Torqi

torqi

Author:
(1) Yuki Koto Table of Links Abstract and Intro
Genus-zero Gromov-Witten Theory
Toric Bundles
Lagrangian cones of Toric bundles
Mirror theorem for a product of projectives bundles
Mirror Theorem for Toric Bundles
Appendix A. Equivariant Fourier Transformation and References 4. Lagrangian cones of toric bundles These sheaves are endowed with T-actions, and all arrows are T-equivariant. By taking the moving parts we obtain the following exact sequence: The moving part can be described as On the other hand, we have These computations give the desired formula. By performing calculations similar to those in the previous proof, we can establish the following formulas. Using the above lemmas, we can compute the contributions of the graphs of type (α, 1). Proposition 4.15. Proof. To begin with, we rewrite the left-hand side using the bijection Φ1 as follows: By using Lemma 4.11, Lemma 4.12 and Lemma 4.13, we have 4.4. Contribution of the (α, 2)-type graphs. The contribution of the (α, 2)-type graphs can be computed as follows. This paper is available on arxiv under CC 4.0 license. Author: (1) Yuki Koto Author: Author: (1) Yuki Koto Table of Links Abstract and Intro Genus-zero Gromov-Witten Theory Toric Bundles Lagrangian cones of Toric bundles Mirror theorem for a product of projectives bundles Mirror Theorem for Toric Bundles Appendix A. Equivariant Fourier Transformation and References Abstract and Intro Abstract and Intro Genus-zero Gromov-Witten Theory Genus-zero Gromov-Witten Theory Toric Bundles Toric Bundles Lagrangian cones of Toric bundles Lagrangian cones of Toric bundles Mirror theorem for a product of projectives bundles Mirror theorem for a product of projectives bundles Mirror Theorem for Toric Bundles Mirror Theorem for Toric Bundles Appendix A. Equivariant Fourier Transformation and References Appendix A. Equivariant Fourier Transformation and References 4. Lagrangian cones of toric bundles These sheaves are endowed with T-actions, and all arrows are T-equivariant. By taking the moving parts we obtain the following exact sequence: The moving part can be described as On the other hand, we have These computations give the desired formula. By performing calculations similar to those in the previous proof, we can establish the following formulas. Using the above lemmas, we can compute the contributions of the graphs of type (α, 1). Proposition 4.15. Proposition 4.15. Proof. To begin with, we rewrite the left-hand side using the bijection Φ1 as follows: By using Lemma 4.11, Lemma 4.12 and Lemma 4.13, we have 4.4. Contribution of the (α, 2)- type graphs . The contribution of the (α, 2)-type graphs can be computed as follows. Contribution of the type graphs This paper is available on arxiv under CC 4.0 license. This paper is available on arxiv under CC 4.0 license. available on arxiv

A Mirror Theorem for Non-split Toric Bundles: Lagrangian Cones of Toric Bundles

অ-বিভক্ত টরিক বান্ডেলের জন্য একটি মিরর উপপাদ্য: টরিক বান্ডেলের ল্যাগ্রাঞ্জিয়ান শঙ্কু

Ein Spiegelsatz für nicht aufgeteilte torische Bündel: Lagrangesche Kegel torischer Bündel

Un teorema del espejo para haces tóricos no divididos: conos lagrangianos de haces tóricos

Un théorème du miroir pour les faisceaux toriques non divisés : les cônes lagrangiens des faisceaux toriques

गैर-विभाजित टॉरिक बंडलों के लिए दर्पण प्रमेय: टॉरिक बंडलों के लैग्रेंजियन शंकु

非分割トーリックバンドルのミラー定理: トーリックバンドルのラグランジアン円錐

비분할 토릭 번들에 대한 거울 정리: 토릭 번들의 라그랑주 원뿔

Um Teorema do Espelho para Pacotes Tóricos Não Divididos: Cones Lagrangianos de Pacotes Tóricos

Зеркальная теорема для нерасщепляемых торических расслоений: лагранжевы конусы торических расслоений

Bölünmemiş Torik Demetler için Ayna Teoremi: Torik Demetlerin Lagrange Konileri

Định lý gương cho các bó Toric không phân chia: Nón Lagrange của các bó Toric

非分裂环面束的镜像定理：环面束的拉格朗日锥

Author:
(1) Yuki Koto Table of Links Abstract and Intro
Genus-zero Gromov-Witten Theory
Toric Bundles
Lagrangian cones of Toric bundles
Mirror theorem for a product of projectives bundles
Mirror Theorem for Toric Bundles
Appendix A. Equivariant Fourier Transformation and References 2. Genus-zero Gromov-Witten theory In this section, we briefly recall the (torus-equivariant/twisted) genus-zero Gromov-Witten theory. We will introduce Gromov-Witten invariants, Givental Lagrangian cones and the quantum Riemann-Roch theorem. 2.1. Gromov-Witten invariant and its variants. We recall the definition of Gromov-Witten invariant. We also introduce an torus-equivariant version and a twisted version of it. 2.3. Quantum Riemann-Roch theorem and twisted theory. We introduce quantum Riemann-Roch theorem [9, Corollary 4], which relates twisted Givental cones via some transcendental operators. We also explain relationships between the Gromow-Witten theory of a vector bundle (resp. a subvariety) and that of a base space (resp. an ambient space) in terms of twisted theories. Note that we will use the material in this subsection only in Section 5. This paper is available on arxiv under CC 4.0 license. Author: (1) Yuki Koto Author: Author: (1) Yuki Koto Table of Links Abstract and Intro Genus-zero Gromov-Witten Theory Toric Bundles Lagrangian cones of Toric bundles Mirror theorem for a product of projectives bundles Mirror Theorem for Toric Bundles Appendix A. Equivariant Fourier Transformation and References Abstract and Intro Abstract and Intro Genus-zero Gromov-Witten Theory Genus-zero Gromov-Witten Theory Toric Bundles Toric Bundles Lagrangian cones of Toric bundles Lagrangian cones of Toric bundles Mirror theorem for a product of projectives bundles Mirror theorem for a product of projectives bundles Mirror Theorem for Toric Bundles Mirror Theorem for Toric Bundles Appendix A. Equivariant Fourier Transformation and References Appendix A. Equivariant Fourier Transformation and References 2. Genus-zero Gromov-Witten theory In this section, we briefly recall the (torus-equivariant/twisted) genus-zero Gromov-Witten theory. We will introduce Gromov-Witten invariants, Givental Lagrangian cones and the quantum Riemann-Roch theorem. 2.1. Gromov-Witten invariant and its variants . We recall the definition of Gromov-Witten invariant. We also introduce an torus-equivariant version and a twisted version of it. Gromov-Witten invariant and its variants 2.3. Quantum Riemann-Roch theorem and twisted theory. We introduce quantum Riemann-Roch theorem [9, Corollary 4], which relates twisted Givental cones via some transcendental operators. We also explain relationships between the Gromow-Witten theory of a vector bundle (resp. a subvariety) and that of a base space (resp. an ambient space) in terms of twisted theories. Note that we will use the material in this subsection only in Section 5. Quantum Riemann-Roch theorem and twisted theory. This paper is available on arxiv under CC 4.0 license. This paper is available on arxiv under CC 4.0 license. available on arxiv

A Mirror Theorem for Non-split Toric Bundles: Genus-zero Gromov-Witten Theory

অ-বিভক্ত টরিক বান্ডেলের জন্য একটি মিরর উপপাদ্য: জেনাস-শূন্য গ্রোমভ-উইটেন তত্ত্ব

Ein Spiegelsatz für nicht-geteilte torische Bündel: Gattung-Null-Gromov-Witten-Theorie

Un teorema del espejo para haces tóricos no divididos: teoría de Gromov-Witten del género cero

Un théorème miroir pour les faisceaux toriques non divisés : théorie de Gromov-Witten du genre zéro

गैर-विभाजित टॉरिक बंडलों के लिए दर्पण प्रमेय: जीनस-शून्य ग्रोमोव-विटन सिद्धांत

非分割トーリック束のミラー定理: 種数ゼロのグロモフ-ウィッテン理論

비분할 토릭 번들에 대한 거울 정리: Genus-zero Gromov-Witten 이론

Um teorema do espelho para pacotes tóricos não divididos: teoria de Gromov-Witten do gênero zero

Зеркальная теорема для нерасщепляемых торических расслоений: теория Громова-Виттена с нулевым родом

Bölünmemiş Torik Demetler için Ayna Teoremi: Cins-sıfır Gromov-Witten Teorisi

Định lý gương cho các gói Toric không phân chia: Lý thuyết Gromov-Witten Genus-zero

非分裂环面束的镜像定理：属零的 Gromov-Witten 理论

Author:
(1) Yuki Koto Table of Links Abstract and Intro
Genus-zero Gromov-Witten Theory
Toric Bundles
Lagrangian cones of Toric bundles
Mirror theorem for a product of projectives bundles
Mirror Theorem for Toric Bundles
Appendix A. Equivariant Fourier Transformation and References 5. Mirror theorem for a product of projective bundles In this section, we construct a twisted I-function for a product of projective bundles each coming from a vector bundle. The proof is based on the proof of the mirror theorem for a projective bundle [21, Theorem 1.1]. This section is independent of the previous section. By combining Theorem 4.2 with the mirror theorem (Theorem 5.1), we will establish the main result in the next section. Remark 5.8. For convenience, we list the rings to which the functions introduced in this section belong. This paper is available on arxiv under CC 4.0 license. Author: (1) Yuki Koto Author: Author: (1) Yuki Koto Table of Links Abstract and Intro Genus-zero Gromov-Witten Theory Toric Bundles Lagrangian cones of Toric bundles Mirror theorem for a product of projectives bundles Mirror Theorem for Toric Bundles Appendix A. Equivariant Fourier Transformation and References Abstract and Intro Abstract and Intro Genus-zero Gromov-Witten Theory Genus-zero Gromov-Witten Theory Toric Bundles Toric Bundles Lagrangian cones of Toric bundles Lagrangian cones of Toric bundles Mirror theorem for a product of projectives bundles Mirror theorem for a product of projectives bundles Mirror Theorem for Toric Bundles Mirror Theorem for Toric Bundles Appendix A. Equivariant Fourier Transformation and References Appendix A. Equivariant Fourier Transformation and References 5. Mirror theorem for a product of projective bundles In this section, we construct a twisted I -function for a product of projective bundles each coming from a vector bundle. The proof is based on the proof of the mirror theorem for a projective bundle [21, Theorem 1.1]. This section is independent of the previous section. By combining Theorem 4.2 with the mirror theorem (Theorem 5.1), we will establish the main result in the next section. I Remark 5.8. For convenience, we list the rings to which the functions introduced in this section belong. Remark 5.8. This paper is available on arxiv under CC 4.0 license. This paper is available on arxiv under CC 4.0 license. available on arxiv

A Mirror Theorem for Non-split Toric Bundles: Mirror Theorem for a Product of Projectives Bundle

অ-বিভক্ত টরিক বান্ডেলের জন্য একটি মিরর উপপাদ্য: প্রজেক্টিভ বান্ডেলের একটি পণ্যের জন্য মিরর উপপাদ্য

Ein Spiegelsatz für nicht aufgeteilte torische Bündel: Spiegelsatz für ein Produkt projektiver Bündel

Un teorema del espejo para haces tóricos no divididos: teorema del espejo para un producto de haces proyectivos

Un théorème miroir pour les bundles toriques non divisés : Théorème miroir pour un produit de bundle projectifs

गैर-विभाजित टॉरिक बंडलों के लिए दर्पण प्रमेय: प्रोजेक्टिव बंडल के उत्पाद के लिए दर्पण प्रमेय

非分割トーリックバンドルのミラー定理: 射影バンドルの積のミラー定理

비분할 토릭 번들에 대한 거울 정리: 투영 번들 제품에 대한 거울 정리

Um Teorema do Espelho para Pacotes Tóricos Não Divididos: Teorema do Espelho para um Produto de Pacotes Projetivos

Зеркальная теорема для нерасщепляемых торических расслоений: зеркальная теорема для произведения пучка проективов

Bölünmemiş Torik Paketler için Ayna Teoremi: Projektifler Paketinin Bir Çarpımı için Ayna Teoremi

Định lý gương cho các gói Toric không phân chia: Định lý gương cho một tích của gói xạ ảnh

非分裂环面束的镜像定理：射影束积的镜像定理

Author:
(1) Yuki Koto Table of Links Abstract and Intro
Genus-zero Gromov-Witten Theory
Toric Bundles
Lagrangian cones of Toric bundles
Mirror theorem for a product of projectives bundles
Mirror Theorem for Toric Bundles
Appendix A. Equivariant Fourier Transformation and References 6. Mirror theorem for toric bundles In this section, we will prove the mirror theorem (Theorem 6.1) for (possibly non-split) toric bundles. Throughout this section, we fix the following data: for any vector bundle V . Using these formulas and the projection formula, it holds that From the calculations above, the right-hand side of (6.2) can be computed as follow: which coincides with (6.3). This paper is available on arxiv under CC 4.0 license. Author: (1) Yuki Koto Author: Author: (1) Yuki Koto Table of Links Abstract and Intro Genus-zero Gromov-Witten Theory Toric Bundles Lagrangian cones of Toric bundles Mirror theorem for a product of projectives bundles Mirror Theorem for Toric Bundles Appendix A. Equivariant Fourier Transformation and References Abstract and Intro Abstract and Intro Genus-zero Gromov-Witten Theory Genus-zero Gromov-Witten Theory Toric Bundles Toric Bundles Lagrangian cones of Toric bundles Lagrangian cones of Toric bundles Mirror theorem for a product of projectives bundles Mirror theorem for a product of projectives bundles Mirror Theorem for Toric Bundles Mirror Theorem for Toric Bundles Appendix A. Equivariant Fourier Transformation and References Appendix A. Equivariant Fourier Transformation and References 6. Mirror theorem for toric bundles In this section, we will prove the mirror theorem (Theorem 6.1) for (possibly non-split) toric bundles. Throughout this section, we fix the following data: for any vector bundle V . Using these formulas and the projection formula, it holds that From the calculations above, the right-hand side of (6.2) can be computed as follow: which coincides with (6.3). This paper is available on arxiv under CC 4.0 license. This paper is available on arxiv under CC 4.0 license. available on arxiv

A Mirror Theorem for Non-split Toric Bundles: Mirror Theorem for Toric Bundles

অ-বিভক্ত টরিক বান্ডেলের জন্য একটি মিরর উপপাদ্য: টরিক বান্ডেলের জন্য মিরর উপপাদ্য

Ein Spiegelsatz für nicht aufgeteilte torische Bündel: Spiegelsatz für torische Bündel

Un teorema del espejo para haces tóricos no divididos: Teorema del espejo para haces tóricos

Un théorème miroir pour les faisceaux toriques non divisés : Théorème miroir pour les faisceaux toriques

गैर-विभाजित टॉरिक बंडलों के लिए दर्पण प्रमेय: टॉरिक बंडलों के लिए दर्पण प्रमेय

非分割トーリックバンドルのミラー定理: トーリックバンドルのミラー定理

비분할 토릭 번들에 대한 거울 정리: 토릭 번들에 대한 거울 정리

Um teorema do espelho para pacotes tóricos não divididos: Teorema do espelho para pacotes tóricos

Зеркальная теорема для нерасщепляемых торических расслоений: Зеркальная теорема для торических расслоений

Bölünmemiş Torik Demetler için Ayna Teoremi: Torik Demetler için Ayna Teoremi

Định lý gương cho các gói Toric không phân chia: Định lý gương cho các gói Toric

非分裂环面束的镜像定理：环面束的镜像定理

Author:
(1) Yuki Koto Table of Links Abstract and Intro
Genus-zero Gromov-Witten Theory
Toric Bundles
Lagrangian cones of Toric bundles
Mirror theorem for a product of projectives bundles
Mirror Theorem for Toric Bundles
Appendix A. Equivariant Fourier Transformation and References Appendix A. Equivariant Fourier transformation Note that this is a straightforward generalization of [20, Conjecture 1.7]. References Dan Abramovich, Tom Graber, and Angelo Vistoli, Gromov-Witten theory of Deligne-Mumford stacks, Amer. J. Math. 130 (2008), no. 5, 1337–1398.


M. F. Atiyah and R. Bott, The moment map and equivariant cohomology, Topology 23 (1984), no. 1, 1–28.


K. Behrend, Gromov-Witten invariants in algebraic geometry, Invent. Math. 127 (1997), no. 3, 601–617.


Nicole Berline and Mich`ele Vergne, Classes caract´eristiques ´equivariantes. Formule de localisation en cohomologie ´equivariante, C. R. Acad. Sci. Paris S´er. I Math. 295 (1982), no. 9, 539–541.


Jeff Brown, Gromov-Witten invariants of toric fibrations, Int. Math. Res. Not. IMRN (2014), no. 19, 5437–5482.


Charles Cadman, Using stacks to impose tangency conditions on curves, Amer. J. Math. 129 (2007), no. 2, 405–427.


Tom Coates, Alessio Corti, Hiroshi Iritani, and Hsian-Hua Tseng, Computing genus-zero twisted GromovWitten invariants, Duke Math. J. 147 (2009), no. 3, 377–438.


_________, A mirror theorem for toric stacks, Compos. Math. 151 (2015), no. 10, 1878–1912.


Tom Coates and Alexander Givental, Quantum Riemann-Roch, Lefschetz and Serre, Ann. of Math. (2) 165 (2007), no. 1, 15–53.


Artur Elezi, A mirror conjecture for projective bundles, Int. Math. Res. Not. (2005), no. 55, 3445–3458.


Honglu Fan and Yuan-Pin Lee, On Gromov-Witten theory of projective bundles, Michigan Math. J. 69 (2020), no. 1, 153–178.


William Fulton, Intersection theory, second ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 2, Springer-Verlag, Berlin, 1998.


Alexander Givental, A mirror theorem for toric complete intersections, Topological field theory, primitive forms and related topics (Kyoto, 1996), Progr. Math., vol. 160, Birkh¨auser Boston, Boston, MA, 1998, pp. 141–175.


Alexander B. Givental, Symplectic geometry of Frobenius structures, Frobenius manifolds, Aspects Math., vol. E36, Friedr. Vieweg, Wiesbaden, 2004, pp. 91–112.


T. Graber and R. Pandharipande, Localization of virtual classes, Invent. Math. 135 (1999), no. 2, 487–518.


Tam´as Hausel and Bernd Sturmfels, Toric hyperK¨ahler varieties, Doc. Math. 7 (2002), 495–534.


Hiroshi Iritani, An integral structure in quantum cohomology and mirror symmetry for toric orbifolds, Adv. Math. 222 (2009), no. 3, 1016–1079.


_________, Quantum cohomology and periods, Ann. Inst. Fourier (Grenoble) 61 (2011), no. 7, 2909–2958.


_________, Shift operators and toric mirror theorem, Geom. Topol. 21 (2017), no. 1, 315–343.


Hiroshi Iritani, Quantum cohomology of blowups, 2023.


Hiroshi Iritani and Yuki Koto, Quantum cohomology of projective bundles, 2023, arXiv:2307.03696 [math.AG].


Hiroshi Iritani and Fumihiko Sanda, private communication.


Yunfeng Jiang, Hsian-Hua Tseng, and Fenglong You, The quantum orbifold cohomology of toric stack bundles, Lett. Math. Phys. 107 (2017), no. 3, 439–465.


Bumsig Kim, Andrew Kresch, and Tony Pantev, Functoriality in intersection theory and a conjecture of Cox, Katz, and Lee, J. Pure Appl. Algebra 179 (2003), no. 1-2, 127–136.


Chiu-Chu Melissa Liu, Localization in Gromov-Witten theory and orbifold Gromov-Witten theory, Handbook of moduli. Vol. II, Adv. Lect. Math. (ALM), vol. 25, Int. Press, Somerville, MA, 2013, pp. 353–425.


Rahul Pandharipande, Rational curves on hypersurfaces (after A. Givental), no. 252, 1998, S´eminaire Bourbaki. Vol. 1997/98, pp. Exp. No. 848, 5, 307–340.


Constantin Teleman, Gauge theory and mirror symmetry, Proceedings of the International Congress of Mathematicians—Seoul 2014. Vol. II, Kyung Moon Sa, Seoul, 2014, pp. 1309–1332.


Valentin Tonita, Twisted orbifold Gromov-Witten invariants, Nagoya Math. J. 213 (2014), 141–187.


Angelo Vistoli, Intersection theory on algebraic stacks and on their moduli spaces, Invent. Math. 97 (1989), no. 3, 613–670. This paper is available on arxiv under CC 4.0 license. Author: (1) Yuki Koto Author: Author: (1) Yuki Koto Table of Links Abstract and Intro Genus-zero Gromov-Witten Theory Toric Bundles Lagrangian cones of Toric bundles Mirror theorem for a product of projectives bundles Mirror Theorem for Toric Bundles Appendix A. Equivariant Fourier Transformation and References Abstract and Intro Abstract and Intro Genus-zero Gromov-Witten Theory Genus-zero Gromov-Witten Theory Toric Bundles Toric Bundles Lagrangian cones of Toric bundles Lagrangian cones of Toric bundles Mirror theorem for a product of projectives bundles Mirror theorem for a product of projectives bundles Mirror Theorem for Toric Bundles Mirror Theorem for Toric Bundles Appendix A. Equivariant Fourier Transformation and References Appendix A. Equivariant Fourier Transformation and References Appendix A. Equivariant Fourier transformation Note that this is a straightforward generalization of [20, Conjecture 1.7]. References Dan Abramovich, Tom Graber, and Angelo Vistoli, Gromov-Witten theory of Deligne-Mumford stacks, Amer. J. Math. 130 (2008), no. 5, 1337–1398. M. F. Atiyah and R. Bott, The moment map and equivariant cohomology, Topology 23 (1984), no. 1, 1–28. K. Behrend, Gromov-Witten invariants in algebraic geometry, Invent. Math. 127 (1997), no. 3, 601–617. Nicole Berline and Mich`ele Vergne, Classes caract´eristiques ´equivariantes. Formule de localisation en cohomologie ´equivariante, C. R. Acad. Sci. Paris S´er. I Math. 295 (1982), no. 9, 539–541. Jeff Brown, Gromov-Witten invariants of toric fibrations, Int. Math. Res. Not. IMRN (2014), no. 19, 5437–5482. Charles Cadman, Using stacks to impose tangency conditions on curves, Amer. J. Math. 129 (2007), no. 2, 405–427. Tom Coates, Alessio Corti, Hiroshi Iritani, and Hsian-Hua Tseng, Computing genus-zero twisted GromovWitten invariants, Duke Math. J. 147 (2009), no. 3, 377–438. _________, A mirror theorem for toric stacks, Compos. Math. 151 (2015), no. 10, 1878–1912. Tom Coates and Alexander Givental, Quantum Riemann-Roch, Lefschetz and Serre, Ann. of Math. (2) 165 (2007), no. 1, 15–53. Artur Elezi, A mirror conjecture for projective bundles, Int. Math. Res. Not. (2005), no. 55, 3445–3458. Honglu Fan and Yuan-Pin Lee, On Gromov-Witten theory of projective bundles, Michigan Math. J. 69 (2020), no. 1, 153–178. William Fulton, Intersection theory, second ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 2, Springer-Verlag, Berlin, 1998. Alexander Givental, A mirror theorem for toric complete intersections, Topological field theory, primitive forms and related topics (Kyoto, 1996), Progr. Math., vol. 160, Birkh¨auser Boston, Boston, MA, 1998, pp. 141–175. Alexander B. Givental, Symplectic geometry of Frobenius structures, Frobenius manifolds, Aspects Math., vol. E36, Friedr. Vieweg, Wiesbaden, 2004, pp. 91–112. T. Graber and R. Pandharipande, Localization of virtual classes, Invent. Math. 135 (1999), no. 2, 487–518. Tam´as Hausel and Bernd Sturmfels, Toric hyperK¨ahler varieties, Doc. Math. 7 (2002), 495–534. Hiroshi Iritani, An integral structure in quantum cohomology and mirror symmetry for toric orbifolds, Adv. Math. 222 (2009), no. 3, 1016–1079. _________, Quantum cohomology and periods, Ann. Inst. Fourier (Grenoble) 61 (2011), no. 7, 2909–2958. _________, Shift operators and toric mirror theorem, Geom. Topol. 21 (2017), no. 1, 315–343. Hiroshi Iritani, Quantum cohomology of blowups, 2023. Hiroshi Iritani and Yuki Koto, Quantum cohomology of projective bundles, 2023, arXiv:2307.03696 [math.AG]. Hiroshi Iritani and Fumihiko Sanda, private communication. Yunfeng Jiang, Hsian-Hua Tseng, and Fenglong You, The quantum orbifold cohomology of toric stack bundles, Lett. Math. Phys. 107 (2017), no. 3, 439–465. Bumsig Kim, Andrew Kresch, and Tony Pantev, Functoriality in intersection theory and a conjecture of Cox, Katz, and Lee, J. Pure Appl. Algebra 179 (2003), no. 1-2, 127–136. Chiu-Chu Melissa Liu, Localization in Gromov-Witten theory and orbifold Gromov-Witten theory, Handbook of moduli. Vol. II, Adv. Lect. Math. (ALM), vol. 25, Int. Press, Somerville, MA, 2013, pp. 353–425. Rahul Pandharipande, Rational curves on hypersurfaces (after A. Givental), no. 252, 1998, S´eminaire Bourbaki. Vol. 1997/98, pp. Exp. No. 848, 5, 307–340. Constantin Teleman, Gauge theory and mirror symmetry, Proceedings of the International Congress of Mathematicians—Seoul 2014. Vol. II, Kyung Moon Sa, Seoul, 2014, pp. 1309–1332. Valentin Tonita, Twisted orbifold Gromov-Witten invariants, Nagoya Math. J. 213 (2014), 141–187. Angelo Vistoli, Intersection theory on algebraic stacks and on their moduli spaces, Invent. Math. 97 (1989), no. 3, 613–670. Dan Abramovich, Tom Graber, and Angelo Vistoli, Gromov-Witten theory of Deligne-Mumford stacks, Amer. J. Math. 130 (2008), no. 5, 1337–1398. Dan Abramovich, Tom Graber, and Angelo Vistoli, Gromov-Witten theory of Deligne-Mumford stacks, Amer. J. Math. 130 (2008), no. 5, 1337–1398. M. F. Atiyah and R. Bott, The moment map and equivariant cohomology, Topology 23 (1984), no. 1, 1–28. M. F. Atiyah and R. Bott, The moment map and equivariant cohomology, Topology 23 (1984), no. 1, 1–28. K. Behrend, Gromov-Witten invariants in algebraic geometry, Invent. Math. 127 (1997), no. 3, 601–617. K. Behrend, Gromov-Witten invariants in algebraic geometry, Invent. Math. 127 (1997), no. 3, 601–617. Nicole Berline and Mich`ele Vergne, Classes caract´eristiques ´equivariantes. Formule de localisation en cohomologie ´equivariante, C. R. Acad. Sci. Paris S´er. I Math. 295 (1982), no. 9, 539–541. Nicole Berline and Mich`ele Vergne, Classes caract´eristiques ´equivariantes. Formule de localisation en cohomologie ´equivariante, C. R. Acad. Sci. Paris S´er. I Math. 295 (1982), no. 9, 539–541. Jeff Brown, Gromov-Witten invariants of toric fibrations, Int. Math. Res. Not. IMRN (2014), no. 19, 5437–5482. Jeff Brown, Gromov-Witten invariants of toric fibrations, Int. Math. Res. Not. IMRN (2014), no. 19, 5437–5482. Charles Cadman, Using stacks to impose tangency conditions on curves, Amer. J. Math. 129 (2007), no. 2, 405–427. Charles Cadman, Using stacks to impose tangency conditions on curves, Amer. J. Math. 129 (2007), no. 2, 405–427. Tom Coates, Alessio Corti, Hiroshi Iritani, and Hsian-Hua Tseng, Computing genus-zero twisted GromovWitten invariants, Duke Math. 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Honglu Fan and Yuan-Pin Lee, On Gromov-Witten theory of projective bundles, Michigan Math. J. 69 (2020), no. 1, 153–178. William Fulton, Intersection theory, second ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 2, Springer-Verlag, Berlin, 1998. William Fulton, Intersection theory, second ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 2, Springer-Verlag, Berlin, 1998. Alexander Givental, A mirror theorem for toric complete intersections, Topological field theory, primitive forms and related topics (Kyoto, 1996), Progr. Math., vol. 160, Birkh¨auser Boston, Boston, MA, 1998, pp. 141–175. Alexander Givental, A mirror theorem for toric complete intersections, Topological field theory, primitive forms and related topics (Kyoto, 1996), Progr. Math., vol. 160, Birkh¨auser Boston, Boston, MA, 1998, pp. 141–175. Alexander B. Givental, Symplectic geometry of Frobenius structures, Frobenius manifolds, Aspects Math., vol. E36, Friedr. Vieweg, Wiesbaden, 2004, pp. 91–112. Alexander B. Givental, Symplectic geometry of Frobenius structures, Frobenius manifolds, Aspects Math., vol. E36, Friedr. Vieweg, Wiesbaden, 2004, pp. 91–112. T. Graber and R. Pandharipande, Localization of virtual classes, Invent. Math. 135 (1999), no. 2, 487–518. T. Graber and R. Pandharipande, Localization of virtual classes, Invent. Math. 135 (1999), no. 2, 487–518. Tam´as Hausel and Bernd Sturmfels, Toric hyperK¨ahler varieties, Doc. Math. 7 (2002), 495–534. Tam´as Hausel and Bernd Sturmfels, Toric hyperK¨ahler varieties, Doc. Math. 7 (2002), 495–534. Hiroshi Iritani, An integral structure in quantum cohomology and mirror symmetry for toric orbifolds, Adv. Math. 222 (2009), no. 3, 1016–1079. Hiroshi Iritani, An integral structure in quantum cohomology and mirror symmetry for toric orbifolds, Adv. Math. 222 (2009), no. 3, 1016–1079. _________, Quantum cohomology and periods, Ann. Inst. Fourier (Grenoble) 61 (2011), no. 7, 2909–2958. _________ , Quantum cohomology and periods, Ann. Inst. Fourier (Grenoble) 61 (2011), no. 7, 2909–2958. _________ _________, Shift operators and toric mirror theorem, Geom. Topol. 21 (2017), no. 1, 315–343. _________ , Shift operators and toric mirror theorem, Geom. Topol. 21 (2017), no. 1, 315–343. _________ Hiroshi Iritani, Quantum cohomology of blowups, 2023. Hiroshi Iritani, Quantum cohomology of blowups, 2023. Hiroshi Iritani and Yuki Koto, Quantum cohomology of projective bundles, 2023, arXiv:2307.03696 [math.AG]. Hiroshi Iritani and Yuki Koto, Quantum cohomology of projective bundles, 2023, arXiv:2307.03696 [ math.AG ]. math.AG Hiroshi Iritani and Fumihiko Sanda, private communication. Hiroshi Iritani and Fumihiko Sanda, private communication. Yunfeng Jiang, Hsian-Hua Tseng, and Fenglong You, The quantum orbifold cohomology of toric stack bundles, Lett. Math. Phys. 107 (2017), no. 3, 439–465. Yunfeng Jiang, Hsian-Hua Tseng, and Fenglong You, The quantum orbifold cohomology of toric stack bundles, Lett. Math. Phys. 107 (2017), no. 3, 439–465. Bumsig Kim, Andrew Kresch, and Tony Pantev, Functoriality in intersection theory and a conjecture of Cox, Katz, and Lee, J. Pure Appl. Algebra 179 (2003), no. 1-2, 127–136. Bumsig Kim, Andrew Kresch, and Tony Pantev, Functoriality in intersection theory and a conjecture of Cox, Katz, and Lee, J. Pure Appl. Algebra 179 (2003), no. 1-2, 127–136. Chiu-Chu Melissa Liu, Localization in Gromov-Witten theory and orbifold Gromov-Witten theory, Handbook of moduli. Vol. II, Adv. Lect. Math. (ALM), vol. 25, Int. Press, Somerville, MA, 2013, pp. 353–425. Chiu-Chu Melissa Liu, Localization in Gromov-Witten theory and orbifold Gromov-Witten theory, Handbook of moduli. Vol. II, Adv. Lect. Math. (ALM), vol. 25, Int. Press, Somerville, MA, 2013, pp. 353–425. Rahul Pandharipande, Rational curves on hypersurfaces (after A. Givental), no. 252, 1998, S´eminaire Bourbaki. Vol. 1997/98, pp. Exp. No. 848, 5, 307–340. Rahul Pandharipande, Rational curves on hypersurfaces (after A. Givental), no. 252, 1998, S´eminaire Bourbaki. Vol. 1997/98, pp. Exp. No. 848, 5, 307–340. Constantin Teleman, Gauge theory and mirror symmetry, Proceedings of the International Congress of Mathematicians—Seoul 2014. Vol. II, Kyung Moon Sa, Seoul, 2014, pp. 1309–1332. Constantin Teleman, Gauge theory and mirror symmetry, Proceedings of the International Congress of Mathematicians—Seoul 2014. Vol. II, Kyung Moon Sa, Seoul, 2014, pp. 1309–1332. Valentin Tonita, Twisted orbifold Gromov-Witten invariants, Nagoya Math. J. 213 (2014), 141–187. Valentin Tonita, Twisted orbifold Gromov-Witten invariants, Nagoya Math. J. 213 (2014), 141–187. Angelo Vistoli, Intersection theory on algebraic stacks and on their moduli spaces, Invent. Math. 97 (1989), no. 3, 613–670. Angelo Vistoli, Intersection theory on algebraic stacks and on their moduli spaces, Invent. Math. 97 (1989), no. 3, 613–670. This paper is available on arxiv under CC 4.0 license. This paper is available on arxiv under CC 4.0 license. available on arxiv

A Mirror Theorem for Non-split Toric Bundles: Appendix a and References

অ-বিভক্ত টরিক বান্ডেলের জন্য একটি মিরর উপপাদ্য: পরিশিষ্ট একটি এবং রেফারেন্স

Ein Spiegelsatz für nicht-geteilte torische Bündel: Anhang a und Referenzen

Un teorema del espejo para haces tóricos no divididos: Apéndice a y referencias

Un théorème du miroir pour les faisceaux toriques non divisés : annexe a et références

गैर-विभाजित टॉरिक बंडलों के लिए दर्पण प्रमेय: परिशिष्ट ए और संदर्भ

非分割トーリック束のミラー定理: 付録 a と参考文献

비분할 토릭 번들에 대한 거울 정리: 부록 a 및 참고 자료

Um Teorema do Espelho para Pacotes Tóricos Não Divididos: Apêndice a e Referências

Зеркальная теорема для нерасщепляемых торических расслоений: приложение а и список литературы

Bölünmemiş Torik Demetler için Ayna Teoremi: Ek a ve Referanslar

Định lý gương cho các gói Toric không phân chia: Phụ lục a và Tài liệu tham khảo

非分裂环面束的镜像定理：附录 a 和参考文献

Author:
(1) Yuki Koto Table of Links Abstract and Intro
Genus-zero Gromov-Witten Theory
Toric Bundles
Lagrangian cones of Toric bundles
Mirror theorem for a product of projectives bundles
Mirror Theorem for Toric Bundles
Appendix A. Equivariant Fourier Transformation and References 3. Toric bundles In this section we introduce toric bundles. We first review toric varieties, and then define toric bundles by doing the construction of toric varieties in a relative setting. Note that they include toric bundles appearing in [5] [21]. We then investigate geometric structures of toric bundles: T-equivariant cohomology ring (3.2), effective curves (3.3), T-fixed loci and one-dimensional orbits (3.4). As explained in Section 1, mirror theorems for split toric bundles [5] and (non-split) projective bundles [21] are already known. We will prove a mirror theorem for (non-split) toric bundles (Theorem 6.1). This paper is available on arxiv under CC 4.0 license. Author: (1) Yuki Koto Author: Author: (1) Yuki Koto Table of Links Abstract and Intro Genus-zero Gromov-Witten Theory Toric Bundles Lagrangian cones of Toric bundles Mirror theorem for a product of projectives bundles Mirror Theorem for Toric Bundles Appendix A. Equivariant Fourier Transformation and References Abstract and Intro Abstract and Intro Genus-zero Gromov-Witten Theory Genus-zero Gromov-Witten Theory Toric Bundles Toric Bundles Lagrangian cones of Toric bundles Lagrangian cones of Toric bundles Mirror theorem for a product of projectives bundles Mirror theorem for a product of projectives bundles Mirror Theorem for Toric Bundles Mirror Theorem for Toric Bundles Appendix A. Equivariant Fourier Transformation and References Appendix A. Equivariant Fourier Transformation and References 3. Toric bundles In this section we introduce toric bundles. We first review toric varieties, and then define toric bundles by doing the construction of toric varieties in a relative setting. Note that they include toric bundles appearing in [5] [21]. We then investigate geometric structures of toric bundles: T-equivariant cohomology ring (3.2), effective curves (3.3), T-fixed loci and one-dimensional orbits (3.4). As explained in Section 1, mirror theorems for split toric bundles [5] and (non-split) projective bundles [21] are already known. We will prove a mirror theorem for (non-split) toric bundles (Theorem 6.1). This paper is available on arxiv under CC 4.0 license. This paper is available on arxiv under CC 4.0 license. available on arxiv

A Mirror Theorem for Non-split Toric Bundles: Toric Bundles

অ-বিভক্ত টরিক বান্ডেলের জন্য একটি মিরর উপপাদ্য: টরিক বান্ডেল

Ein Spiegelsatz für nicht aufgeteilte torische Bündel: Torische Bündel

Un teorema del espejo para haces tóricos no divididos: haces tóricos

Un théorème miroir pour les faisceaux toriques non divisés : les faisceaux toriques

गैर-विभाजित टॉरिक बंडलों के लिए दर्पण प्रमेय: टॉरिक बंडल

非分割トーリックバンドルのミラー定理: トーリックバンドル

비분할 토릭 번들에 대한 거울 정리: 토릭 번들

Um teorema do espelho para pacotes tóricos não divididos: pacotes tóricos

Зеркальная теорема для нерасщепляемых торических расслоений: Торические расслоения

Bölünmemiş Torik Demetler için Ayna Teoremi: Torik Demetler

Định lý gương cho các gói Toric không phân chia: Gói Toric

非分裂环面束的镜像定理：环面束

Author:
(1) Yuki Koto Table of Links Abstract and Intro
Genus-zero Gromov-Witten Theory
Toric Bundles
Lagrangian cones of Toric bundles
Mirror theorem for a product of projectives bundles
Mirror Theorem for Toric Bundles
Appendix A. Equivariant Fourier Transformation and References Abstract We construct an I-function for toric bundles obtained as a fiberwise GIT quotient of a (not necessarily split) vector bundle. This is a generalization of Brown’s I-function for split toric bundles [5] and the I-function for non-split projective bundles [21]. In order to prove the mirror theorem, we establish a characterization of points on the Givental Lagrangian cones of toric bundles and prove a mirror theorem for the twisted Gromov-Witten theory of a fiber product of projective bundles. The former result generalizes Brown’s characterization for split toric bundles [5] to the non-split case. 1. Introduction The genus-zero Gromov-Witten theory of a smooth projective variety X plays a significant role in symplectic geometry, algebraic geometry and mirror symmetry. It can be studied by a mirror theorem [13], that is, by finding a convenient point (called an I-function) on the Givental Lagrangian cone LX [14]. The cone LX is a Lagrangian submanifold of an infinite-dimensional symplectic vector space HX, called the Givental space, and is defined by genus-zero gravitational Gromov-Witten invariants. A mirror theorem for X enables us to compute genus-zero Gromov-Witten invariants of X and study quantum cohomology. This is a generalization of Brown’s result [5, Theorem 2], which gives the same characterization for split toric bundles. There are also similar characterization results for other varieties/stacks; see [8, 23, 11]. This result is a straightforward generalization of the mirror theorem for non-split projective bundles [21, Theorem 3.3]. The key ingredient of the proof is the quantum Riemann-Roch theorem [9, Corollary 4] and the well-known fact [24] that Gromov-Witten invariants of the zero locus of a regular section of a convex vector bundle over a variety X are given by twisted Gromov-Witten invariants of X. The plan of the paper is as follows. In Section 2, we recall the definition of GromovWitten invariants, and introduce the non-equivariant/equivariant/twisted Givental cones and quantum Riemann-Roch theorem. In Section 3, we introduce the notion of split/non-split toric bundles, and summarize the structure of cohomology and the semigroups generated by effective curve classes, which will be needed in the subsequent sections. In Section 4, we establish a characterization theorem (Theorem 4.2) for points on the Lagrangian cone of a toric bundle. In Section 5, we prove a mirror theorem for twisted Gromov-Witten theory of a fiber product of projective bundles over B. In Section 6, we prove the main result (Theorem 6.1) of this paper, that is, a mirror theorem for (possibly non-split) toric bundles. In Appendix A, we briefly explain a Fourier transform of Givental cones, and check that our I-function coincides with the Fourier transform of the I-function of a vector bundle. Acknowledgements. The author is deeply grateful to Hiroshi Iritani for his guidance and enthusiastic support during the writing of this paper. He also would like to thank Yuan-Pin Lee and Fumihiko Sanda for very helpful discussions. This work was supported by JSPS KAKENHI Grant Number 22KJ1717. This paper is available on arxiv under CC 4.0 license. Author: (1) Yuki Koto Author: Author: (1) Yuki Koto Table of Links Abstract and Intro Genus-zero Gromov-Witten Theory Toric Bundles Lagrangian cones of Toric bundles Mirror theorem for a product of projectives bundles Mirror Theorem for Toric Bundles Appendix A. Equivariant Fourier Transformation and References Abstract and Intro Abstract and Intro Genus-zero Gromov-Witten Theory Genus-zero Gromov-Witten Theory Toric Bundles Toric Bundles Lagrangian cones of Toric bundles Lagrangian cones of Toric bundles Mirror theorem for a product of projectives bundles Mirror theorem for a product of projectives bundles Mirror Theorem for Toric Bundles Mirror Theorem for Toric Bundles Appendix A. Equivariant Fourier Transformation and References Appendix A. Equivariant Fourier Transformation and References Abstract We construct an I-function for toric bundles obtained as a fiberwise GIT quotient of a (not necessarily split) vector bundle. This is a generalization of Brown’s I-function for split toric bundles [5] and the I-function for non-split projective bundles [21]. In order to prove the mirror theorem, we establish a characterization of points on the Givental Lagrangian cones of toric bundles and prove a mirror theorem for the twisted Gromov-Witten theory of a fiber product of projective bundles. The former result generalizes Brown’s characterization for split toric bundles [5] to the non-split case. 1. Introduction The genus-zero Gromov-Witten theory of a smooth projective variety X plays a significant role in symplectic geometry, algebraic geometry and mirror symmetry. It can be studied by a mirror theorem [13], that is, by finding a convenient point (called an I-function) on the Givental Lagrangian cone LX [14]. The cone LX is a Lagrangian submanifold of an infinite-dimensional symplectic vector space HX, called the Givental space, and is defined by genus-zero gravitational Gromov-Witten invariants. A mirror theorem for X enables us to compute genus-zero Gromov-Witten invariants of X and study quantum cohomology. This is a generalization of Brown’s result [5, Theorem 2], which gives the same characterization for split toric bundles. There are also similar characterization results for other varieties/stacks; see [8, 23, 11]. This result is a straightforward generalization of the mirror theorem for non-split projective bundles [21, Theorem 3.3]. The key ingredient of the proof is the quantum Riemann-Roch theorem [9, Corollary 4] and the well-known fact [24] that Gromov-Witten invariants of the zero locus of a regular section of a convex vector bundle over a variety X are given by twisted Gromov-Witten invariants of X. The plan of the paper is as follows. In Section 2, we recall the definition of GromovWitten invariants, and introduce the non-equivariant/equivariant/twisted Givental cones and quantum Riemann-Roch theorem. In Section 3, we introduce the notion of split/non-split toric bundles, and summarize the structure of cohomology and the semigroups generated by effective curve classes, which will be needed in the subsequent sections. In Section 4, we establish a characterization theorem (Theorem 4.2) for points on the Lagrangian cone of a toric bundle. In Section 5, we prove a mirror theorem for twisted Gromov-Witten theory of a fiber product of projective bundles over B. In Section 6, we prove the main result (Theorem 6.1) of this paper, that is, a mirror theorem for (possibly non-split) toric bundles. In Appendix A, we briefly explain a Fourier transform of Givental cones, and check that our I-function coincides with the Fourier transform of the I-function of a vector bundle. Acknowledgements . The author is deeply grateful to Hiroshi Iritani for his guidance and enthusiastic support during the writing of this paper. He also would like to thank Yuan-Pin Lee and Fumihiko Sanda for very helpful discussions. This work was supported by JSPS KAKENHI Grant Number 22KJ1717. Acknowledgements This paper is available on arxiv under CC 4.0 license. This paper is available on arxiv under CC 4.0 license. available on arxiv

A Mirror Theorem for Non-split Toric Bundles: Abstract and Intro

অ-বিভক্ত টরিক বান্ডেলের জন্য একটি মিরর উপপাদ্য: বিমূর্ত এবং ভূমিকা

Ein Spiegelsatz für nicht-geteilte torische Bündel: Zusammenfassung und Einführung

Un teorema del espejo para haces tóricos no divididos: resumen e introducción

Un théorème du miroir pour les faisceaux toriques non divisés : résumé et introduction

गैर-विभाजित टॉरिक बंडलों के लिए दर्पण प्रमेय: सार और परिचय

非分割トーリックバンドルのミラー定理: 概要と導入

비분할 토릭 번들을 위한 거울 정리: 요약 및 소개

Um teorema do espelho para pacotes tóricos não divididos: resumo e introdução

Зеркальная теорема для нерасщепимых торических расслоений: Аннотация и введение

Bölünmemiş Torik Paketler için Ayna Teoremi: Özet ve Giriş

Định lý gương cho các gói Toric không phân chia: Tóm tắt và giới thiệu

非分裂环面束的镜像定理：摘要和简介

In the era of widespread adoption of AI models, the data science community exhibits a rich diversity of backgrounds, fostering an environment conducive to innovation. Collaborative efforts stemming from this diversity continue to propel us toward the envisioned future of a decade or two ago. However, innovation often follows incremental patterns and may require decades before yielding tangible commercial benefits. Among industries adept at translating ideas into tangible products, automotive manufacturers stand out, boasting a deep familiarity with research and development processes and advanced algorithms. This proficiency is evident in the innovation pipeline for super and hypercars, drawing inspiration from advancements in rocket science. Leveraging extensive simulations and field tests, these manufacturers consistently push the boundaries of performance and safety in vehicle design. Major manufacturers have achieved breakthroughs with torque vectoring systems by capitalizing on established development pipelines and harnessing the capabilities of reinforcement learning. The late 2010s witnessed a surge in commercial vehicles featuring these systems, attributed partly to their association with electric and hybrid vehicles and partly to the appeal of enhanced safety. While the full extent of their influence on driver experience remains to be realized, these systems have already demonstrated remarkable efficacy, particularly in high-performance vehicles. So, what exactly is a torque vectoring system, and how is this innovation shaping the landscape of automotive manufacturing? What is Torque Vectoring In Torque Vectoring Systems (TVS), torque denotes a rotational force applied to a vehicle’s differentials or generated by independent electric motors on its wheels. These systems aim to regulate this rotational force to enhance speed, stability, handling, balance, and other performance aspects. A compelling illustration of the protracted nature of innovation is evident in the evolution of torque vectoring systems, with the first cars equipped with Torque Vectoring Systems (TVS) debuting in the 1990s, pioneered by Mitsubishi. Initially known as Active Yaw Control, Mitsubishi integrated this technology into vehicles like the 1996 Mitsubishi Lancer Evolution IV GSR models. The initial implementation relied on steering, throttle inputs, and g-force sensors to mitigate torque steer, a phenomenon where engine torque influences steering behavior. Initially geared towards racing applications, these systems have since evolved to cater to broader audiences, with companies like Rimac continuing to employ them for racing. In contrast, others emphasize safety and enhanced handling features for everyday drivers. Furthermore, advancements driven by the pursuit of eco-friendly transportation have led to the utilization of independent electric motors in torque vectoring systems, further expanding their applicability. Commercial Applications Ricardo (Honorable Mention) In 2006, Ricardo, a renowned engineering consultancy with over a century of experience, introduced the concept of torque vectoring. Ricardo rolled out its initial torque vectoring project centered around the Audi A6 4.2L, intending to license its technology to interested automakers. Unlike Mitsubishi’s Active Yaw Control system and similar counterparts, Ricardo’s torque vectoring approach prioritized safety and stability, showcasing a different aspect of the technology. Limited information is currently available on Ricardo’s involvement in assisting manufacturers in developing these systems for commercially available vehicles. However, they have publicly provided engineering support to Chinese students building cars for the global multi-event Formula SAE race in 2019. Rimac When discussing electric vehicles, Tesla typically takes the spotlight regarding market dominance. However, Rimac stands out as a company dedicated to crafting high-performance electric hypercars, where they excel. Just ask Elon… ( t) X pos Rimac burst onto the scene in January 2016 with their inaugural concept car, the Concept_One prototype. This marked the debut of their torque vectoring system (TVS) in an all-wheel-drive fully electric vehicle. Spearheaded by Tomislav Šimunić, Rimac’s TVS development pushed the boundaries by employing individual motors on each wheel. Moreover, they implemented reinforcement algorithms utilizing data from various sensors, including accelerometers, gyroscopes, steering angle, and wheel speed sensors. This data is then used to predict torque distribution to optimize vehicle stability. https://www.youtube.com/shorts/i4jVW29gbDc?embedable=true Currently, Rimac boasts one of the fastest and safest hypercars on the market. Their innovation has led to unprecedented control, resulting in their car achieving the title of the fastest production car ever in 21 acceleration and deceleration categories. Furthermore, their TVS is so reliable that their drivers can cruise hands-free at speeds exceeding 230 mph while enjoying gelato! Toyota In 2018, Toyota introduced their Dynamic Torque Vectoring AWD and Electric 4WD systems, expanding the scope of this technology beyond on-road safety and control to encompass off-road safety features and enhanced fuel efficiency. https://www.youtube.com/watch?v=LCqCa7PP1gE&embedable=true The video above showcases Toyota’s emphasis on “proactive torque transfer,” allowing for seamless adjustment of torque distribution when one or more wheels lose traction, enabling drivers to navigate various terrains more effectively. https://youtu.be/d1UG1a38diw?embedable=true Furthermore, Toyota optimizes torque distribution to enhance fuel-efficiency, shifting from AWD to 2WD when conditions permit, directing torque predominantly to the front wheels and thus cutting fuel consumption. Their limited content suggests the employment of a Dynamic Control System designed for stability, taking into account challenging driving scenarios. Moreover, the system can evaluate tire traction levels. These functionalities resonate with the research findings discussed below, showcasing the seamless transition from research concepts to practical implementation in the automotive industry. Research Review Conceptual Models — Bicycle, Vehicle, Tire Model Following Occam’s razor principle, simplicity reigns supreme when initiating modeling endeavors, with complexity introduced only when its benefits are worthwhile. This philosophy holds for TVS as well. Early TVS adopted a bicycle model, allowing researchers to streamline vehicle dynamics assumptions by confining movements to two wheels. This simplified framework facilitated the study of lateral vehicle dynamics during various events like braking, acceleration, and cornering under typical conditions. Key inputs into the bicycle model include yaw rate, lateral acceleration, steering angle, and velocity. Yaw rate denotes the vehicle’s rotation speed around its vertical axis, aiding in determining its orientation during maneuvers like cornering. Lateral acceleration captures sideways movements perpendicular to the vehicle’s direction of travel. Steering angle refers to the front wheel angle, typically in relation to the rear wheel positioning, while velocity represents the vehicle’s speed. For more details, check out this article explaining a simple bicycle model. https://dingyan89.medium.com/simple-understanding-of-kinematic-bicycle-model-81cac6420357?embedable=true Simplicity in modeling often comes with assumptions that may not hold, especially in challenging driving scenarios. Extreme conditions like rain, snow, or rough off-road terrain expose the limitations of the traditional bicycle model. The assumption of symmetry between the left and right wheels hinders the accurate analysis of lateral dynamics across all four wheels. To address this, researchers upgraded to a more comprehensive 4-wheel vehicle model. One such model, proposed in the 2023 journal article by Hui Deng, Youqun Zhao, Fen Lin, and Qiuwei Wang, boasts seven degrees of freedom. This expanded model captures all four wheels’ lateral, longitudinal, rotational, tilting, leaning, and spinning movements. However, it still falls short in capturing driving dynamics such as suspension behavior and tire-road interactions. Although suspension dynamics may be indirectly represented within the model, a separate tire model is necessary to capture tire-road interactions accurately. Deep Reinforcement Learning-Based Torque Vectoring Control Considering Economy & Safety² Tire models typically include parameters like tire force and slip angles. In the journal article by Michele Vignati, Edoardo Sabbioni, and Federico Cheli, researchers describe a simple tire model that incorporates lateral and longitudinal dynamics, slip angles, and slippage. Like the seven degrees of freedom model indirectly addresses suspension dynamics, the tire model considers slippage influenced by latent variables such as surface texture, conditions, and angles. This comprehensive approach allows researchers to analyze vehicle dynamics across various conditions and terrains, enhancing the torque vectoring system’s adaptability and performance. A Torque Vectoring Control for Enhancing Vehicle Performance in Drifting³ Control Schemes/Implementations In Torque Vectoring Systems (TVS) development, a significant portion of research revolves around employing Reinforcement Learning techniques, leveraging real-time data collected from various measurement devices. These solutions are predominantly implemented in all-wheel drive, four-wheel drive hybrid, and electric vehicles. Typically, the training of these systems takes place within simulated environments. However, in certain instances, researchers conduct road tests and utilize live test outcomes to validate simulation results. Torque Vectoring System Testing While specific details about testing TVS are often limited, researchers generally have three options: entirely computer-based simulations, hardware-in-the-loop testing combining simulation with real parts, and full-scale vehicle road tests. Computer-based simulations offer cost-effective evaluation of TVS with precise control over various driving conditions. Hardware-in-the-loop testing balances simulation and road tests, providing insights aligned with real-world performance. Although the most expensive, road tests offer the most reliable results by implementing TVS fully, enabling the most accurate evaluation of model performance. In the study by Shayan Taherian, Sampo Kuutti, Marco Visca, and Saber Fallah, computer-based simulations were utilized to test their RL algorithm across diverse environments. The researchers employed an adaptive algorithm based on Deep Deterministic Policy Gradient to update TVS control parameters as it learns from the environment. Testing across various velocity and friction environments allowed validation of their adaptive algorithm, particularly in high-speed/low-velocity scenarios. Self-adaptive Torque Vectoring Controller Using Reinforcement Learning⁴ Similarly, in the research by Wei Xu, Hong Chen, Haiyan Zhao, and Bingtao Ren, real-time Simulink simulations were leveraged to validate their model predictive controller. This controller predicts future system states and optimizes for multiple energy efficiency and safety objectives. Tests were conducted using an electric vehicle model with a 120 km/h velocity and an “adhesion coefficient” of .85, measuring tire friction between tire and ground. Torque Optimization Control for Electric Vehicles with Four In-Wheel Motors Equipped with Regenerative Braking System⁵ Powertrain & Fuel Types The specific problem addressed by torque vectoring systems heavily influences the powertrain choice and fuel types utilized in their implementations. Traditional gasoline and diesel vehicles necessitate mechanical implementations due to a single torque source, leading to complex mechanical differentials, challenges in power distribution, and overall less efficiency compared to hybrid and electric vehicle setups, which enable independent motor implementations. Similarly, All-Wheel Drive (AWD) and Four-Wheel Drive (4WD) powertrains offer superior advantages over other options. These configurations enhance stability, versatility, and energy efficiency. The work of Jeongmin Cho and Kunsoo Huh, in their study , underscores the stability and versatility benefits of AWD vehicles. By distributing power to all four wheels, AWD enhances the effectiveness of torque vectoring systems, granting increased control over each wheel. This translates to improved vehicle dynamics during cornering, acceleration, and deceleration. Torque Vectoring System Design for Hybrid Electric–All-Wheel Drive Vehicle⁶ Meta Control Algorithms In Reinforcement Learning (RL), an agent decides actions based on the current situation to maximize a reward function. It learns which actions yield the highest rewards by assessing the outcomes of its actions. In torque vectoring algorithms, auto manufacturers define different reward functions based on the specific goals of their TVS, which vary depending on the type of vehicles used, such as sports cars versus mini-vans. These goals include increased stability, controllability, fuel economy, cornering performance, and optimizing various vehicle dynamics behaviors. One example of a multi-objective implementation is outlined in the paper by Jae-Young Park, Seung-Jin Heo, and Daeoh Kang. Their algorithm features a supervisory controller that selects between an “agile” mode and a “safe” mode based on vehicle and driving conditions. Agile mode prioritizes driver controllability, while safe mode focuses on stability. Controllability refers to the synchronization between driver steering input and vehicle direction, while stability relates to the vehicle’s resistance to deviations or disturbances. Inputs such as vehicle speed, steering angle, yaw rate, lateral acceleration, and wheel slip guide the supervisory controller’s decision-making process. Development of Torque Vectoring Control Algorithm for Front Wheel Driven Dual Motor System and Evaluation of Vehicle Dynamics Performance⁷ Another example is presented in the paper by Hui Deng, Youqun Zhao, Fen Lin, and Qiuwei Wang. Here, the researchers optimize for safety, economy, and driver load. Economy involves assessing motor efficiency and energy savings compared to vehicles without torque vectoring systems. Safety focuses on vehicle stability, while driver load considers the physical and cognitive load based on the correlation between the driver’s steering wheel angle and the vehicle’s longitudinal acceleration. Deep Reinforcement Learning-Based Torque Vectoring Control Considering Economy and Safety⁸ The algorithm includes an active safety controller and a torque allocation layer. The active safety controller calculates total longitudinal acceleration and additional “yaw moment” values based on driving conditions, applying safety and driver load penalties. These values are then fed to the torque allocation layer, which distributes torque to all four wheels. Additionally, the researchers implement a tire model using a Fibonacci tree optimization algorithm and a vehicle model with 7 degrees of freedom to enhance system performance. The , research done by Park et al., aims to enhance the driver’s experience through improved smoothness and convergence. Their TVS, designed for commercial application, underwent rigorous validation through real vehicle experiments conducted under various conditions. Torque Vectoring Algorithm of Electronic-Four-Wheel Drive Vehicles for Enhancement of Cornering Performance⁹ The researchers implemented a smooth sliding controller, GPS, and sensor inputs to generate desired yaw moment values. These values are then utilized by the torque allocation algorithm, which employs a daisy-chaining allocation method to distribute torque efficiently. The goal is to optimize the convergence from current yaw moment values to target values, thereby reducing the instability during this transition. Conclusion The evolution of torque vectoring systems underscores the transformative power of innovation within the automotive industry. From pioneering beginnings to cutting-edge applications, these systems have revolutionized vehicle dynamics, speed, stability, and handling. As automotive manufacturers continue to push the boundaries of technology and leverage advancements in advanced algorithms, the future holds promise for further improvements in safety, performance, and driver experience. However, hurdles remain in integrating this technology into internal combustion engine vehicles, and ongoing research and development efforts are required to overcome them. With continued dedication to innovation, torque vectoring systems are poised to shape the future landscape of automotive manufacturing, driving us toward a new era of mobility. Research Reference Martin Dendaluce Jahnke et al., “Efficient Neural Network Implementations on Parallel Embedded Platforms Applied to Real-Time Torque-Vectoring Optimization Using Predictions for Multi-Motor Electric Vehicles,” 8, no. 2 (February 22, 2019): 250 Electronics Huifan Deng et al., “Deep Reinforcement Learning-Based Torque Vectoring Control Considering Economy and Safety,” 11, no. 4 (April 6, 2023): 459 Machines Michele Vignati, Edoardo Sabbioni, and Federico Cheli, “A Torque Vectoring Control for Enhancing Vehicle Performance in Drifting,” 7, no. 12 (December 5, 2018): 394 Electronics Shayan Taherian et al., “Self-Adaptive Torque Vectoring Controller Using Reinforcement Learning,” , September 19, 2021 2021 IEEE International Intelligent Transportation Systems Conference (ITSC) Wei Xu et al., “Torque Optimization Control for Electric Vehicles with Four In-Wheel Motors Equipped with Regenerative Braking System,” 57 (February 2019): 95–108 Mechatronics Jeongmin Cho and Kunsoo Huh, “Torque Vectoring System Design for Hybrid Electric–All Wheel Drive Vehicle,” 234, no. 10–11 (April 7, 2020): 2680–92 Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering Jae-Young Park, Seung-Jin Heo, and Dae-Oh Kang, “Development of Torque Vectoring Control Algorithm for Front Wheel Driven Dual Motor System and Evaluation of Vehicle Dynamics Performance,” 21, no. 5 (October 2020): 1283–91 International Journal of Automotive Technology Huifan Deng et al., “Deep Reinforcement Learning-Based Torque Vectoring Control Considering Economy and Safety,” 11, no. 4 (April 6, 2023): 459 Machines Giseo Park et al., “Torque Vectoring Algorithm of Electronic-Four-Wheel Drive Vehicles for Enhancement of Cornering Performance,” 69, no. 4 (April 2020): 3668–79 IEEE Transactions on Vehicular Technology

Driving Innovation: The Science of Torque Vectoring Systems

Advancing the Detection of Ultralight Vector Dark Matter through Novel Analytical Strategies

Authors:
(1) Dorian W. P. Amaral, Department of Physics and Astronomy, Rice University and These authors contributed approximately equally to this work;
(2) Mudit Jain, Department of Physics and Astronomy, Rice University, Theoretical Particle Physics and Cosmology, King’s College London and These authors contributed approximately equally to this work;
(3) Mustafa A. Amin, Department of Physics and Astronomy, Rice University;
(4) Christopher Tunnell, Department of Physics and Astronomy, Rice University. Table of Links Abstract and 1 Introduction 2 Calculating the Stochastic Wave Vector Dark Matter Signal 2.1 The Dark Photon Field 2.2 The Detector Signal 3 Statistical Analysis and 3.1 Signal Likelihood 3.2 Projected Exclusions 4 Application to Accelerometer Studies 4.1 Recasting Generalised Limits onto B − L Dark Matter 5 Future Directions 6 Conclusions, Acknowledgments, and References A Equipartition between Longitudinal and Transverse Modes B Derivation of Marginal Likelihood with Stochastic Field Amplitude C Covariance Matrix D The Case of the Gradient of a Scalar 4.1 Recasting Generalised Limits onto B − L Dark Matter To recast our limits on β shown in Fig. 4 into one on the parameter of interest for this model, the gauge coupling strength gB−L, we must define four quantities. Firstly, we must make clear what the quantity A is for this experiment, which will depend on both the model and the signal of interest for this sensor. Secondly, we must choose an observation time, Tobs. Thirdly, we must elaborate on what the concrete noise profile, σ(ω), for this type of experiment is. Lastly, and least importantly following from our discussion above, we must choose a latitude for the experiment. Once these are known, Eq. (2.8) can be re-arranged for gB−L, giving us our model- and experiment-specific 90% CL limit. For B − L coupled dark photon dark matter, the relevant signal is a differential acceleration. This is given by Eq. (2.6) in the time domain, with a now concrete choice for A, Here, gB−L is the gauge coupling strength of the model, ∆ij is the differential B −L charge per nucleon between materials i and j, given by For our observation time, we take Tobs = 10T⊕ to be firmly in the regime where the three peaks can be resolved. For light cavities, which may only be able to remain coherent over the scale of hours rather than days, such a runtime is optimistic. However, since our aim here is merely to showcase how our method can be used concretely and compare to the works of Refs. [36, 38], we do not see this as an issue. Our strategy is general and can be applied to any axial sensor and vector-like DM model, and we have settled on this configuration only for the sake of argument; other sensor technologies, such a magnetically levitated sensors, do not have this issue. Moreover, multiple cavities or data-stacking techniques can be employed to mitigate this issue. Our choices for all of the above parameters except for the laser power, which we expand on below, are summarised in Table 1. These are in keeping with the choices made in Refs. [36, 38]. The choice of where to tune the laser power is critical to give us competitive limits for a wide range of dark matter masses. We have found that we can achieve excellent limits for low dark matter masses, which are of most relevance to our work, by tuning the laser power such that the back-action and shot-noise components are minimised at low We note that this choice of power tuning is different from the strategies usually employed in other studies. Typically, the laser power is tuned so that the measurement-added noise is either minimised on resonance or, as LIGO implements it, well above resonance [88]. However, we found that both of these choices detriments the limit that we can draw at low masses, increasing the background at low frequencies beyond the thermal noise floor. For these other strategies to be beneficial at both the respective frequency targets and at low frequencies, the thermal noise would have to be significantly lower than that achieved with our choice of sensor mass, quality factor, and bath temperature. In the treatment of our backgrounds, we have neglected seismic noise, which can become important at frequencies below ∼ 10 Hz. Since we require the use of two materials to measure the differential acceleration peculiar to B − L DM, we can envision constructing two sensors: one with a moveable mirror made of material i and another of material j. By subtracting their signals, we both isolate the differential acceleration signature and remove backgrounds common to both—this includes the seismic noise component [63] We show our limits in Fig. 5 for three choices of resonance frequency: 1 Hz, 10 Hz and 100 Hz. For all but the last of these frequencies, we are able to exclude new regions of the UDLM B − L parameter space, best excluded by the fifth-force satellite experiment MICROSCOPE [89] and torsion-balance experiment E¨ot-Wash [90]. This showcases the power of such sensors in searching for this DM candidate, as was first pointed out in Ref. [63]. In a single-peak analysis focusing solely on the Compton peak, our limits would be weakened by approximately a factor of 2, as can be seen from Fig. 4. This difference becomes more dramatic for sensors located closer to the equator. We note that σ(ω), though mostly slowly varying in the frequency width ∆ω = ω⊕, does exhibit a large gradient around the resonance frequency of the cavity. In the neighbourhood of this frequency, our assumption that σ(ω) does not vary greatly in the above range is incorrect, and a more careful analysis in which all three peaks take on different noise levels would have to be conducted for more representative limits. However, we do not expect that our limits would differ greatly from our calculation and, at any rate, would still smoothly join to the regimes on either side of the resonance frequency, where our assumption holds. This paper is available on arxiv under CC BY 4.0 DEED license. Authors: (1) Dorian W. P. Amaral, Department of Physics and Astronomy, Rice University and These authors contributed approximately equally to this work; (2) Mudit Jain, Department of Physics and Astronomy, Rice University, Theoretical Particle Physics and Cosmology, King’s College London and These authors contributed approximately equally to this work; (3) Mustafa A. Amin, Department of Physics and Astronomy, Rice University; (4) Christopher Tunnell, Department of Physics and Astronomy, Rice University. Authors: Authors: (1) Dorian W. P. Amaral, Department of Physics and Astronomy, Rice University and These authors contributed approximately equally to this work; (2) Mudit Jain, Department of Physics and Astronomy, Rice University, Theoretical Particle Physics and Cosmology, King’s College London and These authors contributed approximately equally to this work; (3) Mustafa A. Amin, Department of Physics and Astronomy, Rice University; (4) Christopher Tunnell, Department of Physics and Astronomy, Rice University. Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 2 Calculating the Stochastic Wave Vector Dark Matter Signal 2.1 The Dark Photon Field 2.1 The Dark Photon Field 2.2 The Detector Signal 2.2 The Detector Signal 3 Statistical Analysis and 3.1 Signal Likelihood 3 Statistical Analysis and 3.1 Signal Likelihood 3.2 Projected Exclusions 3.2 Projected Exclusions 4 Application to Accelerometer Studies 4 Application to Accelerometer Studies 4.1 Recasting Generalised Limits onto B − L Dark Matter 4.1 Recasting Generalised Limits onto B − L Dark Matter 5 Future Directions 5 Future Directions 6 Conclusions, Acknowledgments, and References 6 Conclusions, Acknowledgments, and References A Equipartition between Longitudinal and Transverse Modes A Equipartition between Longitudinal and Transverse Modes B Derivation of Marginal Likelihood with Stochastic Field Amplitude B Derivation of Marginal Likelihood with Stochastic Field Amplitude C Covariance Matrix C Covariance Matrix D The Case of the Gradient of a Scalar D The Case of the Gradient of a Scalar 4.1 Recasting Generalised Limits onto B − L Dark Matter To recast our limits on β shown in Fig. 4 into one on the parameter of interest for this model, the gauge coupling strength gB−L, we must define four quantities. Firstly, we must make clear what the quantity A is for this experiment, which will depend on both the model and the signal of interest for this sensor. Secondly, we must choose an observation time, Tobs. Thirdly, we must elaborate on what the concrete noise profile, σ(ω), for this type of experiment is. Lastly, and least importantly following from our discussion above, we must choose a latitude for the experiment. Once these are known, Eq. (2.8) can be re-arranged for gB−L, giving us our model- and experiment-specific 90% CL limit. For B − L coupled dark photon dark matter, the relevant signal is a differential acceleration. This is given by Eq. (2.6) in the time domain, with a now concrete choice for A, Here, gB−L is the gauge coupling strength of the model, ∆ij is the differential B −L charge per nucleon between materials i and j, given by For our observation time, we take Tobs = 10T⊕ to be firmly in the regime where the three peaks can be resolved. For light cavities, which may only be able to remain coherent over the scale of hours rather than days, such a runtime is optimistic. However, since our aim here is merely to showcase how our method can be used concretely and compare to the works of Refs. [36, 38], we do not see this as an issue. Our strategy is general and can be applied to any axial sensor and vector-like DM model, and we have settled on this configuration only for the sake of argument; other sensor technologies, such a magnetically levitated sensors, do not have this issue. Moreover, multiple cavities or data-stacking techniques can be employed to mitigate this issue. Our choices for all of the above parameters except for the laser power, which we expand on below, are summarised in Table 1. These are in keeping with the choices made in Refs. [36, 38]. The choice of where to tune the laser power is critical to give us competitive limits for a wide range of dark matter masses. We have found that we can achieve excellent limits for low dark matter masses, which are of most relevance to our work, by tuning the laser power such that the back-action and shot-noise components are minimised at low We note that this choice of power tuning is different from the strategies usually employed in other studies. Typically, the laser power is tuned so that the measurement-added noise is either minimised on resonance or, as LIGO implements it, well above resonance [88]. However, we found that both of these choices detriments the limit that we can draw at low masses, increasing the background at low frequencies beyond the thermal noise floor. For these other strategies to be beneficial at both the respective frequency targets and at low frequencies, the thermal noise would have to be significantly lower than that achieved with our choice of sensor mass, quality factor, and bath temperature. In the treatment of our backgrounds, we have neglected seismic noise, which can become important at frequencies below ∼ 10 Hz. Since we require the use of two materials to measure the differential acceleration peculiar to B − L DM, we can envision constructing two sensors: one with a moveable mirror made of material i and another of material j. By subtracting their signals, we both isolate the differential acceleration signature and remove backgrounds common to both—this includes the seismic noise component [63] We show our limits in Fig. 5 for three choices of resonance frequency: 1 Hz, 10 Hz and 100 Hz. For all but the last of these frequencies, we are able to exclude new regions of the UDLM B − L parameter space, best excluded by the fifth-force satellite experiment MICROSCOPE [89] and torsion-balance experiment E¨ot-Wash [90]. This showcases the power of such sensors in searching for this DM candidate, as was first pointed out in Ref. [63]. In a single-peak analysis focusing solely on the Compton peak, our limits would be weakened by approximately a factor of 2, as can be seen from Fig. 4. This difference becomes more dramatic for sensors located closer to the equator. We note that σ(ω), though mostly slowly varying in the frequency width ∆ω = ω⊕, does exhibit a large gradient around the resonance frequency of the cavity. In the neighbourhood of this frequency, our assumption that σ(ω) does not vary greatly in the above range is incorrect, and a more careful analysis in which all three peaks take on different noise levels would have to be conducted for more representative limits. However, we do not expect that our limits would differ greatly from our calculation and, at any rate, would still smoothly join to the regimes on either side of the resonance frequency, where our assumption holds. This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv

Recasting Three-Peak Exclusion Limits on B − L Dark Matter via Accelerometer Studies

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