An official website of the United States government
Overlap fermion on the lattice has been shown to properly reproduce topological aspects of gauge fields. In this paper, we review the derivation of Overlap fermion formalism in a torus of three space-time dimensions. Using the formalism, we show how to use the Overlap fermion determinants in the massless and infinite mass limits to construct different continuum topological gauge actions, such as the level-k Chern–Simons action, “half-CS” term and the mixed Chern–Simons (BF) coupling, in a gauge-invariant lattice UV regulated manner. Taking special Abelian and non-Abelian background fields, we demonstrate numerically how the lattice formalism beautifully reproduces the continuum expectations, such as the flow of action under large gauge transformations.
more »
« less
Sasakians and the geometry of a mass term
A gauge-invariant mass term for nonabelian gauge fields in two dimensions can be expressed as the Wess–Zumino–Witten (WZW) action. Hard thermal loops in the gauge theory in four dimensions at finite temperatures generate a screening mass for some components of the gauge field. This can be expressed in terms of the WZW action using the bundle of complex structures (for Euclidean signature) or the bundle of lightcones over Minkowski space. We show that a dynamically generated mass term in three dimensions can be put within the same general framework using the bundle of Sasakian structures.
more »
« less
- Award ID(s):
- 2112729
- PAR ID:
- 10348470
- Date Published:
- Journal Name:
- Reviews in Mathematical Physics
- Volume:
- 33
- Issue:
- 08
- ISSN:
- 0129-055X
- Page Range / eLocation ID:
- 2140002
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
A bstract Pure gravity in AdS 3 is a theory of boundary excitations, most simply expressed as a constrained free scalar with an improved stress tensor that is needed to match the Brown-Henneaux central charge. Excising a finite part of AdS gives rise to a static gauge Nambu-Goto action for the boundary graviton. We show that this is the $$ T\overline{T} $$ T T ¯ deformation of the infinite volume theory, as the effect of the improvement term on the deformed action can be absorbed into a field redefinition. The classical gravitational stress tensor is reproduced order by order by the $$ T\overline{T} $$ T T ¯ trace equation. We calculate the finite volume energy spectrum in static gauge and find that the trace equation imposes sufficient constraints on the ordering ambiguities to guarantee agreement with the light-cone gauge prediction. The correlation functions, however, are not completely fixed by the trace equation. We show how both the gravitational action and the $$ T\overline{T} $$ T T ¯ deformation allow for finite improvement terms, and we match these to the undetermined total derivative terms in Zamolodchikov’s point splitting definition of the $$ T\overline{T} $$ T T ¯ operator.more » « less
-
Abstract We define a natural state space and Markov process associated to the stochastic Yang–Mills heat flow in two dimensions. To accomplish this we first introduce a space of distributional connections for which holonomies along sufficiently regular curves (Wilson loop observables) and the action of an associated group of gauge transformations are both well-defined and satisfy good continuity properties. The desired state space is obtained as the corresponding space of orbits under this group action and is shown to be a Polish space when equipped with a natural Hausdorff metric. To construct the Markov process we show that the stochastic Yang–Mills heat flow takes values in our space of connections and use the “DeTurck trick” of introducing a time dependent gauge transformation to show invariance, in law, of the solution under gauge transformations. Our main tool for solving for the Yang–Mills heat flow is the theory of regularity structures and along the way we also develop a “basis-free” framework for applying the theory of regularity structures in the context of vector-valued noise – this provides a conceptual framework for interpreting several previous constructions and we expect this framework to be of independent interest.more » « less
-
null (Ed.)We derive a holomorphic anomaly equation for the Vafa-Wittenpartition function for twisted four-dimensional \mathcal{N} =4 𝒩 = 4 super Yang-Mills theory on \mathbb{CP}^{2} ℂ ℙ 2 for the gauge group SO(3) S O ( 3 ) from the path integral of the effective theory on the Coulomb branch.The holomorphic kernel of this equation, which receives contributionsonly from the instantons, is not modular but ‘mock modular’. Thepartition function has correct modular properties expected from S S -dualityonly after including the anomalous nonholomorphic boundary contributionsfrom anti-instantons. Using M-theory duality, we relate this phenomenonto the holomorphic anomaly of the elliptic genus of a two-dimensionalnoncompact sigma model and compute it independently in two dimensions.The anomaly both in four and in two dimensions can be traced to atopological term in the effective action of six-dimensional (2,0) ( 2 , 0 ) theory on the tensor branch. We consider generalizations to othermanifolds and other gauge groups to show that mock modularity is genericand essential for exhibiting duality when the relevant field space isnoncompact.more » « less
-
A<sc>bstract</sc> Euler hydrodynamics of perfect fluids can be viewed as an effective bosonic field theory. In cases when the underlying microscopic system involves Dirac fermions, the quantum anomalies should be properly described. In 1+1 dimensions the action formulation of hydrodynamics at zero temperature is reconsidered and shown to be equal to standard field-theory bosonization. Furthermore, it can be derived from a topological gauge theory in one extra dimension, which identifies the fluid variables through the anomaly inflow relations. Extending this framework to 3+1 dimensions yields an effective field theory/hydrodynamics model, capable of elucidating the mixed axial-vector and axial-gravitational anomalies of Dirac fermions. This formulation provides a platform for bosonization in higher dimensions. Moreover, the connection with 4+1 dimensional topological theories suggests some generalizations of fluid dynamics involving additional degrees of freedom.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1142/S0129055X2140002X
