from $\mathrm{AdS}_p \times S^q$ spacetimes

    Federico Faedo, Nicolo Petri, and Alessia Segati. Holographic reconstruction for defect CFTs fromAdSp×Sq spacetimes. 6 2026.arXiv:2606.03106

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  51. [53]

    Giuseppe Dibitetto and Nicolo Petri.AdSD×Isolutions in axio-dilaton gravity. 6 2026. arXiv:2606.04933

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  52. [54]

    Giuseppe Dibitetto and Nicolo Petri.AdS 9 solutions in type II supergravities. 6 2026. arXiv:2606.04938

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  53. [55]

    Gluing gravitational blocks for AdS black holes.JHEP, 12:168, 2019.arXiv:1909.10550,doi:10.1007/JHEP12(2019) 168

    Seyed Morteza Hosseini, Kiril Hristov, and Alberto Zaffaroni. Gluing gravitational blocks for AdS black holes.JHEP, 12:168, 2019.arXiv:1909.10550,doi:10.1007/JHEP12(2019) 168

    work page doi:10.1007/jhep12(2019 2019

  54. [56]

    Two-dimensional SCFTs from wrapped branes and c-extremization

    Francesco Benini and Nikolay Bobev. Two-dimensional SCFTs from wrapped branes and c-extremization.JHEP, 06:005, 2013.arXiv:1302.4451,doi:10.1007/JHEP06(2013)005

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep06(2013)005 2013

  55. [57]

    An equivalence between generalized Maxwell model and fractional Zener model, Mechanics of Materials 100:148-153 (2016)

    Antonio Pittelli. Supersymmetric localization of refined chiral multiplets on topologically twisted H2 x S1.Phys. Lett. B, 801:135154, 2020.arXiv:1812.11151,doi:10.1016/j. physletb.2019.135154. 58

    work page doi:10.1016/j 2020

  56. [58]

    Supersymmetry and Localization on Three-Dimensional Orbifolds

    Matteo Inglese, Dario Martelli, and Antonio Pittelli. Supersymmetry and Localization on Three-Dimensional Orbifolds. 12 2023.arXiv:2312.17086

    work page arXiv 2023

  57. [59]

    Orbifold Indices in Four Dimensions

    Antonio Pittelli. Orbifold Indices in Four Dimensions. 3 2024.arXiv:2403.12318

    work page arXiv 2024

  58. [60]

    Sohnius and Peter C

    Martin F. Sohnius and Peter C. West. An alternative minimal off-shell version of N=1 supergravity.Phys. Lett. B, 105:353–357, 1981.doi:10.1016/0370-2693(81)90778-4

    work page doi:10.1016/0370-2693(81)90778-4 1981

  59. [61]

    Cohomological localization of N=2 gauge theories with matter.JHEP, 09:133, 2020.arXiv:2005.12944,doi:10.1007/JHEP09(2020)133

    Guido Festuccia, Anastasios Gorantis, Antonio Pittelli, Konstantina Polydorou, and Lorenzo Ruggeri. Cohomological localization of N=2 gauge theories with matter.JHEP, 09:133, 2020.arXiv:2005.12944,doi:10.1007/JHEP09(2020)133

    work page doi:10.1007/jhep09(2020)133 2020

  60. [62]

    Topological Correlators and Surface Defects from Equivariant Cohomology.JHEP, 09:185, 2020.arXiv:2006.06692, doi:10.1007/JHEP09(2020)185

    Rodolfo Panerai, Antonio Pittelli, and Konstantina Polydorou. Topological Correlators and Surface Defects from Equivariant Cohomology.JHEP, 09:185, 2020.arXiv:2006.06692, doi:10.1007/JHEP09(2020)185

    work page doi:10.1007/jhep09(2020)185 2020

  61. [63]

    A one-dimensional theory for Higgs branch operators

    Mykola Dedushenko, Silviu S. Pufu, and Ran Yacoby. A one-dimensional theory for Higgs branch operators.JHEP, 03:138, 2018.arXiv:1610.00740,doi:10.1007/JHEP03(2018) 138

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep03(2018 2018

  62. [64]

    E. W. Barnes. On the theory of the multiple gamma function.Transactions of the Cam- bridge Philosophical Society, 19:374–425, 1904

    1904

  63. [65]

    On a kronecker limit formula for real quadratic fields.Journal of the Faculty of Science, University of Tokyo

    Takuro Shintani. On a kronecker limit formula for real quadratic fields.Journal of the Faculty of Science, University of Tokyo. Section IA, Mathematics, 24:167–199, 1977

    1977

  64. [66]

    3d & 5d gauge theory partition functions as q-deformed CFT correlators

    Fabrizio Nieri, Sara Pasquetti, and Filippo Passerini. 3d and 5d Gauge Theory Partition Functions asq-deformed CFT Correlators.Lett. Math. Phys., 105(1):109–148, 2015.arXiv: 1303.2626,doi:10.1007/s11005-014-0727-9

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/s11005-014-0727-9 2015

  65. [67]

    5D partition functions, q-Virasoro systems and integrable spin-chains

    Fabrizio Nieri, Sara Pasquetti, Filippo Passerini, and Alessandro Torrielli. 5D partition functions, q-Virasoro systems and integrable spin-chains.JHEP, 12:040, 2014.arXiv: 1312.1294,doi:10.1007/JHEP12(2014)040

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep12(2014)040 2014

  66. [68]

    The $\mathcal{N}=1$ Chiral Multiplet on $T^2\times S^2$ and Supersymmetric Localization

    CyrilClossetandItamarShamir. TheN=1ChiralMultipletonT 2×S2 andSupersymmetric Localization. 11 2013.arXiv:1311.2430

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  67. [69]

    Superconformal Field Theory on Threebranes at a Calabi-Yau Singularity

    Igor R. Klebanov and Edward Witten. Superconformal field theory on threebranes at a Calabi-Yau singularity.Nucl. Phys. B, 536:199–218, 1998.arXiv:hep-th/9807080, doi:10.1016/S0550-3213(98)00654-3

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/s0550-3213(98)00654-3 1998

  68. [70]

    The Large N Limit of Superconformal Field Theories and Supergravity

    Juan Martin Maldacena. The Large N limit of superconformal field theories and su- pergravity.Adv. Theor. Math. Phys., 2:231–252, 1998.arXiv:hep-th/9711200,doi: 10.4310/ATMP.1998.v2.n2.a1. 59

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.4310/atmp.1998.v2.n2.a1 1998