arXiv subjects

Shan Hu

Publications and source records attributed to Shan Hu.

17 recordsLinked to original sources

arXiv2025

S-Duality for Non-Abelian Monopoles

In $\mathcal{N}=4$ super-Yang-Mills theory with gauge group $G$ spontaneously broken to a subgroup $H$, S-duality requires that the BPS monopole spectrum organizes into the same representation as W-bosons in the dual theory, where $G^{\vee}$ is broken to $H^{\vee}$. The expectation has been extensively verified in the maximally broken phase $G\to U(1)^r$. Here we address the non-Abelian regime in which $H$ contains a semisimple factor $H^{s}$. Using the stratified description of monopole moduli space, we give a general proof of this matching for any simple gauge group $G$. Each BPS monopole state is naturally labeled by a weight of the relevant $W$-boson representation of $(H^{\vee})^{s}$. We construct non-Abelian magnetic gauge transformation operators implementing the $(H^{\vee})^{s}$-action on the monopole Hilbert space, which commute with the electric $H^{s}$-transformations and thereby realize the $H^{s}\times (H^{\vee})^{s}$ symmetry at the level of monopole quantum mechanics.

hep-th↗

arXiv2024

Nontrivial bundles and defect operators in $n$-form gauge theories

In $(d+1)$-dimensional $1$-form nonabelian gauge theories, we classify nontrivial $0$-form bundles in $ \mathbb{R}^{d} $, which yield configurations of $D(d-2j)$-branes wrapping $(d-2j)$-cycles $c_{d-2j} $ in $Dd$-branes. We construct the related defect operators $ U^{(2j-1)} ( c_{d-2j} ) $, which are disorder operators carrying the $D(d-2j)$ charge. We compute the commutation relations between the defect operators and Chern-Simons operators on odd-dimensional closed manifolds, and derive the generalized Witten effect for $U^{(2j-1)} ( c_{d-2j} ) $. When $c_{d-2j}$ is not exact, $ U^{(2j-1)} ( c_{d-2j} ) $ and $ U^{(2j-1)} (- c_{d-2j} ) $ can also combine into an electric $(2j-1)$-form global symmetry operator, where the $(2j-1)$-form is the Chern-Simons form. The dual magnetic $(d-2j)$-form global symmetry is generated by the $D(d-2j)$ charge. We also study nontrivial $1$-form bundles in $(d+1)$-dimensional $2$-form nonabelian gauge theories, where the defect operators are $\mathcal{U}^{(2j)} ( c_{d-2j-1} ) $. With the field strength of the $1$-form taken as the flat connection of the $2$-form, we classify the topological sectors in $2$-form theories.

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arXiv2023

Quantum Spectrum of BPS Instanton States in $ 5d $ Guage Theories

We construct $ 1/2 $ BPS 1-instanton states in the Hilbert space of $ 5d $ gauge theories. The states are local with the scale size $ \rho=0 $. With the degeneracy from the gauge orientation taken into account, the BPS spectrum consists of an infinite number of states in the definite representations of the gauge group and the $ (J,0) $ representation of the $ SO(4)\cong SU(2)_{L} \times SU(2)_{R} $ rotation group. In the brane webs realization of the $ 5d $ $ \mathcal{N}=1 $ gauge theory, BPS states are string webs inside the 5-brane webs. For a theory with the gauge group $ SU(N) $ and the Chern-Simons level $ \kappa $, we give a classification of string webs in the Coulomb branch and show that for each web, one can always find an instanton state carrying the same spin and the same electric charge. The action of supercharges on instanton states gives the instanton supermultiplets. For the $ \mathcal{N} =1$ $ SU(2) $ gauge theory with $ N_{f} \leq 4$, we construct the instanton vector and hyper multiplets, which, together with the original multiplets of the theory, furnish the complete representation of the $ E_{N_{f}+1} $ group at UV.

hep-th↗

arXiv2022

Extracting Densest Sub-hypergraph with Convex Edge-weight Functions

The densest subgraph problem (DSG) aiming at finding an induced subgraph such that the average edge-weights of the subgraph is maximized, is a well-studied problem. However, when the input graph is a hypergraph, the existing notion of DSG fails to capture the fact that a hyperedge partially belonging to an induced sub-hypergraph is also a part of the sub-hypergraph. To resolve the issue, we suggest a function $f_e:\mathbb{Z}_{\ge0}\rightarrow \mathbb{R}_{\ge 0}$ to represent the partial edge-weight of a hyperedge $e$ in the input hypergraph $\mathcal{H}=(V,\mathcal{E},f)$ and formulate a generalized densest sub-hypergraph problem (GDSH) as $\max_{S\subseteq V}\frac{\sum_{e\in \mathcal{E}}{f_e(|e\cap S|)}}{|S|}$. We demonstrate that, when all the edge-weight functions are non-decreasing convex, GDSH can be solved in polynomial-time by the linear program-based algorithm, the network flow-based algorithm and the greedy $\frac{1}{r}$-approximation algorithm where $r$ is the rank of the input hypergraph. Finally, we investigate the computational tractability of GDSH where some edge-weight functions are non-convex.

cs.DS↗

arXiv2021

Monopole operators and symmetry enhancement in ABJM theory revisited

We construct monopole operators for 3d Yang-Mills-matter theories and Chern-Simons-matter theories in canonical formalism. In this framework, monopole operators, although as the disorder operators, could be written in terms of the fundamental fields of the theory, and thus could be treated in the same way as the ordinary operators. We study the properties of the constructed monopole operators. In Chern-Simons-matter theories, monopole operators transform as the local operators with the classical conformal dimension $ 0 $ under the action of the dilation and are also covariantly constant. In supersymmetric Chern-Simons-matter theories like the ABJM model, monopole operators commute with all of the supercharges, and thus are SUSY invariant. ABJM model with level $ k=1,2 $ is expected to have enhanced $SO(8) $ R-symmetry due to the existence of the conserved extra R-symmetry currents $ j^{AB}_{\mu} $ involving monopoles. With the explicit form of the monopole operators given, we prove the current conservation equation $\partial^{\mu} j^{AB}_{\mu} =0$ using the equations of motion. We also compute the extra $ \mathcal{N}=2 $ supercharges, derive the extra $ \mathcal{N}=2 $ SUSY transformation rules, and verify the closure of the $ \mathcal{N}=8 $ supersymmetry.

hep-th↗

arXiv2020

S-duality and loop operators in canonical formalism

We study the gauge invariant 't Hooft operator in canonical formalism for Yang-Mills theory as well as the $\mathcal{N} =4 $ super-Yang-Mills theory with the gauge group $ U(N) $. It is shown that the spectrum of the 't Hooft operator labeled by the arbitrary irreducible representation of the gauge group is the same as the spectrum of the Wilson operator labeled by the same representation. So it is possible to construct a unitary operator $ S $ making the two kinds of loop operators transformed into each other. S-duality transformation could be realized by the operator $ S $. We compute the supersymmetry variations of the loop operators with the fermionic couplings turned off. The result is consistent with the expectation that the action of $ S $ should make supercharges transform with a $ U(1)_{Y} $ phase.

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arXiv2019

S-duality transformation of $\mathcal{N}$ $=4$ SYM theory at the operator level

We consider the S-duality transformation of gauge invariant operators and states in $\mathcal{N}$ $=4$ SYM theory. The transformation is realized through an operator $ S $ which is the $ SL(2,Z) $ canonical transformation in loop space with the gauge invariant electric and the magnetic flux operators composing the canonical variables. Based on $ S $, S-duals for all of the physical operators and states can be defined. The criterion for the theory to be S-duality invariant is that the superconformal charges and their S-duals differ by a $ U(1)_{Y} $ phase. The verification can be done by checking the S transformation for supersymmetry and special supersymmetry variations of the loop operators. The fact that supercharges preserved by BPS Wilson operators and the S-dual BPS 't Hooft operators differ by a $4d $ chiral rotation could in some sense serve as a proof.

hep-th↗

arXiv2019

A note on supergravity inflation in braneworld

We discuss supergravity inflation in braneworld cosmology for the class of potentials $V(\phi)=\alpha \phi^n\rm{exp}(-\beta^m \phi^m)$ with $m=1,~2$. These minimal SUGRA models evade the $\eta$ problem due to a broken shift symmetry and can easily accommodate the observational constraints. In the high energy regime $V/\lambda\gg 1$, the numerical predictions and approximate analytic formulas are given for the scalar spectral index $n_s$ and tensor-to-scalar ratio $r$. The models with smaller $n$ are preferred while the models with larger $n$ are out of the $2\sigma$ region. Remarkably, the $\rho^2/\lambda$ correction to the energy density in Friedmann equation results in sub-Planckian inflaton excursions $\Delta\phi <1$.

hep-th↗

arXiv2018

Natural Higgs Inflation, Gauge Coupling Unification, and Neutrino Masses

We present a class of non-supersymmetric models in which so-called critical Higgs inflation ($\xi<100$) naturally can be realized without using specific values for Higgs and top quark masses. In these scenarios, the Standard Model (SM) vacuum stability problem, gauge coupling unification, neutrino mass generation and Higgs inflation mechanism are linked to each other. We adopt in our models Type I seesaw mechanism for neutrino masses. An appropriate choice of the Type I Seesaw scale allows us to have an arbitrarily small but positive value of SM Higgs quartic coupling around the inflation scale. We present a few benchmark points where we show that the scalar spectral indices are around 0.9626 and 0.9685 for the number of e-folding $N=50$ and $N=60$ respectively. The tensor-to-scalar ratios are order of $10^{-3}$. The running of the scalar spectral index is negative and is order of $10^{-4}$.

hep-ph↗

arXiv2018

The Minimal GUT with Inflaton and Dark Matter Unification

Giving up the solutions to the fine-tuning problems, we propose the non-supersymmetric flipped $SU(5)\times U(1)_X$ model based on the minimal particle content principle, which can be constructed from the four-dimensional $SO(10)$ models, five-dimensional orbifold $SO(10)$ models, and local F-theory $SO(10)$ models. To achieve gauge coupling unification, we introduce one pair of vector-like fermions, which form complete $SU(5)\times U(1)_X$ representation. Proton lifetime is around $5\times 10^{35}$ years, neutrino masses and mixing can be explained via seesaw mechanism, baryon asymmetry can be generated via leptogenesis, and vacuum stability problem can be solved as well. In particular, we propose that inflaton and dark matter particle can be unified to a real scalar field with $Z_2$ symmetry, which is not an axion and does not have the non-minimal coupling to gravity. Such kind of scenarios can be applied to the generic scalar dark matter models. Also, we find that the vector-like particle corrections to the $B_s^0$ masses can be about 6.6%, while their corrections to the $K^0$ and $B_d^0$ masses are negligible.

hep-ph↗

arXiv2016

U-duality transformation of membrane on $T^{n}$ revisited

The problem with the U-duality transformation of membrane on $T^{n}$ is recently addressed in [arXiv:1509.02915 [hep-th]]. We will consider the U-duality transformation rule of membrane on $T^{n}\times R$. It turns out that winding modes on $T^{n}$ should be taken into account, since the duality transformation may bring the membrane configuration without winding modes into the one with winding modes. With the winding modes added, the membrane worldvolume theory in lightcone gauge is equivalent to the $n+1$ dimensional super-Yang-Mills (SYM) theory in $\tilde{T}^{n}$, which has $SL(2,Z)\times SL(3,Z)$ and $SL(5,Z)$ symmetries for $n=3$ and $n=4$, respectively. The $SL(2,Z)\times SL(3,Z)$ transformation can be realized classically, making the on-shell field configurations transformed into each other. However, the $SL(5,Z)$ symmetry may only be realized at the quantum level, since the classical $5d$ SYM field configurations cannot form the representation of $SL(5,Z)$.

hep-th↗

arXiv2016

No-Scale $μ$-Term Hybrid Inflation

To solve the fine-tuning problem in $μ$-Term Hybrid Inflation, we will realize the supersymmetry scenario with the TeV-scale supersymmetric particles and intermediate-scale gravitino from anomaly mediation, which can be consistent with the WMAP and Planck experiments. Moreover, we for the first time propose the $μ$-term hybrid inflation in no-scale supergravity. With four Scenarios for the $SU(3)_C\times SU(2)_L\times SU(2)_R\times U(1)_{B-L}$ model, we show that the correct scalar spectral index $n_s$ can be obtained, while the tensor-to-scalar ratio $r$ is prediced to be tiny, about $10^{-10}-10^{-8}$. Also, the $SU(2)_R\times U(1)_{B-L}$ symmetry breaking scale is around $10^{14}$ GeV, and all the supersymmetric particles except gravitino are around TeV scale while gravitino mass is around $10^{9-10}$ GeV. Considering the complete potential terms linear in $S$, we for the first time show that the tadpole term, which is the key for such kind of inflationary models to be consistent with the observed scalar spectral index, vanishes after inflation. Thus, to obtain the $μ$ term, we need to generate the supersymmetry breaking soft term $A^{S ΦΦ'}_κ κS ΦΦ'$ due to $A^{S ΦΦ'}_κ=0 $ in no-scale supergravity, where $Φ$ and $Φ'$ are vector-like Higgs fields at high energy. We show that the proper $A^{S ΦΦ'}_κ κS ΦΦ'$ term can be obtained in the M-theory inspired no-scale supergravity. We also point out that $A^{S ΦΦ'}_κ$ around 700 GeV can be generated via the renormalization group equation running from string scale.

hep-ph↗

arXiv2015

Group manifold approach to higher spin theory

We consider the group manifold approach to higher spin theory. The deformed local higher spin transformation is realized as the diffeomorphism transformation in the group manifold $\textbf{M}$. With the suitable rheonomy condition and the torsion constraint imposed, the unfolded equation can be obtained from the Bianchi identity, by solving which, fields in $\textbf{M}$ are determined by the multiplet at one point, or equivalently, by $(W^{[a(s-1),b(0)]}_μ,H)$ in $AdS_{4}\subset \textbf{M}$. Although the space is extended to $\textbf{M}$ to get the geometrical formulation, the dynamical degrees of freedom are still in $AdS_{4}$. The $4d$ equations of motion for $(W^{[a(s-1),b(0)]}_μ,H)$ are obtained by plugging the rheonomy condition into the Bianchi identity. The proper rheonomy condition allowing for the maximum on-shell degrees of freedom is given by Vasiliev equation. We also discuss the theory with the global higher spin symmetry, which is in parallel with the WZ model in supersymmetry.

hep-th↗

arXiv2014

Radial quantization of the 3d CFT and the higher spin/vector model duality

We study the radial quantization of the 3d O(N) vector model. We calculate the higher spin charges whose commutation relations give the higher spin algebra. The Fock states of higher spin gravity in AdS_{4} are realized as the states in the 3d CFT. The inner products between these states encode dynamical information. The construction of bulk operators from CFT is discussed as well. This serves as the simplest explicit demonstration of the CFT definition for the quantum gravity.

hep-th↗

arXiv2012

Momentum modes of M5-branes in a 2d space

We study M5 branes by considering the selfdual strings parallel to a plane. With the internal oscillation frozen, each selfdual string gives a 5d SYM field. All selfdual strings together give a 6d field with 5 scalars, 3 gauge degrees of freedom and 8 fermionic degrees of freedom in adjoint representation of U(N). Selfdual strings with the same orientation have the SYM-type interaction. For selfdual strings with the different orientations, which could also be taken as the unparallel momentum modes of the 6d field on that plane or the (p,q) (r,s) strings on D3 with (p,q)\neq (r,s), the [i,j]+[j,k]\rightarrow [i,k] relation is not valid, so the coupling cannot be written in terms of the standard N \times N matrix multiplication. 3-string junction, which is the bound state of the unparallel [i,j] [j,k] selfdual strings, may play a role here.

hep-th↗

arXiv2011

Hopf-Wess-Zumino term in the effective action of the 6d, (2, 0) field theory revisted

We discuss the Hopf-Wess-Zumino term in the effective action of the 6d (2, 0) theory of the type A_{N-1} in a generic Coulomb branch. For such terms, the supergravity calculation could be trusted. We calculate the WZ term on supergravity side and show that it could compensate the anomaly deficit, as is required by the anomaly matching condition. In contrast with the SYM theory, in which each WZ term involves one root e_{i}-e_{j}, here, the typical WZ term involves two roots e_{i}-e_{j} and e_{k}-e_{j}. Such kind of triple interaction may come from the integrating out of the massive states carrying three indices. A natural candidate is the recently proposed 1/4 BPS objects in the Coulomb phase of the 6d (2, 0) theories. The WZ term could be derived from the field theory by the integration out of massive degrees of freedom. Without the 6d (2, 0) theory at hand, we take the supersymmetric equations for the 3-algebra valued (2, 0) tensor multiplet as the prototype to see how far we can go. The H_{3}\wedge A_{3} part of the WZ term is obtained, while the A_{3}\wedge F_{4} part, which is the term accounting for the anomaly matching, cannot be produced by the standard fermion loop integration.

hep-th↗

arXiv2011

Type IIB Supersymmetric Flux Vacua

On the Type IIB toroidal T^6 orientifolds with generic flux compactifications, we conjecture that in generic supsersymmetric Minkowski vacua, at least one of the flux contributions to the seven-brane and D3-brane tadpoles is positive if the moduli are stabilized properly, and then the tadpole cancellation conditions can not be relaxed. To study the supsersymmetric Minkowski flux vacua, we simplify the fluxes reasonably and discuss the corresponding superpotential. We show that we can not have simultaneously the positive real parts of all the moduli and the negative/zero flux contributions to all the seven-brane and D3-brane tadpoles. Therefore, we can not construct realistic flux models with the relaxed tadpole cancellation conditions. When studying the supsersymmetric AdS vacua, we obtain flux models with the seven-brane and D3-brane tadpole cancellation conditions relaxed elegantly, and we present a semi-realistic Pati-Salam model as well as its particle spectrum. The lifting from the AdS vacua to the Minkowski/dS vacua remains a great challenge in flux model buildings on toroidal orientifolds.

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