Joel Kamnitzer, Ben Webster, Alex Weekes, Oded Yacobi
We develop a theory of parabolic induction and restriction functors relating modules over Coulomb branch algebras, in the sense of Braverman-Finkelberg-Nakajima. Our functors generalize Bezrukavnikov-Etingof’s induction and restriction functors for Cherednik algebras, but their definition uses different tools.
After this general definition, we focus on quiver gauge theories attached to a quiver $Gamma $. The induction and restriction functors allow us to define a categorical action of the corresponding symmetric Kac-Moody algebra $mathfrak {g}_{Gamma }$ on category $ mathcal {O}$ for these Coulomb branch algebras. When $ Gamma $ is of Dynkin type, the Coulomb branch algebras are truncated shifted Yangians and quantize generalized affine Grassmannian slices. Thus, we regard our action as a categorification of the geometric Satake correspondence.
To establish this categorical action, we define a new class of ‘flavoured’ KLRW algebras, which are similar to the diagrammatic algebras originally constructed by the second author for the purpose of tensor product categorification. We prove an equivalence between the category of Gelfand-Tsetlin modules over a Coulomb branch algebra and the modules over a flavoured KLRW algebra. This equivalence relates the categorical action by induction and restriction functors to the usual categorical action on modules over a KLRW algebra.
{"title":"Lie algebra actions on module categories for truncated shifted yangians","authors":"Joel Kamnitzer, Ben Webster, Alex Weekes, Oded Yacobi","doi":"10.1017/fms.2024.3","DOIUrl":"https://doi.org/10.1017/fms.2024.3","url":null,"abstract":"<p>We develop a theory of parabolic induction and restriction functors relating modules over Coulomb branch algebras, in the sense of Braverman-Finkelberg-Nakajima. Our functors generalize Bezrukavnikov-Etingof’s induction and restriction functors for Cherednik algebras, but their definition uses different tools.</p><p>After this general definition, we focus on quiver gauge theories attached to a quiver <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240130131013942-0166:S2050509424000033:S2050509424000033_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$Gamma $</span></span></img></span></span>. The induction and restriction functors allow us to define a categorical action of the corresponding symmetric Kac-Moody algebra <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240130131013942-0166:S2050509424000033:S2050509424000033_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$mathfrak {g}_{Gamma }$</span></span></img></span></span> on category <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240130131013942-0166:S2050509424000033:S2050509424000033_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$ mathcal {O}$</span></span></img></span></span> for these Coulomb branch algebras. When <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240130131013942-0166:S2050509424000033:S2050509424000033_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$ Gamma $</span></span></img></span></span> is of Dynkin type, the Coulomb branch algebras are truncated shifted Yangians and quantize generalized affine Grassmannian slices. Thus, we regard our action as a categorification of the geometric Satake correspondence.</p><p>To establish this categorical action, we define a new class of ‘flavoured’ KLRW algebras, which are similar to the diagrammatic algebras originally constructed by the second author for the purpose of tensor product categorification. We prove an equivalence between the category of Gelfand-Tsetlin modules over a Coulomb branch algebra and the modules over a flavoured KLRW algebra. This equivalence relates the categorical action by induction and restriction functors to the usual categorical action on modules over a KLRW algebra.</p>","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":"15 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139645141","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The $Delta $-Springer varieties are a generalization of Springer fibers introduced by Levinson, Woo and the author that have connections to the Delta Conjecture from algebraic combinatorics. We prove a positive Hall–Littlewood expansion formula for the graded Frobenius characteristic of the cohomology ring of a $Delta $-Springer variety. We do this by interpreting the Frobenius characteristic in terms of counting points over a finite field $mathbb {F}_q$ and partitioning the $Delta $-Springer variety into copies of Springer fibers crossed with affine spaces. As a special case, our proof method gives a geometric meaning to a formula of Haglund, Rhoades and Shimozono for the Hall–Littlewood expansion of the symmetric function in the Delta Conjecture at $t=0$.
{"title":"Δ–Springer varieties and Hall–Littlewood polynomials","authors":"Sean T. Griffin","doi":"10.1017/fms.2024.1","DOIUrl":"https://doi.org/10.1017/fms.2024.1","url":null,"abstract":"<p>The <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240130123256385-0889:S205050942400001X:S205050942400001X_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$Delta $</span></span></img></span></span>-Springer varieties are a generalization of Springer fibers introduced by Levinson, Woo and the author that have connections to the Delta Conjecture from algebraic combinatorics. We prove a positive Hall–Littlewood expansion formula for the graded Frobenius characteristic of the cohomology ring of a <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240130123256385-0889:S205050942400001X:S205050942400001X_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$Delta $</span></span></img></span></span>-Springer variety. We do this by interpreting the Frobenius characteristic in terms of counting points over a finite field <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240130123256385-0889:S205050942400001X:S205050942400001X_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$mathbb {F}_q$</span></span></img></span></span> and partitioning the <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240130123256385-0889:S205050942400001X:S205050942400001X_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$Delta $</span></span></img></span></span>-Springer variety into copies of Springer fibers crossed with affine spaces. As a special case, our proof method gives a geometric meaning to a formula of Haglund, Rhoades and Shimozono for the Hall–Littlewood expansion of the symmetric function in the Delta Conjecture at <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240130123256385-0889:S205050942400001X:S205050942400001X_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$t=0$</span></span></img></span></span>.</p>","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":"2 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139645010","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper will study almost everywhere behaviors of functions on partition spaces of cardinals possessing suitable partition properties. Almost everywhere continuity and monotonicity properties for functions on partition spaces will be established. These results will be applied to distinguish the cardinality of certain subsets of the power set of partition cardinals. The following summarizes the main results proved under suitable partition hypotheses. <jats:list list-type="bullet"> <jats:list-item> <jats:label>•</jats:label> If <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050509423001305_inline1.png" /> <jats:tex-math> $kappa $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is a cardinal, <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050509423001305_inline2.png" /> <jats:tex-math> $epsilon < kappa $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050509423001305_inline3.png" /> <jats:tex-math> ${mathrm {cof}}(epsilon ) = omega $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050509423001305_inline4.png" /> <jats:tex-math> $kappa rightarrow _* (kappa )^{epsilon cdot epsilon }_2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050509423001305_inline5.png" /> <jats:tex-math> $Phi : [kappa ]^epsilon _* rightarrow mathrm {ON}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, then <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050509423001305_inline6.png" /> <jats:tex-math> $Phi $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> satisfies the almost everywhere short length continuity property: There is a club <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050509423001305_inline7.png" /> <jats:tex-math> $C subseteq kappa $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and a <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050509423001305_inline8.png" /> <jats:tex-math> $delta < epsilon $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> so that for all <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" x
本文将研究具有适当分区性质的红心分区空间上函数的几乎无处不在行为。本文将建立分区空间上函数的几乎无处不在的连续性和单调性。这些结果将被应用于区分分区红心的幂集的某些子集的红心性。下面总结了在合适的分割假设下证明的主要结果。 - If $kappa $ is a cardinal, $epsilon < kappa $ , ${mathrm {cof}}(epsilon ) = omega $ , $kappa rightarrow _* (kappa )^{epsilon cdot epsilon }_2$ and $Phi ：[kappa ]^epsilon _* rightarrow mathrm {ON}$，那么 $Phi $ 满足几乎无处不在的短长连续性属性：There is a club $C subseteq kappa $ and a $delta < epsilon $ so that for all $f,g in [C]^epsilon _*$ , if $f upharpoonright delta = g upharpoonright delta $ and $sup (f) = sup (g)$ , then $Phi (f) = Phi (g)$ . - 如果 $kappa $ 是红心数，$epsilon $ 是可数数，$kappa rightarrow _* (kappa )^{epsiloncdotepsilon}_2$ 成立，并且 $Phi : [kappa ]^epsilon _* rightarrow mathrm {ON}$，那么 $Phi $ 满足强的几乎无处不在的短长连续性属性：There is a club $C subseteq kappa $ and finitely many ordinals $delta _0, ..., delta _k leq epsilon $ 这样对于 [C]^epsilon _*$ 中的所有 $f,g , 如果对于所有 $0 leq i leq k$ , $sup (f upharpoonright delta _i) = sup (g upharpoonright delta _i)$ , 那么 $Phi (f) = Phi (g)$ 。 - 如果 $kappa $ 满足 $kappa rightarrow _* (kappa )^kappa _2$ , $epsilon leq kappa $ 和 $Phi : [kappa ]^epsilon _* rightarrow mathrm {ON}$，那么 $Phi $ 满足几乎无处不在的单调性属性：There is a club $C subseteq kappa $ so that for all $f,g in [C]^epsilon _*$ , if for all $alpha < epsilon $ , $f(alpha ) leq g(alpha )$ , then $Phi (f) leq Phi (g)$ . - 假设从属选择（$mathsf {DC}$ ），${omega _1}rightarrow _* ({omega _1})^{omega _1}_2$ 以及 ${omega _1}$ 的几乎无处不在的短长俱乐部均匀化原则成立。那么每个函数 $Phi : [{omega _1}]^{omega _1}_* rightarrow {omega _1}$都满足关于闭合点的有限连续性：让 $mathfrak {C}_f$ 成为 $alpha < {omega _1}$ 的俱乐部，使得 $sup (f upharpoonright alpha ) = alpha $ 。有一个俱乐部 $C subseteq {omega _1}$ 和有限多个函数 $Upsilon _0, ..., Upsilon _{n -1} ：[C]^{omega _1}_* rightarrow {omega _1}$，这样对于所有 $f in [C]^{omega _1}_*$ , 对于所有 $g in [C]^{omega _1}_*$，如果 $mathfrak {C}_g = mathfrak {C}_f$ 并且对于所有 $i <；n$ , $sup (g upharpoonright Upsilon _i(f)) = sup (f upharpoonright Upsilon _i(f))$ , 那么 $Phi (g) = Phi (f)$ 。 - 假设 $kappa $ 满足 $kappa rightarrow _* (kappa )^epsilon _2$ for all $epsilon < kappa $ .对于所有$chi < kappa $ ，$[kappa ]^{<kappa }$不会注入到${}^chi mathrm {ON}$中，也就是$chi $长度的序数序列的类中，因此，$|[kappa ]^chi|<|[kappa ]^{<kappa}|$也不会注入到${}^chi mathrm {ON}$中。因此，根据确定性公理 $(mathsf {AD})$，当 $kappa $ 是以下确定性弱或强分区红心数之一时，这两个红心数结果成立： ${omega _1}$ , $omega _2$ , $boldsymbol {delta }_n^1$ (for all $1 leq n < omega $ ) 和 $boldsymbol {delta }^2_1$ (另外假设 $mathsf {DC}_{mathbb {R}}$).
{"title":"Almost Everywhere Behavior of Functions According to Partition Measures","authors":"William Chan, Stephen Jackson, Nam Trang","doi":"10.1017/fms.2023.130","DOIUrl":"https://doi.org/10.1017/fms.2023.130","url":null,"abstract":"This paper will study almost everywhere behaviors of functions on partition spaces of cardinals possessing suitable partition properties. Almost everywhere continuity and monotonicity properties for functions on partition spaces will be established. These results will be applied to distinguish the cardinality of certain subsets of the power set of partition cardinals. The following summarizes the main results proved under suitable partition hypotheses. <jats:list list-type=\"bullet\"> <jats:list-item> <jats:label>•</jats:label> If <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509423001305_inline1.png\" /> <jats:tex-math> $kappa $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is a cardinal, <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509423001305_inline2.png\" /> <jats:tex-math> $epsilon < kappa $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509423001305_inline3.png\" /> <jats:tex-math> ${mathrm {cof}}(epsilon ) = omega $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509423001305_inline4.png\" /> <jats:tex-math> $kappa rightarrow _* (kappa )^{epsilon cdot epsilon }_2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509423001305_inline5.png\" /> <jats:tex-math> $Phi : [kappa ]^epsilon _* rightarrow mathrm {ON}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, then <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509423001305_inline6.png\" /> <jats:tex-math> $Phi $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> satisfies the almost everywhere short length continuity property: There is a club <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509423001305_inline7.png\" /> <jats:tex-math> $C subseteq kappa $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and a <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509423001305_inline8.png\" /> <jats:tex-math> $delta < epsilon $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> so that for all <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" x","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":"28 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139578359","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We characterize totally symmetric self-complementary plane partitions (TSSCPP) as bounded compatible sequences satisfying a Yamanouchi-like condition. As such, they are in bijection with certain pipe dreams. Using this characterization and the recent bijection of Gao–Huang between reduced pipe dreams and reduced bumpless pipe dreams, we give a bijection between alternating sign matrices and TSSCPP in the reduced, 1432-avoiding case. We also give a different bijection in the 1432- and 2143-avoiding case that preserves natural poset structures on the associated pipe dreams and bumpless pipe dreams.
{"title":"A Pipe Dream Perspective on Totally Symmetric Self-Complementary Plane Partitions","authors":"Daoji Huang, Jessica Striker","doi":"10.1017/fms.2023.131","DOIUrl":"https://doi.org/10.1017/fms.2023.131","url":null,"abstract":"We characterize totally symmetric self-complementary plane partitions (TSSCPP) as bounded compatible sequences satisfying a Yamanouchi-like condition. As such, they are in bijection with certain pipe dreams. Using this characterization and the recent bijection of Gao–Huang between reduced pipe dreams and reduced bumpless pipe dreams, we give a bijection between alternating sign matrices and TSSCPP in the reduced, 1432-avoiding case. We also give a different bijection in the 1432- and 2143-avoiding case that preserves natural poset structures on the associated pipe dreams and bumpless pipe dreams.","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":"25 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139578356","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove an isomorphism theorem between the canonical denotation systems for large natural numbers and large countable ordinal numbers, linking two fundamental concepts in Proof Theory. The first one is fast-growing hierarchies. These are sequences of functions on $mathbb {N}$ obtained through processes such as the ones that yield multiplication from addition, exponentiation from multiplication, etc. and represent the canonical way of speaking about large finite numbers. The second one is ordinal collapsing functions, which represent the best-known method of describing large computable ordinals. We observe that fast-growing hierarchies can be naturally extended to functors on the categories of natural numbers and of linear orders. The isomorphism theorem asserts that the categorical extensions of binary fast-growing hierarchies to ordinals are isomorphic to denotation systems given by cardinal collapsing functions. As an application of this fact, we obtain a restatement of the subsystem $Pi ^1_1$ - ${mathsf {CA_0}}$ of analysis as a higher-type well-ordering principle asserting that binary fast-growing hierarchies preserve well-foundedness.
{"title":"Functorial Fast-Growing Hierarchies","authors":"J. P. Aguilera, F. Pakhomov, A. Weiermann","doi":"10.1017/fms.2023.128","DOIUrl":"https://doi.org/10.1017/fms.2023.128","url":null,"abstract":"We prove an isomorphism theorem between the canonical denotation systems for large natural numbers and large countable ordinal numbers, linking two fundamental concepts in Proof Theory. The first one is fast-growing hierarchies. These are sequences of functions on <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509423001287_inline1.png\" /> <jats:tex-math> $mathbb {N}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> obtained through processes such as the ones that yield multiplication from addition, exponentiation from multiplication, etc. and represent the canonical way of speaking about large finite numbers. The second one is ordinal collapsing functions, which represent the best-known method of describing large computable ordinals. We observe that fast-growing hierarchies can be naturally extended to functors on the categories of natural numbers and of linear orders. The isomorphism theorem asserts that the categorical extensions of binary fast-growing hierarchies to ordinals are isomorphic to denotation systems given by cardinal collapsing functions. As an application of this fact, we obtain a restatement of the subsystem <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509423001287_inline2.png\" /> <jats:tex-math> $Pi ^1_1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-<jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509423001287_inline3.png\" /> <jats:tex-math> ${mathsf {CA_0}}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> of analysis as a higher-type well-ordering principle asserting that binary fast-growing hierarchies preserve well-foundedness.","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":"26 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139578352","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider manifold-knot pairs $(Y,K)$, where Y is a homology 3-sphere that bounds a homology 4-ball. We show that the minimum genus of a PL surface $Sigma $ in a homology ball X, such that $partial (X, Sigma ) = (Y, K)$ can be arbitrarily large. Equivalently, the minimum genus of a surface cobordism in a homology cobordism from $(Y, K)$ to any knot in $S^3$ can be arbitrarily large. The proof relies on Heegaard Floer homology.
{"title":"PL-Genus of surfaces in homology balls","authors":"Jennifer Hom, Matthew Stoffregen, Hugo Zhou","doi":"10.1017/fms.2023.126","DOIUrl":"https://doi.org/10.1017/fms.2023.126","url":null,"abstract":"<p>We consider manifold-knot pairs <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240124084648793-0779:S2050509423001263:S2050509423001263_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$(Y,K)$</span></span></img></span></span>, where <span>Y</span> is a homology 3-sphere that bounds a homology 4-ball. We show that the minimum genus of a PL surface <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240124084648793-0779:S2050509423001263:S2050509423001263_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$Sigma $</span></span></img></span></span> in a homology ball <span>X</span>, such that <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240124084648793-0779:S2050509423001263:S2050509423001263_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$partial (X, Sigma ) = (Y, K)$</span></span></img></span></span> can be arbitrarily large. Equivalently, the minimum genus of a surface cobordism in a homology cobordism from <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240124084648793-0779:S2050509423001263:S2050509423001263_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$(Y, K)$</span></span></img></span></span> to any knot in <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240124084648793-0779:S2050509423001263:S2050509423001263_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$S^3$</span></span></img></span></span> can be arbitrarily large. The proof relies on Heegaard Floer homology.</p>","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":"143 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139562239","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We establish the asymptotic expansion in $beta $ matrix models with a confining, off-critical potential in the regime where the support of the equilibrium measure is a finite union of segments. We first address the case where the filling fractions of these segments are fixed and show the existence of a $frac {1}{N}$ expansion. We then study the asymptotics of the sum over the filling fractions to obtain the full asymptotic expansion for the initial problem in the multi-cut regime. In particular, we identify the fluctuations of the linear statistics and show that they are approximated in law by the sum of a Gaussian random variable and an independent Gaussian discrete random variable with oscillating center. Fluctuations of filling fractions are also described by an oscillating discrete Gaussian random variable. We apply our results to study the all-order small dispersion asymptotics of solutions of the Toda chain associated with the one Hermitian matrix model ( $beta = 2$ ) as well as orthogonal ( $beta = 1$ ) and skew-orthogonal ( $beta = 4$ ) polynomials outside the bulk.
{"title":"Asymptotic expansion of matrix models in the multi-cut regime","authors":"Gaëtan Borot, Alice Guionnet","doi":"10.1017/fms.2023.129","DOIUrl":"https://doi.org/10.1017/fms.2023.129","url":null,"abstract":"We establish the asymptotic expansion in <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509423001299_inline2.png\" /> <jats:tex-math> $beta $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> matrix models with a confining, off-critical potential in the regime where the support of the equilibrium measure is a finite union of segments. We first address the case where the filling fractions of these segments are fixed and show the existence of a <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509423001299_inline3.png\" /> <jats:tex-math> $frac {1}{N}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> expansion. We then study the asymptotics of the sum over the filling fractions to obtain the full asymptotic expansion for the initial problem in the multi-cut regime. In particular, we identify the fluctuations of the linear statistics and show that they are approximated in law by the sum of a Gaussian random variable and an independent Gaussian discrete random variable with oscillating center. Fluctuations of filling fractions are also described by an oscillating discrete Gaussian random variable. We apply our results to study the all-order small dispersion asymptotics of solutions of the Toda chain associated with the one Hermitian matrix model (<jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509423001299_inline4.png\" /> <jats:tex-math> $beta = 2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>) as well as orthogonal (<jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509423001299_inline5.png\" /> <jats:tex-math> $beta = 1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>) and skew-orthogonal (<jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509423001299_inline6.png\" /> <jats:tex-math> $beta = 4$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>) polynomials outside the bulk.","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":"255 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139553115","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The Manin–Peyre conjecture is established for a class of smooth spherical Fano varieties of semisimple rank one. This includes all smooth spherical Fano threefolds of type T as well as some higher-dimensional smooth spherical Fano varieties.
马宁-佩雷猜想是为一类半简单阶一的光滑球面法诺变种建立的。这包括 T 型的所有光滑球面法诺三折以及一些高维光滑球面法诺变项。
{"title":"The Manin–Peyre conjecture for smooth spherical Fano varieties of semisimple rank one","authors":"Valentin Blomer, Jörg Brüdern, Ulrich Derenthal, Giuliano Gagliardi","doi":"10.1017/fms.2023.123","DOIUrl":"https://doi.org/10.1017/fms.2023.123","url":null,"abstract":"The Manin–Peyre conjecture is established for a class of smooth spherical Fano varieties of semisimple rank one. This includes all smooth spherical Fano threefolds of type <jats:italic>T</jats:italic> as well as some higher-dimensional smooth spherical Fano varieties.","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":"6 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139498885","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A set of reals is <jats:italic>universally Baire</jats:italic> if all of its continuous preimages in topological spaces have the Baire property. <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050509423001275_inline1.png" /> <jats:tex-math> $mathsf {Sealing}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is a type of generic absoluteness condition introduced by Woodin that asserts in strong terms that the theory of the universally Baire sets cannot be changed by forcing. The <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050509423001275_inline2.png" /> <jats:tex-math> $mathsf {Largest Suslin Axiom}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> (<jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050509423001275_inline3.png" /> <jats:tex-math> $mathsf {LSA}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>) is a determinacy axiom isolated by Woodin. It asserts that the largest Suslin cardinal is inaccessible for ordinal definable bijections. Let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050509423001275_inline4.png" /> <jats:tex-math> $mathsf {LSA-over-uB}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> be the statement that in all (set) generic extensions there is a model of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050509423001275_inline5.png" /> <jats:tex-math> $mathsf {LSA}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> whose Suslin, co-Suslin sets are the universally Baire sets. We show that over some mild large cardinal theory, <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050509423001275_inline6.png" /> <jats:tex-math> $mathsf {Sealing}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is equiconsistent with <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050509423001275_inline7.png" /> <jats:tex-math> $mathsf {LSA-over-uB}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. In fact, we isolate an exact large cardinal theory that is equiconsistent with both (see Definition 2.7). As a consequence, we obtain that <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050509423001275_inline8.png" /> <jats:tex-math> $mathsf {Sealing}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is weaker than the theory ‘<jats:inline-
{"title":"The exact consistency strength of the generic absoluteness for the universally Baire sets","authors":"Grigor Sargsyan, Nam Trang","doi":"10.1017/fms.2023.127","DOIUrl":"https://doi.org/10.1017/fms.2023.127","url":null,"abstract":"A set of reals is <jats:italic>universally Baire</jats:italic> if all of its continuous preimages in topological spaces have the Baire property. <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509423001275_inline1.png\" /> <jats:tex-math> $mathsf {Sealing}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is a type of generic absoluteness condition introduced by Woodin that asserts in strong terms that the theory of the universally Baire sets cannot be changed by forcing. The <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509423001275_inline2.png\" /> <jats:tex-math> $mathsf {Largest Suslin Axiom}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> (<jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509423001275_inline3.png\" /> <jats:tex-math> $mathsf {LSA}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>) is a determinacy axiom isolated by Woodin. It asserts that the largest Suslin cardinal is inaccessible for ordinal definable bijections. Let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509423001275_inline4.png\" /> <jats:tex-math> $mathsf {LSA-over-uB}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> be the statement that in all (set) generic extensions there is a model of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509423001275_inline5.png\" /> <jats:tex-math> $mathsf {LSA}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> whose Suslin, co-Suslin sets are the universally Baire sets. We show that over some mild large cardinal theory, <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509423001275_inline6.png\" /> <jats:tex-math> $mathsf {Sealing}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is equiconsistent with <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509423001275_inline7.png\" /> <jats:tex-math> $mathsf {LSA-over-uB}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. In fact, we isolate an exact large cardinal theory that is equiconsistent with both (see Definition 2.7). As a consequence, we obtain that <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509423001275_inline8.png\" /> <jats:tex-math> $mathsf {Sealing}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is weaker than the theory ‘<jats:inline-","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":"212 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139498808","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
<p>We exhibit the Hodge degeneration from nonabelian Hodge theory as a <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240112035023147-0105:S2050509423001226:S2050509423001226_inline1.png"><span data-mathjax-type="texmath"><span>$2$</span></span></img></span></span>-fold delooping of the filtered loop space <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240112035023147-0105:S2050509423001226:S2050509423001226_inline2.png"><span data-mathjax-type="texmath"><span>$E_2$</span></span></img></span></span>-groupoid in formal moduli problems. This is an iterated groupoid object which in degree <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240112035023147-0105:S2050509423001226:S2050509423001226_inline3.png"><span data-mathjax-type="texmath"><span>$1$</span></span></img></span></span> recovers the filtered circle <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240112035023147-0105:S2050509423001226:S2050509423001226_inline4.png"><span data-mathjax-type="texmath"><span>$S^1_{fil}$</span></span></img></span></span> of [MRT22]. This exploits a hitherto unstudied additional piece of structure on the topological circle, that of an <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240112035023147-0105:S2050509423001226:S2050509423001226_inline5.png"><span data-mathjax-type="texmath"><span>$E_2$</span></span></img></span></span>-cogroupoid object in the <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240112035023147-0105:S2050509423001226:S2050509423001226_inline6.png"><span data-mathjax-type="texmath"><span>$infty $</span></span></img></span></span>-category of spaces. We relate this cogroupoid structure with the more commonly studied ‘pinch map’ on <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240112035023147-0105:S2050509423001226:S2050509423001226_inline7.png"><span data-mathjax-type="texmath"><span>$S^1$</span></span></img></span></span>, as well as the Todd class of the Lie algebroid <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240112035023147-0105:S2050509423001226:S2050509423001226_inline8.png"><span data-mathjax-type="texmath"><span>$mathbb {T}_{X}$</span></span></img></span></span>; this is an invariant of a smooth and proper scheme <span>X</span> that arises, for example, in the Grothendieck-Riemann-Roch theorem. In particular, we relate the existence of nontriv
{"title":"Cogroupoid structures on the circle and the Hodge degeneration","authors":"Tasos Moulinos","doi":"10.1017/fms.2023.122","DOIUrl":"https://doi.org/10.1017/fms.2023.122","url":null,"abstract":"<p>We exhibit the Hodge degeneration from nonabelian Hodge theory as a <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240112035023147-0105:S2050509423001226:S2050509423001226_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$2$</span></span></img></span></span>-fold delooping of the filtered loop space <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240112035023147-0105:S2050509423001226:S2050509423001226_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$E_2$</span></span></img></span></span>-groupoid in formal moduli problems. This is an iterated groupoid object which in degree <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240112035023147-0105:S2050509423001226:S2050509423001226_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$1$</span></span></img></span></span> recovers the filtered circle <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240112035023147-0105:S2050509423001226:S2050509423001226_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$S^1_{fil}$</span></span></img></span></span> of [MRT22]. This exploits a hitherto unstudied additional piece of structure on the topological circle, that of an <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240112035023147-0105:S2050509423001226:S2050509423001226_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$E_2$</span></span></img></span></span>-cogroupoid object in the <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240112035023147-0105:S2050509423001226:S2050509423001226_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$infty $</span></span></img></span></span>-category of spaces. We relate this cogroupoid structure with the more commonly studied ‘pinch map’ on <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240112035023147-0105:S2050509423001226:S2050509423001226_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$S^1$</span></span></img></span></span>, as well as the Todd class of the Lie algebroid <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240112035023147-0105:S2050509423001226:S2050509423001226_inline8.png\"><span data-mathjax-type=\"texmath\"><span>$mathbb {T}_{X}$</span></span></img></span></span>; this is an invariant of a smooth and proper scheme <span>X</span> that arises, for example, in the Grothendieck-Riemann-Roch theorem. In particular, we relate the existence of nontriv","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":"46 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139469967","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
$Gamma $. The induction and restriction functors allow us to define a categorical action of the corresponding symmetric Kac-Moody algebra 
$mathfrak {g}_{Gamma }$ on category 
$ mathcal {O}$ for these Coulomb branch algebras. When 
$ Gamma $ is of Dynkin type, the Coulomb branch algebras are truncated shifted Yangians and quantize generalized affine Grassmannian slices. Thus, we regard our action as a categorification of the geometric Satake correspondence.
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