FI Parameters Research Articles

Probing bad theories with the dualization algorithm. Part I

Recently an algorithm to build SL(2, ℤ) duals, including mirror duals, of 3d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 quiver theories and their 4d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 1 uplift has been introduced. In this work we use this new tool to study the so-called bad theories. Our approach allows us to determine exactly indices/partition functions for generic values of fugacities/real mass and FI parameters revealing their surprising feature: the 4d index/3d partition function of a bad theory behaves as a sum of distributions rather than an ordinary function of the deformation parameters. We focus on the bad SQCD, with U(Nc) gauge group in 3d and USp(2Nc) in 4d, while in an upcoming paper we will consider linear quivers which, in the 3d case, have both unitary and special unitary bad nodes.

Remarks on 2d unframed quiver gauge theories

We study 2d mathcal{N} = (2, 2) quiver gauge theories without flavor nodes. There is a special class of quivers whose gauge group ranks stay positive in any duality frame. We illustrate this with the Abelian Kronecker quiver and the Abelian Markov quiver as the simplest examples. In the geometric phase, they engineer an infinite sequence of projective spaces and hypersurfaces in Calabi-Yau spaces, respectively. We show that the Markov quiver provides an Abelianization of SU(3) SQCD. Turning on the FI parameters and the θ angles for the Abelian quiver effectively deform SQCD by such parameters. For an Abelian necklace quiver corresponding to SU(k) SQCD, we find evidence for singular loci supporting non-compact Coulomb branches in the Kähler moduli space.

The 3d twisted index and wall-crossing

We study the twisted index of 3d \mathcal{N}=2𝒩=2 supersymmetric gauge theories on S^1 \times \SigmaS1×Σ in the presence of a real FI parameter deformation. This parameter induces a 1d FI parameter for the effective supersymmetric quantum mechanics on S^1S1. Using supersymmetric localisation, the twisted index can be expressed as a contour integral. We show that the contour prescription is modified in the presence of the 1d FI parameter, leading to wall-crossing phenomena for the twisted index. In particular, we derive a general wall-crossing formula for abelian gauge theories. We also examine the origin of wall-crossing as change of stability condition in the algebro-geometric interpretation of the twisted index. These ideas are illustrated for abelian theories with \mathcal{N}=4𝒩=4 supersymmetry and in a non-abelian example that reproduces wall-crossing phenomena associated to moduli spaces of stable pairs.

Generating functions for Higgs/Coulomb branch operators from 1d–3d cohomological equivalence

We provide a proof for the conjectured equality of the generating function of integrated Higgs and Coulomb branch topological operators in 3d mathcal{N} ≥ 4 gauge theories and the three sphere partition function deformed by mass or FI parameters. The equality is the result of cohomological equivalence and applies to all theories in this class, including ABJM and other generalized Gaiotto-Witten models, and those without an explicit supersymmetric Lagrangian.

Supersymmetric Ground States of 3d $\mathcal{N}=4$ Gauge Theories on a Riemann Surface

This paper studies supersymmetric ground states of 3d \mathcal{N}=4𝒩=4 supersymmetric gauge theories on a Riemann surface of genus gg. There are two distinct spaces of supersymmetric ground states arising from the AA and BB type twists on the Riemann surface, which lead to effective supersymmetric quantum mechanics with four supercharges and supermultiplets of type \mathcal{N}=(2,2)𝒩=(2,2) and \mathcal{N}=(0,4)𝒩=(0,4) respectively. We compute the space of supersymmetric ground states in each case, graded by flavour and R-symmetries and in different chambers for real mass and FI parameters, for a large class of supersymmetric gauge theories. The results are formulated geometrically in terms of the Higgs branch geometry. We perform extensive checks of compatibility with the twisted index and mirror symmetry.

Boundaries, Vermas and factorisation

We revisit the factorisation of supersymmetric partition functions of 3d mathcal{N} = 4 gauge theories. The building blocks are hemisphere partition functions of a class of UV mathcal{N} = (2, 2) boundary conditions that mimic the presence of isolated vacua at infinity in the presence of real mass and FI parameters. These building blocks can be unambiguously defined and computed using supersymmetric localisation. We show that certain limits of these hemisphere partition functions coincide with characters of lowest weight Verma mod- ules over the quantised Higgs and Coulomb branch chiral rings. This leads to expressions for the superconformal index, twisted index and S3 partition function in terms of such characters. On the way we uncover new connections between boundary ’t Hooft anomalies, hemisphere partition functions and lowest weights of Verma modules.

Tests of Seiberg-like dualities in three dimensions

We use localization techniques to study several duality proposals for supersymmetric gauge theories in three dimensions reminiscent of Seiberg duality. We compare the partition functions of dual theories deformed by real mass terms and FI parameters. We find that Seiberg-like duality for mathcal{N} = 3 Chern-Simons gauge theories proposed by Giveon and Kutasov holds on the level of partition functions and is closely related to level-rank duality in pure Chern-Simons theory. We also clarify the relationship between the Giveon-Kutasov duality and a duality in theories of fractional M2 branes and propose a generalization of the latter. Our analysis also confirms previously known results concerning decoupled free sectors in mathcal{N} = 4 gauge theories realized by monopole operators.

Quantized Coulomb branches, monopole bubbling and wall-crossing phenomena in 3d mathcal{N} = 4 theories

To study the quantized Coulomb branch of 3d mathcal{N} = 4 unitary SQCD theories, we propose a new method to compute correlators of monopole and Casimir operators that are inserted in the mathrm{mathbb{R}}times {mathrm{mathbb{R}}}_{upepsilon}^2 Omega background. This method combines results from supersymmetric localization with inputs from the brane realisation of the correlators in type IIB string theory. The main challenge is the computation of the partition functions of certain Super-Matrix-Models (SMMs), which appear in the contribution of monopole bubbling sectors and are realised as the theory living on the D1 strings in the brane construction. We find that the non-commutativity arising in the monopole operator insertions is related to a wall-crossing phenomenon in the FI parameter space of the SMM. We illustrate our method in various examples and we provide explicit results for arbitrary correlators of non-bubbling bare monopole operators. We also discuss the realisation of the non-commutative product as a Moyal (star) product and use it to successfully test our results.

Exact WKB analysis of \u2102\u21191 holomorphic blocks

We study holomorphic blocks in the three dimensional mathcal{N} = 2 gauge theory that describes the ℂℙ1 model. We apply exact WKB methods to analyze the line operator identities associated to the holomorphic blocks and derive the analytic continuation formulae of the blocks as the twisted mass and FI parameter are varied. The main technical result we utilize is the connection formula for the 1\U0001d7191q-hypergeometric function. We show in detail how the q-Borel resummation methods reproduce the results obtained previously by using block-integral methods.

Twisted indices of 3d mathcal{N} = 4 gauge theories and enumerative geometry of quasi-maps

We explore the geometric interpretation of the twisted index of 3d mathcal{N} = 4 gauge theories on S1 × Σ where Σ is a closed Riemann surface. We focus on a rich class of supersymmetric quiver gauge theories that have isolated vacua under generic mass and FI parameter deformations. We show that the path integral localises to a moduli space of solutions to generalised vortex equations on Σ, which can be understood algebraically as quasi-maps to the Higgs branch. We show that the twisted index reproduces the virtual Euler characteristic of the moduli spaces of twisted quasi-maps and demonstrate that this agrees with the contour integral representation introduced in previous work. Finally, we investigate 3d mathcal{N} = 4 mirror symmetry in this context, which implies an equality of enumerative invariants associated to mirror pairs of Higgs branches under the exchange of equivariant and degree counting parameters.

Quantitative effects of urban spatial characteristics on outdoor thermal comfort based on the LCZ scheme

Based on the local climate zone (LCZ) scheme, nine typical urban areas in Shenzhen, China were chosen to conduct a field survey to quantitatively analyze the outdoor thermal comfort levels under different urban spatial characteristics. Four integrated indicators, F1, F2, F3, and F4, were defined based on the Fuzzy-AHP method by considering four different considerations for expressing nine different LCZs. Through physical meteorological measurements and simultaneous human responses on local environments, thermal “neutral” value ranges of three thermal comfort indicators (TCI) were obtained, varying from 27 °C to 29.5 °C in PET, from 21 °C to 24 °C in OUT-SET*, and from 28 °C to 29.5 °C in UTCI. Quantitative equations between Fi and TCI were expressed by applying Linear, Logarithmic and Cubic models to the calculated Fi and TCI values in the nine LCZs. Sampling experiments were conducted to generate near-random sampling Fi parameters from a four-dimensional distribution. By referring to thermal “neutral” TCI ranges, six optimal Fi combinations were determined and results show that the LCZ 1 B appears a preferable LCZ class for optimizing local-scale thermal comfort conditions in the high-temperature environments in subtropical regions of China. Additionally, the optimal Fi combinations provide quantitative theoretical guidance in sustainable urban design and planning.

An alternative analysis of indirect tensile test results for evaluating fatigue characteristics of asphalt mixes

Fatigue cracking is one of the major distresses responsible for the failure of asphalt pavements. The widely-accepted Superpave® volumetric mix design method does not consider screening the asphalt mixes based on their fatigue resistance. Based on a survey conducted in this study, it was found that many state Departments of Transportations (DOTs) do not perform a fatigue test during mix design, mainly due to lack of specialized equipment, trained personnel and consensus about the most appropriate test method. This present study was undertaken to suggest a simple, quick and effective fatigue test method and the corresponding data analysis procedure. It was found that the indirect tension test, which is usually conducted in the DOTs on a regular basis, can be used to characterize the fatigue resistance of asphalt mixes as well. A simplified data analysis approach has been proposed. The fatigue resistance of asphalt mixes can be determined by using a newly derived parameter called Fatigue Index (fi). Fatigue resistance of five different asphalt mixes were evaluated using this new parameter, fi. It was found that the fi parameter were able to statistically discriminate five selected asphalt mixes with respect to their fatigue resistance. The effectiveness of the fi parameter was verified by investigating its correlations with the results of the semi-circular bend and four-point beam fatigue test results.

Cluster-enriched Yang–Baxter equation from SUSY gauge theories

We propose a new generalization of the Yang-Baxter equation, where the R-matrix depends on cluster $y$-variables in addition to the spectral parameters. We point out that we can construct solutions to this new equation from the recently-found correspondence between Yang-Baxter equations and supersymmetric gauge theories. The $S^2$ partition function of a certain 2d $\mathcal{N}=(2,2)$ quiver gauge theory gives an R-matrix, whereas its FI parameters can be identified with the cluster $y$-variables.

Boundaries and defects of N = 4 $$ \mathcal{N}=4 $$ SYM with 4 supercharges. Part II: Brane constructions and 3d N = 2 $$ \mathcal{N}=2 $$ field theories

We study the vacuum moduli spaces of 3d $$ \mathcal{N}=2 $$ supersymmetric quantum field theories by applying the formalism developed in our previous paper [1]. The 3d theories can be realized by branes in type IIB string theory, which in a decoupling limit reduce to 4d $$ \mathcal{N}=4 $$ super-Yang-Mills theory on an interval with BPS defects inserted. The moduli space of a given 3d theory is obtained by solving a generalization of Nahm’s equations with appropriate boundary/junction conditions, along with help from the S-duality of type IIB string theory. Our classical computations reproduce many known results about the quantum-corrected moduli spaces of 3d theories, e.g. U(N c ) theories with N f flavors with mass and FI parameters turned on. In particular, our methods give first-principles derivations of several results in the literature, including the s-rule, quantum splitting of classical Coulomb branches, the lifting of the Coulomb branch by non-Abelian instantons, quantum merging of Coulomb and Higgs branches, and phase transitions from re-ordering 5-branes.

Multivariate optimization of a solid phase extraction system employing l-tyrosine immobilized on carbon nanotubes applied to molybdenum analysis by inductively coupled plasma optical emission spectrometry with ultrasound nebulization

A method for Mo solid phase extraction on l-tyrosine immobilized on carbon nanotubes (l-tyr-CNTs) is presented. l-tyr-CNTs were used to fill a minicolumn and introduced into a FI system employing inductively coupled plasma optical emission spectrometry with ultrasound nebulization (USN-ICP OES). Five FI parameters such as buffer flow rate (BFR) and concentration (BC); sample flow rate (SFR); eluent flow rate (EFR) and concentration (EC); and pH were chosen for optimization employing a half fraction composite design (HFFD). Multivariate optimization through central composite design (CCD) allowed establishing the statistical ideal parameter values to reach maximum Mo signal. From HFFD and CCD it was established that SFR was not affecting the system significantly and that the optimal experimental conditions were: pH, 4.0; BC, 5mM ammonium acetate; EC, 15% (vv−1) and EFR, 2mLmin−1. Under these conditions an enhancement factor of 750-fold (25 for preconcentration system and 30 for USN) was obtained reaching a detection limit of 40ngL−1 with a precision of 1.32%. The system was successfully applied to a certified reference material NIST CRM 1643e (trace elements in water) and river, thermal, mine and tap water samples.

Three dimensional mirror symmetry and partition function on S 3

We provide non-trivial checks of $ \mathcal{N} $ = 4, D = 3 mirror symmetry in a large class of quiver gauge theories whose Type IIB (Hanany-Witten) descriptions involve D3 branes ending on orbifold/orientifold 5-planes at the boundary. From the M-theory perspective, such theories can be understood in terms of coincident M2 branes sitting at the origin of a product of an A-type and a D-type ALE (Asymtotically Locally Euclidean) space with G-fluxes. Families of mirror dual pairs, which arise in this fashion, can be labeled as (A m−1 , D n ), where m and n are integers. For a large subset of such infinite families of dual theories, corresponding to generic values of n ≥ 4, arbitrary ranks of the gauge groups and varying m, we test the conjectured duality by proving the precise equality of the S 3 partition functions for dual gauge theories in the IR as functions of masses and FI parameters. The mirror map for a given pair of mirror dual theories can be read off at the end of this computation and we explicitly present these for the aforementioned examples. The computation uses non-trivial identities of hyperbolic functions including certain generalizations of Cauchy determinant identity and Schur’s Pfaffian identity, which are discussed in the paper.

Aspects of 3d $ \mathcal{N}=2 $ Chern-Simons-Matter theories

We comment on various aspects of the the dynamics of 3d N=2 Chern-Simons gauge theories and their possible phases. Depending on the matter content, real masses and FI parameters, there can be non-compact Higgs or Coulomb branches, compact Higgs or Coulomb branches, and isolated vacua. We compute the Witten index of the theories, and show that it does not change when the system undergoes a phase transition. We study aspects of monopole operators and solitons in these theories, and clarify subtleties in the soliton collective coordinate quantization. We show that solitons are compatible with a mirror symmetry exchange of Higgs and Coulomb branches, with BPS solitons on one branch related to the modulus of the other. Among other results, we show how to derive Aharony duality from Giveon-Kutasov duality.

Wall-crossing, Toric divisor and Seiberg duality

We study the wall-crossing phenomena of BPS D4-D2-D0 states on the conifold and orbifold $ {{{{{\mathbb{C}}^2}}} \left/ {{{{\mathbb{Z}}_2}}} \right.} $ , from the viewpoint of the quiver quantum mechanics on the D-branes. The Kahler moduli dependence of the BPS index is translated into the FI parameter dependence of the Witten index. The wall-crossing phenomena are related to the Seiberg dualities of the quiver quantum mechanics. All the differences from the D6-D2-D0 case arise from the additional superpotential and “anti-quark” induced by the D4-brane. When the D-branes are on the conifold, the flop transition changes the duality cascade. When the D-branes are on the orbifold $ {{{{{\mathbb{C}}^2}}} \left/ {{{{\mathbb{Z}}_2}}} \right.} $ , the generating function of the Witten index is always given by a character of the affine SU(2) algebra. Both are consistent with the wall-crossing formula for BPS indices.

Acting Gay: Male Actors Shift the Frequency Components of Their Voices Towards Female Values When Playing Homosexual Characters

The purpose of this study was to investigate whether actors playing homosexual male characters in North-American television shows speak with a feminized voice, thus following longstanding stereotypes that attribute feminine characteristics to male homosexuals. We predicted that when playing homosexual characters, actors would raise the frequency components of their voice towards more stereotypically feminine values. This study compares fundamental frequency (F0) and formant frequencies (F i ) parameters in the speech of fifteen actors playing homosexual and heterosexual characters in North-American television shows. Our results reveal that the voices of actors playing homosexual male characters are characterized by a raised F0 (corresponding to a higher pitch), and raised formant frequencies (corresponding to a less baritone timbre), approaching values typical of female voices. Besides providing further evidence of the existence of an “effeminacy” stereotype in portraying male homosexuals in the media, these results show that actors perform pitch and vocal tract length adjustments in order to alter their perceived sexual orientation, emphasizing the role of these frequency components in the behavioral expression of gender attributes in the human voice.

Relation between the 4d superconformal index and the S 3 partition function

A relation between the 4d superconformal index and the S^3 partition function is studied with focus on the 4d and 3d actions used in localization. In the case of vanishing Chern-Simons levels and round S^3 we explicitly show that the 3d action is obtained from the 4d action by dimensional reduction up to terms which do not affect the exact results. By combining this fact and a recent proposal concerning a squashing of S^3 and SU(2) Wilson line, we obtain a formula which gives the partition function depending on the Weyl weight of chiral multiplets, real mass parameters, FI parameters, and a squashing parameter as a limit of the index of a parent 4d theory.