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All reductive spherical subgroups of the group SL(n) are found for which the intersections with every parabolic subgroup of SL(n) are connected. This condition guarantees that open equivariant embeddings of the corresponding homogeneous spaces into Moishezon spaces are algebraic. Bibliography: 6 titles.
In this paper we present a beautifully consistent web of evidence for the existence of interacting 4d rank-1 $ \mathcal{N} $ = 2 SCFTs obtained from gauging discrete subgroups of global symmetries of other existing 4d rank-1 $ \mathcal{N} $ = 2 SCFTs. The global symmetries that can be gauged involve a non-trivial combination of discrete subgroups of the U(1) R , low-energy EM duality group SL(2,Z), and the outer automorphism group of the flavor symmetry algebra, Out(F ). The theories that we construct are remarkable in many ways: (i) two of them have exceptional F 4 and G 2 flavor groups; (ii) they substantially complete the picture of the landscape of rank-1 $ \mathcal{N} $ = 2 SCFTs as they realize all but one of the remaining consistent rank-1 Seiberg-Witten geometries that we previously constructed but were not associated to known SCFTs; and (iii) some of them have enlarged $ \mathcal{N} $ = 3 SUSY, and have not been previously constructed. They are also examples of SCFTs which violate the ShapereTachikawa relation between the conformal central charges and the scaling dimension of the Coulomb branch vev. Here, we propose a modification of the formulas computing these central charges from the topologically twisted Coulomb branch partition function which correctly compute them for discretely gauged theories.
We study the cobordism conjecture of McNamara and Vafa which asserts that the bordism group of quantum gravity is trivial. In the context of type IIB string theory compactified on a circle, this predicts the presence of D7-branes. On the other hand, the non-Abelian structure of the IIB duality group SL (2, Z) implies the existence of additional [p, q] 7-branes. We find that this additional information is instead captured by the space of closed paths on the moduli space of elliptic curves parameterizing distinct values of the type IIB axio-dilaton. This description allows to recover the full structure of non-Abelian braid statistics for 7-branes. Combining the cobordism conjecture with an earlier Swampland conjecture by Ooguri and Vafa, we argue that only certain congruence subgroups $\Gamma\subset$ SL(2, Z) specifying genus zero modular curves can appear in 8D F-theory vacua. This leads to a successful prediction for the allowed Mordell–Weil torsion groups for 8D F-theory vacua.
Let p > 3 be a prime number and F{sub p} be a field of p elements. Let K be a commutative and associative ring obtained by adjoining to F{sub p} an element α such that α satisfies a polynomial over F{sub p} and a polynomial of the least degree whose root is α can be written as a product of distinct polynomials irreducible over F{sub p}. We prove that the special linear group SL{sub 2}(K) is generated by two elementary transvections ( (table) ), ( (table) )
For G = G(ℝ), a split, simply connected, semisimple Lie group of rank n and K the maximal compact subgroup of G, we give a method for computing Iwasawa coordinates of K∖G using the Chevalley generators and the Steinberg presentation. When K∖G is a scalar coset for a supergravity theory in dimensions ≥3, we determine the action of the integral form G(ℤ) on K∖G. We give explicit results for the action of the discrete U-duality groups SL{sub 2}(ℤ) and E{sub 7}(ℤ) on the scalar cosets SO(2)∖SL{sub 2}(ℝ) and [SU(8)/( ± Id)]∖E{sub 7(+7)}(ℝ) for type IIB supergravity in ten dimensions and 11-dimensional supergravity reduced to D = 4 dimensions, respectively. For the former, we use this to determine the discrete U-duality transformations on the scalar sector in the Borel gauge and we describe the discrete symmetries of the dyonic charge lattice. We determine the spectrum-generating symmetry group for fundamental BPS solitons of type IIB supergravity in D = 10 dimensions at the classical level and we propose an analog of this symmetry at the quantum level. We indicate how our methods can be used to study the orbits of discrete U-duality groups in general.
We provide an explicit twistorial construction of quaternion-Kähler manifolds obtained by deformation of c-map spaces and carrying an isometric action of the modular group SL(2,Z). The deformation is not assumed to preserve any continuous isometry and therefore this construction presents a general framework for describing NS5-brane instanton effects in string compactifications with N= 2 supersymmetry. In this context the modular invariant parametrization of twistor lines found in this work yields the complete non-perturbative mirror map between type IIA and type IIB physical fields.
In this paper we study supersymmetric co-dimension 2 and 4 defects in the compactification of the 6d (2, 0) theory of type A N-1 on a 3-manifold M . The so-called 3d-3d correspondence is a relation between complexified Chern-Simons theory (with gauge group SL(N,C) ) on M and a 3d N=2 theory T N [M ]. We study this correspondence in the presence of supersymmetric defects, which are knots/links inside the 3-manifold. Our study employs a number of different methods: state-integral models for complex Chern-Simons theory, cluster algebra techniques, domain wall theory T [SU(N )], 5d N=2 SYM, and also supergravity analysis through holography. These methods are complementary and we find agreement between them. In some cases the results lead to highly non-trivial predictions on the partition function. Our discussion includes a general expression for the cluster partition function, which can be used to compute in the presence of maximal and certain class of non-maximal punctures when N > 2. We also highlight the non-Abelian description of the 3d N=2 T N [M ] theory with defect included, when such a description is available. This paper is a companion to our shorter paper, which summarizes our main results.
Here, we study knots in 3d Chern-Simons theory with complex gauge group $SL(N,\mathbb{C})$, in the context of its relation with 3d $\mathcal{N}=2$ theory (the so-called 3d-3d correspondence). The defect has either co-dimension 2 or co-dimension 4 inside the 6d $(2,0)$ theory, which is compactified on a 3-manifold $\hat{M}$. We identify such defects in various corners of the 3d-3d correspondence, namely in 3d $SL(N,\mathbb{C})$ Chern-Simons theory, in 3d $\mathcal{N}=2$ theory, in 5d $\mathcal{N}=2$ super Yang-Mills theory, and in the M-theory holographic dual. We can make quantitative checks of the 3d-3d correspondence by computing partition functions at each of these theories. This Letter is a companion to a longer paper, which contains more details and more results.
The Quantum Modularity Conjecture of Zagier predicts the existence of a formal power series with arithmetically interesting coefficients that appears in the asymptotics of the Kashaev invariant at each root of unity. Our goal is to construct a power series from a Neumann-Zagier datum (i.e., an ideal triangulation of the knot complement and a geometric solution to the gluing equations) and a complex root of unity $\zeta$. We prove that the coefficients of our series lie in the trace field of the knot, adjoined a complex root of unity. We conjecture that our series are those that appear in the Quantum Modularity Conjecture and confirm that they match the numerical asymptotics of the Kashaev invariant (at various roots of unity) computed by Zagier and the first author. Our construction is motivated by the analysis of singular limits in Chern-Simons theory with gauge group $SL(2,C)$ at fixed level $k$, where $\zeta^k=1$.
We develop the generalized Cartan Calculus for the groups G = SL(2,R) × R + , SL(5,R) and SO(5,5). They are the underlying algebraic structures of d=9,7,6 exceptional field theory, respectively. These algebraic identities are needed for the "tensor hierarchy" structure in exceptional field theory. The validity of Poincar\'e lemmas in this new differential geometry is also discussed. Lastly, we explore some possible extension of the generalized Cartan calculus beyond the exceptional series.
We explicitly determine all Rota-Baxter operators (of weight zero) on sl (2,C) under the Cartan-Weyl basis. For the skew-symmetric operators, we give the corresponding skew-symmetric solutions of the classical Yang-Baxter equation in sl (2,C), confirming the related study by Semenov-Tian-Shansky. In general, these Rota-Baxter operators give a family of solutions of the classical Yang-Baxter equation in the six-dimensional Lie algebra sl (2,C)⋉{sub ad{sup *}} sl (2,C){sup *}. They also give rise to three-dimensional pre-Lie algebras which in turn yield solutions of the classical Yang-Baxter equation in other six-dimensional Lie algebras.
Liouville's theorem holds for Hamiltonian systems with complete Hamiltonian fields which possess a complete involutive system of first integrals; such systems are called Liouville-integrable. In this paper integrable systems with incomplete Hamiltonian fields are investigated. It is shown that Liouville's theorem remains valid in the case of a single incomplete field, while if the number of incomplete fields is greater, a certain analogue of the theorem holds. An integrable system on the algebra sl(3) is taken as an example. Bibliography: 11 titles.
Here we present a sawtooth model that explains observations where the central safety factor, q 0 , stays well below one, which is irreconcilable with current models that predict a reset to q 0 = 1 after the crash. We identify the structure of the field around the magnetic axis with elements of the Lie group $\mathrm{SL}(2,\ \mathbb{R})$ and find a transition to an alternating-hyperbolic geometry when q 0 = 2/3. This transition is driven by an ideal MHD instability and leads to a chaotic magnetic field near the axis.
The general form of a 2D conformal field theory (CFT) correlator on a Euclidean Riemann surface, Lorentzian plane or Lorentzian cylinder is well known. This paper describes the general form of 2- and 3-point CFT correlators on the Lorentzian torus LT 2 which arises as the conformal boundary of the group manifold SL ( 2 , R ) ≃ AdS 3 / Z . We consider only generic points, thereby omitting an analysis of contact terms, which already exhibits a surprisingly rich structure. The results are relevant to celestial holography, for which the LT 2 at the boundary of Klein space is the home of the putative celestial CFT. Published by the American Physical Society 2024
Perturbations of the classical Bateman Lagrangian preserving a certain subalgebra of Noether symmetries are studied, and conservative perturbations are characterized by the Lie algebra sl(2, ℝ) ⊕ so(2). Non-conservative albeit integrable perturbations are determined by the simple Lie algebra sl(2,ℝ), showing further the relation of the corresponding non-linear systems with the notion of generalized Ermakov systems.
To analyze the reducibility of the Verma modules one often needs to find the extremal vectors of the given representations. On the example of algebra U{sub q}(sl(3, ℂ)) we study how the set of extremal vectors is affected when we factorize the original representation and give their explicit formula.
We review the geometry of the space of quantum states S(H) of a finite-level quantum system with Hilbert space H from a group-theoretical point of view. This space carries two stratifications generated by the action of two different Lie groups, namely, the special unitary group SU(H) and its complexification SL(H), the complex special linear group. A stratum of the stratification generated by SU(H) is composed of isospectral states, that is, density operators with the same spectrum. A stratum of the stratification generated by SL(H) is composed of quantum states with the same rank. We prove that on every submanifold of isospectral quantum states there is also a canonical left action of SL(H) which is related with the canonical Kähler structure on isospectral quantum states. The fundamental vector fields of this SL(H)-action are divided into Hamiltonian and gradient vector fields. The former give rise to invertible maps on S that preserve the von Neumann entropy and the convex structure of S(H), while the latter give rise to invertible maps on S(H) that preserve the von Neumann entropy but not the convex structure of S(H). A similar decomposition is given for the fundamental vector fields of the SL(H)-action generating the stratification of S(H) into manifolds of quantum states with the same rank. However, in this case, the gradient vector fields preserve the rank but do not preserve entropy. Finally, some comments on multipartite quantum systems are made, and it is proved that the sets of product states of a multipartite quantum system are homogeneous manifolds for the local action of the complex special linear group associated with the partition.
The fundamental difference between bosons and fermions is that they obey two alternative representations of the Z{sub 2} group, resulting in symmetric or anti-symmetric binary commutation relations. Our aim is to explore possibilities offered by ternary Z{sub 3} generalization commutation relations. This leads to cubic and ternary algebras which are a direct generalization of usual commutation relations, with Z{sub 3}-grading replacing the usual Z{sub 2}-grading. Properties and structure of such algebras are discussed, with special interest in a low-dimensional one, with two generators. Invariant cubic forms on such algebras are introduced, and it is shown how the SL(2,C) group arises naturally as the symmetry group preserving these forms. In the case of lowest dimension, with only two generators, it is shown how the cubic combinations of elements of the same Z{sub 3} grade behave like Lorentz spinors, while binary products of elements of this algebra with an element of the conjugate algebra behave like Lorentz vectors. The wave equation generalizing the Dirac operator to the Z{sub 3}-graded case is introduced, whose diagonalization leads to a third-order equation. The solutions of this equation cannot propagate because their exponents always contain non-oscillating real damping factor. We show how certain cubic products can propagate nevertheless. The model suggests the origin of the color SU(3) symmetry obeyed by quark states.