The quantum mechanics of the supersymmetric nonlinear σ-model

Davis, A.C.; Macfarlane, Alan; Popat, P C; Holten, J.W. van

doi:10.1088/0305-4470/17/15/012

Cited by 47 publications

(33 citation statements)

References 13 publications

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“…See also the discussion in [131,132]. It follows from the identification (4.84) that the super-covariant momentum operator,π m , is represented by the e 1/2 -twisted covariant derivative on spinors: …”

Section: Quantizationmentioning

confidence: 99%

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Semiclassical framed BPS states

Moore

Royston

Bleeken

2016

J. High Energ. Phys.

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We provide a semiclassical description of framed BPS states in fourdimensional N = 2 super Yang-Mills theories probed by 't Hooft defects, in terms of a supersymmetric quantum mechanics on the moduli space of singular monopoles. Framed BPS states, like their ordinary counterparts in the theory without defects, are associated with the L 2 kernel of certain Dirac operators on moduli space, or equivalently with the L 2 cohomology of related Dolbeault operators. The Dirac/Dolbeault operators depend on two Cartan-valued Higgs vevs. We conjecture a map between these vevs and the SeibergWitten special coordinates, consistent with a one-loop analysis and checked in examples. The map incorporates all perturbative and nonperturbative corrections that are relevant for the semiclassical construction of BPS states, over a suitably defined weak coupling regime of the Coulomb branch. We use this map to translate wall crossing formulae and the no-exotics theorem to statements about the Dirac/Dolbeault operators. The no-exotics theorem, concerning the absence of nontrivial SU(2) R representations in the BPS spectrum, implies that the kernel of the Dirac operator is chiral, and further translates into a statement that all L 2 cohomology of the Dolbeault operator is concentrated in the middle degree. Wall crossing formulae lead to detailed predictions for where the Dirac operators fail to be Fredholm and how their kernels jump. We explore these predictions in nontrivial examples. This paper explains the background and arguments behind the results announced in the short note [1].

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Section: Quantizationmentioning

confidence: 99%

“…These ambiguities exist only because of our 'truncation then quantization' approach; see footnote 41. However in the cases at hand it is easy to resolve them by demanding that the corresponding operator be Hermitian [131,132]…”

Section: Quantizationmentioning

confidence: 99%

Semiclassical framed BPS states

Moore

Royston

Bleeken

2016

J. High Energ. Phys.

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“…Solving Eq. (5·5) with respect to GO, we obtain Since za, za*, ijA(-)7ra, ijA(-)7ra* are commutable with qA, the product-formula of POM readily yields pA({za*, 7ra*}_{za, 7r a })=({pA z a*, pA 7ra *}_{pA z a, p A 7ra}) , (5)(6)(7)(8)(9)(10)(11) where Similarly, the other condition…”

Section: [¢Aa(x) Itb(y)]=(g2/2i){¢aa(x) Zb*(y)}8(x-y)mentioning

confidence: 97%

“…Using the product-formula, further, the projection of{¢aa(x), cjJ/(y)} becomes PA{ ¢aa(x), cjJ/(y)}={cosZ[8A(x, y)] -sin Z [ 8 A (x, y)]( y2)ar( y2)pS}{ pA ¢ra(X), gJAcjJl(y)} + (1/ 4)sin[2 8 A (x, y) ]rJ/[ p A ¢ra(X),· PAcjJl(y)] ,(5 -12) where rJ/=Oa/yZ)ps-(yZ)arOPs.Since 8 A (x, y)becomes zero when x=y, we obtain From Eqs. (5-10),(5)(6)(7)(8)(9)(10)(11) and(5)(6)(7)(8)(9)(10)(11)(12)(13), thus, we obtain the projection-condition…”

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confidence: 99%

Commutators in Supersymmetric CPn-12 Model

Nakamura

Minowa

1991

Progress of Theoretical Physics

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The supersymmetric extension of cp,n-l model is investigated under the several gauge-fixing conditions by using the proiection-operator-method of constrained dynamics. It is shown that, when the gauge-freedom is fixed, the commutation relations among the field operators contain certain quantum corrections, which can never be obtained by the quantization with the Dirac bracket. The physical interpretation for the difference between these two approaches to the quantization of constrained systems is given. § 1. IntroductionNonlinear sigma models and .their supersymmetric extensions have played the important roles in the investigations of the gauge theories in four dimensions and the string theories.!) The recent field theoretical approach 2 ) to the high-Tc superconductivity has caused a renewed interest in these models. They are the constrained systems in which the constraints are in the second-class and nonlinear with respect to the field variables. The Hamiltonian formulation of such systems are accomplished by using the generalized Hamiltonian formalism due to Dirac,3) and there have been many investigations 4 )-8) of nonlinear sigma models with Dirac's formalism. Now; there are two standard approaches to the quantization of constrained systems. The first approach is to impose the constraints first and then to carry out the quantization on the reduced phase space. In the second, one first performs the quantization on the whole phase space and then imposes the constraints. Then, one often encounters the situations in which the two are not equivalent. In the case of first-class constraints, recently, this problem has been discussed in the investigations of the quantization of the Chern-Simons theories. 9 ),IO) There, it has been show~ that there arises the explicit difference between the two approaches in the calculation of the quantum holonomy.9) In the case of second-class ones, the first approach corresponds to the quantization with the Dirac brackets because they are equivalent to the Poisson brackets on the reduced phase space spanned by the new independent variables constructed with solving the constraints.ll) Although we have proposed the alternative approach to quantize constrained systems, which we have called the projection-operator-method (POM)/2) on the other hand, it is in the second approach. Then, there also arises the situation in which the various differences between these two approaches occur: When the first approach is used, for example, the Heisenberg equations of motion in several nonlinear constrained systems miss certain quantum effects, which are derived by using the second one. 13 ) Since the canonical structures are equivalent in these cases, however, it may be possible to avoid this problem by using the Dirac brackets to only setting up the commutation relations among the

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“…Hats are not placed over the operator quantities to simplify the notation 3. Further discussion of ambiguities in the definition of Hermitian operators in supersymmetric models can be found, e.g., in[23],[24].…”

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confidence: 99%

Massless spinning particle and null-string on AdS_d: projective-space approach

Uvarov¹

2018

J. Phys. A: Math. Theor.

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Massless spinning particle and tensionless string models on AdS d background in the projective-space realization are proposed as constrained Hamiltonian systems. Various forms of particle and string Lagrangians are derived and classical mechanics is studied including the Lax-type representation of the equations of motion. After that transition to the quantum theory is discussed. Analysis of potential anomalies in the tensionless string model necessitates introduction of ghosts and BRST charge. It is shown that quantum BRST charge is nilpotent for any d if coordinate-momentum ordering for the phase-space bosonic variables, Weyl ordering for the fermions and cb (γβ) ordering for ghosts is chosen, while conformal reparametrizations and space-time dilatations turn out to be anomalous for the ordering in terms of positive and negative Fourier modes of the phase-space variables and ghosts.

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The quantum mechanics of the supersymmetric nonlinear σ-model

Cited by 47 publications

References 13 publications

Semiclassical framed BPS states

Semiclassical framed BPS states

Commutators in Supersymmetric CPn-12 Model

Massless spinning particle and null-string on AdS_d: projective-space approach

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The quantum mechanics of the supersymmetric nonlinear σ-model

Cited by 47 publications

References 13 publications

Semiclassical framed BPS states

Semiclassical framed BPS states

Commutators in Supersymmetric CPn-12 Model

Massless spinning particle and null-string on AdS d : projective-space approach

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Product

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Massless spinning particle and null-string on AdS_d: projective-space approach