204 research outputs found

    Blowup Equations for Topological Strings and Supersymmetric Gauge Theories

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    Blowup equations and their K-theoretic version were proposed by Nakajima-Yoshioka and G"{o}ttsche as functional equations for Nekrasov partition functions of supersymmetric gauge theories in 4d and 5d. We generalize the blowup equations in two directions: one is for the refined topological string theory on arbitrary local Calabi-Yau threefolds, the other is an elliptic version for arbitrary 6d (1,0)(1,0) superconformal field theories (SCFTs) in the "atomic classification" of Heckman-Morrison-Rudelius-Vafa. In general, blowup equations fall into two types: the unity and the vanishing. We find the unity part of generalized blowup equations can be used to efficiently solve all refined BPS invariants of local Calabi-Yau geometries and the elliptic genera of 6d (1,0)(1,0) SCFTs, while the vanishing part can derive the compatibility formulas between two quantization schemes of algebraic curves, which are the exact Nekrasov-Shatashivili quantization conditions and the Grassi-Hatsuda-Mari~no conjecture. Blowup equations also give many interesting identities among modular forms and Jacobi forms. Furthermore, we study the relation between the elliptic genera of pure gauge 6d (1,0)(1,0) SCFTs and the superconformal indices of certain 4d mathcalN=2mathcal{N}=2 SCFTs. At last, we study the K-theoretic blowup equations on mathbbZ2mathbb{Z}_2 orbifold space and their connection with the bilinear relations of qq-deformed periodic Toda systems

    Blowup Equations for Refined Topological Strings

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    G\"{o}ttsche-Nakajima-Yoshioka K-theoretic blowup equations characterize the Nekrasov partition function of five dimensional N=1\mathcal{N}=1 supersymmetric gauge theories compactified on a circle, which via geometric engineering correspond to the refined topological string theory on SU(N)SU(N) geometries. In this paper, we study the K-theoretic blowup equations for general local Calabi-Yau threefolds. We find that both vanishing and unity blowup equations exist for the partition function of refined topological string, and the crucial ingredients are the r\bf r fields introduced in our previous paper. These blowup equations are in fact the functional equations for the partition function and each of them results in infinite identities among the refined free energies. Evidences show that they can be used to determine the full refined BPS invariants of local Calabi-Yau threefolds. This serves an independent and sometimes more powerful way to compute the partition function other than the refined topological vertex in the A-model and the refined holomorphic anomaly equations in the B-model. We study the modular properties of the blowup equations and provide a procedure to determine all the vanishing and unity r\bf r fields from the polynomial part of refined topological string at large radius point. We also find that certain form of blowup equations exist at generic loci of the moduli space.Comment: 85 pages. v2: Journal versio

    Functional Analysis of Dehydratase Domains of a Polyunsaturated fatty acid Synthase from Thraustochytrium sp. by Mutagenesis

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    Very long chain polyunsaturated fatty acids (VLCPUFAs), such as docosahexaenoic acid (DHA, 22:6-4,7,10,13,16,19) and eicosapentaenoic acid (EPA, 20:5-5,8,11,14,17), are essential for humans and animals since they are parts of the cell membrane and are involved in mediating various physiological processes. Thraustochytrium sp. ATCC 26185 is a marine protist that can produce large amounts of VLCPUFAs such as DHA and DPA (docosapentaenoic acid, 22:5-4,7,10,13,16) for human food and animal feed. Biosynthesis of these fatty acids in Thraustochytrium is catalyzed by a polyunsaturated fatty acid (PUFA) synthase that comprises three subunits, each with multiple catalytic domains. Three dehydratases (DH) domains in the PUFA synthase are believed to be responsible for coordinately introducing multiple double bonds in VLCPUFAs; however, the exact function of these domains remains to be determined. In this research, two DH domains (DH1 and DH2) in subunit-C of the PUFA synthase that have sequence similarity to E. coli FabA were functionally analyzed by site-directed mutagenesis and domain deletion analyses. Site-directed mutagenesis analysis showed that mutation of a histidine residue at calalytic site into alanine in DH1 of the PUFA synthase resulted in the complete loss of activity in the biosynthesis of all VLCPUFAs. Mutation of catalytic residue histidine into alanine in DH2 resulted in the production of a small amount of DPA, but not DHA. In addition, the deletion of DH1 domain also led to the complete loss of function while deletion of DH2 domain resulted in the production of only a very small amount of DPA. These results indicate that two FabA-like domains of the PUFA synthase possess distinct functions. DH1 domain, but not DH2 domain, is essential for the biosynthesis of VLCPUFAs, and DH2 domain is required for the biosynthesis of DHA. The PUFA synthase must have both for the efficient production of VLCPUFAs. Next, expression and purification of the PUFA synthase were attempted for future structure analysis. Partial purification of these subunits was accomplished using a His-tagged protein purification system and verified with western blot analysis. Successful purification of the subunit of the PUFA synthase expressed in E. coli would be an important step forwards for studying the structure and activity of each subunits of this enzyme and offer strategies for elucidating the molecular mechanism for the biosynthesis of VLCPUFAs

    Blowup Equations for 6d SCFTs. I

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    We propose novel functional equations for the BPS partition functions of 6d (1,0) SCFTs, which can be regarded as an elliptic version of Gottsche-Nakajima-Yoshioka's K-theoretic blowup equations. From the viewpoint of geometric engineering, these are the generalized blowup equations for refined topological strings on certain local elliptic Calabi-Yau threefolds. We derive recursion formulas for elliptic genera of self-dual strings on the tensor branch from these functional equations and in this way obtain a universal approach for determining refined BPS invariants. As examples, we study in detail the minimal 6d SCFTs with SU(3) and SO(8) gauge symmetry. In companion papers, we will study the elliptic blowup equations for all other non-Higgsable clusters.Comment: 52 pages, 3 figure

    Hecke Relations among 2d Fermionic RCFTs

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    Recently, Harvey and Wu proposed a suitable Hecke operator for vector-valued SL(2,Z)SL(2,\mathbb{Z}) modular forms to connect the characters of different 2d rational conformal field theories (RCFTs). We generalize such an operator to the 2d fermionic RCFTs and call it fermionic Hecke operator. The new Hecke operator naturally maps the Neveu-Schwarz (NS) characters of a fermionic theory to the NS characters of another fermionic theory. Mathematically, it is the natural Hecke operator on vector-valued Γθ\Gamma_\theta modular forms of weight zero. We find it can also be extended to NS~\mathrm{\widetilde{NS}} and Ramond (R) sectors by combining the characters of the two sectors together. We systematically study the fermionic Hecke relations among 2d fermionic RCFTs with up to five NS characters and find that almost all known supersymmetric RCFTs can be realized as fermionic Hecke images of some simple theories such as supersymmetric minimal models. We also study the coset relations between fermionic Hecke images with respect to c=12kc=12k holomorphic SCFTs

    A novel interaction between the 5′ untranslated region of the virus genome and Musashi homolog 2 is essential for Chikungunya virus genome replication

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    Chikungunya virus (CHIKV) is a single-stranded, positive-sense alphavirus of the Togaviridae family and is transmitted among humans via Aedes spp. mosquitos. Typical symptoms of CHIKV infection include debilitating arthralgia which can persist for months or years. The recent re-emergence of CHIKV raises serious global health concerns due to high rates of morbidity and the lack of licensed antiviral drugs or clinically approved vaccines. Current knowledge about the molecular mechanisms controlling CHIKV replication and virus-host interactions is limited. Previous studies from our group have mapped six stem-loops within the 5′ untranslated region (5′ UTR) and the first ~200nt of ORF-1. Phenotypic analysis demonstrated that they are RNA replication elements (RREs) required for virus genome replication through structuredependent mechanisms, which involve vertebrate and invertebrate-specific factors. However, the aspect of molecular virology of how the RREs function or through what interactions are yet to be investigated. In this study, reverse genetics and biochemical approaches were used to identify and confirm a specific interaction between cellular RNA binding protein Musashi homolog 2 and this structured region of the CHIKV genome. Using electromobility shift assay, I confirmed the direct interaction between MSI2 and the 5′ UTR of the CHIKV genome, with the binding site being the single-stranded region upstream of the AUG start codon. Using infectious virus and sub-genomic replicon systems, combined with RNA silencing and drug inhibition assays, it was demonstrated for the first time that MSI2 is required for CHIKV genome replication. A CHIKV trans-complementation system and strand-specific qRT-PCR were used to show that MSI2 is required for the initiation of negative-strand synthesis, possibly by functioning as a molecular switch for translation and replication as MSI2 also interacts directly or indirectly with viral non-structural proteins nsP1 and nsP3 – both are essential components of the viral replication complex. These findings provide novel insights into how CHIKV exploits cellular components for its replication and identify potential targets for antiviral therapy

    Elliptic Blowup Equations for 6d SCFTs. III: E-strings, M-strings and Chains

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    We establish the elliptic blowup equations for E-strings and M-strings and solve elliptic genera and refined BPS invariants from them. Such elliptic blowup equations can be derived from a path integral interpretation. We provide toric hypersurface construction for the Calabi-Yau geometries of M-strings and those of E-strings with up to three mass parameters turned on, as well as an approach to derive the perturbative prepotential directly from the local description of the Calabi-Yau threefolds. We also demonstrate how to systematically obtain blowup equations for all rank one 5d SCFTs from E-string by blow-down operations. Finally, we present blowup equations for E-M and M string chains.Comment: 54 pages, 10 tables, references adde

    The Benefits of Being Distributional: Small-Loss Bounds for Reinforcement Learning

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    While distributional reinforcement learning (RL) has demonstrated empirical success, the question of when and why it is beneficial has remained unanswered. In this work, we provide one explanation for the benefits of distributional RL through the lens of small-loss bounds, which scale with the instance-dependent optimal cost. If the optimal cost is small, our bounds are stronger than those from non-distributional approaches. As warmup, we show that learning the cost distribution leads to small-loss regret bounds in contextual bandits (CB), and we find that distributional CB empirically outperforms the state-of-the-art on three challenging tasks. For online RL, we propose a distributional version-space algorithm that constructs confidence sets using maximum likelihood estimation, and we prove that it achieves small-loss regret in the tabular MDPs and enjoys small-loss PAC bounds in latent variable models. Building on similar insights, we propose a distributional offline RL algorithm based on the pessimism principle and prove that it enjoys small-loss PAC bounds, which exhibit a novel robustness property. For both online and offline RL, our results provide the first theoretical benefits of learning distributions even when we only need the mean for making decisions
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