scholarly journals Understanding the index theorems with massive fermions

2021 ◽  
Vol 36 (26) ◽  
Author(s):  
Hidenori Fukaya
Keyword(s):  
Boundary Condition ◽  
Condensed Matter ◽  
Index Theorem ◽  
Chiral Anomaly ◽  
Special Focus ◽  
Original Formulation ◽  
Fermion System ◽  
Massive Fermion ◽  
Index Theorems ◽  
Mathematical Relations

The index theorems relate the gauge field and metric on a manifold to the solution of the Dirac equation on it. In the standard approach, the Dirac operator must be massless to make the chirality operator well defined. In physics, however, the index theorem appears as a consequence of chiral anomaly, which is an explicit breaking of the symmetry. It is then natural to ask if we can understand the index theorems in a massive fermion system which does not have chiral symmetry. In this review, we discuss how to reformulate the chiral anomaly and index theorems with massive Dirac operators, where we find nontrivial mathematical relations between massless and massive fermions. A special focus is placed on the Atiyah–Patodi–Singer index, whose original formulation requires a physicist-unfriendly boundary condition, while the corresponding massive domain-wall fermion reformulation does not. The massive formulation provides a natural understanding of the anomaly inflow between the bulk and edge in particle and condensed matter physics.

Micro-Deformation Mechanism of Granular Materials under Different Boundary Conditions Based on DEM Simulation

2012 ◽  
Vol 170-173 ◽  
pp. 3361-3366
Author(s):  
Zhao Xia Tong ◽  
Min Zhou ◽  
Yang Ping Yao
Keyword(s):  
Boundary Condition ◽  
Granular Materials ◽  
Biaxial Compression ◽  
Special Focus ◽  
Particle Rotation ◽  
Axis Orientation ◽  
Dem Simulation ◽  
Steel Bars ◽  
Strain Relationship ◽  
Micro Deformation

Series of biaxial compression simulations are carried out to investigate the effects of boundary condition on the deformation of granular materials by using DEM. The parameters used in DEM are validated by the biaxial compression experiments on elliptical steel bars. The effects of boundary condition on the stress-strain relationship are analyzed. And special focus are put in the analysis of particle displacement, particle rotation, void distribution, particle long axis orientation and contact force with the development of deformation.

scholarly journals Index theorem for non-supersymmetric fermions coupled to a non-Abelian string and electric charge quantization

2018 ◽  
Vol 33 (09) ◽  
pp. 1850053
Author(s):  
M. Shifman ◽  
A. Yung
Keyword(s):  
Gauge Group ◽  
Electric Charge ◽  
Index Theorem ◽  
Index Theorems ◽  
Zero Modes ◽  
Left Handed ◽  
Charge Quantization ◽  
Supersymmetric Theories ◽  
String Background

Non-Abelian strings are considered in non-supersymmetric theories with fermions in various appropriate representations of the gauge group U[Formula: see text]. We derive the electric charge quantization conditions and the index theorems counting fermion zero modes in the string background both for the left-handed and right-handed fermions. In both cases we observe a non-trivial [Formula: see text] dependence.

scholarly journals Atiyah-Patodi-Singer index theorem for domain-wall fermion Dirac operator

2018 ◽  
Vol 175 ◽  
pp. 11009 ◽  
Author(s):  
Hidenori Fukaya ◽  
Tetsuya Onogi ◽  
Satoshi Yamaguchi
Keyword(s):  
Boundary Condition ◽  
Domain Wall ◽  
Dirac Operator ◽  
Index Theorem ◽  
Topological Phases ◽  
Domain Wall Fermions ◽  
Physical Boundary Condition ◽  
Domain Wall Fermion ◽  
Set Up ◽  
Physical Boundary

Recently, the Atiyah-Patodi-Singer(APS) index theorem attracts attention for understanding physics on the surface of materials in topological phases. Although it is widely applied to physics, the mathematical set-up in the original APS index theorem is too abstract and general (allowing non-trivial metric and so on) and also the connection between the APS boundary condition and the physical boundary condition on the surface of topological material is unclear. For this reason, in contrast to the Atiyah-Singer index theorem, derivation of the APS index theorem in physics language is still missing. In this talk, we attempt to reformulate the APS index in a "physicist-friendly" way, similar to the Fujikawa method on closed manifolds, for our familiar domain-wall fermion Dirac operator in a flat Euclidean space. We find that the APS index is naturally embedded in the determinant of domain-wall fermions, representing the so-called anomaly descent equations.

MASSIVE FERMION AND CHIRAL GAUGE SYMMETRY

1990 ◽  
Vol 05 (31) ◽  
pp. 2607-2613 ◽  
Author(s):  
SINYA AOKI
Keyword(s):  
Gauge Symmetry ◽  
Effective Action ◽  
Gauge Theories ◽  
Opposite Sign ◽  
Chiral Anomaly ◽  
Fermion Mass ◽  
Mass Limit ◽  
Massive Fermion ◽  
Dependent Term ◽  
Mass Dependent

A mass-dependent odd-parity term in the effective action of chiral gauge theories as well as the usual mass-independent chiral anomaly have been shown to exist. In particular, this mass-dependent term is equal to the anomaly with opposite sign in the limit of infinite fermion mass, and therefore no odd-parity term remains in this limit. We also consider the different regularizations which produce the Wess-Zumino term in the infinite fermion mass limit.

scholarly journals Chiral Anomaly in Interacting Condensed Matter Systems

2021 ◽  
Vol 126 (18) ◽  
Author(s):  
Colin Rylands ◽  
Alireza Parhizkar ◽  
Anton A. Burkov ◽  
Victor Galitski
Keyword(s):  
Condensed Matter ◽  
Chiral Anomaly

INDEX THEOREM AND CHIRAL ANOMALY IN D = 4, N = 1 SUPERGRAVITY

1990 ◽  
Vol 05 (14) ◽  
pp. 1097-1102 ◽  
Author(s):  
SATOSHI YAJIMA
Keyword(s):  
Index Theorem ◽  
Flat Space ◽  
Integral Approach ◽  
Chiral Anomaly ◽  
Gauge Fields ◽  
Minimal Coupling ◽  
Curved Space ◽  
Yang Mills ◽  
Anomalous Term ◽  
Fermion Fields

The chiral anomaly in D = 4 supergravity coupled to supersymmetric Yang-Mills theory is evaluated by using the path integral approach. We consider not only the minimal coupling between the gravitational and gauge fields and the fermion fields but also the interaction term which mixes the gravitino and gaugino. The explicit form of the new anomalous term is given by rewriting the resultant term obtained in flat space in a form corresponding to curved space.

scholarly journals GENERALIZED GINSPARG–WILSON ALGEBRA AND INDEX THEOREM ON THE LATTICE

2002 ◽  
Vol 16 (14n15) ◽  
pp. 1931-1941
Author(s):  
KAZUO FUJIKAWA
Keyword(s):  
General Class ◽  
Dirac Operators ◽  
Index Theorem ◽  
Explicit Construction ◽  
Negative Integer ◽  
Chiral Anomaly ◽  
Gauge Fields ◽  
Topological Properties

Recent studies of the topological properties of a general class of lattice Dirac operators are reported. This is based on a specific algebraic realization of the Ginsparg-Wilson relation in the form γ5 (γ5D) + (γ5D)γ5 = 2a2k+1(γ5D)2k+2 where k stands for a non-negative integer. The choice k = 0 corresponds to the commonly discussed Ginsparg-Wilson relation and thus to the overlap operator. It is shown that local chiral anomaly and the instanton-related index of all these operators are identical. The locality of all these Dirac operators for vanishing gauge fields is proved on the basis of explicit construction, but the locality with dynamical gauge fields has not been fully established yet.

A note on K -theoretical index theorems for orbifolds

1992 ◽  
Vol 437 (1900) ◽  
pp. 429-431 ◽  
Keyword(s):  
Index Theory ◽  
Index Theorem ◽  
Index Formula ◽  
Index Theorems ◽  
Cohomological Index

In this note we study index theory for orbifolds. We deduce from the K -theoretical index theorem of Farsi ( Q . J1 Math. 43, 183-200 (1992)) the orbifold index theorem of T. Kawasaki using cyclic theory (A. Connes and G. Yu). Our main result is an extension of Theorem 14 in the Farsi paper, namely we derive the cohomological index formula for orbifolds of T. Kawasaki from our K -theoretical index theorem by using the methods introduced by A. Connes and G. Yu. In §1 we review some notation and describe the K -theoretical index theorems for orbifolds and in §2 we prove our main result.

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