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1. A Buchsbaum theory for tight closure

   https://doi.org/10.1090/tran/8762  

   Ma, Linquan; Quy, Pham (November 2022, Transactions of the American Mathematical Society)

   A Noetherian local ring ( R , m ) (R,\frak {m}) is called Buchsbaum if the difference ℓ ( R / q ) − e ( q , R ) \ell (R/\mathfrak {q})-e(\mathfrak {q}, R) , where q \mathfrak {q} is an ideal generated by a system of parameters, is a constant independent of q \mathfrak {q} . In this article, we study the tight closure analog of this condition. We prove that in an unmixed excellent local ring ( R , m ) (R,\frak {m}) of prime characteristic p > 0 p>0 and dimension at least one, the difference e ( q , R ) − ℓ ( R / q ∗ ) e(\mathfrak {q}, R)-\ell (R/\mathfrak {q}^*) is independent of q \mathfrak {q} if and only if the parameter test ideal τ p a r ( R ) \tau _{\mathrm {par}}(R) contains m \frak {m} . We also provide a characterization of this condition via derived category which is analogous to Schenzel’s criterion for Buchsbaum rings. 

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2. The holographic contributions to the sphere free energy

   https://doi.org/10.1007/JHEP01(2022)171  

   Binder, Damon J.; Freedman, Daniel Z.; Pufu, Silviu S.; Zan, Bernardo (January 2022, Journal of High Energy Physics)

   A bstract We study which bulk couplings contribute to the S 3 free energy F ( $$ \mathfrak{m} $$ m ) of three-dimensional $$ \mathcal{N} $$ N = 2 superconformal field theories with holographic duals, potentially deformed by boundary real-mass parameters m. In particular, we show that F ( $$ \mathfrak{m} $$ m ) is independent of a large class of bulk couplings that include non-chiral F-terms and all D-terms. On the other hand, in general, F ( $$ \mathfrak{m} $$ m ) does depend non-trivially on bulk chiral F-terms, such as prepotential interactions, and on bulk real-mass terms. These conclusions can be reached solely from properties of the AdS super-algebra, $$ \mathfrak{osp} $$ osp (2|4). We also consider massive vector multiplets in AdS, which in the dual field theory correspond to long single-trace superconformal multiplets of spin zero. We provide evidence that F ( $$ \mathfrak{m} $$ m ) is insensitive to the vector multiplet mass and to the interaction couplings between the massive vector multiplet and massless ones. In particular, this implies that F ( $$ \mathfrak{m} $$ m ) does not contain information about scaling dimensions or OPE coefficients of single-trace long scalar $$ \mathcal{N} $$ N = 2 superconformal multiplets. 

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3. COUNTABLE MODELS OF THE THEORIES OF BALDWIN–SHI HYPERGRAPHS AND THEIR REGULAR TYPES

   https://doi.org/10.1017/jsl.2019.28  

   GUNATILLEKA, DANUL K. (September 2019, The Journal of Symbolic Logic)

   Abstract We continue the study of the theories of Baldwin–Shi hypergraphs from [5]. Restricting our attention to when the rank δ is rational valued, we show that each countable model of the theory of a given Baldwin–Shi hypergraph is isomorphic to a generic structure built from some suitable subclass of the original class used in the construction. We introduce a notion of dimension for a model and show that there is a an elementary chain $$\left\{ {\mathfrak{M}_\beta :\beta \leqslant \omega } \right\}$$ of countable models of the theory of a fixed Baldwin–Shi hypergraph with $$\mathfrak{M}_\beta \preccurlyeq \mathfrak{M}_\gamma $$ if and only if the dimension of $$\mathfrak{M}_\beta $$ is at most the dimension of $$\mathfrak{M}_\gamma $$ and that each countable model is isomorphic to some $$\mathfrak{M}_\beta $$ . We also study the regular types that appear in these theories and show that the dimension of a model is determined by a particular regular type. Further, drawing on a large body of work, we use these structures to give an example of a pseudofinite, ω -stable theory with a nonlocally modular regular type, answering a question of Pillay in [11]. 

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4. Contravariant forms on Whittaker modules

   https://doi.org/10.1090/proc/15205  

   Brown, Adam; Romanov, Anna (January 2021, Proceedings of the American Mathematical Society)

   Let g \mathfrak {g} be a complex semisimple Lie algebra. We give a classification of contravariant forms on the nondegenerate Whittaker g \mathfrak {g} -modules Y ( χ , η ) Y(\chi , \eta ) introduced by Kostant. We prove that the set of all contravariant forms on Y ( χ , η ) Y(\chi , \eta ) forms a vector space whose dimension is given by the cardinality of the Weyl group of g \mathfrak {g} . We also describe a procedure for parabolically inducing contravariant forms. As a corollary, we deduce the existence of the Shapovalov form on a Verma module, and provide a formula for the dimension of the space of contravariant forms on the degenerate Whittaker modules M ( χ , η ) M(\chi , \eta ) introduced by McDowell. 

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5. Superspin chains from superstring theory

   https://doi.org/10.21468/SciPostPhys.13.4.083  

   Ishtiaque, Nafiz; Moosavian, Seyed Faroogh; Raghavendran, Surya; Yagi, Junya (January 2022, SciPost Physics)

   We present a correspondence between two-dimensional \mathcal{N} = (2,2) 𝒩 = ( 2 , 2 ) supersymmetric gauge theories and rational integrable \mathfrak{gl}(m|n) 𝔤 𝔩 ( m | n ) spin chains with spin variables taking values in Verma modules. Toexplain this correspondence, we realize the gauge theories asconfigurations of branes in string theory and map them by dualities tobrane configurations that realize line defects in four-dimensionalChern–Simons theory with gauge group GL(m|n) G L ( m | n ) .The latter configurations embed the superspin chains into superstringtheory. We also provide a string theory derivation of a similarcorrespondence, proposed by Nekrasov, for rational \mathfrak{gl}(m|n) 𝔤 𝔩 ( m | n ) spin chains with spins valued in finite-dimensional representations. 

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