scholarly journals From quantum link models to D-theory: a resource efficient framework for the quantum simulation and computation of gauge theories

2021 ◽  
Vol 380 (2216) ◽  
Author(s):  
Uwe-Jens Wiese
Keyword(s):  
Hilbert Space ◽  
Gauge Theory ◽  
Dimensional Reduction ◽  
Gauge Theories ◽  
Quantum Simulation ◽  
Quantum Spin ◽  
Coulomb Phase ◽  
Finite Dimensional ◽  
Link Model ◽  
Quantum Link

Quantum link models provide an extension of Wilson’s lattice gauge theory in which the link Hilbert space is finite-dimensional and corresponds to a representation of an embedding algebra. In contrast to Wilson’s parallel transporters, quantum links are intrinsically quantum degrees of freedom. In D-theory, these discrete variables undergo dimensional reduction, thus giving rise to asymptotically free theories. In this way ( 1 + 1 ) -d C P ( N − 1 ) models emerge by dimensional reduction from ( 2 + 1 ) -d S U ( N ) quantum spin ladders, the ( 2 + 1 ) -d confining U ( 1 ) gauge theory emerges from the Abelian Coulomb phase of a ( 3 + 1 ) -d quantum link model, and ( 3 + 1 ) -d QCD arises from a non-Abelian Coulomb phase of a ( 4 + 1 ) -d S U ( 3 ) quantum link model, with chiral quarks arising naturally as domain wall fermions. Thanks to their finite-dimensional Hilbert space and their economical mechanism of reaching the continuum limit by dimensional reduction, quantum link models provide a resource efficient framework for the quantum simulation and computation of gauge theories. This article is part of the theme issue ‘Quantum technologies in particle physics’.

scholarly journals a-MAXIMIZATION IN ${\mathcal N}=1$ SUPERSYMMETRIC Spin(10) GAUGE THEORIES

2010 ◽  
Vol 25 (31) ◽  
pp. 5595-5645
Author(s):  
TERUHIKO KAWANO ◽  
FUTOSHI YAGI
Keyword(s):  
Field Theory ◽  
Gauge Theory ◽  
Conformal Field Theory ◽  
Gauge Theories ◽  
Unitarity Bound ◽  
Conformal Field ◽  
Coulomb Phase ◽  
Vector Representations ◽  
Chiral Superfields

A summary is reported on our previous publications about four-dimensional [Formula: see text] supersymmetric Spin(10) gauge theory with chiral superfields in the spinor and vector representations in the non-Abelian Coulomb phase. Carrying out the method of a-maximization, we studied decoupling operators in the infrared and the renormalization flow of the theory. We also give a brief review on the non-Abelian Coulomb phase of the theory after recalling the unitarity bound and the a-maximization procedure in four-dimensional conformal field theory.

scholarly journals On Gauge Invariance of the Bosonic Measure in Chiral Gauge Theories

Universe ◽  
2021 ◽  
Vol 7 (8) ◽  
pp. 283
Author(s):  
Gabriel de Lima e Silva ◽  
Thalis José Girardi ◽  
Sebastião Alves Dias
Keyword(s):  
Hilbert Space ◽  
Gauge Theory ◽  
Path Integral ◽  
Gauge Invariance ◽  
Gauge Theories ◽  
Direct Calculation ◽  
Identity Operator ◽  
Anomaly Cancellation ◽  
Gauge Anomaly ◽  
General Gauge

Gauge invariance of the measure associated with the gauge field is usually taken for granted, in a general gauge theory. We furnish a proof of this invariance, within Fujikawa’s approach. To stress the importance of this fact, we briefly review gauge anomaly cancellation as a consequence of gauge invariance of the bosonic measure and compare this cancellation to usual results from algebraic renormalization, showing that there are no potential inconsistencies. Then, using a path integral argument, we show that a possible Jacobian for the gauge transformation has to be the identity operator, in the physical Hilbert space. We extend the argument to the complete Hilbert space by a direct calculation.

scholarly journals Quantum spin systems and supersymmetric gauge theories. Part I

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
Norton Lee ◽  
Nikita Nekrasov
Keyword(s):  
Gauge Theory ◽  
Spin Chain ◽  
Gauge Theories ◽  
Surface Defects ◽  
Toda Chain ◽  
Quantum Spin ◽  
Spin Systems ◽  
Quantum Spin Systems ◽  
Supersymmetric Gauge Theories ◽  
Supersymmetric Gauge

Abstract The relation between supersymmetric gauge theories in four dimensions and quantum spin systems is exploited to find an explicit formula for the Jost function of the N site $$ \mathfrak{sl} $$ sl 2X X X spin chain (for infinite dimensional complex spin representations), as well as the SLN Gaudin system, which reduces, in a limiting case, to that of the N-particle periodic Toda chain. Using the non-perturbative Dyson-Schwinger equations of the supersymmetric gauge theory we establish relations between the spin chain commuting Hamiltonians with the twisted chiral ring of gauge theory. Along the way we explore the chamber dependence of the supersymmetric partition function, also the expectation value of the surface defects, giving new evidence for the AGT conjecture.

scholarly journals U(∞) GAUGE THEORY FROM HIGHER DIMENSIONS

1992 ◽  
Vol 07 (24) ◽  
pp. 6025-6037
Author(s):  
KIYOSHI SHIRAISHI
Keyword(s):  
Gauge Theory ◽  
Dimensional Reduction ◽  
Gauge Theories ◽  
Higher Dimensions ◽  
Gauge Fields ◽  
Degenerate Metric ◽  
Infinite Dimensional ◽  
Higher Derivative ◽  
Exact Symmetry

We show that classical U (∞) gauge theories can be obtained from the dimensional reduction of a certain class of higher-derivative theories. In general, the exact symmetry is attained in the limit of degenerate metric; otherwise, the infinite-dimensional symmetry can be taken as spontaneously broken. Monopole solutions are examined in the model for scalar and gauge fields. An extension to gravity is also discussed.

A note on normal operators

1963 ◽  
Vol 59 (4) ◽  
pp. 727-729 ◽  
Author(s):  
S. J. Bernau ◽  
F. Smithies
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Adjoint Operator ◽  
Spectral Theorem ◽  
Unitary Operators ◽  
Unitary Space ◽  
Bounded Linear ◽  
Finite Dimensional ◽  
Normal Operators ◽  
Finite Dimensional Case

We recall that a bounded linear operator T in a Hilbert space or finite-dimensional unitary space is said to be normal if T commutes with its adjoint operator T*, i.e. TT* = T*T. Most of the proofs given in the literature for the spectral theorem for normal operators, even in the finite-dimensional case, appeal to the corresponding results for Hermitian or unitary operators.

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 Effective String Description of the Confining Flux Tube at Finite Temperature

Universe ◽  
2021 ◽  
Vol 7 (6) ◽  
pp. 170
Author(s):  
Michele Caselle
Keyword(s):  
Flux Tube ◽  
Dimensional Reduction ◽  
Conformal Invariance ◽  
Finite Temperature ◽  
Gauge Theories ◽  
Linear Increase ◽  
General Introduction ◽  
Interquark Potential ◽  
Effective String Theory ◽  
Good Agreement

In this review, after a general introduction to the effective string theory (EST) description of confinement in pure gauge theories, we discuss the behaviour of EST as the temperature is increased. We show that, as the deconfinement point is approached from below, several universal features of confining gauge theories, like the ratio Tc/σ0, the linear increase of the squared width of the flux tube with the interquark distance, or the temperature dependence of the interquark potential, can be accurately predicted by the effective string. Moreover, in the vicinity of the deconfinement point the EST behaviour turns out to be in good agreement with what was predicted by conformal invariance or by dimensional reduction, thus further supporting the validity of an EST approach to confinement.

scholarly journals When are maps preserving semi-inner products linear?

2021 ◽  
Author(s):  
Paweł Wójcik
Keyword(s):  
Hilbert Space ◽  
Infinite Dimensions ◽  
Normed Spaces ◽  
Linear Isometry ◽  
Inner Products ◽  
Finite Dimensional ◽  
Uniformly Smooth ◽  
Non Linear ◽  
Extra Assumption ◽  
Continuous Injection

AbstractWe observe that every map between finite-dimensional normed spaces of the same dimension that respects fixed semi-inner products must be automatically a linear isometry. Moreover, we construct a uniformly smooth renorming of the Hilbert space $$\ell _2$$ ℓ 2 and a continuous injection acting thereon that respects the semi-inner products, yet it is non-linear. This demonstrates that there is no immediate extension of the former result to infinite dimensions, even under an extra assumption of uniform smoothness.

scholarly journals META-STABLE BRANE CONFIGURATIONS WITH MULTIPLE NS5-BRANES

2009 ◽  
Vol 24 (27) ◽  
pp. 5051-5120
Author(s):  
CHANGHYUN AHN
Keyword(s):  
Gauge Theory ◽  
String Theory ◽  
Gauge Group ◽  
Gauge Theories ◽  
The Other ◽  
Multiple Product

Starting from an [Formula: see text] supersymmetric electric gauge theory with the multiple product gauge group and the bifundamentals, we apply Seiberg dual to each gauge group, obtain the [Formula: see text] supersymmetric dual magnetic gauge theories with dual matters including the gauge singlets. Then we describe the intersecting brane configurations, where there are NS-branes and D4-branes (and anti-D4-branes), of type IIA string theory corresponding to the meta-stable nonsupersymmetric vacua of this gauge theory. We also discuss the case where the orientifold 4-planes are added into the above brane configuration. Next, by adding an orientifold 6-plane, we apply to an [Formula: see text] supersymmetric electric gauge theory with the multiple product gauge group (where a single symplectic or orthogonal gauge group is present) and the bifundamentals. Finally, we describe the other cases where the orientifold 6-plane intersects with NS-brane.

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