scholarly journals Two-dimensional difference Dirac operator in the strip

2015 ◽  
Vol 25 (1) ◽  
pp. 93-100
Author(s):  
T.S. Tinyukova ◽  
Keyword(s):  
Dirac Operator ◽  
Two Dimensional

scholarly journals Spectrum of the Dirac operator coupled to two-dimensional quantum gravity

2002 ◽  
Vol 630 (1-2) ◽  
pp. 339-358 ◽  
Author(s):  
L. Bogacz ◽  
Z. Burda ◽  
C. Petersen ◽  
B. Petersson
Keyword(s):  
Quantum Gravity ◽  
Dirac Operator ◽  
Two Dimensional

FERMIONIC ZERO MODES IN SELF-DUAL VORTEX BACKGROUND ON A SPHERE

2007 ◽  
Vol 22 (21) ◽  
pp. 3643-3653 ◽  
Author(s):  
YU-XIAO LIU ◽  
LI ZHAO ◽  
LI-JIE ZHANG ◽  
YI-SHI DUAN
Keyword(s):  
Riemann Surface ◽  
Dirac Equation ◽  
Dirac Operator ◽  
Topological Structure ◽  
The Self ◽  
Two Dimensional ◽  
Zero Modes ◽  
Self Duality ◽  
Effective Lagrangian ◽  
Topological Term

We study fermionic zero modes in the self-dual vortex background on an extra two-dimensional Riemann surface in 5+1 dimensions. Using the generalized Abelian Higgs model, we obtain the inner topological structure of the self-dual vortex and establish the exact self-duality equation with topological term. Then we analyze the Dirac operator on an extra sphere and the effective Lagrangian of four-dimensional fermions with the self-dual vortex background. Solving the Dirac equation, the fermionic zero modes on a sphere in two simple cases are obtained.

scholarly journals A Note on the Three Dimensional Dirac Operator with Zigzag Type Boundary Conditions

2021 ◽  
Vol 15 (3) ◽  
Author(s):  
Markus Holzmann
Keyword(s):  
Dirac Operator ◽  
Boundary Conditions ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Dirichlet Laplacian ◽  
Open Set ◽  
Type Boundary ◽  
Infinite Multiplicity ◽  
Zigzag Type

AbstractIn this note the three dimensional Dirac operator $$A_m$$ A m with boundary conditions, which are the analogue of the two dimensional zigzag boundary conditions, is investigated. It is shown that $$A_m$$ A m is self-adjoint in $$L^2(\Omega ;{\mathbb {C}}^4)$$ L 2 ( Ω ; C 4 ) for any open set $$\Omega \subset {\mathbb {R}}^3$$ Ω ⊂ R 3 and its spectrum is described explicitly in terms of the spectrum of the Dirichlet Laplacian in $$\Omega $$ Ω . In particular, whenever the spectrum of the Dirichlet Laplacian is purely discrete, then also the spectrum of $$A_m$$ A m consists of discrete eigenvalues that accumulate at $$\pm \infty $$ ± ∞ and one additional eigenvalue of infinite multiplicity.

scholarly journals From $$ \mathcal{N} $$ = 4 Super-Yang-Mills on ℝℙ4 to bosonic Yang-Mills on ℝℙ2

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
Yifan Wang
Keyword(s):  
Partition Function ◽  
Dirac Operator ◽  
Large Class ◽  
Integral Form ◽  
Two Dimensional ◽  
Expectation Values ◽  
Yang Mills ◽  
Spacetime Manifold ◽  
Agt Correspondence ◽  
Potential Applications

Abstract We study the four-dimensional $$ \mathcal{N} $$ N = 4 super-Yang-Mills (SYM) theory on the unorientable spacetime manifold ℝℙ4. Using supersymmetric localization, we find that for a large class of local and extended SYM observables preserving a common supercharge $$ \mathcal{Q} $$ Q , their expectation values are captured by an effective two-dimensional bosonic Yang-Mills (YM) theory on an ℝℙ2 submanifold. This paves the way for understanding $$ \mathcal{N} $$ N = 4 SYM on ℝℙ4 using known results of YM on ℝℙ2. As an illustration, we derive a matrix integral form of the SYM partition function on ℝℙ4 which, when decomposed into discrete holonomy sectors, contains subtle phase factors due to the nontrivial η-invariant of the Dirac operator on ℝℙ4. We also comment on potential applications of our setup for AGT correspondence, integrability and bulk-reconstruction in AdS/CFT that involve cross-cap states on the boundary.

scholarly journals Spectra of graphene nanoribbons with armchair and zigzag boundary conditions

2014 ◽  
Vol 26 (10) ◽  
pp. 1450018 ◽  
Author(s):  
Pedro Freitas ◽  
Petr Siegl
Keyword(s):  
Dirac Operator ◽  
Boundary Conditions ◽  
Spectral Properties ◽  
Graphene Nanoribbons ◽  
Continuous Model ◽  
Bounded Domains ◽  
Two Dimensional ◽  
Edge States ◽  
Finite Multiplicity

We study the spectral properties of the two-dimensional Dirac operator on bounded domains together with the appropriate boundary conditions which provide a (continuous) model for graphene nanoribbons. These are of two types, namely, the so-called armchair and zigzag boundary conditions, depending on the line along which the material was cut. In the former case, we show that the spectrum behaves in what might be called a classical way; while in the latter, we prove the existence of a sequence of finite multiplicity eigenvalues converging to zero and which correspond to edge states.

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