Abstract: Allison Fink

Mentor: Dr. Beckmann

It is a widespread expectation among theoretical physicists that the universe exhibits supersymmetry. A useful way to represent a supersymmetric algebra describing the relation between bosons and fermions is to use a geometrical representation called an adinkra. A basic adinkra is formed from a cube in N dimensions that has vertices that alternate between two colors to represent fermionic and bosonic functions. Each set of parallel lines connecting the vertices is given a different color to correspond to a distinct super-differential operator relating the fermionic and bosonic functions that it connects. Also, the lines may be either dashed or undashed, but each closed path must have an odd number of dashed lines; a dashed line corresponds to a minus sign under the differential operator. To form an adinkra one then rearranges the vertices so that no two vertical levels contain both bosonic and fermionic functions. The action of the super-differential operator on each fermionic or bosonic functions in terms of the bosonic or fermionic function connected by the line corresponding to the operator is then determined by whether one goes up or down from one to another, thus establishing the algebra for that adinkra. Given an adinkra and using the anticommutativity of two different super-differential operators DB and DC we can then evaluate the expression for the relation DA {DB, DC}Ψk where Ψk represents a fermionic function in the adinkra, set this expression equal to zero, and apply on-shell conditions (for which we set an arbitrary number of the functions for fermionic or bosonic functions equal to zero) to potentially obtain useful equations that reflect supersymmetry.

Adinkras are also powerful representations because from their algebras one may derive general physics equations. Recently, my advisor and his colleagues have been using the algebra for an adinkra formed from a 4 dimensional cube and hope to use it to obtain Maxwell’s equations. One can obtain information about this adinkra by folding it upon itself so that fermionic functions are fused together with fermionic functions, bosonic functions are fused with bosonic functions, and identical lines are fused together, in order to preserve the properties of supersymmetry, resulting in the number of bosonic and fermionic functions being reduced from 8 of each to 4 of each. Our team this summer is in charge of extracting the equations out of a particular rearrangement, known as a valise, of this folded adinkra, in which all the bosonic functions appear on the bottom and all fermionic functions appear on the top. We must do this for the two ways that this can occur: the vector multiplet (VM) and chiral multiplet (CM). In this summer’s research, our team will represent in matrix form the calculations for DA {DB, DC}Ψk for the two possible valise adinkras and applying the possible on-shell conditions. We may also examine further calculations.