Pub Date : 2024-07-15DOI: 10.4310/cntp.2024.v18.n2.a2
Daniele Agostini, Claudia Fevola, Anna-Laura Sattelberger, Simon Telen
We study vector spaces associated to a family of generalized Euler integrals. Their dimension is given by the Euler characteristic of a very affine variety. Motivated by Feynman integrals from particle physics, this has been investigated using tools from homological algebra and the theory of $D$-modules. We present an overview and uncover new relations between these approaches. We also provide new algorithmic tools.
{"title":"Vector spaces of generalized Euler integrals","authors":"Daniele Agostini, Claudia Fevola, Anna-Laura Sattelberger, Simon Telen","doi":"10.4310/cntp.2024.v18.n2.a2","DOIUrl":"https://doi.org/10.4310/cntp.2024.v18.n2.a2","url":null,"abstract":"We study vector spaces associated to a family of generalized Euler integrals. Their dimension is given by the Euler characteristic of a very affine variety. Motivated by Feynman integrals from particle physics, this has been investigated using tools from homological algebra and the theory of $D$-modules. We present an overview and uncover new relations between these approaches. We also provide new algorithmic tools.","PeriodicalId":55616,"journal":{"name":"Communications in Number Theory and Physics","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141624633","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-15DOI: 10.4310/cntp.2024.v18.n2.a4
Jan-Willem M. van Ittersum, Giulio Ruzza
Dubrovin $href{https://doi.org/10.1007/s00023-015-0449-2}{[10]}$ has shown that the spectrum of the quantization (with respect to the first Poisson structure) of the dispersionless Korteweg–de Vries (KdV) hierarchy is given by shifted symmetric functions; the latter are related by the Bloch–Okounkov Theorem $href{https://doi.org/10.1007/JHEP07(2014)141}{[1]}$ to quasimodular forms on the full modular group. We extend the relation to quasimodular forms to the full quantum KdV hierarchy (and to the more general quantum Intermediate Long Wave hierarchy). These quantum integrable hierarchies have been defined by Buryak and Rossi $href{https://doi.org/10.1007/s11005-015-0814-6}{[6]}$ in terms of the double ramification cycle in the moduli space of curves. The main tool and conceptual contribution of the paper is a general effective criterion for quasimodularity.
{"title":"Quantum KdV hierarchy and quasimodular forms","authors":"Jan-Willem M. van Ittersum, Giulio Ruzza","doi":"10.4310/cntp.2024.v18.n2.a4","DOIUrl":"https://doi.org/10.4310/cntp.2024.v18.n2.a4","url":null,"abstract":"Dubrovin $href{https://doi.org/10.1007/s00023-015-0449-2}{[10]}$ has shown that the spectrum of the quantization (with respect to the first Poisson structure) of the dispersionless Korteweg–de Vries (KdV) hierarchy is given by shifted symmetric functions; the latter are related by the Bloch–Okounkov Theorem $href{https://doi.org/10.1007/JHEP07(2014)141}{[1]}$ to quasimodular forms on the full modular group. We extend the relation to quasimodular forms to the full quantum KdV hierarchy (and to the more general quantum Intermediate Long Wave hierarchy). These quantum integrable hierarchies have been defined by Buryak and Rossi $href{https://doi.org/10.1007/s11005-015-0814-6}{[6]}$ in terms of the double ramification cycle in the moduli space of curves. The main tool and conceptual contribution of the paper is a general effective criterion for quasimodularity.","PeriodicalId":55616,"journal":{"name":"Communications in Number Theory and Physics","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141624636","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-15DOI: 10.4310/cntp.2024.v18.n2.a3
Yuya Murakami
In this paper, we prove a conjecture by Gukov–Pei–Putrov–Vafa for a wide class of plumbed $3$-manifolds. Their conjecture states that Witten–Reshetikhin–Turaev (WRT) invariants are radial limits of homological blocks, which are $q$-series introduced by them for plumbed $3$-manifolds with negative definite linking matrices. The most difficult point in our proof is to prove the vanishing of weighted Gauss sums that appear in coefficients of negative degree in asymptotic expansions of homological blocks. To deal with it, we develop a new technique for asymptotic expansions, which enables us to compare asymptotic expansions of rational functions and false theta functions related to WRT invariants and homological blocks, respectively. In our technique, our vanishing results follow from holomorphy of such rational functions.
{"title":"Witten–Reshetikhin–Turaev invariants and homological blocks for plumbed homology spheres","authors":"Yuya Murakami","doi":"10.4310/cntp.2024.v18.n2.a3","DOIUrl":"https://doi.org/10.4310/cntp.2024.v18.n2.a3","url":null,"abstract":"In this paper, we prove a conjecture by Gukov–Pei–Putrov–Vafa for a wide class of plumbed $3$-manifolds. Their conjecture states that Witten–Reshetikhin–Turaev (WRT) invariants are radial limits of homological blocks, which are $q$-series introduced by them for plumbed $3$-manifolds with negative definite linking matrices. The most difficult point in our proof is to prove the vanishing of weighted Gauss sums that appear in coefficients of negative degree in asymptotic expansions of homological blocks. To deal with it, we develop a new technique for asymptotic expansions, which enables us to compare asymptotic expansions of rational functions and false theta functions related to WRT invariants and homological blocks, respectively. In our technique, our vanishing results follow from holomorphy of such rational functions.","PeriodicalId":55616,"journal":{"name":"Communications in Number Theory and Physics","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141624634","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-15DOI: 10.4310/cntp.2024.v18.n2.a5
Daniel Bump, Slava Naprienko
We develop the theory of colored bosonic models (initiated by Borodin and Wheeler). We will show how a family of such models can be used to represent the values of Iwahori vectors in the “spherical model” of representations of $mathrm{GL}_r (F)$, where $F$ is a nonarchimedean local field. Among our results are a monochrome factorization, which is the realization of the Boltzmann weights by fusion of simpler weights, a local lifting property relating the colored models with uncolored models, and an action of the Iwahori–Hecke algebra on the partition functions of a particular family of models by Demazure–Lusztig operators. As an application of the local lifting property we reprove a theorem of Korff evaluating the partition functions of the uncolored models in terms of Hall–Littlewood polynomials. Our results are very closely parallel to the theory of fermionic models representing Iwahori–Whittaker functions developed by Brubaker, Buciumas, Bump and Gustafsson, with many striking relationships between the two theories, confirming the philosophy that the spherical and Whittaker models of principal series representations are dual.
{"title":"Colored Bosonic models and matrix coefficients","authors":"Daniel Bump, Slava Naprienko","doi":"10.4310/cntp.2024.v18.n2.a5","DOIUrl":"https://doi.org/10.4310/cntp.2024.v18.n2.a5","url":null,"abstract":"We develop the theory of colored bosonic models (initiated by Borodin and Wheeler). We will show how a family of such models can be used to represent the values of Iwahori vectors in the “spherical model” of representations of $mathrm{GL}_r (F)$, where $F$ is a nonarchimedean local field. Among our results are a <i>monochrome factorization</i>, which is the realization of the Boltzmann weights by fusion of simpler weights, a <i>local lifting</i> property relating the colored models with uncolored models, and an action of the Iwahori–Hecke algebra on the partition functions of a particular family of models by Demazure–Lusztig operators. As an application of the local lifting property we reprove a theorem of Korff evaluating the partition functions of the uncolored models in terms of Hall–Littlewood polynomials. Our results are very closely parallel to the theory of fermionic models representing Iwahori–Whittaker functions developed by Brubaker, Buciumas, Bump and Gustafsson, with many striking relationships between the two theories, confirming the philosophy that the spherical and Whittaker models of principal series representations are dual.","PeriodicalId":55616,"journal":{"name":"Communications in Number Theory and Physics","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141624637","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-15DOI: 10.4310/cntp.2024.v18.n2.a1
Matija Tapušković
In the first part of this paper we study the coaction dual to the action of the cosmic Galois group on the motivic lift of the sunrise Feynman integral with generic masses and momenta, and we express its conjugates in terms of motivic lifts of Feynman integrals associated to related Feynman graphs. Only one of the conjugates of the motivic lift of the sunrise, other than itself, can be expressed in terms of motivic lifts of Feynman integrals of subquotient graphs. To relate the remaining conjugates to Feynman integrals we introduce a general tool: subdividing edges of a graph. We show that all motivic lifts of Feynman integrals associated to graphs obtained by subdividing edges from a graph $G$ are motivic periods of $G$ itself. This was conjectured by Brown in the case of graphs with no kinematic dependence. We also look at the single-valued periods associated to the functions on the motivic Galois group, i.e. the ‘de Rham periods’, which appear in the coaction on the sunrise, and show that they are generalisations of Brown’s non-holomorphic modular forms with two weights. In the second part of the paper we consider the relative completion of the torsor of paths on a modular curve and its periods, the theory of which is due to Brown and Hain. Brown studied the motivic periods of the relative completion of $mathcal{M}_{1,1}$ with respect to the tangential basepoint at infinity, and we generalise this to the case of the torsor of paths on any modular curve. We apply this to reprove the claim that the sunrise Feynman integral in the equal-mass case can be expressed in terms of Eichler integrals, periods of the underlying elliptic curve defined by one of the associated graph hypersurfaces, and powers of $2pi i$.
{"title":"The cosmic Galois group, the sunrise Feynman integral, and the relative completion of $Gamma^1(6)$","authors":"Matija Tapušković","doi":"10.4310/cntp.2024.v18.n2.a1","DOIUrl":"https://doi.org/10.4310/cntp.2024.v18.n2.a1","url":null,"abstract":"In the first part of this paper we study the coaction dual to the action of the cosmic Galois group on the motivic lift of the sunrise Feynman integral with generic masses and momenta, and we express its conjugates in terms of motivic lifts of Feynman integrals associated to related Feynman graphs. Only one of the conjugates of the motivic lift of the sunrise, other than itself, can be expressed in terms of motivic lifts of Feynman integrals of subquotient graphs. To relate the remaining conjugates to Feynman integrals we introduce a general tool: subdividing edges of a graph. We show that all motivic lifts of Feynman integrals associated to graphs obtained by subdividing edges from a graph $G$ are motivic periods of $G$ itself. This was conjectured by Brown in the case of graphs with no kinematic dependence. We also look at the single-valued periods associated to the functions on the motivic Galois group, i.e. the ‘de Rham periods’, which appear in the coaction on the sunrise, and show that they are generalisations of Brown’s non-holomorphic modular forms with two weights. In the second part of the paper we consider the relative completion of the torsor of paths on a modular curve and its periods, the theory of which is due to Brown and Hain. Brown studied the motivic periods of the relative completion of $mathcal{M}_{1,1}$ with respect to the tangential basepoint at infinity, and we generalise this to the case of the torsor of paths on any modular curve. We apply this to reprove the claim that the sunrise Feynman integral in the equal-mass case can be expressed in terms of Eichler integrals, periods of the underlying elliptic curve defined by one of the associated graph hypersurfaces, and powers of $2pi i$.","PeriodicalId":55616,"journal":{"name":"Communications in Number Theory and Physics","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141624632","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-06-07DOI: 10.4310/cntp.2024.v18.n1.a4
Asghar Ghorbanpour, Masoud Khalkhali
We introduce a new family of metrics, called functional metrics, on noncommutative tori and study their spectral geometry. We define a class of Laplace type operators for these metrics and study their spectral invariants obtained from the heat trace asymptotics. A formula for the second density of the heat trace is obtained. In particular, the scalar curvature density and the total scalar curvature of functional metrics are explicitly computed in all dimensions for certain classes of metrics including conformally flat metrics and twisted product of flat metrics. Finally a Gauss-Bonnet type theorem for a noncommutative two torus equipped with a general functional metric is proved.
{"title":"Spectral geometry of functional metrics on noncommutative tori","authors":"Asghar Ghorbanpour, Masoud Khalkhali","doi":"10.4310/cntp.2024.v18.n1.a4","DOIUrl":"https://doi.org/10.4310/cntp.2024.v18.n1.a4","url":null,"abstract":"We introduce a new family of metrics, called functional metrics, on noncommutative tori and study their spectral geometry. We define a class of Laplace type operators for these metrics and study their spectral invariants obtained from the heat trace asymptotics. A formula for the second density of the heat trace is obtained. In particular, the scalar curvature density and the total scalar curvature of functional metrics are explicitly computed in all dimensions for certain classes of metrics including conformally flat metrics and twisted product of flat metrics. Finally a Gauss-Bonnet type theorem for a noncommutative two torus equipped with a general functional metric is proved.","PeriodicalId":55616,"journal":{"name":"Communications in Number Theory and Physics","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2024-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141292708","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-06-07DOI: 10.4310/cntp.2024.v18.n1.a1
Younes Nikdelan
$defM{mathscr{M}}defRscr{mathscr{R}}defRsf{mathsf{R}}defTsf{mathsf{T}}deftildeM{widetilde{M}}$For any positive integer $n$, we introduce a modular vector field $Rsf$ on a moduli space $Tsf$ of enhanced Calabi–Yau $n$-folds arising from the Dwork family. By Calabi–Yau quasi-modular forms associated to $Rsf$ we mean the elements of the graded $mathbb{C}$-algebra $tildeM$ generated by solutions of $Rsf$, which are provided with natural weights. The modular vector field $Rsf$ induces the derivation $Rscr$ and the Ramanujan–Serre type derivation $partial$ on $tildeM$. We show that they are degree $2$ differential operators and there exists a proper subspace $M subset tildeM$, called the space of Calabi–Yau modular forms associated to $Rsf$, which is closed under $partial$. Using the derivation $Rscr$, we define the Rankin–Cohen brackets for $tildeM$ and prove that the subspace generated by the positive weight elements of $M$ is closed under the Rankin–Cohen brackets. We find the mirror map of the Dwork family in terms of the Calabi–Yau modular forms.
{"title":"Rankin–Cohen brackets for Calabi–Yau modular forms","authors":"Younes Nikdelan","doi":"10.4310/cntp.2024.v18.n1.a1","DOIUrl":"https://doi.org/10.4310/cntp.2024.v18.n1.a1","url":null,"abstract":"$defM{mathscr{M}}defRscr{mathscr{R}}defRsf{mathsf{R}}defTsf{mathsf{T}}deftildeM{widetilde{M}}$For any positive integer $n$, we introduce a modular vector field $Rsf$ on a moduli space $Tsf$ of enhanced Calabi–Yau $n$-folds arising from the Dwork family. By Calabi–Yau quasi-modular forms associated to $Rsf$ we mean the elements of the graded $mathbb{C}$-algebra $tildeM$ generated by solutions of $Rsf$, which are provided with natural weights. The modular vector field $Rsf$ induces the derivation $Rscr$ and the Ramanujan–Serre type derivation $partial$ on $tildeM$. We show that they are degree $2$ differential operators and there exists a proper subspace $M subset tildeM$, called the space of Calabi–Yau modular forms associated to $Rsf$, which is closed under $partial$. Using the derivation $Rscr$, we define the Rankin–Cohen brackets for $tildeM$ and prove that the subspace generated by the positive weight elements of $M$ is closed under the Rankin–Cohen brackets. We find the mirror map of the Dwork family in terms of the Calabi–Yau modular forms.","PeriodicalId":55616,"journal":{"name":"Communications in Number Theory and Physics","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2024-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141292722","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-06-07DOI: 10.4310/cntp.2024.v18.n1.a2
Sergei Alexandrov, Soheyla Feyzbakhsh, Albrecht Klemm, Boris Pioline, Thorsten Schimannek
related to Gopakumar-Vafa (GV) invariants, and rank 0 Donaldson-Thomas (DT) invariants countingD4-D2-D0 BPS bound states, we rigorously compute the first few terms in the generating series of Abelian D4-D2-D0 indices for compact one-parameter Calabi-Yau threefolds of hypergeometric type. In all cases where GV invariants can be computed to sufficiently high genus, we find striking confirmation that the generating series is modular, and predict infinite series of Abelian D4-D2-D0 indices. Conversely, we use these results to provide new constraints for the direct integration method, which allows to compute GV invariants (and therefore the topological string partition function) to higher genus than hitherto possible. The triangle of relations between GV/PT/DT invariants is powered by a new explicit formula relating PT and rank 0 DT invariants, which is proven in an Appendix by the second named author. As a corollary, we obtain rigorous Castelnuovo-type bounds for PT and GV invariants for CY threefolds with Picard rank one.
{"title":"Quantum geometry, stability and modularity","authors":"Sergei Alexandrov, Soheyla Feyzbakhsh, Albrecht Klemm, Boris Pioline, Thorsten Schimannek","doi":"10.4310/cntp.2024.v18.n1.a2","DOIUrl":"https://doi.org/10.4310/cntp.2024.v18.n1.a2","url":null,"abstract":"related to Gopakumar-Vafa (GV) invariants, and rank 0 Donaldson-Thomas (DT) invariants countingD4-D2-D0 BPS bound states, we rigorously compute the first few terms in the generating series of Abelian D4-D2-D0 indices for compact one-parameter Calabi-Yau threefolds of hypergeometric type. In all cases where GV invariants can be computed to sufficiently high genus, we find striking confirmation that the generating series is modular, and predict infinite series of Abelian D4-D2-D0 indices. Conversely, we use these results to provide new constraints for the direct integration method, which allows to compute GV invariants (and therefore the topological string partition function) to higher genus than hitherto possible. The triangle of relations between GV/PT/DT invariants is powered by a new explicit formula relating PT and rank 0 DT invariants, which is proven in an Appendix by the second named author. As a corollary, we obtain rigorous Castelnuovo-type bounds for PT and GV invariants for CY threefolds with Picard rank one.","PeriodicalId":55616,"journal":{"name":"Communications in Number Theory and Physics","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2024-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141292673","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-06-07DOI: 10.4310/cntp.2024.v18.n1.a3
Felipe Espreafico, Johannes Walcher
In recent years, a version of enumerative geometry over arbitrary fields has been developed and studied by Kass-Wickelgren, Levine, and others, in which the counts obtained are not integers but quadratic forms. Aiming to understand the relation to other “refined invariants”, and especially their possible interpretation in quantum theory, we explain how to obtain a quadratic version of Donaldson-Thomas invariants from the motivic invariants defined in the work of Kontsevich and Soibelman and pose some questions. We calculate these invariants in a few simple examples that provide standard tests for these questions, including degree zero invariants of A3 and higher-genus Gopakumar-Vafa invariants recently studied by Liu and Ruan. The comparison with known real and complex counts plays a central role throughout.
{"title":"On motivic and arithmetic refinements of Donaldson-Thomas invariants","authors":"Felipe Espreafico, Johannes Walcher","doi":"10.4310/cntp.2024.v18.n1.a3","DOIUrl":"https://doi.org/10.4310/cntp.2024.v18.n1.a3","url":null,"abstract":"In recent years, a version of enumerative geometry over arbitrary fields has been developed and studied by Kass-Wickelgren, Levine, and others, in which the counts obtained are not integers but quadratic forms. Aiming to understand the relation to other “refined invariants”, and especially their possible interpretation in quantum theory, we explain how to obtain a quadratic version of Donaldson-Thomas invariants from the motivic invariants defined in the work of Kontsevich and Soibelman and pose some questions. We calculate these invariants in a few simple examples that provide standard tests for these questions, including degree zero invariants of A3 and higher-genus Gopakumar-Vafa invariants recently studied by Liu and Ruan. The comparison with known real and complex counts plays a central role throughout.","PeriodicalId":55616,"journal":{"name":"Communications in Number Theory and Physics","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2024-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141292704","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-06-07DOI: 10.4310/cntp.2024.v18.n1.a5
Louisa Liles, Eleanor McSpirit
One of the first key examples of a quantum modular form, which unifies the Witten-Reshetikhin-Turaev (WRT) invariants of the Poincaré homology sphere, appears in work of Lawrence and Zagier. We show that the series they construct is one instance in an infinite family of quantum modular invariants of negative definite plumbed 3‑manifolds whose radial limits toward roots of unity may be thought of as a deformation of the WRT invariants. We use a recently developed theory of Akhmechet, Johnson, and Krushkal (AJK) which extends lattice cohomology and BPS $q$‑series of 3‑manifolds. As part of this work, we provide the first calculation of the AJK series for an infinite family of 3‑manifolds. Additionally, we introduce a separate but related infinite family of invariants which also exhibit quantum modularity properties.
{"title":"Infinite families of quantum modular 3-manifold invariants","authors":"Louisa Liles, Eleanor McSpirit","doi":"10.4310/cntp.2024.v18.n1.a5","DOIUrl":"https://doi.org/10.4310/cntp.2024.v18.n1.a5","url":null,"abstract":"One of the first key examples of a quantum modular form, which unifies the Witten-Reshetikhin-Turaev (WRT) invariants of the Poincaré homology sphere, appears in work of Lawrence and Zagier. We show that the series they construct is one instance in an infinite family of quantum modular invariants of negative definite plumbed 3‑manifolds whose radial limits toward roots of unity may be thought of as a deformation of the WRT invariants. We use a recently developed theory of Akhmechet, Johnson, and Krushkal (AJK) which extends lattice cohomology and BPS $q$‑series of 3‑manifolds. As part of this work, we provide the first calculation of the AJK series for an infinite family of 3‑manifolds. Additionally, we introduce a separate but related infinite family of invariants which also exhibit quantum modularity properties.","PeriodicalId":55616,"journal":{"name":"Communications in Number Theory and Physics","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2024-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141292691","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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