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1. Characterizing 4-string contact interaction using machine learning

   https://doi.org/10.1007/JHEP04(2024)016  

   Erbin, Harold; Fırat, Atakan Hilmi (April 2024, Journal of High Energy Physics)

   A<sc>bstract</sc> The geometry of 4-string contact interaction of closed string field theory is characterized using machine learning. We obtain Strebel quadratic differentials on 4-punctured spheres as a neural network by performing unsupervised learning with a custom-built loss function. This allows us to solve for local coordinates and compute their associated mapping radii numerically. We also train a neural network distinguishing vertex from Feynman region. As a check, 4-tachyon contact term in the tachyon potential is computed and a good agreement with the results in the literature is observed. We argue that our algorithm is manifestly independent of number of punctures and scaling it to characterize the geometry ofn-string contact interaction is feasible. 

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2. On the classification of 5d SCFTs

   https://doi.org/10.1007/JHEP09(2020)007  

   Bhardwaj, Lakshya (September 2019, ArXivorg)

   We determine all 5d SCFTs upto rank three by studying RG flows of 5d KK theories. Our analysis reveals the existence of new rank one and rank two 5d SCFTs not captured by previous classifications. In addition to that, we provide for the first time a systematic and conjecturally complete classification of rank three 5d SCFTs. Our methods are based on a recently studied geometric description of 5d KK theories, thus demonstrating the utility of these geometric descriptions. It is straightforward, though computationally intensive, to extend this work and systematically classify 5d SCFTs of higher ranks (greater than or equal to four) by using the geometric description of 5d KK theories. 

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3. Twisted Circle Compactifications of 6d SCFTs

   Bhardwaj, Lakshya; Jefferson, Patrick; Kim, Hee-Cheol; Tarazi, Houri-Christina; Vafa, Cumrun (September 2019, ArXivorg)

   We study 6d superconformal field theories (SCFTs) compactified on a circle with arbitrary twists. The theories obtained after compactification, often referred to as 5d Kaluza-Klein (KK) theories, can be viewed as starting points for RG flows to 5d SCFTs. According to a conjecture, all 5d SCFTs can be obtained in this fashion. We compute the Coulomb branch prepotential for all 5d KK theories obtainable in this manner and associate to these theories a smooth local genus one fibered Calabi-Yau threefold in which is encoded information about all possible RG flows to 5d SCFTs. These Calabi-Yau threefolds provide hitherto unknown M-theory duals of F-theory configurations compactified on a circle with twists. For certain exceptional KK theories that do not admit a standard geometric description we propose an algebraic description that appears to retain the properties of the local Calabi-Yau threefolds necessary to determine RG flows to 5d SCFTs, along with other relevant physical data. 

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4. Stable homotopy groups of spheres

   https://doi.org/10.1073/pnas.2012335117  

   Isaksen, Daniel C.; Wang, Guozhen; Xu, Zhouli (September 2020, Proceedings of the National Academy of Sciences)

   We discuss the current state of knowledge of stable homotopy groups of spheres. We describe a computational method using motivic homotopy theory, viewed as a deformation of classical homotopy theory. This yields a streamlined computation of the first 61 stable homotopy groups and gives information about the stable homotopy groups in dimensions 62 through 90. As an application, we determine the groups of homotopy spheres that classify smooth structures on spheres through dimension 90, except for dimension 4. The method relies more heavily on machine computations than previous methods and is therefore less prone to error. The main mathematical tool is the Adams spectral sequence. 

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5. Ruijsenaars wavefunctions as modular group matrix coefficients

   https://doi.org/10.1007/s11005-024-01881-1  

   Di_Francesco, Philippe; Kedem, Rinat; Khoroshkin, Sergey; Schrader, Gus; Shapiro, Alexander (November 2024, Letters in Mathematical Physics)

   Abstract We give a description of the Hallnäs–Ruijsenaars eigenfunctions of the 2-particle hyperbolic Ruijsenaars system as matrix coefficients for the order 4 element$$S\in SL(2,{\mathbb {Z}})$$ $S\in SL(2,Z)$acting on the Hilbert space ofGL(2) quantum Teichmüller theory on the punctured torus. TheGL(2) Macdonald polynomials are then obtained as special values of the analytic continuation of these matrix coefficients. The main tool used in the proof is the cluster structure on the moduli space of framedGL(2)-local systems on the punctured torus, and an$$SL(2,{\mathbb {Z}})$$ $SL(2,Z)$-equivariant embedding of theGL(2) spherical DAHA into the quantized coordinate ring of the corresponding cluster Poisson variety. 

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