Adinkras, Supersymmetry, and Spectral Triples

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Ma148b Spring 2016 Topics in Mathematical Physics Different possible approach to SUSY and Spectral Triples Reference: M. Marcolli, Nick Zolman, work in preparation. C. Doran, K. Iga, G. Landweber, S. Méndez-Diez,
Ma148b Spring 2016 Topics in Mathematical Physics Different possible approach to SUSY and Spectral Triples Reference: M. Marcolli, Nick Zolman, work in preparation. C. Doran, K. Iga, G. Landweber, S. Méndez-Diez, Geometrization of N-extended 1-dimensional supersymmetry algebras, arxiv: Yan X. Zhang, Adinkras for Mathematicians, Trans. Amer. Math. Soc., Vol.366 (2014) N.6, Hyungrok Kim, Ingmar Saberi, Real homotopy theory and supersymmetric quantum mechanics, arxiv: Based on Nick Zolman s SURF project Supersymmetry algebras focus fon 1-dim spacetime: time direction t, zero-dim space off-shell supersymmetry algebras: operators Q 1, Q 2,..., Q N and t commutation relations: H = i t Hamiltonian [Q i, H] = 0 (1) {Q i, Q j } = 2δ ij H (2) representations of these operators acting on bosonic and fermionic fields Representations {φ 1,..., φ m } (bosonic fields) real commuting {ψ 1,..., ψ m } (fermionic fields) real anticommuting off-shell: no other equation satisfied except commutation relations (1) and (2) above operators acting as with c { 1, 1} and λ {0, 1} Q k φ a = c λ t ψ b (3) Q k ψ b = i c 1 λ t φ a (4) classify these by graphical combinatorial data (Adinkras) Adinkras (introduced by Faux and Gates) N-dimensional chromotopology: finite connected graph A with N-regular (all valences N) and bipartite edges E(A) are colored by colors in set {1, 2,..., N} every vertex incident to exactly one edge of each color colors i j, edges in E(A) with colors i and j form a disjoint union of 4-cycles bipartite: bosons and fermions (black/white colored vertices) ranking: function h : V (A) Z that defines partial ordering Can be represented by height: vertical placement of vertices dashing: function d : E(A) Z/2Z values 0/1 edge solid or dashed odd-dashing: when a 4-cycle has an odd number of dashed edges well-dashed: a colored graph whose 2-colored 4-cycles all have an odd-dashing Adinkra: a well-dashed, N-chromotopology with a ranking on its bipartition such that bosons have even ranking and fermions have odd ranking Adinkraizable: a chromotopology that admits well-dashing and ranking as above Main example: the N-cube 2 N vertices labelled with binary codewords of length N connect two verteces with an edge of color i if Hamming distance 1 (number of differing digits), differing at index i ranking h : V (A) Z via h(v) = # of 1 s in v bipartion bosons/fermions: even ranking bosons, odd ranking fermions ranked N-cube chromotopology 2 2N 1 possible dashings for the N-cube 2-cube Adinkra Adinkras and binary codes L linear binary code dimension k (k-dim subspace of Z/2Z N ) codeword c L: weight wt(c) number of 1 s in word L even if every c L has even weight L doubly-even if weight of every codeword divisible by 4 quotient Z/2Z N /L N-cube chromotopology A N and new graph A = A N /L: vertices = equivalence class of vertices, an edge of color i between classes [v] and [w] V (A) iff at least an edge of color i between a v [v] and a w [w] A has a loop iff L has a codeword of weight 1 A has a double edge iff L has a codeword of weight 2 A can be ranked iff A is bipartite iff L even code A can be well-dashed iff L doubly-even code Coding theory a general source of constructions of Adinkras Adinkras and Supersymmetry Algebras Given {φ 1,..., φ m } (bosonic fields) and {ψ 1,..., ψ m } (fermionic fields) with a representation Q k φ a = c λ t ψ b Q k ψ b = i c 1 λ t φ a with c { 1, 1} and λ {0, 1} white vertices: bosonic fields and time derivatives; black vertices: fermionic fields and time derivatives edge structure: white to black (λ = 0), black to white (λ = 1), dashed (c = 1), solid (c = 1) Grothendieck s theory of dessins d enfant Characterizing Riemann surfaces given by algebraic curves defined over number fields in terms of branched coverings of Ĉ = P 1 (C) Belyi maps: X compact Riemann surface, meromorphic function f : X Ĉ unramified outside the points {0, 1, } dessin: bipartite graph Γ embedded on the surface X with white vertices at points f 1 (1), black vertices at points f 1 (0), edges along preimage f 1 (I) of interval I = (0, 1) Belyi pair (X, f ) Riemann surface X with Belyi map f : every Belyi pair defines a dessin and every dessin defines a Belyi pair Riemann surfaces X that admit a Belyi map f : algebraic curves defined over a number field Example dessin corresponding to f (x) = (x 1)3 (x 9) 64x shown that Adinkras determine dessins d enfant: - C. Doran, K. Iga, G. Landweber, S. Méndez-Diez, Geometrization of N-extended 1-dimensional supersymmetry algebras, arxiv: embed Adinkra graph A N,k in a Riemann surface: attach 2-cells to consecutively colored 4-cycles (all 2-colored 4-cycles with color pairs {i, i + 1}, {N, 1}) get oriented Riemann surface X N,k genus g = N k 3 (N 4) if N 2 and genus g = 0 if N 2 Dessins, Adinkras, and Spin Curves D. Cimasoni, N. Reshetikhin, Dimers on surface graphs and spin structures, I, Comm. Math. Phys., Vol. 275 (2007) a dimer configuration on an embedded graph A on a Riemann surface X determines an isomorphism between Kasteleyn orientations on A (up to equivalence) and spin structures on X dimer configuration bipartite graph A, perfect matching : edges such that every vertex incident to exactly one edge perfect matchings = dimer configurations taking edges of a fixed color on an Adinkra determines a dimer configuration Kasteleyn orientation graph embedded on a Riemann surface: orientation of edges so that when going around the boundary of a face counterclockwise going against orientation of an odd number of edges an odd dashing of an Adinkra determines a Kasteleyn orientation dashings equivalent if obtained by sequence of vertex changes: dash/solid of each edge incident to a vertex is changed equivalent dashings give equivalent orientation and same spin structure on X Super Riemann Surfaces - Yu.I. Manin, Topics in Noncommutative Geometry, Princeton University Press, Yu.I. Manin, Gauge Field Theory and Complex Geometry, Springer, M locally modeled on C 1 1 local coordinates z (bosonic) theta (fermionic); subbundle D T C 1 1 defined by D θ = θ + θ z gives D D TM/D [D θ, D θ ] = 2 z D related to spinor bundle S on underlying Riemann surface X An odd dashing on an Adinkra determines a Super Riemann Surface structure on X Spectral Triples start with a Supersymmetry Algebra encode as an Adinkra associated dessin d enfant, Belyi map, and Riemann surface X additional data: Super Riemann Surface, and spin structure spectral triple for the Super Riemann Surface with Dirac operator associated to the assigned spin structure (C (X ), L 2 (X, S E), /D E ) Dirac on Super Riemann Surface is a twisted Dirac on underlying X Spectral Action Various methods for studying explicit spectra of (twisted) Dirac operators on Riemann surfaces: Christian Grosche, Selberg supertrace formula for super Riemann surfaces, analytic properties of Selberg super zeta-functions and multiloop contributions for the fermionic string, Comm. Math. Phys. 133 (1990), no. 3, A. López Almorox, C. Tejero Prieto, Holomorphic spectrum of twisted Dirac operators on compact Riemann surfaces, J. Geom. Phys. 56 (2006), no. 10, still work in progress! also additional work using origami curves: analog of dessins but branched coverings of elliptic curves instead of P 1 (C)
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