## supersymmetry and number theory

A. Kapustin and E. Witten, "Electric-Magnetic Duality And The Geometric Langlands Program", Communications in Number Theory and Physics 1 no. 1 (2007) 1-236

[abstract:] "The geometric Langlands program can be described in a natural way by compactifying on a Riemann surface C a twisted version of $N=4$ super Yang-Mills theory in four dimensions. The key ingredients are electric-magnetic duality of gauge theory, mirror symmetry of sigma-models, branes, Wilson and 't Hooft operators, and topological field theory. Seemingly esoteric notions of the geometric Langlands program, such as Hecke eigensheaves and D-modules, arise naturally from the physics."

S.K. Ashok, F. Cachazo, E. Dell'Aquila, "Children's drawings from Seiberg-Witten curves", Communications in Number Theory and Physics 1 no. 2 (2007) 237-305

[abstract:] "We consider $N=2$ supersymmetric gauge theories perturbed by tree level superpotential terms near isolated singular points in the Coulomb moduli space. We identify the Seiberg-Witten curve at these points with polynomial equations used to construct what Grothendieck called "dessins d'enfants" or "children's drawings" on the Riemann sphere. From a mathematical point of view, the dessins are important because the absolute Galois group $Gal(\bar{Q}/Q)$ acts faithfully on them. We argue that the relation between the dessins and Seiberg-Witten theory is useful because gauge theory criteria used to distinguish branches of $N=1$ vacua can lead to mathematical invariants that help to distinguish dessins belonging to different Galois orbits. For instance, we show that the confinement index defined in hep-th/0301006 is a Galois invariant. We further make some conjectures on the relation between Grothendieck's programme of classifying dessins into Galois orbits and the physics problem of classifying phases of $N=1$ gauge theories. "

S. Bose, J. Gundry and Y.-H. He, "Gauge theories and dessins d'enfants: Beyond the torus" (preprint /2014)

[abstract:] "Dessin d'enfants on elliptic curves are a powerful way of encoding doubly-periodic brane tilings, and thus, of four-dimensional supersymmetric gauge theories whose vacuum moduli space is toric, providing an interesting interplay between physics, geometry, combinatorics and number theory. We discuss and provide a partial classification of the situation in genera other than one by computing explicit Belyi pairs associated to the gauge theories. Important also is the role of the Igusa and Shioda invariants that generalise the elliptic $j$-invariant."

Y.-H. He, "Graph zeta function and gauge theories" (preprint 02/2011)

[abstract:] "Along the recently trodden path of studying certain number theoretic properties of gauge theories, especially supersymmetric theories whose vacuum manifolds are non-trivial, we investigate Ihara's Graph Zeta Function for large classes of quiver theories and periodic tilings by bi-partite graphs. In particular, we examine issues such as the spectra of the adjacency and whether the gauge theory satisfies the strong and weak versions of the graph theoretical analogue of the Riemann Hypothesis."

G. Chalmers, "Comment on the Riemann hypothesis" (preprint 03/05)

[abstract:] "The Riemann hypothesis is identified with zeros of ${\cal N}=4$ supersymmetric gauge theory four-point amplitude. The zeros of the $\zeta(s)$ function are identified with the complex dimension of the spacetime, or the dimension of the toroidal compactification. A sequence of dimensions are identified in order to map the zeros of the amplitude to the Riemann hypothesis."

D. Spector, "Duality, partial supersymmetry, and arithmetic number theory", Journal of Mathematical Physics 39(4) (1998) 1919-1927

"We find examples of duality among quantum theories that are related to arithmetic functions by identifying distinct Hamiltonians that have identical partition functions at suitably related coupling constants or temperatures. We are led to this after first developing the notion of partial supersymmetry, in which some, but not all, of the operators of a theory have superpartners, and using it to construct fermionic and parafermionic thermal partition functions, and to derive some number theoretic identities. In the process, we also find a bosonic analog of the Witten index, and use this, too, to obtain some number theoretic results related to the Riemann zeta function."

D. Spector, "Supersymmetry and the Mobius inversion function", Communications in Mathematical Physics 127 (1990) 239.

"We show that the Mobius inversion function of number theory can be interpreted as the operator (-1)F in quantum field theory...We will see in this paper that the function...has a very natural interpretation. In the proper context, it is equivalent to (-1)F, the operator that distinguishes fermionic from bosonic states and operators, with the fact that mu(n) = 0 when n is not squarefree being equivalent to the Pauli exclusion principle...One of the results we obtain is equivalent to the prime number theorem, one of the central achievements of number theory, in which the asymptotic density of prime numbers is computed."

C. Castro, "On two strategies towards the Riemann Hypothesis: Fractal Supersymmetric QM and a trace formula" (preprint 06/06)

[abstract:] "The Riemann Hypothesis (RH) states that the nontrivial zeros of the Riemann zeta-function are of the form $s_n =1/2+i lambda_n$. An improvement of our previous construction to prove the RH is presented by implementing the Hilbert-Pólya proposal and furnishing the Fractal Supersymmetric Quantum Mechanical (SUSY-QM) model whose spectrum reproduces the imaginary parts of the zeta zeros. We model the fractal fluctuations of the smooth Wu-Sprung potential (that capture the average level density of zeros) by recurring to a weighted superposition of Weierstrass functions $W(x,p,D)$ and where the summation has to be performed over all primes $p$ in order to recapture the connection between the distribution of zeta zeros and prime numbers. We proceed next with the construction of a smooth version of the fractal QM wave equation by writing an ordinary Schrödinger equation whose fluctuating potential (relative to the smooth Wu-Sprung potential) has the same functional form as the fluctuating part of the level density of zeros. The second approach to prove the RH relies on the existence of a continuous family of scaling-like operators involving the Gauss-Jacobi theta series. An explicit trace formula related to a superposition of eigenfunctions of these scaling-like operators is defined. If the trace relation is satisfied this could be another test of the Riemann Hypothesis."

C. Castro (Perelman), "p-Adic stochastic dynamics, supersymmetry and the Riemann conjecture"

"Supersymmetry,p-adic stochastic dynamics, Brownian motion, Fokker-Planck equation, Langevin equation, prime number random distribution, random matrices, p-adic fractal strings, the adelic condition, etc...are all deeply interconnected in this paper."

C. Castro, A. Granik, and J. Mahecha, "On SUSY-QM, fractal strings and steps towards a proof of the Riemann hypothesis"

"The steps towards a proof of Riemann's conjecture using spectral analysis are rigorously provided. We prove that the only zeroes of the Rieamnn zeta-function are of the form s = 1/2 + i lambdan. A supersymmetric quantum mechanical model is proposed as an alternative way to prove the Riemann conjecture, inspired in the Hilbert–Pólya proposal; it uses an inverse scattering approach associated with a system of p-adic harmonic oscillators. An interpretation of the Riemann's fundamental relation Z(s) = Z(1 - s) as a duality relation, from one fractal string L to another dual fractal string L' is proposed."

C. Castro and J. Mahecha, "A fractal SUSY-QM model and the Riemann hypothesis" (preprint 06/03)

[abstract:] "The Riemann hypothesis (RH) states that the nontrivial zeros of the Riemann zeta-function are of the form $s = 1/2 + i\lambda_n$. Hilbert-Polya argued that if a Hermitian operator exists whose eigenvalues are the imaginary parts of the zeta zeros, $\lambda_n$, then the RH is true. In this paper a fractal supersymmetric quantum mechanical (SUSY-QM) model is proposed to prove the RH. It is based on a quantum inverse scattering method related to a fractal potential given by a Weierstrass function (continuous but nowhere differentiable) that is present in the fractal analog of the CBC (Comtet, Bandrauk, Campbell) formula in SUSY QM. It requires using suitable fractal derivatives and integrals of irrational order whose parameter $\beta$ is one-half the fractal dimension of the Weierstrass function. An ordinary SUSY-QM oscillator is constructed whose eigenvalues are of the form $\lambda_n = n\pi$, and which coincide with the imaginary parts of the zeros of the funciton sin(iz). This sine function obeys a trivial analog of the RH. A review of our earlier proof of the RH based on a SUSY QM model whose potential is related ot the Gauss-Jacobi theta series is also included. The spectrum is given by s(1 - s) which is real in the critical line (location of the nontrivial zeros) and in the real axis (location of the trivial zeros)."

C. Castro and J. Mahecha, "Fractal supersymmetric quantum mechanics, geometric probability and the Riemann Hypothesis", International Journal of Geometric Methods in Modern Physics 1 no. 6 (2004) 751-793

[abstract:] "The Riemann Hypothesis (RH) states that the nontrivial zeros of the Riemann zeta-function are of the form $s = 1/2 + i\lambda_{n}$. Earlier work on the RH based on Supersymmetric QM, whose potential was related to the Gauss-Jacobi theta series, allows to provide the proper framework to construct the well defined algorithm to compute the probability to find a zero (an infinity of zeros) in the critical line. Geometric Probability Theory furnishes the answer to the very difficult question whether the probability that the RH is true is indeed equal to unity or not. To test the validity of this Geometric Probabilistic framework to compute the probability if the RH is true, we apply it directly to the hyperbolic sine function sinh(s) case which obeys a trivial analog of the RH. Its zeros are equally spaced in the imaginary axis $s_n = 0 + in\pi$. The Geometric Probability to find a zero (and an infinity of zeros) in the imaginary axis is exactly unity. We proceed with a fractal supersymmetric quantum mechanical (SUSY-QM) model to implement the Hilbert-Pólya proposal to prove the RH by postulating a Hermitian operator that reproduces all the $\lambda_n$'s for its spectrum. Quantum inverse scattering methods related to a fractal potential given by a Weierstrass function (continuous but nowhere differentiable) are applied to the analog of the fractal analog of the CBC (Comtet-Bandrauk-Campbell) formula in SUSY QM. It requires using suitable fractal derivatives and integrals of irrational order whose parameter $\beta$ is one-half the fractal dimension (D = 1.5) of the Weierstrass function. An ordinary SUSY-QM oscillator is also constructed whose eigenvalues are of the form $\lambda_n = n\pi$ and which coincide which the imaginary parts of the zeros of the function sinh(s). Finally, we discuss the relationship to the theory of 1/f noise."

P.B. Slater, "A numerical examination of the Castro-Mahecha supersymmetric model of the Riemann zeros" (preprint 11/05)

[abstract:] "The unknown parameters of the recently-proposed (Int J. Geom. Meth. Mod. Phys. 1, 751 [2004]) Castro-Mahecha model of the imaginary parts (lambda_{j}) of the nontrivial Riemann zeros are the phases (alpha_{k}) and the frequency parameter (gamma) of the Weierstrass function of fractal dimension D=3/2 and the turning points (x_{j}) of the supersymmetric potential-squared Phi^2(x) - which incorporates the smooth Wu-Sprung potential (Phys. Rev. E 48, 2595 [1993]), giving the average level density of the Riemann zeros. We conduct numerical investigations to estimate/determine these parameters - as well as a parameter (sigma) we introduce to scale the fractal contribution. Our primary analyses involve two sets of coupled equations: one set being of the form Phi^{2}(x_{j}) = lambda_{j}, and the other set corresponding to the fractal extension - according to an ansatz of Castro and Mahecha - of the Comtet-Bandrauk-Campbell (CBC) quasi-classical quantization conditions for good supersymmetry. Our analyses suggest the possibility strongly that gamma converges to its theoretical lower bound of 1, and the possibility that all the phases (alpha_{k}) should be set to zero."

Y. Fyodorov, "Negative moments of characteristic polynomials of random matrices: Ingham-Siegel integral as an alternative to Hubbard-Stratonovich transformation", Nuclear Physics B 621 (2002) 643-674.

"We reconsider the problem of calculating arbitrary negative integer moments of the (regularized) characteristic polynomial for N x N random matrices taken from the Gaussian Unitary Ensemble (GUE). A very compact and convenient integral representation is found via the use of a matrix integral close to that considered by Ingham and Siegel. We find the asymptotic expression for the discussed moments in the limit of large N. The latter is of interest because of a conjectured relation to properties of the Riemann zeta-function zeroes. Our method reveals a striking similarity between the structure of the negative and positive integer moments which is usually obscured by the use of the Hubbard-Stratonovich transformation. This sheds a new light on "bosonic" versus "fermionic" replica trick and has some implications for the supersymmetry method. We briefly discuss the case of the chiral GUE model from that perspective."

M. Bordag, A. S. Goldhaber, P. van Nieuwenhuizen and D. Vassilevich, "Heat kernels and zeta-function regularization for the mass of the SUSY kink"

[abstract:] "We apply zeta-function regularization to the kink and susy kink and compute its quantum mass. We fix ambiguities by the renormalization condition that the quantum mass vanishes as one lets the mass gap tend to infinity while keeping scattering data fixed. As an alternative we write the regulated sum over zero point energies in t erms of the heat kernel and apply standard heat kernel subtractions. Finally we discuss to what extent these procedures are equivalent to the usual renormalization conditions that tadpoles vanish."

A. Pérez, M.R. De Traubenberg and P. Simon, "2D-fractional supersymmetry: from rational to irrational conformal field theory"

[abstract:] "Supersymmetry can be consistently generalized in one and two dimensional spaces, fractional supersymmetry being one of the possible extension. Fractional supersymmetry of arbitrary order F is explicitly constructed using an adapted superspace formalism. This symmetry connects the fractional spin states (0,{1 \over F}, \cdots,{ F-1 \over F}). Besides the stress momentum tensor, we obtain a conserved current of spin (1 + { 1 \over F}). The central charges are generally irrational numbers except for the particular cases F=2,3,4,6. A natural classification emerges according to the decomposition of F into its product of prime numbers leading to sub-systems with smaller symmetries. The limit F goes to the infinity is also considered."

D.B. Grunberg, "Integrality of open instanton numbers" (preprint, 05/03)

[abstract:] "We prove the integrality of the open instanton numbers in two examples of counting holomorphic disks on local Calabi-Yau threefolds: the resolved conifold and the degenerate P x P. Given the B-model superpotential, we extract by hand all Gromow-Witten invariants in the expansion of the A-model superpotential. The proof of their integrality relies on enticing congruences of binomial coefficients modulo powers of a prime. We also derive an expression for the factorial (pk-1)! modulo powers of the prime p. We generalise two theorems of elementary number theory, by Wolstenholme and by Wilson."

V. Di Clemente, S. F. King and D.A.J. Rayner, "Supersymmetry and electroweak breaking with large and small extra dimensions", Nucl. Phys. B 617 (2001) 71-100

[abstract:] "We consider the problem of supersymmetry and electroweak breaking in a 5d theory compactified on an $S^{1}/Z_{2}$ orbifold, where the extra dimension may be large or small. We consider the case of a supersymmetry breaking 4d brane located at one of the orbifold fixed points with the Standard Model gauge sector, third family and Higgs fields in the 5d bulk, and the first two families on a parallel 4d matter brane located at the other fixed point. We compute the Kaluza-Klein mass spectrum in this theory using a matrix technique which allows us to interpolate between large and small extra dimensions. We also consider the problem of electroweak symmetry breaking in this theory and localize the Yukawa couplings on the 4d matter brane spatially separated from the brane where supersymmetry is broken. We calculate the 1-loop effective potential using a zeta-function regularization technique, and find that the dominant top and stop contributions are separately finite. Using this result we find consistent electroweak symmetry breaking for a compactification scale {$1/R \approx 830$ GeV} and a lightest Higgs boson mass $m_{h} \approx 170$ GeV."

G. Moore, "Arithmetic and attractors" (preprint 07/03)

[abstract:] "We study relations between some topics in number theory and supersymmetric black holes. These relations are based on the "attractor mechanism" of N=2 supergravity. In IIB string compactification this mechanism singles out certain "attractor varieties". ' We show that these attractor varieties are constructed from products of elliptic curves with complex multiplication for N=4 and N=8 compactifications. The heterotic dual theories are related to rational conformal field theories. In the case of N=4 theories U-duality inequivalent backgrounds with the same horizon area are counted by the class number of a quadratic imaginary field. The attractor varieties are defined over fields closely related to class fields of the quadratic imaginary field. We discuss some extensions to more general Calabi-Yau compactifications and explore further connections to arithmetic including connections to Kronecker's Jugendtraum and the theory of modular heights. The paper also includes a short review of the attractor mechanism. A much shorter version of the paper summarizing the main points is the companion note entitled "Attractors and Arithmetic""

[abstract:] "This paper is a review of some interesting results that has been obtained in the study of the quantum gravity partition functions in three-dimensions, in the Selberg zeta function, in the vanishing of cosmological constant and in the ten-dimensional anomaly cancellations. In the Section 1, we have described some equations concerning the pure three-dimensional quantum gravity with a negative cosmological constant and the pure three-dimensional supergravity partition functions. In the Section 2, we have described some equations concerning the Selberg super-trace formula for Super-Riemann surfaces, some analytic properties of Selberg super zeta-functions and multiloop contributions for the fermionic strings. In the Section 3, we have described some equations concerning the ten-dimensional anomaly cancellations and the vanishing of cosmological constant. In the Section 4, we have described some equations concerning p-adic strings, p-adic and adelic zeta functions and zeta strings. In conclusion, in the Section 5, we have described the possible and very interesting mathematical connections obtained between some equations regarding the various sections and some sectors of number t heory (Riemann zeta functions, Ramanujan modular equations, etc...) and some interesting mathematical applications concerning the Selberg super-zeta functions and some equations regarding the Section 1."

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