Pub Date : 2016-12-15DOI: 10.4310/cntp.2019.v13.n3.a2
Michael E. Hoffman
For positive integers $i_1,...,i_k$ with $i_1 > 1$, we define the multiple $t$-value $t(i_1,...,i_k)$ as the sum of those terms in the usual infinite series for the multiple zeta value $zeta(i_1,...,i_k)$ with odd denominators. Like the multiple zeta values, the multiple $t$-values can be multiplied according to the rules of the harmonic algebra. Using this fact, we obtain explicit formulas for multiple $t$-values of repeated arguments analogous to those known for multiple zeta values. Multiple $t$-values can be written as rational linear combinations of the alternating or "colored" multiple zeta values. Using known results for colored multiple zeta values, we obtain tables of multiple $t$-values through weight 7, suggesting some interesting conjectures, including one that the dimension of the rational vector space generated by weight-$n$ multiple $t$-values has dimension equal to the $n$th Fibonacci number. We express the generating function of the height one multiple $t$-values $t(n,1,...,1)$ in terms of a generalized hypergeometric function. We also define alternating multiple $t$-values and prove some results about them.
{"title":"An odd variant of multiple zeta values","authors":"Michael E. Hoffman","doi":"10.4310/cntp.2019.v13.n3.a2","DOIUrl":"https://doi.org/10.4310/cntp.2019.v13.n3.a2","url":null,"abstract":"For positive integers $i_1,...,i_k$ with $i_1 > 1$, we define the multiple $t$-value $t(i_1,...,i_k)$ as the sum of those terms in the usual infinite series for the multiple zeta value $zeta(i_1,...,i_k)$ with odd denominators. Like the multiple zeta values, the multiple $t$-values can be multiplied according to the rules of the harmonic algebra. Using this fact, we obtain explicit formulas for multiple $t$-values of repeated arguments analogous to those known for multiple zeta values. Multiple $t$-values can be written as rational linear combinations of the alternating or \"colored\" multiple zeta values. Using known results for colored multiple zeta values, we obtain tables of multiple $t$-values through weight 7, suggesting some interesting conjectures, including one that the dimension of the rational vector space generated by weight-$n$ multiple $t$-values has dimension equal to the $n$th Fibonacci number. We express the generating function of the height one multiple $t$-values $t(n,1,...,1)$ in terms of a generalized hypergeometric function. We also define alternating multiple $t$-values and prove some results about them.","PeriodicalId":55616,"journal":{"name":"Communications in Number Theory and Physics","volume":"1 1","pages":""},"PeriodicalIF":1.9,"publicationDate":"2016-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70422911","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 : 2016-12-07DOI: 10.4310/CNTP.2018.V12.N4.A2
Matthew Krauel, Christopher Marks
Using the language of vertex operator algebras (VOAs) and vector-valued modular forms we study the modular group representations and spaces of 1-point functions associated to intertwining operators for Virasoro minimal model VOAs. We examine all representations of dimension less than four associated to irreducible modules for minimal models, and determine when the kernel of these representations is a congruence or noncongruence subgroup of the modular group. Arithmetic criteria are given on the indexing of the irreducible modules for minimal models that imply the associated modular group representation has a noncongruence kernel, independent of the dimension of the representation. The algebraic structure of the spaces of 1-point functions for intertwining operators is also studied, via a comparison with the associated spaces of holomorphic vector-valued modular forms.
{"title":"Intertwining operators and vector-valued modular forms for minimal models","authors":"Matthew Krauel, Christopher Marks","doi":"10.4310/CNTP.2018.V12.N4.A2","DOIUrl":"https://doi.org/10.4310/CNTP.2018.V12.N4.A2","url":null,"abstract":"Using the language of vertex operator algebras (VOAs) and vector-valued modular forms we study the modular group representations and spaces of 1-point functions associated to intertwining operators for Virasoro minimal model VOAs. We examine all representations of dimension less than four associated to irreducible modules for minimal models, and determine when the kernel of these representations is a congruence or noncongruence subgroup of the modular group. Arithmetic criteria are given on the indexing of the irreducible modules for minimal models that imply the associated modular group representation has a noncongruence kernel, independent of the dimension of the representation. The algebraic structure of the spaces of 1-point functions for intertwining operators is also studied, via a comparison with the associated spaces of holomorphic vector-valued modular forms.","PeriodicalId":55616,"journal":{"name":"Communications in Number Theory and Physics","volume":"12 1","pages":"657-686"},"PeriodicalIF":1.9,"publicationDate":"2016-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70423335","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 : 2016-11-29DOI: 10.4310/CNTP.2020.V14.N3.A3
V. Przyjalkowski, C. Shramov
We prove that if a smooth variety with non-positive canonical class can be embedded into a weighted projective space of dimension $n$ as a well formed complete intersection and it is not an intersection with a linear cone therein, then the weights of the weighted projective space do not exceed $n+1$. Based on this bound we classify all smooth Fano complete intersections of dimensions $4$ and $5$, and compute their invariants.
{"title":"Bounds for smooth Fano weighted complete intersections","authors":"V. Przyjalkowski, C. Shramov","doi":"10.4310/CNTP.2020.V14.N3.A3","DOIUrl":"https://doi.org/10.4310/CNTP.2020.V14.N3.A3","url":null,"abstract":"We prove that if a smooth variety with non-positive canonical class can be embedded into a weighted projective space of dimension $n$ as a well formed complete intersection and it is not an intersection with a linear cone therein, then the weights of the weighted projective space do not exceed $n+1$. Based on this bound we classify all smooth Fano complete intersections of dimensions $4$ and $5$, and compute their invariants.","PeriodicalId":55616,"journal":{"name":"Communications in Number Theory and Physics","volume":"1 1","pages":""},"PeriodicalIF":1.9,"publicationDate":"2016-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70422980","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 : 2016-07-01DOI: 10.4310/cntp.2020.v14.n1.a1
Imma G'alvez-Carrillo, R. Kaufmann, A. Tonks
We consider three a priori totally different setups for Hopf algebras from number theory, mathematical physics and algebraic topology. These are the Hopf algebra of Goncharov for multiple zeta values, that of Connes-Kreimer for renormalization, and a Hopf algebra constructed by Baues to study double loop spaces. We show that these examples can be successively unified by considering simplicial objects, co-operads with multiplication and Feynman categories at the ultimate level. These considerations open the door to new constructions and reinterpretations of known constructions in a large common framework, which is presented step-by-step with examples throughout. In this first part of two papers, we concentrate on the simplicial and operadic aspects.
{"title":"Three Hopf algebras from number theory, physics & topology, and their common background I: operadic & simplicial aspects","authors":"Imma G'alvez-Carrillo, R. Kaufmann, A. Tonks","doi":"10.4310/cntp.2020.v14.n1.a1","DOIUrl":"https://doi.org/10.4310/cntp.2020.v14.n1.a1","url":null,"abstract":"We consider three a priori totally different setups for Hopf algebras from number theory, mathematical physics and algebraic topology. These are the Hopf algebra of Goncharov for multiple zeta values, that of Connes-Kreimer for renormalization, and a Hopf algebra constructed by Baues to study double loop spaces. We show that these examples can be successively unified by considering simplicial objects, co-operads with multiplication and Feynman categories at the ultimate level. These considerations open the door to new constructions and reinterpretations of known constructions in a large common framework, which is presented step-by-step with examples throughout. In this first part of two papers, we concentrate on the simplicial and operadic aspects.","PeriodicalId":55616,"journal":{"name":"Communications in Number Theory and Physics","volume":"1 1","pages":""},"PeriodicalIF":1.9,"publicationDate":"2016-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70422959","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 : 2016-05-15DOI: 10.4310/cntp.2020.v14.n3.a5
Zhan Li
We show that general Clifford double mirrors constructed in "On Clifford double mirrors of toric complete intersections" are derived equivalent.
在“关于环面完全交点的Clifford双镜”中构造的一般Clifford双镜是等价的。
{"title":"On derived equivalence of general Clifford double mirrors","authors":"Zhan Li","doi":"10.4310/cntp.2020.v14.n3.a5","DOIUrl":"https://doi.org/10.4310/cntp.2020.v14.n3.a5","url":null,"abstract":"We show that general Clifford double mirrors constructed in \"On Clifford double mirrors of toric complete intersections\" are derived equivalent.","PeriodicalId":55616,"journal":{"name":"Communications in Number Theory and Physics","volume":"1 1","pages":""},"PeriodicalIF":1.9,"publicationDate":"2016-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70422989","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 : 2016-04-08DOI: 10.4310/CNTP.2019.V13.N1.A4
Ben Brubaker, Valentin Buciumas, D. Bump
We will give new applications of quantum groups to the study of spherical Whittaker functions on the metaplectic $n$-fold cover of $GL(r,F)$, where $F$ is a nonarchimedean local field. Earlier Brubaker, Bump, Friedberg, Chinta and Gunnells had shown that these Whittaker functions can be identified with the partition functions of statistical mechanical systems. They postulated that a Yang-Baxter equation underlies the properties of these Whittaker functions. We confirm this, and identify the corresponding Yang-Baxter equation with that of the quantum affine Lie superalgebra $U_{sqrt{v}}(widehat{mathfrak{gl}}(1|n))$, modified by Drinfeld twisting to introduce Gauss sums. (The deformation parameter $v$ is specialized to the inverse of the residue field cardinality.) For principal series representations of metaplectic groups, the Whittaker models are not unique. The scattering matrix for the standard intertwining operators is vector valued. For a simple reflection, it was computed by Kazhdan and Patterson, who applied it to generalized theta series. We will show that the scattering matrix on the space of Whittaker functions for a simple reflection coincides with the twisted $R$-matrix of the quantum group $U_{sqrt{v}}(widehat{mathfrak{gl}}(n))$. This is a piece of the twisted $R$-matrix for $U_{sqrt{v}}(widehat{mathfrak{gl}}(1|n))$, mentioned above.
{"title":"A Yang–Baxter equation for metaplectic ice","authors":"Ben Brubaker, Valentin Buciumas, D. Bump","doi":"10.4310/CNTP.2019.V13.N1.A4","DOIUrl":"https://doi.org/10.4310/CNTP.2019.V13.N1.A4","url":null,"abstract":"We will give new applications of quantum groups to the study of spherical Whittaker functions on the metaplectic $n$-fold cover of $GL(r,F)$, where $F$ is a nonarchimedean local field. Earlier Brubaker, Bump, Friedberg, Chinta and Gunnells had shown that these Whittaker functions can be identified with the partition functions of statistical mechanical systems. They postulated that a Yang-Baxter equation underlies the properties of these Whittaker functions. We confirm this, and identify the corresponding Yang-Baxter equation with that of the quantum affine Lie superalgebra $U_{sqrt{v}}(widehat{mathfrak{gl}}(1|n))$, modified by Drinfeld twisting to introduce Gauss sums. (The deformation parameter $v$ is specialized to the inverse of the residue field cardinality.) \u0000For principal series representations of metaplectic groups, the Whittaker models are not unique. The scattering matrix for the standard intertwining operators is vector valued. For a simple reflection, it was computed by Kazhdan and Patterson, who applied it to generalized theta series. We will show that the scattering matrix on the space of Whittaker functions for a simple reflection coincides with the twisted $R$-matrix of the quantum group $U_{sqrt{v}}(widehat{mathfrak{gl}}(n))$. This is a piece of the twisted $R$-matrix for $U_{sqrt{v}}(widehat{mathfrak{gl}}(1|n))$, mentioned above.","PeriodicalId":55616,"journal":{"name":"Communications in Number Theory and Physics","volume":"1 1","pages":""},"PeriodicalIF":1.9,"publicationDate":"2016-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70423342","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 : 2016-01-31DOI: 10.4310/CNTP.2017.v11.n2.a4
I. Florakis, B. Pioline
Closed string amplitudes at genus $hleq 3$ are given by integrals of Siegel modular functions on a fundamental domain of the Siegel upper half-plane. When the integrand is of rapid decay near the cusps, the integral can be computed by the Rankin-Selberg method, which consists of inserting an Eisenstein series $E_h(s)$ in the integrand, computing the integral by the orbit method, and finally extracting the residue at a suitable value of $s$. String amplitudes, however, typically involve integrands with polynomial or even exponential growth at the cusps, and a renormalization scheme is required to treat infrared divergences. Generalizing Zagier's extension of the Rankin-Selberg method at genus one, we develop the Rankin-Selberg method for Siegel modular functions of degree 2 and 3 with polynomial growth near the cusps. In particular, we show that the renormalized modular integral of the Siegel-Narain partition function of an even self-dual lattice of signature $(d,d)$ is proportional to a residue of the Langlands-Eisenstein series attached to the $h$-th antisymmetric tensor representation of the T-duality group $O(d,d,Z)$.
{"title":"On the Rankin-Selberg method for higher genus string amplitudes","authors":"I. Florakis, B. Pioline","doi":"10.4310/CNTP.2017.v11.n2.a4","DOIUrl":"https://doi.org/10.4310/CNTP.2017.v11.n2.a4","url":null,"abstract":"Closed string amplitudes at genus $hleq 3$ are given by integrals of Siegel modular functions on a fundamental domain of the Siegel upper half-plane. When the integrand is of rapid decay near the cusps, the integral can be computed by the Rankin-Selberg method, which consists of inserting an Eisenstein series $E_h(s)$ in the integrand, computing the integral by the orbit method, and finally extracting the residue at a suitable value of $s$. String amplitudes, however, typically involve integrands with polynomial or even exponential growth at the cusps, and a renormalization scheme is required to treat infrared divergences. Generalizing Zagier's extension of the Rankin-Selberg method at genus one, we develop the Rankin-Selberg method for Siegel modular functions of degree 2 and 3 with polynomial growth near the cusps. In particular, we show that the renormalized modular integral of the Siegel-Narain partition function of an even self-dual lattice of signature $(d,d)$ is proportional to a residue of the Langlands-Eisenstein series attached to the $h$-th antisymmetric tensor representation of the T-duality group $O(d,d,Z)$.","PeriodicalId":55616,"journal":{"name":"Communications in Number Theory and Physics","volume":"11 1","pages":"337-404"},"PeriodicalIF":1.9,"publicationDate":"2016-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70423276","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 : 2016-01-01DOI: 10.4310/CNTP.2016.V10.N4.A3
Peter Overholser
{"title":"A descendent tropical Landau–Ginzburg potential for $mathbb{P}^2$","authors":"Peter Overholser","doi":"10.4310/CNTP.2016.V10.N4.A3","DOIUrl":"https://doi.org/10.4310/CNTP.2016.V10.N4.A3","url":null,"abstract":"","PeriodicalId":55616,"journal":{"name":"Communications in Number Theory and Physics","volume":"10 1","pages":"739-803"},"PeriodicalIF":1.9,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70423214","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 : 2015-01-01DOI: 10.4310/CNTP.2015.V9.N1.A2
G. Borot, B. Eynard, N. Orantin
loop equations, topological recursion and applications Gaetan Borot, Bertrand Eynard, Nicolas Orantin Abstract We formulate a notion of ”abstract loop equations”, and show that their solution is provided by a topological recursion under some assumptions, in particular the result takes a universal form. The Schwinger-Dyson equation of the one and two hermitian matrix models, and of the Opnq model appear as special cases. We study applications to repulsive particles systems, and explain how our notion of loop equations are related to Virasoro constraints. Then, as a special case, we study in detail applications to enumeration problems in a general class of non-intersecting loop models on the random lattice of all topologies, to SUpNq Chern-Simons invariants of torus knots in the large N expansion. We also mention an application to Liouville theory on surfaces of positive genus.We formulate a notion of ”abstract loop equations”, and show that their solution is provided by a topological recursion under some assumptions, in particular the result takes a universal form. The Schwinger-Dyson equation of the one and two hermitian matrix models, and of the Opnq model appear as special cases. We study applications to repulsive particles systems, and explain how our notion of loop equations are related to Virasoro constraints. Then, as a special case, we study in detail applications to enumeration problems in a general class of non-intersecting loop models on the random lattice of all topologies, to SUpNq Chern-Simons invariants of torus knots in the large N expansion. We also mention an application to Liouville theory on surfaces of positive genus.
{"title":"Abstract loop equations, topological recursion and new applications","authors":"G. Borot, B. Eynard, N. Orantin","doi":"10.4310/CNTP.2015.V9.N1.A2","DOIUrl":"https://doi.org/10.4310/CNTP.2015.V9.N1.A2","url":null,"abstract":"loop equations, topological recursion and applications Gaetan Borot, Bertrand Eynard, Nicolas Orantin Abstract We formulate a notion of ”abstract loop equations”, and show that their solution is provided by a topological recursion under some assumptions, in particular the result takes a universal form. The Schwinger-Dyson equation of the one and two hermitian matrix models, and of the Opnq model appear as special cases. We study applications to repulsive particles systems, and explain how our notion of loop equations are related to Virasoro constraints. Then, as a special case, we study in detail applications to enumeration problems in a general class of non-intersecting loop models on the random lattice of all topologies, to SUpNq Chern-Simons invariants of torus knots in the large N expansion. We also mention an application to Liouville theory on surfaces of positive genus.We formulate a notion of ”abstract loop equations”, and show that their solution is provided by a topological recursion under some assumptions, in particular the result takes a universal form. The Schwinger-Dyson equation of the one and two hermitian matrix models, and of the Opnq model appear as special cases. We study applications to repulsive particles systems, and explain how our notion of loop equations are related to Virasoro constraints. Then, as a special case, we study in detail applications to enumeration problems in a general class of non-intersecting loop models on the random lattice of all topologies, to SUpNq Chern-Simons invariants of torus knots in the large N expansion. We also mention an application to Liouville theory on surfaces of positive genus.","PeriodicalId":55616,"journal":{"name":"Communications in Number Theory and Physics","volume":"13 1","pages":"51-187"},"PeriodicalIF":1.9,"publicationDate":"2015-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85411098","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 : 2015-01-01DOI: 10.4310/CNTP.2015.V9.N4.A4
Minxian Zhu
Mirror symmetry was originally formulated as a correspondence between the N = (2, 2) superconformal field theories constructed for a Calabi-Yau n-fold X and for its mirror partner X∨. On the level of cohomology groups, there is a 90-degree rotation of the Hodge diamond, i.e. hp,q(X,C) = hn−p,q(X∨,C). Batyrev’s construction of Calabi-Yau hypersurfaces in Gorenstein Fano toric varieties associated to a pair of reflexive polytopes ([B]) is a prolific source of examples of mirror Calabi-Yau varieties. This construction was later generalized by Borisov to Calabi-Yau complete intersections in Gorenstein Fano toric varieties ([B1]), and further by Batyrev and Borisov to the mirror duality of reflexive Gorenstein cones ([BB1]). They proved that the stringtheoretic Hodge numbers of (singular) Calabi-Yau varieties arising from their constructions satisfy the expected mirror duality ([BB2]). Around the same time, physicists Berglund and Hübsch proposed a way to construct mirror pairs of (2, 2)-superconformal field theories in the formalism of orbifold Landau-Ginzburg theories ([BH]). They considered a nondegenerate invertible polynomial potential W whose transpose W∨ is again a non-degenerate invertible potential. They claimed that there exists a suitable group H such that the Landau-Ginzburg orbifolds W and W∨/H form a mirror pair. Recently, Krawitz found a general construction of the dual group G∨ for any subgroup G of diagonal symmetries of W , and proved an “LG-to-LG” mirror symmetry theorem for the pair (W/G,W∨/G∨) at the level of double-graded state spaces ([K]). Under a certain CY condition, the polynomials W , W∨ define CalabiYau hypersurfacesXW ,XW∨ in (usually different) weighted projective spaces.
{"title":"Elliptic genera of Berglund–Hübsch Landau–Ginzburg orbifolds","authors":"Minxian Zhu","doi":"10.4310/CNTP.2015.V9.N4.A4","DOIUrl":"https://doi.org/10.4310/CNTP.2015.V9.N4.A4","url":null,"abstract":"Mirror symmetry was originally formulated as a correspondence between the N = (2, 2) superconformal field theories constructed for a Calabi-Yau n-fold X and for its mirror partner X∨. On the level of cohomology groups, there is a 90-degree rotation of the Hodge diamond, i.e. hp,q(X,C) = hn−p,q(X∨,C). Batyrev’s construction of Calabi-Yau hypersurfaces in Gorenstein Fano toric varieties associated to a pair of reflexive polytopes ([B]) is a prolific source of examples of mirror Calabi-Yau varieties. This construction was later generalized by Borisov to Calabi-Yau complete intersections in Gorenstein Fano toric varieties ([B1]), and further by Batyrev and Borisov to the mirror duality of reflexive Gorenstein cones ([BB1]). They proved that the stringtheoretic Hodge numbers of (singular) Calabi-Yau varieties arising from their constructions satisfy the expected mirror duality ([BB2]). Around the same time, physicists Berglund and Hübsch proposed a way to construct mirror pairs of (2, 2)-superconformal field theories in the formalism of orbifold Landau-Ginzburg theories ([BH]). They considered a nondegenerate invertible polynomial potential W whose transpose W∨ is again a non-degenerate invertible potential. They claimed that there exists a suitable group H such that the Landau-Ginzburg orbifolds W and W∨/H form a mirror pair. Recently, Krawitz found a general construction of the dual group G∨ for any subgroup G of diagonal symmetries of W , and proved an “LG-to-LG” mirror symmetry theorem for the pair (W/G,W∨/G∨) at the level of double-graded state spaces ([K]). Under a certain CY condition, the polynomials W , W∨ define CalabiYau hypersurfacesXW ,XW∨ in (usually different) weighted projective spaces.","PeriodicalId":55616,"journal":{"name":"Communications in Number Theory and Physics","volume":"9 1","pages":"741-761"},"PeriodicalIF":1.9,"publicationDate":"2015-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70423189","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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