scholarly journals Searching for gauge theories with the conformal bootstrap

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
Zhijin Li ◽  
David Poland
Keyword(s):  
Fixed Points ◽  
Gauge Theories ◽  
Free Field ◽  
Infinite Family ◽  
Preliminary Evidence ◽  
Upper Bounds ◽  
Free Fermion ◽  
Conformal Bootstrap ◽  
General Dimension ◽  
Show Evidence

Abstract Infrared fixed points of gauge theories provide intriguing targets for the modern conformal bootstrap program. In this work we provide some preliminary evidence that a family of gauged fermionic CFTs saturate bootstrap bounds and can potentially be solved with the conformal bootstrap. We start by considering the bootstrap for SO(N) vector 4-point functions in general dimension D. In the large N limit, upper bounds on the scaling dimensions of the lowest SO(N) singlet and traceless symmetric scalars interpolate between two solutions at ∆ = D/2 − 1 and ∆ = D − 1 via generalized free field theory. In 3D the critical O(N) vector models are known to saturate the bootstrap bounds and correspond to the kinks approaching ∆ = 1/2 at large N. We show that the bootstrap bounds also admit another infinite family of kinks $$ {\mathcal{T}}_D $$ T D , which at large N approach solutions containing free fermion bilinears at ∆ = D − 1 from below. The kinks $$ {\mathcal{T}}_D $$ T D appear in general dimensions with a D-dependent critical N* below which the kink disappears. We also study relations between the bounds obtained from the bootstrap with SO(N) vectors, SU(N) fundamentals, and SU(N) × SU(N) bi-fundamentals. We provide a proof for the coincidence between bootstrap bounds with different global symmetries. We show evidence that the proper symmetries of the underlying theories of $$ {\mathcal{T}}_D $$ T D are subgroups of SO(N), and we speculate that the kinks $$ {\mathcal{T}}_D $$ T D relate to the fixed points of gauge theories coupled to fermions.

scholarly journals Circular quiver gauge theories, isomonodromic deformations and $$W_N$$ fermions on the torus

2021 ◽  
Vol 111 (3) ◽  
Author(s):  
Giulio Bonelli ◽  
Fabrizio Del Monte ◽  
Pavlo Gavrylenko ◽  
Alessandro Tanzini
Keyword(s):  
Integrable System ◽  
Gauge Theories ◽  
The Self ◽  
Free Fermion ◽  
Two Dimensional ◽  
Isomonodromic Deformation ◽  
Specific Time ◽  
Dimensional Torus ◽  
Isomonodromic Deformations ◽  
Deformation Problem

AbstractWe study the relation between class $$\mathcal {S}$$ S theories on punctured tori and isomonodromic deformations of flat SL(N) connections on the two-dimensional torus with punctures. Turning on the self-dual $$\Omega $$ Ω -background corresponds to a deautonomization of the Seiberg–Witten integrable system which implies a specific time dependence in its Hamiltonians. We show that the corresponding $$\tau $$ τ -function is proportional to the dual gauge theory partition function, the proportionality factor being a nontrivial function of the solution of the deautonomized Seiberg–Witten integrable system. This is obtained by mapping the isomonodromic deformation problem to $$W_N$$ W N free fermion correlators on the torus.

Auditory response properties of neurons in deep layers of cat superior colliculus

1983 ◽  
Vol 49 (3) ◽  
pp. 674-685 ◽  
Author(s):  
L. Z. Wise ◽  
D. R. Irvine
Keyword(s):  
Superior Colliculus ◽  
Free Field ◽  
Receptive Fields ◽  
Great Majority ◽  
Noise Burst ◽  
Preliminary Evidence ◽  
Excitatory Input ◽  
Response Properties ◽  
Deep Layers ◽  
Spatial Fields

1. The auditory responses of 207 single neurons in the intermediate and deep layers of the superior colliculus (SC) of barbiturate -or chloralose-anesthetized cats were recorded extracellularly. Sealed stimulating systems incorporating calibrated probe microphone assemblies were employed to present tone- and noise-burst stimuli. 2. All acoustically activated neurons responded with onset responses to noise bursts. Of those neurons also tested with tonal stimuli, approximately 30% were unresponsive over the frequency range tested (0.1-40 kHz), while the others had higher thresholds to tones than to noise. 3. Details of frequency responsiveness were obtained for 55 neurons; 21 were broadly tuned, while 34 were sharply tuned with clearly defined characteristic frequencies (CFs). All sharply tuned neurons had CFs greater than or equal to 10 kHz. 4. The majority of neurons (81%) responded with latencies in the range 8-20 ms; only 11% of neurons had latencies greater than 30 ms. 5. Binaural response properties were examined for 165 neurons. The great majority (79%) received monaural excitatory input only from the contralateral ear (EO). However, most EO cells were binaurally influenced, the contralateral response being either inhibited (EO/I; 96 of 131 units) or facilitated (EO/F; 33 of 131 units) by simultaneous ipsilateral stimulation. Small subgroups were monaurally excited by either ear (EE cells; 8%) or were unresponsive monaurally but responded strongly to binaural stimulation (OO/F cells; 7%). 6. EO/I, EO/F, and OO/F neurons showed characteristic forms of sensitivity to interaural intensity differences (IIDs). The IID functions of EO/I neurons would be expected to produce large contralateral spatial receptive fields with clearly defined medial borders, such as have been described in studies of deep SC neurons employing free-field stimuli. 7. Preliminary evidence suggests a possible topographic organization of IID sensitivity in deep SC, such that the steeply sloping portion of the function (corresponding to the medial edge of the receptive field) is shifted laterally for EO/I neurons located more caudally in the nucleus. 8. The auditory properties of deep SC neurons are compared with previous reports and implications for the organization of auditory input are considered. The binaural properties and auditory spatial fields of deep SC neurons suggest that any representation of auditory space in this structure is unlikely to be based on restricted spatial fields.

scholarly journals CONFORMAL HOUSE

2010 ◽  
Vol 25 (24) ◽  
pp. 4603-4621 ◽  
Author(s):  
THOMAS A. RYTTOV ◽  
FRANCESCO SANNINO
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Gauge Group ◽  
Gauge Theories ◽  
Simultaneous Presence ◽  
Consistency Check ◽  
Effective Lagrangian ◽  
Anomalous Dimensions ◽  
Mass Terms

We investigate the gauge dynamics of nonsupersymmetric SU (N) gauge theories featuring the simultaneous presence of fermionic matter transforming according to two distinct representations of the underlying gauge group. We bound the regions of flavors and colors which can yield a physical infrared fixed point. As a consistency check we recover the previously investigated bounds of the conformal windows when restricting to a single matter representation. The earlier conformal windows can be imagined to be part now of the new conformal house. We predict the nonperturbative anomalous dimensions at the infrared fixed points. We further investigate the effects of adding mass terms to the condensates on the conformal house chiral dynamics and construct the simplest instanton induced effective Lagrangian terms.

scholarly journals An infinite series of triangles and conics with a common pole and polar

1911 ◽  
Vol 30 ◽  
pp. 13-30
Author(s):  
J. A. Donaldson
Keyword(s):  
Fixed Points ◽  
Infinite Series ◽  
Infinite Family

An infinite family of triangles, having a common pole (determined by three fixed lines through it), and polar (determined by three fixed points on it), and an allied family of conics with imaginary double contact.Construction for pole of a line with reference to a triangle.

scholarly journals Iterative Algorithm for Common Fixed Points of Infinite Family of Nonexpansive Mappings in Banach Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Songnian He ◽  
Jun Guo
Keyword(s):  
Fixed Points ◽  
Iterative Algorithm ◽  
Nonexpansive Mappings ◽  
Infinite Family ◽  
Common Fixed Points ◽  
Nonempty Closed Convex Subset ◽  
Duality Mapping ◽  
Normalized Duality Mapping ◽  
Uniformly Smooth ◽  
Computational Work

LetCbe a nonempty closed convex subset of a real uniformly smooth Banach spaceX,{Tk}k=1∞:C→Can infinite family of nonexpansive mappings with the nonempty set of common fixed points⋂k=1∞Fix⁡(Tk), andf:C→Ca contraction. We introduce an explicit iterative algorithmxn+1=αnf(xn)+(1-αn)Lnxn, whereLn=∑k=1n(ωk/sn)Tk,Sn=∑k=1nωk,  andwk>0with∑k=1∞ωk=1. Under certain appropriate conditions on{αn}, we prove that{xn}converges strongly to a common fixed pointx*of{Tk}k=1∞, which solves the following variational inequality:〈x*-f(x*),J(x*-p)〉≤0,    p∈⋂k=1∞Fix(Tk), whereJis the (normalized) duality mapping ofX. This algorithm is brief and needs less computational work, since it does not involveW-mapping.

IR fixed points in lattice field theories

2014 ◽  
Vol 29 (25) ◽  
pp. 1445006 ◽  
Author(s):  
L. Del Debbio
Keyword(s):  
Fixed Points ◽  
Predictive Models ◽  
Gauge Theories ◽  
Quantitative Description ◽  
Electroweak Symmetry Breaking ◽  
Field Theories ◽  
Electroweak Symmetry ◽  
Lattice Field ◽  
Starting Point ◽  
Theoretical Foundations

The existence of strongly-interacting infrared (IR) fixed points in gauge theories can be studied using field theories defined on a lattice. Besides their academic interest, it has been argued that these theories could be used as a starting point to construct viable models of electroweak symmetry breaking (EWSB). A quantitative description of the large-distance dynamics is mandatory in order to develop predictive models for phenomenology. In these notes, we discuss the theoretical foundations of IR conformality, and present possible ways to study the problem using the lattice formulation of gauge theories. An overview of current results and methods is discussed.

scholarly journals Solving q-Virasoro constraints

2019 ◽  
Vol 110 (1) ◽  
pp. 179-210 ◽  
Author(s):  
Rebecca Lodin ◽  
Aleksandr Popolitov ◽  
Shamil Shakirov ◽  
Maxim Zabzine
Keyword(s):  
Wilson Loop ◽  
Gauge Theories ◽  
Free Field ◽  
Point Of View ◽  
Difference Operators ◽  
Highest Weight ◽  
Virasoro Constraints ◽  
Zero Momentum ◽  
Complete Analogy ◽  
Full Derivative

Abstract We show how q-Virasoro constraints can be derived for a large class of (q, t)-deformed eigenvalue matrix models by an elementary trick of inserting certain q-difference operators under the integral, in complete analogy with full-derivative insertions for $$\beta $$ β -ensembles. From free field point of view, the models considered have zero momentum of the highest weight, which leads to an extra constraint $$T_{-1} \mathcal {Z} = 0$$ T - 1 Z = 0 . We then show how to solve these q-Virasoro constraints recursively and comment on the possible applications for gauge theories, for instance calculation of (supersymmetric) Wilson loop averages in gauge theories on $$D^2 \times S^1$$ D 2 × S 1 and $$S^3$$ S 3 .

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