Journal articles: 'Twisted elliptic genus'

1

Eguchi, Tohru, and Kazuhiro Hikami. "Note on twisted elliptic genus of K3 surface." Physics Letters B 694, no. 4-5 (January 2011): 446–55. http://dx.doi.org/10.1016/j.physletb.2010.10.017.

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Eguchi, Tohru, and Kazuhiro Hikami. "Twisted Elliptic Genus for K3 and Borcherds Product." Letters in Mathematical Physics 102, no. 2 (May 26, 2012): 203–22. http://dx.doi.org/10.1007/s11005-012-0569-2.

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Eager, Richard, and Ingmar Saberi. "Holomorphic field theories and Calabi–Yau algebras." International Journal of Modern Physics A 34, no. 16 (June 10, 2019): 1950071. http://dx.doi.org/10.1142/s0217751x19500714.

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We consider the holomorphic twist of the worldvolume theory of flat D[Formula: see text]-branes transversely probing a Calabi–Yau manifold. A chain complex, constructed using the BV formalism, computes the local observables in the holomorphically twisted theory. Generalizing earlier work in the case [Formula: see text], we find that this complex can be identified with the Ginzburg dg algebra associated to the Calabi–Yau. However, the identification is subtle; the complex is the space of fields contributing to the holomorphic twist of the free theory, and its differential arises from interactions. For [Formula: see text], this holomorphically twisted theory is related to the elliptic genus. We give a general description for D1-branes probing a Calabi–Yau fourfold singularity, and for [Formula: see text] quiver gauge theories. In addition, we propose a relation between the equivariant Hirzebruch [Formula: see text] genus of large-[Formula: see text] symmetric products and cyclic homology.

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YADAV, REKHA, SHAILJA TRIPATHI, DILESHWAR PRASAD, SHUBHAM JAISWAL, VIRENDRA K. MADHUKAR, and PRIYANKA AGNIHOTRI. "Lectotypification of names in Duthiea (Poaceae)." Phytotaxa 494, no. 1 (March 31, 2021): 173–76. http://dx.doi.org/10.11646/phytotaxa.494.1.15.

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The genus, Duthiea Hackel (1895: 200) consists of three species, viz. D. brachypodium (P.Candargy 1901: 65) Keng & Keng (1965: 182), D. bromoides Hackel (1895: 200) and D. oligostachya (Munro ex Aitchison 1880: 108) Stapf (1896: sub Pl. 2474), distributed mainly in mountainous zones of Afghanistan to western China (Kellogg 2015). Duthiea is characterized by having pedicelled spikelets arranged to congested one-sided racemes; rachilla disarticulating above the glumes and between the florets; glumes equal to sub-equal, elliptic or lanceolate, with 5–7 nerves, persistent; lemma hirsute or villous with bifid apex and geniculate single awn with twisted column arising from the sinus of lemma; lodicules absent; ovary obovoid; style single, tomentose, longer or shorter than the stigmas; stigmas 2, terminally exerted from the floret; caryopsis cylindrical, covered with forwardly directed bristles (Bor 1953, 1960).

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Bruinier, Jan Hendrik, Stephan Ehlen, and Tonghai Yang. "CM values of higher automorphic Green functions for orthogonal groups." Inventiones mathematicae 225, no. 3 (March 17, 2021): 693–785. http://dx.doi.org/10.1007/s00222-021-01038-0.

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AbstractGross and Zagier conjectured that the CM values (of certain Hecke translates) of the automorphic Green function $$G_s(z_1,z_2)$$ G s ( z 1 , z 2 ) for the elliptic modular group at positive integral spectral parameter s are given by logarithms of algebraic numbers in suitable class fields. We prove a partial average version of this conjecture, where we sum in the first variable $$z_1$$ z 1 over all CM points of a fixed discriminant $$d_1$$ d 1 (twisted by a genus character), and allow in the second variable the evaluation at individual CM points of discriminant $$d_2$$ d 2 . This result is deduced from more general statements for automorphic Green functions on Shimura varieties associated with the group $${\text {GSpin}}(n,2)$$ GSpin ( n , 2 ) . We also use our approach to prove a Gross–Kohnen–Zagier theorem for higher Heegner divisors on Kuga–Sato varieties over modular curves.

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Shnidman, Ari, and Ariel Weiss. "Rank growth of elliptic curves over 𝑁-th root extensions." Transactions of the American Mathematical Society, Series B 10, no. 16 (April 14, 2023): 482–506. http://dx.doi.org/10.1090/btran/149.

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Fix an elliptic curve E E over a number field F F and an integer n n which is a power of 3 3 . We study the growth of the Mordell–Weil rank of E E after base change to the fields K d = F ( d 2 n ) K_d = F(\!\sqrt [2n]{d}) . If E E admits a 3 3 -isogeny, then we show that the average “new rank” of E E over K d K_d , appropriately defined, is bounded as the height of d d goes to infinity. When n = 3 n = 3 , we moreover show that for many elliptic curves E / Q E/\mathbb {Q} , there are no new points on E E over Q ( d 6 ) \mathbb {Q}(\sqrt [6]d) , for a positive proportion of integers d d . This is a horizontal analogue of a well-known result of Cornut and Vatsal [Nontriviality of Rankin-Selberg L-functions and CM points, L-functions and Galois representations, vol. 320, Cambridge Univ. Press, Cambridge, 2007, pp. 121–186]. As a corollary, we show that Hilbert’s tenth problem has a negative solution over a positive proportion of pure sextic fields Q ( d 6 ) \mathbb {Q}(\sqrt [6]{d}) . The proofs combine our recent work on ranks of abelian varieties in cyclotomic twist families with a technique we call the “correlation trick”, which applies in a more general context where one is trying to show simultaneous vanishing of multiple Selmer groups. We also apply this technique to families of twists of Prym surfaces, which leads to bounds on the number of rational points in sextic twist families of bielliptic genus 3 curves.

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ZHANG, JIAWEI, ZHEN WANG, SUQING ZHUO, YAHUI GAO, XUESONG LI, JUN ZHANG, LIN SUN, JUNRONG LIANG, LANG LI, and CHANGPING CHEN. "Scoliolyra elliptica gen. et sp. nov. (Bacillariophyceae), a new marine genus from sandy beach in Southern China." Phytotaxa 472, no. 1 (November 18, 2020): 1–12. http://dx.doi.org/10.11646/phytotaxa.472.1.1.

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Observation of sand samples collected from an estuarine sandy beach in Southern China revealed the presence of a new biraphid diatom. It is characterized by the presence of a twisted raphe system, apically elongate areolae, and striae interrupted by longitudinal H-shaped lateral areas in a form of valve face depressions, as well as girdle composed of open, plain copulae. The structure of valve outline, the raphe sternum and striae bears some resemblance to some established genera including e.g., Lyrella, Fallacia, Scolioneis, Scoliopleura and Scoliotropis, however, these characters are uniquely combined in this novel taxon. The new species, Scoliolyra elliptica, belongs to a new biraphid genus, Scoliolyra, and it is tentatively placed within the family Scolioneidaceae.

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Bruin, Peter, and Filip Najman. "Hyperelliptic modular curves and isogenies of elliptic curves over quadratic fields." LMS Journal of Computation and Mathematics 18, no. 1 (2015): 578–602. http://dx.doi.org/10.1112/s1461157015000157.

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We study elliptic curves over quadratic fields with isogenies of certain degrees. Let $n$ be a positive integer such that the modular curve $X_{0}(n)$ is hyperelliptic of genus ${\geqslant}2$ and such that its Jacobian has rank $0$ over $\mathbb{Q}$. We determine all points of $X_{0}(n)$ defined over quadratic fields, and we give a moduli interpretation of these points. We show that, with a finite number of exceptions up to $\overline{\mathbb{Q}}$-isomorphism, every elliptic curve over a quadratic field $K$ admitting an $n$-isogeny is $d$-isogenous, for some $d\mid n$, to the twist of its Galois conjugate by a quadratic extension $L$ of $K$. We determine $d$ and $L$ explicitly, and we list all exceptions. As a consequence, again with a finite number of exceptions up to $\overline{\mathbb{Q}}$-isomorphism, all elliptic curves with $n$-isogenies over quadratic fields are in fact $\mathbb{Q}$-curves.

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9

Ashok, Sujay K., and Jan Troost. "A twisted non-compact elliptic genus." Journal of High Energy Physics 2011, no. 3 (March 2011). http://dx.doi.org/10.1007/jhep03(2011)067.

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10

Duan, Zhihao, Kimyeong Lee, June Nahmgoong, and Xin Wang. "Twisted 6d (2, 0) SCFTs on a circle." Journal of High Energy Physics 2021, no. 7 (July 2021). http://dx.doi.org/10.1007/jhep07(2021)179.

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Abstract We study twisted circle compactification of 6d (2, 0) SCFTs to 5d $$ \mathcal{N} $$ N = 2 supersymmetric gauge theories with non-simply-laced gauge groups. We provide two complementary approaches towards the BPS partition functions, reflecting the 5d and 6d point of view respectively. The first is based on the blowup equations for the instanton partition function, from which in particular we determine explicitly the one-instanton contribution for all simple Lie groups. The second is based on the modular bootstrap program, and we propose a novel modular ansatz for the twisted elliptic genera that transform under the congruence subgroups Γ0(N) of SL(2, ℤ). We conjecture a vanishing bound for the refined Gopakumar-Vafa invariants of the genus one fibered Calabi-Yau threefolds, upon which one can determine the twisted elliptic genera recursively. We use our results to obtain the 6d Cardy formulas and find universal behaviour for all simple Lie groups. In addition, the Cardy formulas remain invariant under the twist once the normalization of the compact circle is taken into account.

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11

Álvarez-García, Rafael, and Lorenz Schlechter. "Analytic periods via twisted symmetric squares." Journal of High Energy Physics 2022, no. 7 (July 2022). http://dx.doi.org/10.1007/jhep07(2022)024.

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Abstract We study the symmetric square of Picard-Fuchs operators of genus one curves and the thereby induced generalized Clausen identities. This allows the computation of analytic expressions for the periods of all one-parameter K3 manifolds in terms of elliptic integrals. The resulting expressions are globally valid throughout the moduli space and allow the explicit inversion of the mirror map and the exact computation of distances, useful for checks of the Swampland Distance Conjecture. We comment on the generalization to multi-parameter models and provide a two-parameter example.

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Cota, Cesar Fierro, Albrecht Klemm, and Thorsten Schimannek. "State counting on fibered CY 3-folds and the non-Abelian weak gravity conjecture." Journal of High Energy Physics 2021, no. 5 (May 2021). http://dx.doi.org/10.1007/jhep05(2021)030.

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Abstract We extend the dictionary between the BPS spectrum of Heterotic strings and the one of F-/M-theory compactifications on K3 fibered Calabi-Yau 3-folds to cases with higher rank non-Abelian gauge groups and in particular to dual pairs between Heterotic CHL orbifolds and compactifications on Calabi-Yau 3-folds with a compatible genus one fibration. We show how to obtain the new supersymmetric index purely from the Calabi-Yau geometry by reconstructing the Noether-Lefschetz generators, which are vector-valued modular forms. There is an isomorphism between the latter objects and vector-valued lattice Jacobi forms, which relates them to the elliptic genera and twisted-twined elliptic genera of six- and five-dimensional Heterotic strings. The meromorphic Jacobi forms generate the dimensions of the refined cohomology of the Hilbert schemes of symmetric products of the fiber and allow us to refine the BPS indices in the fiber and therefore to obtain, conjecturally, actual state counts. Using the properties of the vector-valued lattice Jacobi forms we also provide a mathematical proof of the non-Abelian weak gravity conjecture for F-/M-theory compactified on this general class of fibered Calabi-Yau 3-folds.

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13

Hong, Junho, and James T. Liu. "The topologically twisted index of N $$ \mathcal{N} $$ = 4 super-Yang-Mills on T 2 × S2 and the elliptic genus." Journal of High Energy Physics 2018, no. 7 (July 2018). http://dx.doi.org/10.1007/jhep07(2018)018.

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14

Fischbach, Fabian, Albrecht Klemm, and Christoph Nega. "Lost chapters in CHL black holes: untwisted quarter-BPS dyons in the ℤ2 model." Journal of High Energy Physics 2021, no. 1 (January 2021). http://dx.doi.org/10.1007/jhep01(2021)157.

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Abstract Motivated by recent advances in Donaldson-Thomas theory, four-dimensional $$ \mathcal{N} $$ N = 4 string-string duality is examined in a reduced rank theory on a less studied BPS sector. In particular we identify candidate partition functions of “untwisted” quarter-BPS dyons in the heterotic ℤ2 CHL model by studying the associated chiral genus two partition function, based on the M-theory lift of string webs argument by Dabholkar and Gaiotto. This yields meromorphic Siegel modular forms for the Iwahori subgroup B(2) ⊂ Sp4(ℤ) which generate BPS indices for dyons with untwisted sector electric charge, in contrast to twisted sector dyons counted by a multiplicative lift of twisted-twining elliptic genera known from Mathieu moonshine. The new partition functions are shown to satisfy the expected constraints coming from wall-crossing and S-duality symmetry as well as the black hole entropy based on the Gauss-Bonnet term in the effective action. In these aspects our analysis confirms and extends work of Banerjee, Sen and Srivastava, which only addressed a subset of the untwisted sector dyons considered here. Our results are also compared with recently conjectured formulae of Bryan and Oberdieck for the partition functions of primitive DT invariants of the CHL orbifold X = (K3 × T2)/ℤ2, as suggested by string duality with type IIA theory on X.

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Dabholkar, Atish, Pavel Putrov, and Edward Witten. "Duality and mock modularity." SciPost Physics 9, no. 5 (November 13, 2020). http://dx.doi.org/10.21468/scipostphys.9.5.072.

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We derive a holomorphic anomaly equation for the Vafa-Witten partition function for twisted four-dimensional \mathcal{N} =4𝒩=4 super Yang-Mills theory on \mathbb{CP}^{2}ℂℙ2 for the gauge group SO(3)SO(3) from the path integral of the effective theory on the Coulomb branch. The holomorphic kernel of this equation, which receives contributions only from the instantons, is not modular but ‘mock modular’. The partition function has correct modular properties expected from SS-duality only after including the anomalous nonholomorphic boundary contributions from anti-instantons. Using M-theory duality, we relate this phenomenon to the holomorphic anomaly of the elliptic genus of a two-dimensional noncompact sigma model and compute it independently in two dimensions. The anomaly both in four and in two dimensions can be traced to a topological term in the effective action of six-dimensional (2,0)(2,0) theory on the tensor branch. We consider generalizations to other manifolds and other gauge groups to show that mock modularity is generic and essential for exhibiting duality when the relevant field space is noncompact.

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Dierigl, Markus, Paul-Konstantin Oehlmann, and Thorsten Schimannek. "The discrete Green-Schwarz mechanism in 6D F-theory and elliptic genera of non-critical strings." Journal of High Energy Physics 2023, no. 3 (March 14, 2023). http://dx.doi.org/10.1007/jhep03(2023)090.

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Abstract We study global anomalies of discrete gauge symmetries in six-dimensional supergravities and their realizations in F-theory. We explicitly construct a discrete Green-Schwarz mechanism that depends on the choice of a coupling constant and on a certain quadratic refinement in differential cohomology. By geometrically engineering theories with G = ℤ3 gauge symmetry and no tensor multiplets, we observe that a particular choice of the quadratic refinement is singled out in F-theory. This implies new Swampland constraints on the discrete charge spectra of 6d supergravities. On the other hand, the discrete Green-Schwarz coupling depends on the geometry of the Calabi-Yau. We use anomaly inflow to relate this to a ’t Hooft anomaly of the induced global symmetry in the worldsheet theories of non-critical strings. Using topological symmetry lines, we further relate this anomaly to the modular properties of twisted-twined elliptic genera. We then argue that the latter are encoded in the A-model topological string partition functions on different torus fibrations that are equipped with a flat torsional B-field. This allows us to derive a geometric expression for the global discrete anomaly in terms of the height-pairing of a multi-section on a genus one fibered Calabi-Yau.

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Karemaker, Valentijn, Sophie Marques, and Jeroen Sijsling. "Cubic function fields with prescribed ramification." International Journal of Number Theory, April 20, 2021, 1–35. http://dx.doi.org/10.1142/s1793042121500755.

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This paper describes cubic function fields [Formula: see text] with prescribed ramification, where [Formula: see text] is a rational function field. We give general equations for such extensions, an explicit procedure to obtain a defining equation when the purely cubic closure [Formula: see text] of [Formula: see text] is of genus zero, and a description of the twists of [Formula: see text] up to isomorphism over [Formula: see text]. For cubic function fields of genus at most one, we also describe the twists and isomorphism classes obtained when one allows Möbius transformations on [Formula: see text]. The paper concludes by studying the more general case of covers of elliptic and hyperelliptic curves that are ramified above exactly one point.

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Kim, Hee-Cheol, Minsung Kim, and Sung-Soo Kim. "Topological vertex for 6d SCFTs with ℤ2-twist." Journal of High Energy Physics 2021, no. 3 (March 2021). http://dx.doi.org/10.1007/jhep03(2021)132.

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Abstract We compute the partition function for 6d $$ \mathcal{N} $$ N = 1 SO(2N) gauge theories compactified on a circle with ℤ2 outer automorphism twist. We perform the computation based on 5-brane webs with two O5-planes using topological vertex with two O5-planes. As representative examples, we consider 6d SO(8) and SU(3) gauge theories with ℤ2 twist. We confirm that these partition functions obtained from the topological vertex with O5-planes indeed agree with the elliptic genus computations.

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19

Ciceri, Axel, Imtak Jeon, and Sameer Murthy. "Localization on AdS3 × S2. Part I. The 4d/5d connection in off-shell Euclidean supergravity." Journal of High Energy Physics 2023, no. 7 (July 28, 2023). http://dx.doi.org/10.1007/jhep07(2023)218.

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Abstract We begin to develop the formalism of localization for the functional integral of supergravity on AdS3× S2. We show how the condition of supersymmetry in the Euclidean ℍ3/ℤ × S2 geometry naturally leads to a twist of the S2 around the time direction of AdS3. The twist gives us a five-dimensional Euclidean supergravity background dual to the elliptic genus of (0, 4) SCFT2 at the semiclassical level. On this background we set up the off-shell BPS equations for one of the Killing spinors, such that the functional integral of five-dimensional Euclidean supergravity on ℍ3/ℤ × S2 localizes to its space of solutions. We obtain a class of solutions to these equations by lifting known off-shell BPS solutions of 4-dimensional supergravity on AdS2× S2. In order to do this consistently, we construct and use a Euclidean version of the off-shell 4d/5d lift of arXiv:1112.5371, which could be of independent interest.

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20

Chakravarti, Srinjay. "Striptease." Neke. The New Zealand Journal of Translation Studies 5, no. 1 (September 19, 2022). http://dx.doi.org/10.26686/neke.v5i1.7968.

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Parashuram’s short story ‘Striptease’, titled ‘Nirmok Nritya’ in Bengali (Bangla), is rather popular for its novel treatment of the female body vis-à-vis the male gaze. In this story, Parashuram (alias Rajshekhar Basu) spotlights the objectification of the female body, but with a wicked twist. However, the dynamics of gender and the binaries of female/male sexuality are expressed in a matrix of Hindu mythology, which initially make the story, especially some of its referents, somewhat inaccessible to not just Western readers but anyone unfamiliar with the Indian milieu. Yet, given Basu’s genius, his treatment of the theme is such that the appeal of the short story is universal, irrespective of the culture the reader belongs to. The gentleness of his satire—without being titillating or obscene—is especially alluring. The Bengali title can be literally translated as ‘The Dance of the Shedding of Shells’, or ‘The Dance in which Skins are Sloughed Off’. This is typical of Parashuram’s understated, elliptical, implicit sense of humour. The Bengali title does contribute to the overall impact of the original story, but it would not be an appropriate one in an English translation. Ergo, the title that naturally suggested itself was ‘Striptease’.

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Journal articles on the topic 'Twisted elliptic genus'

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