Modular Invariance of Field Theories and String Compactifications

Self-dual models with theta terms

Modular flavor symmetries have been proposed as a new way to address the flavor problem. It is known that they can emerge from string compactifications. We discuss this connection in detail, and show how the congruence subgroups of SL(2, ℤ), which underlie many modular flavor symmetries, emerge from stringy duality symmetries by orbifolding. This requires an analysis of massive states, which reveals a picture that is more intricate than the well-known situation on the torus. It involves towers of states of different quantum numbers, related by modular transformations. Members of different towers become massless at different points in moduli space. We also show that, at least in the ℤ3 orbifold, the string selection rules can be understood as discrete remnants of continuous gauge symmetries. Non-Abelian discrete flavor symmetries arise as relics of various, relatively misaligned, continuous Abelian gauge symmetries. The generators of these U(1) symmetries give rise to $$ \mathcal{CP} $$ CP -violating Clebsch-Gordan coefficients. If the modulus settles close to a critical point, the corresponding gauge bosons may be light enough to be searched for at future colliders.

In this paper, we want to better understand the causality reflection that arises under a subset of Abelian local gauge transformations in geometrodynamics. We proved in previous papers that in Einstein–Maxwell spacetimes, there exist two local orthogonal planes of gauge symmetry at every spacetime point for non-null electromagnetic fields. Every vector in these planes is an eigenvector of the Einstein–Maxwell stress–energy tensor. The vectors that span these local orthogonal planes are dependent on electromagnetic gauge. The local group of Abelian electromagnetic gauge transformations has been proved isomorphic to the local groups of tetrad transformations in these planes. We called LB1 the local group of tetrad transformations made up of SO(1, 1) plus two different kinds of discrete transformations. One of the discrete transformations is the full inversion two by two which is a Lorentz transformation. The other discrete transformation is given by a matrix with zeroes on the diagonal and ones off-diagonal two by two, a reflection. The group LB1 is realized on this plane, we call this plane one, and is spanned by the time-like and one space-like vectors. The other local orthogonal plane is plane two and the local group of tetrad transformations, we call this LB2, which is just SO(2). The local group of Abelian electromagnetic gauge transformations is isomorphic to both LB1 and LB2, independently. It has already been proved that a subset of local electromagnetic gauge transformations that leave the electromagnetic tensor invariant induces a change in sign in the norm of the tetrad vectors that span the local plane one. The reason is that one of the discrete transformations on the local plane one that belongs to the group LB1 is not a Lorentz transformation, it is a flip or reflection. It is precisely on this kind of discrete transformation that we have an interest since it has the effect of changing the signature and the causality. This effect has never been noticed before.

In this paper we study particle-vortex duality and the effect of theta terms from the point of view of AdS/CMT constructions. We can construct the duality in 2+1 dimensional field theories with or without a Chern-Simons term, and derive an effect on conductivities, when the action is viewed as a response action. We can find its effect on 3+1 dimensional theories, with or without a theta term, coupled to gravity in asymptotically AdS space, and derive the resulting effect on conductivities defined in the spirit of AdS/CFT. AdS/CFT then relates the 2+1 dimensional and the 3+1 dimensional cases naturally. Quantum gravity corrections, as well as more general effective actions for the abelian vector, can be treated similarly. We can use the fluid/gravity correspondence, and the membrane paradigm, to define shear and bulk viscosities η and ζ for a gravity plus abelian vector plus scalar system near a black hole, and define the effect of the S-duality on it.

Abelian lattice gauge theories in 3 + 1 dimensions: The linked cluster approach

Infinite discrete symmetry group for the Yang-Baxter equations: Spin models

Let ( X , Γ ) (X, \Gamma ) be a free and minimal topological dynamical system, where X X is a separable compact Hausdorff space and Γ \Gamma is a countable infinite discrete amenable group. It is shown that if ( X , Γ ) (X, \Gamma ) has the Uniform Rokhlin Property (URP) and Cuntz comparison of open sets (COS), then m d i m ( X , Γ ) = 0 \mathrm {mdim}(X, \Gamma )=0 implies that ( C ( X ) ⋊ Γ ) ⊗ Z ≅ C ( X ) ⋊ Γ (\mathrm {C}(X) \rtimes \Gamma )\otimes \mathcal Z \cong \mathrm {C}(X) \rtimes \Gamma , where m d i m \mathrm {mdim} is the mean dimension of ( X , Γ ) (X, \Gamma ) , Z \mathcal Z is the Jiang-Su algebra, and C ( X ) ⋊ Γ \mathrm {C}(X) \rtimes \Gamma is the transformation group C*-algebra of ( X , Γ ) (X, \Gamma ) . In particular, in this case, m d i m ( X , Γ ) = 0 \mathrm {mdim}(X, \Gamma )=0 implies that the C*-algebra C ( X ) ⋊ Γ \mathrm {C}(X) \rtimes \Gamma is classified by the Elliott invariant.

This monograph introduces a framework for genuine proper equivariant stable homotopy theory for Lie groups. The adjective ‘proper’ alludes to the feature that equivalences are tested on compact subgroups, and that the objects are built from equivariant cells with compact isotropy groups; the adjective ‘genuine’ indicates that the theory comes with appropriate transfers and Wirthmüller isomorphisms, and the resulting equivariant cohomology theories support the analog of an R O ( G ) R O(G) -grading. Our model for genuine proper G G -equivariant stable homotopy theory is the category of orthogonal G G -spectra; the equivalences are those morphisms that induce isomorphisms of equivariant stable homotopy groups for all compact subgroups of G G . This class of π ∗ \pi _* -isomorphisms is part of a symmetric monoidal stable model structure, and the associated tensor triangulated homotopy category is compactly generated. Consequently, every orthogonal G G -spectrum represents an equivariant cohomology theory on the category of G G -spaces. These represented cohomology theories are designed to only depend on the ‘proper G G -homotopy type’, tested by fixed points under all compact subgroups. An important special case of our theory are infinite discrete groups. For these, our genuine equivariant theory is related to finiteness properties in the sense of geometric group theory; for example, the G G -sphere spectrum is a compact object in our triangulated equivariant homotopy category if the universal space for proper G G -actions has a finite G G -CW-model. For discrete groups, the represented equivariant cohomology theories on finite proper G G -CW-complexes admit a more explicit description in terms of parameterized equivariant homotopy theory, suitably stabilized by G G -vector bundles. Via this description, we can identify the previously defined G G -cohomology theories of equivariant stable cohomotopy and equivariant K-theory as cohomology theories represented by specific orthogonal G G -spectra.

A connection between integrability properties and general statistical properties of the spectra of symmetric transfer matrices of the asymmetric eight-vertex model is studied using random matrix theory (eigenvalue spacing distribution and spectral rigidity). For Yang-Baxter integrable cases, including free-fermion solutions, we have found a Poissonian behavior, whereas level repulsion close to the Wigner distribution is found for non-integrable models. For the asymmetric eight-vertex model, however, the level repulsion can also disappearand the Poisson distribution be recovered on (non Yang--Baxter integrable) algebraic varieties, the so-called disorder varieties. We also present an infinite set of algebraic varieties which are stable under the action of an infinite discrete symmetry group of the parameter space. These varieties are possible loci for free parafermions. Using our numerical criterion we have tested the generic calculability of the model on these algebraic varieties.

The chiral anomaly in lattice abelian gauge theory is investigated by applying noncommutative differential geometry (NCDG). A new kind of double complex and descent equation is proposed on an infinite hypercubic lattice in arbitrary even-dimensional Euclidean space in the framework of NCDG. By using the general solutions of descent equation, we derive the chiral anomaly in Abelian lattice gauge theory. The topological origin of the anomaly is nothing but the Chern classes in NCDG.

The completion theorem in K-theory for proper actions of a discrete group

We show that every infinite discrete group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has an infinite minimal subflow in its Bernoulli flow <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace 0 comma 1 right-brace Superscript upper G"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:msup> <mml:mo fence="false" stretchy="false">}</mml:mo> <mml:mi>G</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">\{0,1\}^G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . A countably infinite group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has an effective minimal subflow in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace 0 comma 1 right-brace Superscript upper G"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:msup> <mml:mo fence="false" stretchy="false">}</mml:mo> <mml:mi>G</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">\{0,1\}^G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . If <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is countable and residually finite, then it has such a subflow which is free. We do not know whether there are groups <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with no free subflows in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace 0 comma 1 right-brace Superscript upper G"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:msup> <mml:mo fence="false" stretchy="false">}</mml:mo> <mml:mi>G</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">\{0,1\}^G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .

Taking the simple examples of an Abelian 1-form gauge theory in two (1+1)-dimensions, a 2-form gauge theory in four (3+1)-dimensions and a 3-form gauge theory in six (5+1)-dimensions of space–time, we establish that such gauge theories respect, in addition to the gauge symmetry transformations that are generated by the first-class constraints of the theory, additional continuous symmetry transformations. We christen the latter symmetry transformations as the dual-gauge transformations. We generalize the above gauge and dual-gauge transformations to obtain the proper (anti-)BRST and (anti-)dual-BRST transformations for the Abelian 3-form gauge theory within the framework of BRST formalism. We concisely mention such symmetries for the 2D free Abelian 1-form and 4D free Abelian 2-form gauge theories and briefly discuss their topological aspects in our present endeavor. We conjecture that any arbitrary Abelian p-form gauge theory would respect the above cited additional symmetry in D = 2p(p = 1, 2, 3, …) dimensions of space–time. By exploiting the above inputs, we establish that the Abelian 3-form gauge theory, in six (5+1)-dimensions of space–time, is a perfect model for the Hodge theory whose discrete and continuous symmetry transformations provide the physical realizations of all aspects of the de Rham cohomological operators of differential geometry. As far as the physical utility of the above nilpotent symmetries is concerned, we demonstrate that the 2D Abelian 1-form gauge theory is a perfect model of a new class of topological theory and 4D Abelian 2-form as well as 6D Abelian 3-form gauge theories are the field theoretic models for the quasi-topological field theory.

The purpose of this paper is to investigate the topological and analytical restrictions on a domain D in euclidean n-space R n on which an infinite discrete quasiconformal group can act. We will see that the restrictions are indeed severe, unlike the case of a discrete group of topological or differentiable homeomorphisms.KeywordsUnit BallLimit PointUniversal CoverTangent PlaneQuasiconformal MappingThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this paper, we classify non-freely acting discrete symmetries of complete intersection Calabi–Yau manifolds and their quotients by freely-acting symmetries. These non-freely acting symmetries can appear as symmetries of low-energy theories resulting from string compactifications on these Calabi–Yau manifolds, particularly in the context of the heterotic string. Hence, our results are relevant for four-dimensional model building with discrete symmetries and they give an indication which symmetries of this kind can be expected from string theory. For the 1695 known quotients of complete intersection manifolds by freely-acting discrete symmetries, non-freely-acting, generic symmetries arise in 381 cases and are, therefore, a relatively common feature of these manifolds. We find that 9 different discrete groups appear, ranging in group order from 2 to 18, and that both regular symmetries and R-symmetries are possible.