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Register a new userFusion algebra at a rational level and cohomology of nilpotent subalgebras of sl_2
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We define and calculate the fusion algebra of WZW model at a rational level by cohomological methods. As a byproduct we obtain a cohomological characterization of admissible representations of $widehat{gtsl}_{2}$.
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The mutual consistency of boundary conditions twisted by an automorphism group G of the chiral algebra is studied for general modular invariants of rational conformal field theories. We show that a consistent set of twisted boundary states associated with any modular invariant realizes a non-negative integer matrix representation (NIM-rep) of the generalized fusion algebra, an extension of the fusion algebra by representations of the twisted chiral algebra associated with the automorphism group G. We check this result for several concrete cases. In particular, we find that two NIM-reps of the fusion algebra for $su(3)_k (k=3,5)$ are organized into a NIM-rep of the generalized fusion algebra for the charge-conjugation automorphism of $su(3)_k$. We point out that the generalized fusion algebra is non-commutative if G is non-abelian and provide some examples for $G = S_3$. Finally, we give an argument that the graph fusion algebra associated with simple current extensions coincides with the generalized fusion algebra for the extended chiral algebra, and thereby explain that the graph fusion algebra contains the fusion algebra of the extended theory as a subalgebra.
We compute the braiding for the `principal gradation of $U_q(hat{{it sl}_2})$ for $|q|=1$ from first principles, starting from the idea of a rigid braided tensor category. It is not necessary to assume either the crossing or the unitarity condition from S-matrix theory. We demonstrate the uniqueness of the normalisation of the braiding under certain analyticity assumptions, and show that its convergence is critically dependent on the number-theoretic properties of the number $tau$ in the deformation parameter $q=e^{2pi itau}$. We also examine the convergence using probability, assuming a uniform distribution for $q$ on the unit circle.
Let $SL_2$ be the rank one simple algebraic group defined over an algebraically closed field $k$ of characteristic $p>0$. The paper presents a new method for computing the dimension of the cohomology spaces $text{H}^n(SL_2,V(m))$ for Weyl $SL_2$-modules $V(m)$. We provide a closed formula for $text{dim}text{H}^n(SL_2,V(m))$ when $nle 2p-3$ and show that this dimension is bounded by the $(n+1)$-th Fibonacci number. This formula is then used to compute $text{dim}text{H}^n(SL_2, V(m))$ for $n=1, 2,$ or $3$. For $n>2p-3$, an exponential bound, only depending on $n$, is obtained for $text{dim}text{H}^n(SL_2,V(m))$. Analogous results are also established for the extension spaces $text{Ext}^n_{SL_2}(V(m_2),V(m_1))$ between Weyl modules $V(m_1)$ and $V(m_2)$. In particular, we determine the degree three extensions for all Weyl modules of $SL_2$. As a byproduct, our results and techniques give explicit upper bounds for the dimensions of the cohomology of the Specht modules of symmetric groups, the cohomology of simple modules of $SL_2$, and the finite group of Lie type $SL_2(p^s)$.
We use factorization homology and higher Hochschild (co)chains to study various problems in algebraic topology and homotopical algebra, notably brane topology, centralizers of $E_n$-algebras maps and iterated bar constructions. In particular, we obtain an $E_{n+1}$-algebra model on the shifted integral chains of the mapping space of the n-sphere into an orientable closed manifold $M$. We construct and use $E_infty$-Poincare duality to identify higher Hochschild cochains, modeled over the $n$-sphere, with the chains on the above mapping space, and then relate Hochschild cochains to the deformation complex of the $E_infty$-algebra $C^*(M)$, thought of as an $E_n$-algebra. We invoke (and prove) the higher Deligne conjecture to furnish $E_n$-Hochschild cohomology, and all that is naturally equivalent to it, with an $E_{n+1}$-algebra structure. We prove that this construction recovers the sphere product. In fact, our approach to the Deligne conjecture is based on an explicit description of the $E_n$-centralizers of a map of $E_infty$-algebras $f:Ato B$ by relating it to the algebraic structure on Hochschild cochains modeled over spheres, which is of independent interest and explicit. More generally, we give a factorization algebra model/description of the centralizer of any $E_n$-algebra map and a solution of Deligne conjecture. We also apply similar ideas to the iterated bar construction. We obtain factorization algebra models for (iterated) bar construction of augmented $E_m$-algebras together with their $E_n$-coalgebras and $E_{m-n}$-algebra structures, and discuss some of its features. For $E_infty$-algebras we obtain a higher Hochschild chain model, which is an $E_n$-coalgebra. In particular, considering an n-connected topological space $Y$, we obtain a higher Hochschild cochain model of the natural $E_n$-algebra structure of the chains of the iterated loop space of $Y$.
We summarize some recent results obtained in collaboration with J. McCarthy on the spectrum of physical states in $W_3$ gravity coupled to $c=2$ matter. We show that the space of physical states, defined as a semi-infinite (or BRST) cohomology of the $W_3$ algebra, carries the structure of a BV-algebra. This BV-algebra has a quotient which is isomorphic to the BV-algebra of polyvector fields on the base affine space of $SL(3,C)$. Details have appeared elsewhere. [Published in the proceedings of Gursey Memorial Conference I: Strings and Symmetries, Istanbul, June 1994, eds. G. Aktas et al., Lect. Notes in Phys. 447, (Springer Verlag, Berlin, 1995)]
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