non-Abelian Strings Research Articles

Flowing between string vacua for the critical non-Abelian vortex with a deformation of N=2 Liouville theory

It has been shown that non-Abelian solitonic vortex string supported in four-dimensional (4D) N=2 supersymmetric QCD (SQCD) with the U(2) gauge group and Nf=4 quark flavors becomes a critical superstring. This string propagates in the ten-dimensional space formed by a product of the flat 4D space and an internal space given by a Calabi-Yau noncompact threefold, namely, the conifold. The spectrum of low-lying closed string states was found and interpreted as a spectrum of hadrons in 4D N=2 SQCD. In particular, the lowest string state appears to be a massless Bogomol’nyi-Prasad-Sommerfield (BPS) baryon associated with the deformation of the complex structure modulus b of the conifold. It was recently shown that the Coulomb branch of the associated string sigma model which opens up at strong coupling can be described by N=2 Liouville theory. Building on these results we switch on quark masses in 4D N=2 SQCD and study the interpolation of the initial U(2) SQCD with Nf=4 quarks to the final SQCD with the U(4) gauge group and Nf=8 quarks. To find the true string vacuum which arises due to the mass deformation we solve the effective supergravity equations of motion associated with the deformed world sheet Liouville theory. We show that the massless BPS baryon b survives the deformation and that finding of the spectrum of low-lying massive hadrons in the final SQCD is linked to the Calogero problem. Published by the American Physical Society 2024

First-order formalism for Alice string

We apply the first-order formalism method to obtaining BPS equations for Alice string. This is done by generalizing the well-known first-order formalism to the case of non-Abelian strings. We do not assume any specific gauge group nor the shape of the kinetic term function, but require only that the fields are axially-symmetric and static. With this formalism we reproduce the BPS equations of SU(2)×U(1) Alice strings (Chatterjee and Nitta, 2017), and present their corresponding numerical solutions.

Construction of non-Abelian electric strings

We detail the construction of electric string solutions in $SU(2)$ Yang-Mills-Higgs theory with a scalar in the fundamental representation and discuss the properties of the solution. We show that Schwinger gluon pair production in the electric string background is absent. A similar construction in other models, such as with an adjoint scalar field and the electroweak model, does not yield solutions.

NS three-form flux deformation for the critical non-Abelian vortex string

It has been shown that the non-Abelian solitonic vortex string supported in four-dimensional (4D) $\mathcal{N}=2$ supersymmetric QCD (SQCD) with the U(2) gauge group and ${N}_{f}=4$ quark flavors becomes a critical superstring. This string propagates in the ten-dimensional space formed by a product of the flat 4D space and an internal space given by a Calabi-Yau noncompact threefold, namely, the conifold. The spectrum of low-lying closed string states in the associated type-IIA string theory was found and interpreted as a spectrum of hadrons in 4D $\mathcal{N}=2$ SQCD. In particular, the lowest string state appears to be a massless BPS baryon associated with the deformation of the complex structure modulus $b$ of the conifold. In the previous work the deformation of the ten-dimensional background with nonzero Neveu-Schwarz 3-form flux was considered and interpreted as a switching on a particular choice of quark masses in 4D SQCD. This deformation was studied to leading order at small 3-form flux. In this paper we study the back reaction of the nonzero 3-form flux on the metric and the dilaton introducing the ansatz with several warp factors and solving the gravity equations of motion. We show that 3-form flux produces a potential for the conifold complex structure modulus $b$, which leads to the runaway vacuum. At the runaway vacuum warp factors disappear, while the conifold degenerates. In 4D SQCD we relate this to the flow to the U(1) gauge theory upon switching on quark masses and decoupling of two flavors.

Critical non-Abelian vortices and holography for little string theory

It has been shown that non-Abelian vortex string supported in four dimensional (4D) ${\mathcal N}=2$ supersymmetric QCD (SQCD) with the U(2) gauge group and $N_f = 4$ quark flavors becomes a critical superstring. This string propagates in the ten dimensional space formed by a product of the flat 4D space and an internal space given by a Calabi-Yau non-compact threefold, namely, the conifold. The spectrum of closed string states of the associated string theory was obtained using the equivalence between the critical string on the conifold and the non-critical string on the semi-infinite cigar described by SL($2, \mathbb{R}$)/U(1) Wess-Zumino-Novikov-Witten model. This spectrum was identified with the spectrum of hadrons in 4D ${\mathcal N}=2$ SQCD. In order to describe effective interactions of these 4D hadrons in this paper we study correlation functions of normalizable vertex operators localized near the tip of the SL($2, \mathbb{R}$)/U(1) cigar. We also compare our solitonic string approach to the gauge-string duality to the AdS/CFT-type holography for little string theories (LSTs). The latter relates off mass-shell correlation functions on the field theory side to correlation functions of non-normalizable vertex operators on the cigar. We show that in most channels holographic approach works in our theory because normalizable and non-normalizable vertex operators with the same conformal dimension are related due to the reflection from the tip of the cigar. However, we find that holography does not work for lightest hadrons with given baryonic charge.

Flux compactification for the critical non-Abelian vortex and quark masses

It has been shown that non-Abelian solitonic vortex strings supported in four-dimensional (4D) ${\mathcal N}=2\;$ supersymmetric QCD (SQCD) with the U($N=2$) gauge group and $N_f=4$ quark flavors behave as critical superstrings. In addition to four translational moduli non-Abelian strings under consideration carry six orientational and size moduli. Together they form a ten-dimensional space required for a superstring to be critical. The target space of the string sigma model is a product of the flat four-dimensional space $\mathbb{R}^4$ and a Calabi-Yau non-compact threefold $Y_6$, namely, the conifold. The spectrum of low lying closed string states in the associated type IIA string theory was found and interpreted as a spectrum of hadrons in 4D ${\mathcal N}=2\;$ SQCD. In particular, the lowest string state appears to be a massless BPS baryon associated with the deformation of the complex structure modulus $b$ of the conifold. Here we address a problem of switching on quark masses in 4D SQCD, which classically breaks the world sheet conformal invariance in the string sigma model. To avoid this problem we follow a standard string theory approach and use a flux "compactification" to lift the complex structure modulus of the conifold. Namely, we find a solution of supergravity equations of motion with non-zero NS 3-form flux. It produces a potential for the baryon $b$, which leads to the run-away vacuum. Using field theory arguments we interpret 3-form flux in terms of a particular choice of quark masses in 4D SQCD. At the run-away vacuum the conifold degenerates to lower dimensions. We interpret this as a flow from a non-Abelian string to an Abelian one.

Vortices penetrating two-flavor quark-hadron continuity

Alice and Boojums are both representative characters created by Lewis Carroll. We show that they possibly meet in cores of rotating neutron stars. Recent studies of quark-hadron continuity suggest that neutron superfluid matter can connect smoothly to two-flavor symmetric quark matter at high densities. We study how this can be maintained in the presence of the vortices. In the neutron matter, quantized superfluid vortices arise. In the two-flavor dense quark matter, vortices carrying color magnetic fluxes together with fractionally quantized superfluid circulations appear as the most stable configuration, and we call these as the non-Abelian Alice strings. We show that three integer neutron superfluid vortices and three non-Abelian Alice strings of different color magnetic fluxes with total color flux canceled out are joined at a junction called a Boojum.

Non-Abelian Alice strings in two-flavor dense QCD

Quark-hadron continuity with two-flavor quarks that was proposed recently connects hadronic matter with neutron $^3P_2$ superfluidity and two-flavor dense quark matter. This two-flavor dense quark phase consists of the coexistence of the 2SC condensates and the $P$-wave diquark condensates of $d$-quarks, which gives rise to color superconductivity as well as superfluidity. We classify vortices in this phase. The most stable vortices are what we call the non-Abelian Alice strings, which are superfluid vortices with non-Abelian color magnetic fluxes therein, exhibiting so-called topological obstruction, or a non-Abelian generalization of the Alice property. We show that a single Abelian superfluid vortex is unstable against decay into three non-Abelian Alice strings. We discover that a non-Abelian Alice string carries orientational moduli of the real projective space $\mathbb{R}P^2$ corresponding to the color flux therein in the presence of the $P$-wave condensates alone. We calculate Aharanov-Bohm (AB) phases around the non-Abelian Alice string, and find that the 2SC condensates and string's orientational moduli must be aligned with each other because of single-valuedness of the AB phases of the 2SC condensates.

New Abelian-like monopoles and the dual Meissner effect

Violation of non-Abelian Bianchi identity can be regarded as $N^2-1$ Abelian-like monopole currents in the continuum SU(N) QCD. Three Abelian-like monopoles, when defined in SU(2) gluodynamics on the lattice \`{a} la DeGrand-Toussaint, are shown to have the continuum limit with respect to the color-invariant monopole density and the effective monopole action. Since each Abelian-like monopole is not gauge invariant, we have introduced various partial gauge fixing for the purpose of reducing lattice artifact monopoles in the thermalized vacuum. Here we investigate Abelian and monopole dominances and the Abelian dual Meissner effects adopting the same gauges like the maximal center gauge (MCG) in comparison with the maximal Abelian gauge (MAG). Abelian and monopole contributions to the string tension in these gauges are observed to be a little smaller than the non-Abelian string tension. However, we find that the monopole dominance is improved well when use is made of the block-spin transformations with respect to Abelian-like monopoles. We find each electric field is squeezed by the corresponding colored Abelian-like monopole in such gauges and the Abelian dual Meissner effect is observed independently for each color. Moreover, we confirm the dual Amp\`{e}re's law in these new gauges as well as in MAG. The SU(2) vacuum is shown to be near the border between the type 1 and type 2 dual superconductors. The penetration length is almost equal for the four gauge fixings and the vacuum type in MCG is almost the same value as the previous results. These results are consistent with the previous results suggesting the continuum limit and the gauge-independence of Abelian monopoles.

Determining non-Abelian topological order from infinite projected entangled pair states

We generalize the method introduced in Phys. Rev. B 101, 041108 (2020) of extracting information about topological order from the ground state of a strongly correlated two-dimensional system represented by an infinite projected entangled pair state (iPEPS) to non-Abelian topological order. When wrapped on a torus the unique iPEPS becomes a superposition of degenerate and locally indistinguishable ground states. We find numerically symmetries of the iPEPS, represented by infinite matrix product operators (MPO), and their fusion rules. The rules tell us how to combine the symmetries into projectors onto states with well defined anyon flux. A linear structure of the MPO projectors allows for efficient determination for each state its second Renyi topological entanglement entropy on an infinitely long cylinder directly in the limit of infinite cylinder's width. The same projectors are used to compute topological $S$ and $T$ matrices encoding mutual- and self-statistics of emergent anyons. The algorithm is illustrated by examples of Fibonacci and Ising non-Abelian string net models.

Long way to Ricci flatness

We study two-dimensional weighted mathcal{N} = (2) supersymmetric ℂℙ models with the goal of exploring their infrared (IR) limit. \U0001d54eℂℙ(N, tilde{N} ) are simplified versions of world-sheet theories on non-Abelian strings in four-dimensional mathcal{N} = 2 QCD. In the gauged linear sigma model (GLSM) formulation, \U0001d54eℂℙ(N, tilde{N} ) has N charges +1 and tilde{N} charges −1 fields. As well-known, at tilde{N} = N this GLSM is conformal. Its target space is believed to be a non-compact Calabi-Yau manifold. We mostly focus on the N = 2 case, then the Calabi-Yau space is a conifold.On the other hand, in the non-linear sigma model (NLSM) formulation the model has ultra-violet logarithms and does not look conformal. Moreover, its metric is not Ricci-flat. We address this puzzle by studying the renormalization group (RG) flow of the model. We show that the metric of NLSM becomes Ricci-flat in the IR. Moreover, it tends to the known metric of the resolved conifold. We also study a close relative of the \U0001d54eℂℙ model — the so called zn model — which in actuality represents the world sheet theory on a non-Abelian semilocal string and show that this zn model has similar RG properties.

String baryon in four-dimensional N=2 supersymmetric QCD from the 2D-4D correspondence

We study non-Abelian vortex strings in four-dimensional (4D) $\mathcal{N} = 2$ supersymmetric QCD with U$(N=2)$ gauge group and $N_f=4$ flavors of quark hypermultiplets. It has been recently shown that these vortices behave as critical superstrings. The spectrum of closed string states in the associated string theory was found and interpreted as a spectrum of hadrons in 4D $\mathcal{N} = 2$ supersymmetric QCD. In particular, the lowest string state appears to be a massless BPS "baryon." Here we show the occurrence of this stringy baryon using a purely field-theoretic method. To this end we study the conformal world-sheet theory on the non-Abelian string -- the so called weighted $\mathcal{N} = (2,2)$ supersymmetric $\mathbb{CP}$ model. Its target space is given by the six-dimensional non-compact Calabi-Yau space $Y_6$, the conifold. We use mirror description of the model to study the BPS kink spectrum and its transformations on curves (walls) of marginal stability. Then we use the 2D-4D correspondence to show that the deformation of the complex structure of the conifold is associated with the emergence of a non-perturbative Higgs branch in 4D theory which opens up at strong coupling. The modulus parameter on this Higgs branch is the vacuum expectation value of the massless BPS "baryon" previously found in string theory.

Dynamics of non-Abelian strings in the theory interpolating from N=2 to N=1 supersymmetric QCD

We study the dynamics of non-Abelian vortex strings supported in ${\mathcal N}=2$ supersymmetric QCD with the U$(N)$ gauge group and $N_f=N$ quark flavors deformed by the mass $\mu$ of the adjoint matter. In the limit of large $\mu$ the bulk four-dimensional theory flows to ${\mathcal N}=1$ supersymmetric QCD. The dynamics of orientational zero modes of the non-Abelian string is described by the world sheet CP$(N-1)$ model. At $\mu=0$ this model has ${\mathcal N}= \left(2,2\right) $ supersymmetry while at large $\mu$ it flows to the non-supersymmetric CP$(N-1)$ model. We solve the world sheet model in the large $N$ approximation and find a rich phase structure with respect to the deformation parameter $\mu$ and quark mass differences. The phases include two strong coupling phases and two Higgs phases. In particular, the Higgs phase at small $\mu$ supports CP$(N-1)$ model kinks representing confined monopoles of the bulk QCD, while in the large-$\mu$ Higgs phase monopoles disappear.

General composite non-Abelian strings and flag manifold sigma models

We fully investigate the symmetry breaking patterns occurring upon creation of composite non-Abelian strings: vortex strings in non-Abelian theories where different sets of colours have different amounts of flux. After spontaneous symmetry breaking, there remains some internal colour degrees of freedom attached to these objects, which we argue must exist in a Flag manifold, a more general kind of projective space than both $\mathbb{CP}(N)$ and the Grassmannian manifold. These strings are expected to be BPS, since its constituents are. We demonstrate that this is true and construct a low-energy effective action for the fluctuations of the internal Flag moduli, which we then re-write it in two different ways for the dynamics of these degrees of freedom: a gauged linear sigma model with auxiliary fields and a non-linear sigma model with an explicit target space metric for the Flag Manifolds, both of which $\mathcal{N}=(2,2)$ supersymmetric. We finish by performing some groundwork analysis of the resulting theory.

Gauging fractons: Immobile non-Abelian quasiparticles, fractals, and position-dependent degeneracies

The study of gapped quantum many-body systems in three spatial dimensions has uncovered the existence of quantum states hosting quasiparticles that are confined, not by energetics but by the structure of local operators, to move along lower dimensional submanifolds. These so-called "fracton" phases are beyond the usual topological quantum field theory description, and thus require new theoretical frameworks to describe them. Here we consider coupling fracton models to topological quantum field theories in (3+1) dimensions by starting with two copies of a known fracton model and gauging the $\mathbb{Z}_2$ symmetry that exchanges the two copies. This yields a class of exactly solvable lattice models that we study in detail for the case of the X-cube model and Haah's cubic code. The resulting phases host finite-energy non-Abelian immobile quasiparticles with robust degeneracies that depend on their relative positions. The phases also host non-Abelian string excitations with robust degeneracies that depend on the string geometry. Applying the construction to Haah's cubic code in particular provides an exactly solvable model with finite energy yet immobile non-Abelian quasiparticles that can only be created at the corners of operators with fractal support.

Composite non-Abelian strings with Grassmannian models on the worldsheet

Most of the non-Abelian string-vortices studied so far are characterized by two-dimensional \cpn models with various degrees of supersymmetry on their world sheet. We generalize this construction to "composite" non-Abelian strings supporting the Grassmann $\mathcal{G}(L,M)$ models (here $L+M=N$). The generalization is straightforward and provides, among other results, a simple and transparent way for counting the number of vacua in ${\mathcal N}=(2,2)$ Grassmannian model.

Heterotically deformed sigma models on the world sheet of semilocal strings in supersymmetric QED

A new two dimensional $\mathcal{N}=(0,2)$ Supersymmetric Non-Linear Sigma Model describes the dynamics of internal moduli of the BPS semi-local vortex string supported in four dimensional $\mathcal{N}=2$ SQED. While the core of these strings is very similar to Abrikosov-Nielsen-Olesen vortices, they are defined with a characteristic size modulus, much like the instanton lump size. This entails that the constituting fields of the vortex do not decay exponentially, as one goes far away from the core of the string, but as a rational function. The appearance of an extra scale in the problem also allows for an explicit, analytic, approximate solution to be written for the BPS equation, surprisingly. Despite the conceptually large differences between semi-local and non-Abelian vortices, it appears that the moduli structures have one main common feature, both undergo the same kind of heterotic deformation when a supersymmetry breaking potential term is added to the spacetime theory, moving from $\mathcal{N}=2$ to $\mathcal{N}=1$. By adding a mass term for the gauge scalar multiplet, a heterotic deformation develops on the worldsheet, which breaks supersymmetry down to $(0,2)$ by coupling supertranslational fermionic zero modes to supersize ones. Such an interaction between zero modes of two different sectors was already hypothesized and subsequently found for non-Abelian strings, providing a neat way of circumventing accidental supersymmetry enhancement via Zumino's theorem. We find that, for small values of the spacetime mass term, an entirely analogous term develops on the worldsheet of semi-local strings.

What becomes of semilocal non-Abelian strings in N=1 supersymmetric QCD

We study non-Abelian vortex strings in ${\mathcal N}=2$ supersymmetric QCD with the gauge group U$(N)$ deformed by the mass $\mu$ of the adjoint matter. This deformation breaks ${\mathcal N}=2$ supersymmetry down to ${\mathcal N}=1$ and in the limit of large $\mu$ the theory flows to ${\mathcal N}=1$ QCD. Non-Abelian strings in addition to translational zero modes have orientation moduli. In the ${\mathcal N}=2$ limit of small $\mu$ the dynamics of orientational moduli is described by the two dimensional $CP(N-1)$ model for QCD with $N_f=N$ flavors of quark hypermultiplets. For the case of $N_f>N$ the non-Abelian string becomes semilocal developing additional size moduli which modify the effective two dimensional $\sigma$-model on the string making its target space non-compact. In this paper we consider the $\mu$-deformed theory with $N_f>N$ eventually making $\mu$ large. We show that size moduli develop a potential that forces the string transverse size to shrink. Eventually in the large $\mu$ limit size moduli decouple and the effective theory on the string reduces to the $CP(N-1)$ model. We also comment on physics of confined monopoles.

Hadrons of N=2 supersymmetric QCD in four dimensions from little string theory

It was recently shown that non-Abelian vortex strings supported in a version of four-dimensional N=2 supersymmetric QCD (SQCD) become critical superstrings. In addition to four translational moduli, non-Abelian strings under consideration have six orientational and size moduli. Together they form a ten-dimensional target space required for a superstring to be critical, namely, the product of the flat four-dimensional space and conifold -- a non-compact Calabi-Yau threefold. In this paper we report on further studies of low-lying closed string states which emerge in four dimensions and identify them as hadrons of our four-dimensional N=2 SQCD. We use the approach based on "little string theory," describing critical string on the conifold as a non-critical $c=1$ string with the Liouville field and a compact scalar at the self-dual radius. In addition to massless hypermultiplet found earlier we observe several massive vector multiplets and a massive spin-2 multiplet, all belonging to the long (non-BPS) representations of N=2 supersymmetry in four dimensions. All the above states are interpreted as baryons formed by a closed string with confined monopoles attached. Our construction presents an example of a "reverse holography."

Non-Abelian strings and domain walls in two Higgs doublet models

Contrary to the standard model that does not admit topologically nontrivial solitons, two Higgs doublet models admit topologically stable vortex strings and domain walls. We numerically confirm the existence of a topological Z-string confining fractional Z-flux inside. We show that topological strings at sin θW = 0 limit reduce to non-Abelian strings which possess non-Abelian moduli S2 associated with spontaneous breakdown of the SU(2) custodial symmetry. We numerically solve the equations of motion for various parameter choices. It is found that a gauging U(1)Y always lowers the tension of the Z-string while it keeps that of the W-string. On the other hand, a deformation of the Higgs potential is either raising or lowering the tensions of the Z-string and W-string. We numerically obtain an effective potential for the non-Abelian moduli S2 for various parameter deformations under the restriction tan β = 1. It is the first time to show that there exists a certain parameter region where the topological W-string can be the most stable topological excitation, contrary to conventional wisdom of electroweak theories. We also obtain numerical solutions of composites of the string and domain walls in a certain condition.