We review cohomology theories corresponding to the chiral and classical operads. The first one is the cohomology theory of vertex algebras, while the second one is the classical cohomology of Poisson vertex algebras (PVA), and we construct a spectral sequence relating them. Since in "good" cases the classical PVA cohomology coincides with the variational PVA cohomology and there are well-developed methods to compute the latter, this enables us to compute the cohomology of vertex algebras in many interesting cases. Finally, we describe a unified approach to integrability through vanishing of the first cohomology, which is applicable to both classical and quantum systems of Hamiltonian PDEs.
We construct a canonical map from the Poisson vertex algebra cohomology complex to the differential Harrison cohomology complex, which restricts to an isomorphism on the top degree. This is an important step in the computation of Poisson vertex algebra and vertex algebra cohomologies.
We develop methods for computation of Poisson vertex algebra cohomology. This cohomology is computed for the free bosonic and fermionic Poisson vertex (super)algebras, as well as for the universal affine and Virasoro Poisson vertex algebras. We establish finite dimensionality of this cohomology for conformal Poisson vertex (super)algebras that are finitely and freely generated by elements of positive conformal weight.
This paper generalizes Huang's cohomology theory of grading-restricted vertex algebras to meromorphic open-string vertex algebras (MOSVAs hereafter), which are noncommutative generalizations of grading-restricted vertex algebras introduced by Huang. The vertex operators for these algebras satisfy associativity but do not necessarily satisfy the commutativity. Moreover, the MOSVA and its bimodules considered in this paper do not necessarily have finite-dimensional homogeneous subspaces, though we do require that they have lower-bounded gradings. The construction and results in this paper will b…
We introduce a cohomology theory of grading-restricted vertex algebras. To construct the {\it correct} cohomologies, we consider linear maps from tensor powers of a grading-restricted vertex algebra to "rational functions valued in the algebraic completion of a module for the algebra," instead of linear maps from tensor powers of the algebra to a module for the algebra. One subtle complication arising from such functions is that we have to carefully address the issue of convergence when we compose these linear maps with vertex operators. In particular, for each $n\in \mathbb{N}$, we have an in…
It is well known that the validity of the so called Lenard-Magri scheme of integrability of a bi-Hamiltonian PDE can be established if one has some precise information on the corresponding 1st variational Poisson cohomology for one of the two Hamiltonian operators. In the first part of the paper we explain how to introduce various cohomology complexes, including Lie superalgebra and Poisson cohomology complexes, and basic and reduced Lie conformal algebra and Poisson vertex algebra cohomology complexes, by making use of the corresponding universal Lie superalebra or Lie conformal superalgebra.…
We give a criterion for the complete reducibility of modules satisfying a composability condition for a meromorphic open-string vertex algebra $V$ using the first cohomology of the algebra. For a $V$-bimodule $M$, let $\hat{H}^{1}_{\infty}(V, M)$ be the first cohomology of $V$ with the coefficients in $M$. Let $\hat{Z}^{1}_{\infty}(V, M)$ be the subspace of $\hat{H}^{1}_{\infty}(V, M)$ canonically isomorphic to the space of derivations obtained from the zero mode of the right vertex operators of weight $1$ elements such that the difference between the skew-symmetric opposite action of the left…
We construct the quantum group $GL_q(2)$ as the semi-infinite cohomology of the tensor product of two braided vertex operator algebras based on the algebra $W_2$ with complementary central charges $c+\bar{c}=28$. The conformal field theory version of the Laplace operator on the quantum group is also obtained.
We define complexes of vector bundles on products of moduli spaces of framed rank r torsion-free sheaves on the complex projective plane. The top non-vanishing Chern classes of the cohomology of these complexes yield actions of the r-colored Heisenberg and Clifford algebras on the equivariant cohomology of the moduli spaces. In this way we obtain a geometric realization of the boson-fermion correspondence and related vertex operators.
Let $V$ be a grading-restricted vertex algebra and $W$ a $V$-module. We show that for any $m\in \mathbb{Z}_{+}$, the first cohomology $H^{1}_{m}(V, W)$ of $V$ with coefficients in $W$ introduced by the author is linearly isomorphic to the space of derivations from $V$ to $W$. In particular, $H^{1}_{m}(V, W)$ for $m\in \mathbb{N}$ are equal (and can be denoted using the same notation $H^{1}(V, W)$). We also show that the second cohomology $H^{2}_{\frac{1}{2}}(V, W)$ of $V$ with coefficients in $W$ introduced by the author corresponds bijectively to the set of equivalence classes of square-zero …
We obtain the quantum group $SL_q(2)$ as semi-infinite cohomology of the Virasoro algebra with values in a tensor product of two braided vertex operator algebras with complementary central charges $c+\bar{c}=26$. Each braided VOA is constructed from the free Fock space realization of the Virasoro algebra with an additional q-deformed harmonic oscillator degree of freedom. The braided VOA structure arises from the theory of local systems over configuration spaces and it yields an associative algebra structure on the cohomology. We explicitly provide the four cohomology classes that serve as the…
The moduli stack of representations of a quiver, or coherent sheaves on a proper curve, carries two structures on its cohomology: a Hall algebra and braided vertex coalgebra. We show that they are compatible, by developing a formulation of abelian localisation which works in the derived/singular setting. An application of these techniques gives an explicit formula for the cohomological Hall algebra products.
In this paper we study the first cohomologies for the following three examples of vertex operator algebras: (i) the simple affine VOA associated to a simple Lie algebra with positive integral level; (ii) the Virasoro VOA corresponding to minimal models; (iii) the lattice VOA associated to a positive definite even lattice. We prove that in all these cases, the first cohomology $H^1(V, W)$ are given by the zero-mode derivations when $W$ is any $V$-module with an $\N$-grading (not necessarily by the operator $L(0)$). This agrees with the conjecture made by Yi-Zhi Huang and the author in 2018. For…
The integral singular cohomology ring of the Grassmann variety parametrizing $r$-dimensional subspaces in the $n$-dimensional complex vector space is naturally an irreducible representation of the Lie algebra of all the $n\times n$ matrices with integral entries. Using the notion of Schubert derivation, a distinguished Hasse-Schmidt derivation on an exterior algebra, we describe explicitly such a representation, indicating its relationship with the celebrated bosonic vertex representation of the Lie algebra of infinite matrices due to Date, Jimbo, Kashiwara and Miwa.
The Hochschild cohomology ring of a group algebra is an object that has received recent attention, but is difficult to compute, in even the simplest of cases. In this paper, we use the product formula due to Witherspoon and Siegel to extend some of their computations. In particular, we compute the Hochschild cohomology algebra of group algebras kG where |G| is less than 16, and we provide an alternative computation of the ring $HH^*(k(E \ltimes P))$ considered by Kessar and Linckelmann.
Let $G$ be the group $SL(2,\mathbb{R})$, $P\subset G$ be the parabolic subgroup of upper triangular matrices and $Γ\subset G$ be a cocompact lattice. A right action of $P$ on $Γ\backslash G$ defines an orbit foliation $\mathcal{F}_P$. We compute the leafwise cohomology ring $H^*(\mathcal{F}_P)$ by exploiting non-abelian harmonic analysis on $G$.
For the split group $G_{2}$ defined over $\mathbb{Z},$ we show that the de Rham cohomology ring of $B(G_{2})_{\mathbb{F}_{2}}$ is isomorphic to the singular cohomology ring with $\mathbb{F}_{2}$-coefficients of $B(G_{2})_{\mathbb{C}}.$ For the spin groups $\textrm{Spin}(n)$ defined over $\mathbb{Z},$ we show that the de Rham cohomology ring of $B\textrm{Spin}(n)_{\mathbb{F}_{2}}$ is isomorphic to the singular cohomology ring with $\mathbb{F}_{2}$-coefficients of $B\textrm{Spin}(n)_{\mathbb{C}}$ for $n \leq 10.$ For $n=11,$ we make a full computation of the de Rham cohomology ring of $B\textrm{…
The Baues--Wirsching cohomology is one of the cohomologies of a small category. Our aim is to describe the first Baues--Wirsching cohomology of the small category generated by a finite quiver freely. We consider the case where the coefficient is a natural system obtained by the composition of a functor and the target functor. We give an algorithm to obtain generators of the vector space of inner derivations. It is known that there exists a surjection from the vector space of derivations of the small category to the first Baues--Wirsching cohomology whose kernel is the vector space of inner der…
There are several examples in which algebraic properties of Jacobian algebras from (unpunctured) Riemann surfaces can be computed from the geometry of the Riemann surface. In this work, we compute the dimension of the Hochschild cohomology groups of any Jacobian algebra from unpunctured Riemann surfaces. In those expressions appear geometric objects of the triangulated surface, namely: the number of internal triangles and certain types of boundaries. Moreover, we give geometric conditions on the triangulated surface $(S,M,\mathbb T)$ such that the Gerstenhaber algebra $\operatorname{HH}^*(A_{\…
We define an action of the (double of) Cohomological Hall algebra of Kontsevich and Soibelman on the cohomology of the moduli space of spiked instantons of Nekrasov. We identify this action with the one of the affine Yangian of $\mathfrak{gl}(1)$. Based on that we derive the vertex algebra at the corner $\mathcal{W}_{r_1,r_2,r_3}$ of Gaiotto and Rapcak. We conjecture that our approach works for a big class of Calabi-Yau categories, including those associated with toric Calabi-Yau $3$-folds.
We apply the sheaf cohomology BGG method developed by the authors and Lachowska-Qi to the computation of Hochschild cohomology groups of various blocks of the small quantum group. All our computations of the center of the corresponding block agree with the conjectures of Lachowska-Qi. In the case of the nontrivial singular block for $\mathfrak{g} = \mathfrak{sl}_3$, we obtain the H*$(\mathfrak{u}, \mathbb{C}) = \mathbb{C}[\mathcal{N}]$-module structure of HH*$(\mathfrak{u}_λ)$.
