scholarly journals Yang–Mills theory and the ABC conjecture

2018 ◽  
Vol 33 (13) ◽  
pp. 1850053
Author(s):  
Yang-Hui He ◽  
Zhi Hu ◽  
Malte Probst ◽  
James Read
Keyword(s):  
Gauge Theory ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Moduli Space ◽  
Fundamental Domain ◽  
Abc Conjecture ◽  
Dimer Model ◽  
Yang Mills ◽  
Large Samples ◽  
Brane Tiling

We establish a precise correspondence between the ABC Conjecture and [Formula: see text] super-Yang–Mills theory. This is achieved by combining three ingredients: (i) Elkies’ method of mapping ABC-triples to elliptic curves in his demonstration that ABC implies Mordell/Faltings; (ii) an explicit pair of elliptic curve and associated Belyi map given by Khadjavi–Scharaschkin; and (iii) the fact that the bipartite brane-tiling/dimer model for a gauge theory with toric moduli space is a particular dessin d’enfant in the sense of Grothendieck. We explore this correspondence for the highest quality ABC-triples as well as large samples of random triples. The conjecture itself is mapped to a statement about the fundamental domain of the toroidal compactification of the string realization of [Formula: see text] SYM.

scholarly journals Truncation of lattice N = 4 super Yang-Mills

2018 ◽  
Vol 175 ◽  
pp. 11008 ◽  
Author(s):  
Joel Giedt ◽  
Simon Catterall ◽  
Raghav Govind Jha
Keyword(s):  
Gauge Theory ◽  
Gauge Symmetry ◽  
Moduli Space ◽  
Lattice Spacing ◽  
Continuum Limit ◽  
Power Counting ◽  
Yang Mills ◽  
Free Theory ◽  
Lattice Gauge ◽  
Two Sectors

In twisted and orbifold formulations of lattice N = 4 super Yang-Mills, the gauge group is necessarily U(1) × SU(N), in order to be consistent with the exact scalar supersymmetry Q. In the classical continuum limit of the theory, where one expands the link fields around a point in the moduli space and sends the lattice spacing to zero, the diagonal U(1) modes decouple from the SU(N) sector, and give an uninteresting free theory. However, lattice artifacts (described by irrelevant operators according to naive power-counting) couple the two sectors, so removing the U(1) modes is a delicate issue. We describe how this truncation to an SU(N) gauge theory can be obtained in a systematic way, with violations of Q that fall off as powers of 1=N2. We are able to achieve this while retaining exact SU(N) lattice gauge symmetry at all N, and provide both theoretical arguments and numerical evidence for the 1=N2 suppression of Q violation.

scholarly journals CANONICAL LIFTS OF FAMILIES OF ELLIPTIC CURVES

2017 ◽  
Vol 233 ◽  
pp. 193-213 ◽  
Author(s):  
JAMES BORGER ◽  
LANCE GURNEY
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Moduli Space ◽  
Level Structure ◽  
Point Of View ◽  
Universal Property ◽  
Formal Proofs ◽  
Witt Vectors ◽  
Formal Schemes

We show that the canonical lift construction for ordinary elliptic curves over perfect fields of characteristic $p>0$ extends uniquely to arbitrary families of ordinary elliptic curves, even over $p$-adic formal schemes. In particular, the universal ordinary elliptic curve has a canonical lift. The existence statement is largely a formal consequence of the universal property of Witt vectors applied to the moduli space of ordinary elliptic curves, at least with enough level structure. As an application, we show how this point of view allows for more formal proofs of recent results of Finotti and Erdoğan.

scholarly journals POISSON GEOMETRY OF PARABOLIC BUNDLES ON ELLIPTIC CURVES

2008 ◽  
Vol 19 (03) ◽  
pp. 339-367
Author(s):  
DAVID BALDUZZI
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Poisson Structure ◽  
Moduli Space ◽  
Loop Group ◽  
Poisson Geometry ◽  
Double Loop ◽  
Sheaf Cohomology ◽  
Parabolic Bundles

The moduli space of G-bundles on an elliptic curve with additional flag structure admits a Poisson structure. The bivector can be defined using double loop group, loop group and sheaf cohomology constructions. We investigate the links between these methods and for the case SL2 perform explicit computations, describing the bracket and its leaves in detail.

scholarly journals Novel wall-crossing behaviour in rank one $$ \mathcal{N} $$ = 2* gauge theory

2021 ◽  
Vol 2021 (9) ◽  
Author(s):  
Philipp Rüter ◽  
Richard J. Szabo
Keyword(s):  
Gauge Theory ◽  
Gauge Group ◽  
Moduli Space ◽  
A Priori ◽  
Field Theories ◽  
Numerical Techniques ◽  
Yang Mills ◽  
Wall Crossing ◽  
Rank One ◽  
Bps Spectrum

Abstract We study the BPS spectrum of four-dimensional $$ \mathcal{N} $$ N = 2 supersymmetric Yang-Mills theory with gauge group SU(2) and a massive adjoint hypermultiplet, which has an extremely intricate structure with infinite spectrum in all chambers of its Coulomb moduli space, and is not well understood. We build on previous results by employing the BPS quiver description of the spectrum, and explore the qualitative features in detail using numerical techniques. We find novel and unexpected behaviour in the form of wall-crossings involving interactions between BPS particles with negative electric-magnetic pairings, which we interpret in terms of the reverse orderings of the usual wall-crossing formulas for rank one $$ \mathcal{N} $$ N = 2 field theories. This identifies new a priori unrelated states in the spectrum.

PERSPECTIVES OF ELLIPTICAL CRYPTOGRAPHY TO ENSURE THE INTEGRITY AND CONFIDENTIALITY OF INFORMATION

2019 ◽  
Author(s):  
Anna ILYENKO ◽  
Sergii ILYENKO ◽  
Yana MASUR
Keyword(s):  
Information Systems ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Finite Fields ◽  
Computational Efficiency ◽  
Digital Signature ◽  
Discrete Logarithm ◽  
Algebraic Groups ◽  
Schnorr Signature ◽  
Stability Of Algorithms

In this article, the main problems underlying the current asymmetric crypto algorithms for the formation and verification of electronic-digital signature are considered: problems of factorization of large integers and problems of discrete logarithm. It is noted that for the second problem, it is possible to use algebraic groups of points other than finite fields. The group of points of the elliptical curve, which satisfies all set requirements, looked attractive on this side. Aspects of the application of elliptic curves in cryptography and the possibilities offered by these algebraic groups in terms of computational efficiency and crypto-stability of algorithms were also considered. Information systems using elliptic curves, the keys have a shorter length than the algorithms above the finite fields. Theoretical directions of improvement of procedure of formation and verification of electronic-digital signature with the possibility of ensuring the integrity and confidentiality of information were considered. The proposed method is based on the Schnorr signature algorithm, which allows data to be recovered directly from the signature itself, similarly to RSA-like signature systems, and the amount of recoverable information is variable depending on the information message. As a result, the length of the signature itself, which is equal to the sum of the length of the end field over which the elliptic curve is determined, and the artificial excess redundancy provided to the hidden message was achieved.

scholarly journals The Volume of the Quiver Vortex Moduli Space

2021 ◽  
Author(s):  
Kazutoshi Ohta ◽  
Norisuke Sakai
Keyword(s):  
Gauge Theory ◽  
Moduli Space ◽  
Riemann Surfaces ◽  
Gauge Theories ◽  
Target Space ◽  
Contour Integral ◽  
Quiver Gauge Theory ◽  
Volume Formula ◽  
Volume Operator ◽  
Space Volume

Abstract We study the moduli space volume of BPS vortices in quiver gauge theories on compact Riemann surfaces. The existence of BPS vortices imposes constraints on the quiver gauge theories. We show that the moduli space volume is given by a vev of a suitable cohomological operator (volume operator) in a supersymmetric quiver gauge theory, where BPS equations of the vortices are embedded. In the supersymmetric gauge theory, the moduli space volume is exactly evaluated as a contour integral by using the localization. Graph theory is useful to construct the supersymmetric quiver gauge theory and to derive the volume formula. The contour integral formula of the volume (generalization of the Jeffrey-Kirwan residue formula) leads to the Bradlow bounds (upper bounds on the vorticity by the area of the Riemann surface divided by the intrinsic size of the vortex). We give some examples of various quiver gauge theories and discuss properties of the moduli space volume in these theories. Our formula are applied to the volume of the vortex moduli space in the gauged non-linear sigma model with CPN target space, which is obtained by a strong coupling limit of a parent quiver gauge theory. We also discuss a non-Abelian generalization of the quiver gauge theory and “Abelianization” of the volume formula.

scholarly journals Primitive divisors of sequences associated to elliptic curves with complex multiplication

2021 ◽  
Vol 7 (2) ◽  
Author(s):  
Matteo Verzobio
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Number Field ◽  
Complex Multiplication ◽  
Dedekind Domain ◽  
Torsion Point ◽  
Integral Ideal ◽  
Primitive Divisors ◽  
Primitive Divisor ◽  
The Ideal

AbstractLet P and Q be two points on an elliptic curve defined over a number field K. For $$\alpha \in {\text {End}}(E)$$ α ∈ End ( E ) , define $$B_\alpha $$ B α to be the $$\mathcal {O}_K$$ O K -integral ideal generated by the denominator of $$x(\alpha (P)+Q)$$ x ( α ( P ) + Q ) . Let $$\mathcal {O}$$ O be a subring of $${\text {End}}(E)$$ End ( E ) , that is a Dedekind domain. We will study the sequence $$\{B_\alpha \}_{\alpha \in \mathcal {O}}$$ { B α } α ∈ O . We will show that, for all but finitely many $$\alpha \in \mathcal {O}$$ α ∈ O , the ideal $$B_\alpha $$ B α has a primitive divisor when P is a non-torsion point and there exist two endomorphisms $$g\ne 0$$ g ≠ 0 and f so that $$f(P)= g(Q)$$ f ( P ) = g ( Q ) . This is a generalization of previous results on elliptic divisibility sequences.

scholarly journals DISCRETE AND CONTINUUM APPROACHES TO THREE-DIMENSIONAL QUANTUM GRAVITY

1991 ◽  
Vol 06 (39) ◽  
pp. 3591-3600 ◽  
Author(s):  
HIROSI OOGURI ◽  
NAOKI SASAKURA
Keyword(s):  
Gauge Theory ◽  
Moduli Space ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Analogue Model ◽  
Gauge Invariant ◽  
Invariant Functions ◽  
Dimensional Lattice ◽  
Chern Simons ◽  
Dimensional Surface

It is shown that, in the three-dimensional lattice gravity defined by Ponzano and Regge, the space of physical states is isomorphic to the space of gauge-invariant functions on the moduli space of flat SU(2) connections over a two-dimensional surface, which gives physical states in the ISO(3) Chern–Simons gauge theory. To prove this, we employ the q-analogue of this model defined by Turaev and Viro as a regularization to sum over states. A recent work by Turaev suggests that the q-analogue model itself may be related to an Euclidean gravity with a cosmological constant proportional to 1/k2, where q=e2πi/(k+2).

scholarly journals Equidistribution Among Cosets of Elliptic Curve Points in Intervals

2020 ◽  
Vol 14 (1) ◽  
pp. 339-345
Author(s):  
Taechan Kim ◽  
Mehdi Tibouchi
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Character Sums ◽  
Fault Analysis ◽  
Signature Schemes ◽  
Large Subgroup

AbstractIn a recent paper devoted to fault analysis of elliptic curve-based signature schemes, Takahashi et al. (TCHES 2018) described several attacks, one of which assumed an equidistribution property that can be informally stated as follows: given an elliptic curve E over 𝔽q in Weierstrass form and a large subgroup H ⊂ E(𝔽q) generated by G(xG, yG), the points in E(𝔽q) whose x-coordinates are obtained from xG by randomly flipping a fixed, sufficiently long substring of bits (and rejecting cases when the resulting value does not correspond to a point in E(𝔽q)) are close to uniformly distributed among the cosets modulo H. The goal of this note is to formally state, prove and quantify (a variant of) that property, and in particular establish sufficient bounds on the size of the subgroup and on the length of the substring of bits for it to hold. The proof relies on bounds for character sums on elliptic curves established by Kohel and Shparlinski (ANTS–IV).

Self-dual propagating wave solutions in Yang-Mills gauge theory

1979 ◽  
Vol 19 (12) ◽  
pp. 3649-3652 ◽  
Author(s):  
Eve Kovacs ◽  
Shui-Yin Lo
Keyword(s):  
Gauge Theory ◽  
Yang Mills ◽  
Wave Solutions
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