Invariant Bilinear Research Articles

All point correlation functions in SYK

Large N melonic theories are characterized by two-point function Feynman diagrams built exclusively out of melons. This leads to conformal invariance at strong coupling, four-point function diagrams that are exclusively ladders, and higher-point functions that are built out of four-point functions joined together. We uncover an incredibly useful property of these theories: the six-point function, or equivalently, the three-point function of the primary O(N ) invariant bilinears, regarded as an analytic function of the operator dimensions, fully determines all correlation functions, to leading nontrivial order in 1/N , through simple Feynman-like rules. The result is applicable to any theory, not necessarily melonic, in which higher-point correlators are built out of four-point functions. We explicitly calculate the bilinear three-point function for q-body SYK, at any q. This leads to the bilinear four-point function, as well as all higher-point functions, expressed in terms of higher-point conformal blocks, which we discuss. We find universality of correlators of operators of large dimension, which we simplify through a saddle point analysis. We comment on the implications for the AdS dual of SYK.

Leibniz Algebras with Invariant Bilinear Forms and Related Lie Algebras

In ([11]), we have studied quadratic Leibniz algebras that are Leibniz algebras endowed with symmetric, nondegenerate, and associative (or invariant) bilinear forms. The nonanticommutativity of the Leibniz product gives rise to other types of invariance for a bilinear form defined on a Leibniz algebra: the left invariance, the right invariance. In this article, we study the structure of Leibniz algebras endowed with nondegenerate, symmetric, and left (resp. right) invariant bilinear forms. In particular, the existence of such a bilinear form on a Leibniz algebra 𝔏 gives rise to a new algebra structure ☆ on the underlying vector space 𝔏. In this article, we study this new algebra, and we give information on the structure of this type of algebras by using some extensions introduced in [11]. In particular, we improve the results obtained in [22].

Non-degenerate Invariant Bilinear Forms on Lie Color Algebras

In this paper, we study Lie color algebras 𝔤 with a non-degenerate color-symmetric, 𝔤-invariant bilinear form B, such a (𝔤,B) is called a quadratic Lie color algebra. Our first result generalizes the notion of double extensions to quadratic Lie color algebras. This notion was introduced by Medina and Revoy to study quadratic Lie algebras. In the second theorem, we give a sufficient condition for a quadratic Lie color algebra to be a quadratic Lie color algebra by double extension. At last, we generalize the notion of T*-extensions to Lie color algebras.

Supertransvectants and Symplectic Geometry

The 1|1-supertransvectants are the osp(1|2)-invariant bilinear operations on weighted densities on the supercircle S1|1, the projective version of ⁠. These operations are analogues of the famous Gordan transvectants (or Rankin–Cohen brackets). We prove that supertransvectants coincide with the iterated Poisson and ghost Poisson brackets on and apply this result to construct star-products.

Proof of a symmetrized trace conjecture for the Abelian Born–Infeld Lagrangian

In this paper we prove a conjecture regarding the form of the Born-Infeld Lagrangian with a U(1)^2n gauge group after the elimination of the auxiliary fields. We show that the Lagrangian can be written as a symmetrized trace of Lorentz invariant bilinears in the field strength. More generally we prove a theorem regarding certain solutions of unilateral matrix equations of arbitrary order. For solutions which have perturbative expansions in the matrix coefficients, the solution and all its positive powers are sums of terms which are symmetrized in all the matrix coefficients and of terms which are commutators.

Decompositions in categories of highest weight modules

Let g be a semisimple finite-dimensional subalgebra of a Lie algebra a defined over C. To analyze an a-module X we often restrict the action from a to g and decompose X as a g-module. For example, if X is U(g)-locally finite then X is completely reducible as a g-module. In this article we focus on this decomposition problem not in cases when X is U(g)-locally finite but instead when X admits a nondegenerate g-invariant bilinear form. Irreducible a-modules often admit invariant bilinear or Hermitian forms which by irreducibility are nondegenerate (or zero). This leads us to study a category 9 of g-modules X that are highest weight modules and admit nondegenerate invariant bilinear forms. Our first main result, Theorem 1.8, gives an essentially unique decomposition of X into indecomposable selfdual submodules and classifies these indecomposable self-dual modules. The choice of category studied here is further suggested by the role of the Zuckerman derived functors ri [12]. These functors are fundamental to representation theory and the theory of Harish-Chandra modules. Let A be a real semisimple Lie group with Lie algebra a,, and complex Lie algebra a Let G be a maximal compact subgroup of A with complex Lie algebra g. Let p = m@u be a parabolic subalgebra of g with reductive component m and nilradical u and set s = dim(u). The functors r’ are the right derived functors of the g-finite submodule functor defined on the category of U(m)-locally finite g-modules. Theorem 1.8, recast using derived functors, would assert that any X in 9 has an essentially unique decomposition into special indecomposable submodules A4 which have cohomology PA4 = 0 for all i except possibly i = s.

On thed 2-type representation of the conformal group extended by reflections

In this paper thed2-type representation of the conformal group is examined. We calculate the explicit basis functions of the irreducible spaces and we have an invariant bilinear functional for them. We extend the representation to include reflections as well, and examine the representation of the extended group.

On Spinors in n Dimensions

The matrix which enters in the charge conjugation transformation of the usual spinors in 4-space is an invariant matrix and is skew symmetric. It is shown that there exists such an invariant matrix C for any number of dimensions (and independent of the number of time like dimensions). Its symmetry properties depend on the dimension number n modulo 8. With the help of the C matrix one can construct, for n = 1, 2, 7, 8 mod 8, an n-dimensional invariant bilinear in the components of a single n-dimensional spinor. Some examples are given for n = 2, 3, 7. A bilinear baryon invariant is formed for a theory with high symmetry. Its existence is closely related to the triality property of 8-space.