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Physical review letters 130(7), 071602 (2023). doi:10.1103/PhysRevLett.130.071602, We study coupled unitary Virasoro minimal models in the large rank (m→∞) limit. In large m perturbation theory, we find two nontrivial IR fixed points which exhibit irrational coefficients in several anomalous dimensions and the central charge. For N>4 copies, we show that the IR theory breaks all possible currents that would otherwise enhance the Virasoro algebra, up to spin 10. This provides strong evidence that the IR fixed points are examples of compact, unitary, irrational conformal field theories with the minimal amount of chiral symmetry. We also analyze anomalous dimension matrices for a family of degenerate operators with increasing spin. These display further evidence of irrationality and begin to reveal the form of the leading quantum Regge trajectory., Published by APS, College Park, Md.

Journal of high energy physics 11(11), 18 (2022). doi:10.1007/JHEP11(2022)018, We use the numerical bootstrap to study conformal line defects with $O(2)$ global symmetry. Our results are very general and capture in particular line defects originating from bulk CFTs with a global symmetry, which can either be preserved or partially broken by the presence of the defect. We begin with an agnostic approach and perform a systematic bootstrap study of correlation functions between two canonical operators on the defect: the displacement and the tilt. We then focus on two interesting theories: a monodromy line defect and a localized magnetic field line defect. To this end, we combine the numerical bootstrap with the $\varepsilon$-expansion, where we complement existing results in the literature with additional calculations. For the monodromy defect our numerical results are consistent with expectations, with known analytic solutions sitting inside our numerical bounds. For the localized magnetic field line defect our plots show a series of intriguing cusps which we explore., Published by SISSA, [Trieste]

Journal of high energy physics 04(4), 093 (2022). doi:10.1007/JHEP04(2022)093, We study two-point functions of single-trace half-BPS operators in the presence of a supersymmetric Wilson line in $ \mathcal{N} $ = 4 SYM. We use modern analytic bootstrap in the form of an inversion formula, that reconstructs the CFT data starting from a single discontinuity of the correlator. In the planar strong coupling limit only a finite number of conformal blocks contributes to the discontinuity, which allows us to obtain elegant closed-form expressions for two-point functions of single-trace operators $ \mathcal{O} _{J}$ of weight J = 2, 3, 4. Our bootstrap approach guarantees that the correlator satisfies all the expected properties: it has the correct discontinuity, it satisfies the superconformal Ward identities, and it has a sensible expansion in both defect and bulk OPEs. The result is also consistent with available results coming from localization. The method is completely algorithmic and can be implemented to calculate correlators of arbitrary weight., Published by SISSA, [Trieste]

Journal of high energy physics 08(8), 290 (2022). doi:10.1007/JHEP08(2022)290, We use the RG framework set up in [1] to explore the ϕ$^{3}$ theory with a random field interaction. According to the Parisi-Sourlas conjecture this theory admits a fixed point with emergent supersymmetry which is related to the pure Lee-Yang CFT in two less dimensions. We study the model using replica trick and Cardy variables in d = 8 − ϵ where the RG flow is perturbative. Allowed perturbations are singlets under the S$_{n}$ symmetry that permutes the n replicas. These are decomposed into operators with different scaling dimensions: the lowest dimensional part, ‘leader’, controls the RG flow in the IR; the other operators, ‘followers’, can be neglected. The leaders are classified into: susy-writable, susy-null and non-susy-writable according to their mixing properties. We construct low lying leaders and compute the anomalous dimensions of a number of them. We argue that there is no operator that can destabilize the SUSY RG flow in d ≤ 8. This agrees with the well known numerical result for critical exponents of Branched Polymers (which are in the same universality class as the random field ϕ$^{3}$ model) that match the ones of the pure Lee-Yang fixed point according to dimensional reduction in all 2 ≤ d ≤ 8. Hence this is a second strong check of the RG framework that was previously shown to correctly predict loss of dimensional reduction in random field Ising model., Published by SISSA, [Trieste]

Physical review / D 106(6), 066003 (2022). doi:10.1103/PhysRevD.106.066003, The center-flavor symmetry of a gauge theory specifies the global form of consistent gauge and flavor bundle background field configurations. For 6D gauge theories which arise from a tensor branch deformation of a superconformal field theory (SCFT), we determine the global structure of such background field configurations, including possible continuous Abelian symmetry and R-symmetry bundles. Proceeding to the conformal fixed point, this provides a prescription for reading off the global form of the continuous factors of the zero-form symmetry, including possible nontrivial mixing between flavor and R-symmetry. As an application, we show that this global structure leads to a large class of 4D $\mathcal{N}$=2 SCFTs obtained by compactifying on a $T^2$ in the presence of a topologically nontrivial flat flavor bundle characterized by an ’t Hooft magnetic flux. The resulting “Stiefel-Whitney twisted” compactifications realize several new infinite families of 4D $\mathcal{N}$=2 SCFTs, and also furnish a 6D origin for a number of recently discovered rank one and two 4D $\mathcal{N}$=2 SCFTs., Published by Inst., Melville, NY

Physical review / D 107(6), 066018 (2023). doi:10.1103/PhysRevD.107.066018, We study a rich set of four-dimensional superconformal field theories (SCFTs) with both central charges identical: $a = c$. These are constructed via the diagonal $\mathcal{N}=2$ or $\mathcal{N}=1$ gauging of the flavor symmetry $G$ of a collection of $\mathcal{N}=2$ Argyres-Douglas theories of type $\mathcal{D}_p(G)$, with or without adjoint chiral multiplets, in arXiv:2106.12579 and arXiv:2111.12092. We compute superconformal indices of some theories where the rank of $G$ is low, performing a refined test for unitarity, and further determine the relevant and marginal operator content in detail. We find that most of these theories flow to interacting SCFTs with $a=c$ in the infrared., Published by Inst., Woodbury, NY

We study a rich set of four-dimensional $\mathcal{N}=1$ superconformal field theories (SCFTs) with both central charges identical: $a = c$. We construct them via the diagonal $\mathcal{N}=1$ gauging of the flavor symmetry $G$ of a collection of $\mathcal{N}=2$ Argyres--Douglas theories of type $\mathcal{D}_p(G)$, with or without additional adjoint chiral multiplets. In this way, we construct infinitely-many theories that flow to interacting SCFTs with $a = c$ in the infrared. Finally, we briefly highlight the features of the SCFTs without $a = c$ that arise from generalizing this construction.

Physical review letters 129(4), 045701 (2022). doi:10.1103/PhysRevLett.129.045701, By the Parisi-Sourlas conjecture, the critical point of a theory with random field (RF) disorder is described by a supersymmeric (SUSY) conformal field theory (CFT), related to a $d-2$ dimensional CFT without SUSY. Numerical studies indicate that this is true for the RF $ϕ^3$ model but not for the RF $ϕ^4$ model in $d < 5$ dimensions. Here we argue that the SUSY fixed point is not reached because of new relevant SUSY-breaking interactions. We use a perturbative renormalization group in a judiciously chosen field basis, allowing systematic exploration of the space of interactions. Our computations agree with the numerical results for both cubic and quartic potential., Published by APS, College Park, Md.

Journal of high energy physics 03, 133 (2023). doi:10.1007/JHEP03(2023)133, Conformal blocks are a central analytic tool for higher dimensional conformal field theory. We employ Harish-Chandra's radial component map to construct universal Casimir differential equations for spinning conformal blocks in any dimension $d$ of Euclidean space. Furthermore, we also build a set of differential ``shifting'' operators that allow to construct solutions of the Casimir equations from certain seeds. In the context of spinning four-point blocks of bulk conformal field theory, our formulas provide an elegant and far reaching generalisation of existing expressions to arbitrary tensor fields and arbitrary dimension $d$. The power of our new universal approach to spinning blocks is further illustrated through applications to defect conformal field theory. In the case of defects of co-dimension $q=2$ we are able to construct conformal blocks for two-point functions of symmetric traceless bulk tensor fields in both the defect and the bulk channel. This opens an interesting avenue for applications to the defect bootstrap. Finally, we also derive the Casimir equations for bulk-bulk-defect three-point functions in the bulk channel., Published by SISSA, [Trieste]

Journal of high energy physics 08(8), 067 (2022). doi:10.1007/JHEP08(2022)067, We study multipoint correlators of protected scalars on the Maldacena-Wilson line in $ \mathcal{N} $ = 4 SYM. Working at weak coupling in the planar limit, we derive an explicit recursion relation that captures next-to-leading order correlators with an arbitrary number of insertions of the fundamental scalar field. By pinching fundamental scalars together, we can build composite protected operators with higher values of the R-charge. Our result then encompasses arbitrary n-point correlators of protected operators with arbitrary weight. As a demonstration of our method, we give explicit formulae for correlators with up to six points. Using these results we observe that all our correlators are annihilated by a special class of differential operators. We conjecture that these differential operators are non-perturbative constraints and can be considered a multipoint extension of the superconformal Ward identities satisfied by four-point functions., Published by SISSA, [Trieste]

Physical review / D 105(4), 046006 (2022). doi:10.1103/PhysRevD.105.046006, We use the numerical conformal bootstrap to study six-dimensional N=(1,0) superconformal field theories with flavor symmetry so4k. We present evidence that minimal (Dk,Dk) conformal matter saturates the unitarity bounds for arbitrary k. Furthermore, using the extremal-functional method, we check that the chiral-ring relations are correctly reproduced, extract the anomalous dimensions of low-lying long superconformal multiplets, and find hints for novel operator product expansion selection rules involving type-B multiplets., Published by Inst., Melville, NY

Journal of high energy physics 01(1), 005 (2022). doi:10.1007/JHEP01(2022)005, Many CFT problems, e.g. ones with global symmetries, have correlation functions with a crossing antisymmetric sector. We show that such a crossing antisymmetric function can be expanded in terms of manifestly crossing antisymmetric objects, which we call the ‘+ type Polyakov blocks’. These blocks are built from AdS$_{d+1}$ Witten diagrams. In 1d they encode the ‘+ type’ analytic functionals which act on crossing antisymmetric functions. In general d we establish this Witten diagram basis from a crossing antisymmetric dispersion relation in Mellin space. Analogous to the crossing symmetric case, the dispersion relation imposes a set of independent ‘locality constraints’ in addition to the usual CFT sum rules given by the ‘Polyakov conditions’. We use the Polyakov blocks to simplify more general analytic functionals in d > 1 and global symmetry functionals., Published by SISSA, [Trieste]

SciPost physics 13(2), 014 (2022). doi:10.21468/SciPostPhys.13.2.014, We apply the conformal bootstrap technique to study the $U(1)$ Dirac spin liquid (i.e. $N_f=4$ QED$_3$) and the newly proposed $N=7$ Stiefel liquid (i.e. a conjectured 3d non-Lagrangian CFT without supersymmetry). For the $N_f=4$ QED$_3$, we focus on the monopole operator and ($SU(4)$ adjoint) fermion bilinear operator. We bootstrap their single correlators as well as the mixed correlators between them. We first discuss the bootstrap kinks from single correlators. Some exponents of these bootstrap kinks are close to the expected values of QED$_3$, but we provide clear evidence that they should not be identified as the QED$_3$. By requiring the critical phase to be stable on the triangular and the kagome lattice, we obtain rigorous numerical bounds for the $U(1)$ Dirac spin liquid and the Stiefel liquid. For the triangular and kagome Dirac spin liquid, the rigorous lower bounds of the monopole operator's scaling dimension are $1.046$ and $1.105$, respectively. These bounds are consistent with the latest Monte Carlo results., Published by SciPost Foundation, Amsterdam

We continue the exploration of multipoint scalar comb channel blocks for conformal field theories in 3D and 4D. The central goal here is to construct novel comb channel cross ratios that are well adapted to perform projections onto all intermediate primary fields. More concretely, our new set of cross ratios includes three for each intermediate mixed symmetry tensor exchange. These variables are designed such that the associated power series expansion coincides with the sum over descendants. The leading term of this expansion is argued to factorise into a product of lower point blocks. We establish this remarkable factorisation property by studying the limiting behaviour of the Gaudin Hamiltonians that are used to characterise multipoint conformal blocks. For six points we can map the eigenvalue equations for the limiting Gaudin differential operators to Casimir equations of spinning four-point blocks., Journal of High Energy Physics, 144 (6), ISSN:1126-6708, ISSN:1029-8479

Journal of high energy physics 10(10), 139 (2021). doi:10.1007/JHEP10(2021)139, The construction of conformal blocks for the analysis of multipoint correlation functions with N > 4 local field insertions is an important open problem in higher dimensional conformal field theory. This is the first in a series of papers in which we address this challenge, following and extending our short announcement in [1]. According to Dolan and Osborn, conformal blocks can be determined from the set of differential eigenvalue equations that they satisfy. We construct a complete set of commuting differential operators that characterize multipoint conformal blocks for any number N of points in any dimension and for any choice of OPE channel through the relation with Gaudin integrable models we uncovered in [1]. For 5-point conformal blocks, there exist five such operators which are worked out smoothly in the dimension d., Published by SISSA, [Trieste]

Journal of high energy physics 05(5), 007 (2021). doi:10.1007/JHEP05(2021)007, We develop a group theoretical formalism to study correlation functions in defect conformal field theory, with multiple insertions of bulk and defect fields. This formalism is applied to construct the defect conformal blocks for three-point functions of scalar fields. Starting from a configuration with one bulk and one defect field, for which the correlation function is determined by conformal symmetry, we explore two possibilities, adding either one additional defect or bulk field. In both cases it is possible to express the blocks in terms of classical hypergeometric functions, though the case of two bulk and one defect field requires Appell’s function F$_{4}$., Published by SISSA, [Trieste]

Journal of high energy physics 05(5), 195 (2021). doi:10.1007/JHEP05(2021)195, We study the two-point function of the stress-tensor multiplet of $ \mathcal{N} $ = 4 SYM in the presence of a line defect. To be more precise, we focus on the single-trace operator of conformal dimension two that sits in the 20′ irrep of the $ \mathfrak{so} $(6)$_{R}$ R-symmetry, and add a Maldacena-Wilson line to the configuration which makes the two-point function non-trivial. We use a combination of perturbation theory and defect CFT techniques to obtain results up to next-to-leading order in the coupling constant. Being a defect CFT correlator, there exist two (super)conformal block expansions which capture defect and bulk data respectively. We present a closed-form formula for the defect CFT data, which allows to write an efficient Taylor series for the correlator in the limit when one of the operators is close to the line. The bulk channel is technically harder and closed-form formulae are particularly challenging to obtain, nevertheless we use our analysis to check against well-known data of $ \mathcal{N} $ = 4 SYM. In particular, we recover the correct anomalous dimensions of a famous tower of twist-two operators (which includes the Konishi multiplet), and successfully compare the one-point function of the stress-tensor multiplet with results obtained using matrix-model techniques., Published by SISSA, [Trieste]

Journal of high energy physics 2021(3), 51 (1-26) (2021). doi:10.1007/JHEP03(2021)051, Using some techniques of conformal field theories, we find a closed expression for the contribution of leading twist operators and their descendants, obtained by adding total derivatives, to the operator product expansion (OPE) of two electromagnetic currents in QCD. Our expression resums contributions of all twists and to all orders in perturbation theory up to corrections proportional to the QCD $��$-function. At tree level and to twist-four accuracy, our result agrees with the expression derived earlier by a different method. The results are directly applicable to deeply-virtual Compton scattering and, e.g., $����^���$ annihilation in two mesons. As a byproduct, we derive a simple representation for the OPE of two scalar currents that is convenient for applications., Published by SISSA, [Trieste]

Symmetry, integrability and geometry: methods and applications 17, 38 (2021). doi:10.3842/SIGMA.2021.007, Superconformal blocks and crossing symmetry equations are among central ingredients in any superconformal field theory. We review the approach to these objects rooted in harmonic analysis on the superconformal group that was put forward in [J. High Energy Phys. 2020 (2020), no. 1, 159, 40 pages, arXiv:1904.04852] and [J. High Energy Phys. 2020 (2020), no. 10, 147, 44 pages, arXiv:2005.13547]. After lifting conformal four-point functions to functions on the superconformal group, we explain how to obtain compact expressions for crossing constraints and Casimir equations. The later allow to write superconformal blocks as finite sums of spinning bosonic blocks., Published by [S.l.]

Physical review letters 126(2), 021602 (2021). doi:10.1103/PhysRevLett.126.021602, In this work, we initiate an integrability-based approach to multipoint conformal blocks for higher-dimensional conformal field theories. Our main observation is that conformal blocks for N-point functions may be considered as eigenfunctions of integrable Gaudin Hamiltonians. This provides us with a complete set of differential equations that can be used to evaluate multipoint blocks., Published by APS, College Park, Md.

Journal of high energy physics 10(10), 116 (2021). doi:10.1007/JHEP10(2021)116, We revisit the problem of bootstrapping CFT correlators of charged fields. After discussing in detail how bounds for uncharged fields can be recycled to the charged case, we introduce two sets of analytic functional bases for correlators on the line. The first, which we call ���simple���, is essentially a direct sum of analytic functionals for the uncharged case. We use it to establish very general bounds on the OPE density appearing in charged correlators. The second basis is dual to generalized free fields and we explain how it is related to a charged version of the Polyakov bootstrap. We apply these functionals to map out the space of correlators and obtain new improved bounds on the 3d Ising twist defect., Published by SISSA, [Trieste]

SISSA 1-54 (2020). doi:10.1007/JHEP02(2021)068, At finite density, the spontaneous breakdown of an internal non-Abelian symmetry dictates, along with gapless modes, modes whose gap is fixed by the algebra and proportional to the chemical potential: the gapped Goldstones. Generically the gap of these states is comparable to that of other non-universal excitations or to the energy scale where the dynamics is strongly coupled. This makes it non-straightforward to derive a universal effective field theory (EFT) description realizing all the symmetries. Focusing on the illustrative example of a fully broken SU(2) group, we demonstrate that such an EFT can be constructed by carving out around the Goldstones, gapless and gapped, at small 3-momentum. The rules governing the EFT, where the gapless Goldstones are soft while the gapped ones are slow, are those of standard nonrelativistic EFTs, like for instance nonrelativistic QED. In particular, the EFT Lagrangian formally preserves gapped Goldstone number, and processes where such number is not conserved are described inclusively by allowing for imaginary parts in the Wilson coefficients. Thus, while the symmetry is manifestly realized in the EFT, unitarity is not. We comment on the application of our construction to the study of the large charge sector of conformal field theories with non-Abelian symmetries., Published by SISSA

Journal of high energy physics 01(1), 175 (2021). doi:10.1007/JHEP01(2021)175, We apply the numerical conformal bootstrap to correlators of Coulomb and Higgs branch operators in 4d$ \mathcal{N} $ = 2 superconformal theories. We start by revisiting previous results on single correlators of Coulomb branch operators. In particular, we present improved bounds on OPE coefficients for some selected Argyres-Douglas models, and compare them to recent work where the same cofficients were obtained in the limit of large r charge. There is solid agreement between all the approaches. The improved bounds can be used to extract an approximate spectrum of the Argyres-Douglas models, which can then be used as a guide in order to corner these theories to numerical islands in the space of conformal dimensions. When there is a flavor symmetry present, we complement the analysis by including mixed correlators of Coulomb branch operators and the moment map, a Higgs branch operator which sits in the same multiplet as the flavor current. After calculating the relevant superconformal blocks we apply the numerical machinery to the mixed system. We put general constraints on CFT data appearing in the new channels, with particular emphasis on the simplest Argyres-Douglas model with non-trivial flavor symmetry., Published by SISSA, [Trieste]

Journal of high energy physics 08(8), 127 (1-86) (2021). doi:10.1007/JHEP08(2021)127, This is the first in a series of papers devoted to the study of spin chains capturing the spectral problem of 4d $ \mathcal{N} $ = 2 SCFTs in the planar limit. At one loop and in the quantum plane limit, we discover a quasi-Hopf symmetry algebra, defined by the R-matrix read off from the superpotential. This implies that when orbifolding the $ \mathcal{N} $ = 4 symmetry algebra down to the $ \mathcal{N} $ = 2 one and then marginaly deforming, the broken generators are not lost, but get upgraded to quantum generators. Importantly, we demonstrate that these chains are dynamical, in the sense that their Hamiltonian depends on a parameter which is dynamically determined along the chain. At one loop we map the holomorphic SU(3) scalar sector to a dynamical 15-vertex model, which corresponds to an RSOS model, whose adjacency graph can be read off from the gauge theory quiver/brane tiling. One scalar SU(2) sub-sector is described by an alternating nearest-neighbour Hamiltonian, while another choice of SU(2) sub-sector leads to a dynamical dilute Temperley-Lieb model. These sectors have a common vacuum state, around which the magnon dispersion relations are naturally uniformised by elliptic functions. Concretely, for the ���$_{2}$ quiver theory we study these dynamical chains by solving the one- and two-magnon problems with the coordinate Bethe ansatz approach. We confirm our analytic results by numerical comparison with the explicit diagonalisation of the Hamiltonian for short closed chains., Published by SISSA, [Trieste]

We study the quantum quench in two coupled Tomonaga-Luttinger Liquids (TLLs), from the off-critical to the critical regime, relying on the conformal field theory approach and the known solutions for single TLLs. We consider a squeezed form of the initial state, whose low energy limit is fixed in a way to describe a massive and a massless mode, and we encode the non-equilibrium dynamics in a proper rescaling of the time. In this way, we compute several correlation functions, which at leading order factorize into multipoint functions evaluated at different times for the two modes. Depending on the observable, the contribution from the massive or from the massless mode can be the dominant one, giving rise to exponential or power-law decay in time, respectively. Our results find a direct application in all the quench problems where, in the scaling limit, there are two independent massless fields: these include the Hubbard model, the Gaudin-Yang gas, and tunnel-coupled tubes in cold atoms experiments., Comment: New section with goal and main results included (1.1), further changes and explanations in sections 2 and 4

Journal of high energy physics 2008(8), 1-29 (2020). doi:10.1007/JHEP08(2020)163, We present a Lorentzian inversion formula valid for any defect CFT that extracts the bulk channel CFT data as an analytic function of the spin variable. This result complements the already obtained inversion formula for the corresponding defect channel, and makes it now possible to implement the analytic bootstrap program for defect CFT, by going back and forth between bulk and defect expansions. A crucial role in our derivation is played by the Calogero-Sutherland description of defect blocks which we review. As first applications we obtain the large-spin limit of bulk CFT data necessary to reproduce the defect identity, and also calculate one-point functions of the twist defect of the 3$d$ Ising model to first order in the $ϵ$-expansion., Published by SISSA, [Trieste]

Communications in mathematical physics -, - (2019). doi:10.1007/s00220-019-03617-y, This paper investigates the relations between the Toda conformal field theories,quantum group theory and the quantisation of moduli spaces of flat connections.We use the free field representation of theW-algebras to define natural bases for spacesof conformal blocks of the Toda conformal field theory associated to the Lie algebrasl3 on the three-punctured sphere with representations of generic type associated to thethree punctures. The operator-valued monodromies of degenerate fields can be used todescribe the quantisation of the moduli spaces of flat SL(3)-connections. It is shown thatthe matrix elements of the monodromies can be expressed as Laurent polynomials ofmore elementary operatorswhich have a simple definition in the free field representation.These operators are identified as quantised counterparts of natural higher rank analogsof the Fenchel–Nielsen coordinates from Teichmüller theory. Possible applications tothe study of the non-Lagrangian SUSY field theories are briefly outlined., Published by Springer, Heidelberg

The reflection operators are the simplest examples of the non-local integrals of motion, which appear in many interesting problems in integrable CFT. For the so-called Fateev, quantum AKNS, paperclip and KdV integrable structures, they are built from the (chiral) reflection S-matrices for the Liouville and cigar CFTs. Here we give the full spectrum of the reflection operators associated with these integrable structures. We also obtained a relation between the reflection S-matrices of the cigar and Liouville CFTs. The results of this work are applicable for the description of the scaling behaviour of the Bethe states in exactly solvable lattice systems and may be of interest to the study of the Generalized Gibbs Ensemble associated with the above mentioned integrable structures., Comment: 33 pages, published version

Journal of high energy physics 10(10), 147 (2020). doi:10.1007/JHEP10(2020)147, Crossing symmetry provides a powerful tool to access the non-perturbative dynamics of conformal and superconformal field theories. Here we develop the mathematical formalism that allows to construct the crossing equations for arbitrary four-point functions in theories with superconformal symmetry of type I, including all superconformal field the- ories in d = 4 dimensions. Our advance relies on a supergroup theoretic construction of tensor structures that generalizes an approach which was put forward in [1] for bosonic theories. When combined with our recent construction of the relevant superblocks, we are able to derive the crossing symmetry constraint in particular for four-point functions of arbitrary long multiplets in all 4-dimensional superconformal field theories., Published by SISSA, [Trieste]

Journal of high energy physics 04(4), 151 (2020). doi:10.1007/JHEP04(2020)151, We study the scalar $\phi^{3}$ theory above six dimensions. The beta function $ \beta (g)=-\in g-\frac{3}{4}{g}^3 $ in d = 6 − 2ϵ dimensions has a UV fixed point when ϵ < 0. Like the O(N) vector models above four dimensions, such a fixed point observed perturbatively in fact corresponds to a pair of complex CFTs separated by a branch cut. Using both the numerical bootstrap method and Gliozzi’s fusion rule truncation method, we argue that the fixed points of the $\phi^{3}$66 theory above six dimensions exist., Published by SISSA, [Trieste]

Journal of physics / A 53(45), 453002 (2020). doi:10.1088/1751-8121/abb0fe, A review of Boundary and defect conformal field theory: open problems and applications, following a workshop held at Chicheley Hall, Buckinghamshire, UK, 7���8 Sept. 2017. We attempt to provide a broad, bird���s-eye view of the latest progress in boundary and defect conformal field theory in various sub-fields of theoretical physics, including the renormalization group, integrability, conformal bootstrap, topological field theory, supersymmetry, holographic duality, and more. We also discuss open questions and promising research directions in each of these sub-fields, and combinations thereof., Published by IOP Publ., Bristol

Journal of high energy physics 01(1), 106 (2021). doi:10.1007/JHEP01(2021)106, Building on arXiv:1911.05827, we uncover new properties of type-B conformal anomalies for Coulomb-branch operators in continuous families of 4D $\mathcal{N}=2$ SCFTs. We study a large class of such anomalies on the Higgs branch, where conformal symmetry is spontaneously broken, and compare them with their counterpart in the CFT phase. In Lagrangian theories, the non-perturbative matching of the anomalies can be determined with a weak coupling Feynman diagram computation involving massive multi-loop banana integrals. We extract the part corresponding to the anomalies of interest. Our calculations support the general conjecture that the Coulomb-branch type-B conformal anomalies always match on the Higgs branch when the IR Coulomb-branch chiral ring is empty. In the opposite case, there are anomalies that do not match. An intriguing implication of the mismatch is the existence of a second covariantly constant metric on the conformal manifold (other than the Zamolodchikov metric), which imposes previously unknown restrictions on its holonomy group., Published by SISSA, [Trieste]

SciPost physics 10(5), 115 (1-17) (2021). doi:10.21468/SciPostPhys.10.5.115, It is well established that the $O(N)$ Wilson-Fisher (WF) CFT sits at a kink of the numerical bounds from bootstrapping four point function of $O(N)$ vector. Moving away from the WF kinks, there indeed exists another family of kinks (dubbed non-WF kinks) on the curve of $O(N)$ numerical bounds. Different from the $O(N)$ WF kinks that exist for arbitary $N$ in 2 < d < 4 dimensions, the non-WF kinks exist in arbitrary dimensions but only for a large enough $N$ > $N_c(d)$ in a given dimension $d$. In this paper we have achieved a thorough understanding for few special cases of these non-WF kinks, which already hints interesting physics. The first case is the $O$(4) bootstrap in 2d, where the non-WF kink turns out to be the $SU$(2)$_1$ Wess-Zumino-Witten (WZW) model, and all the $SU$(2)$_{k>2}$ WZW models saturate the numerical bound on the left side of the kink. This is a mirror version of the $Z_2$ bootstrap, where the 2d Ising CFT sits at a kink while all the other minimal models saturating the bound on the right. We further carry out dimensional continuation of the 2d $SU$(2)$_1$ kink towards the 3d $SO$(5) deconfined phase transition. We find the kink disappears at around $d$ = 2.7 dimensions indicating the $SO$(5) deconfined phase transition is weakly first order. The second interesting observation is, the $O$(2) bootstrap bound does not show any kink in 2d ($N_c$ = 2), but is surprisingly saturated by the 2d free boson CFT (also called Luttinger liquid) all the way on the numerical curve. The last case is the $N$ = $∞$ limit, where the non-WF kink sits at ($∆_φ$,$∆_T$) = ($d$ −1,2$d$) in $d$ dimensions. We manage to write down its analytical four point function in arbitrary dimensions, which equals to the subtraction of correlation functions of a free fermion theory and generalized free theory. An important feature of this solution is the existence of a full tower of conserved higher spin current. We speculate that a new family of CFTs will emerge at non-WF kinks for finite $N$, in a similar fashion as $O(N)$ WF CFTs originating from free boson at $N$ = $∞$., Published by SciPost Foundation, Amsterdam

Journal of physics / A Mathematical and theoretical A53(27), 275401 - (2020). doi:10.1088/1751-8121/ab9275, We construct an ϵ-deformation of W algebras, corresponding to the additive version of quiver algebras which feature prominently in the 5D version of the BPS/CFT correspondence and refined topological strings on toric Calabi–Yau’s. This new type of algebras fill in the missing intermediate level between q-deformed and ordinary W algebras. We show that ϵ-deformed W algebras are spectral duals of conventional W algebras, in particular the ϵ-deformed conformal blocks manifestly reproduce instanton partition functions of 4D quiver gauge theories in the full Ω-background and give dual integral representations of ordinary W conformal blocks., Published by IOP Publ., Bristol

Journal of high energy physics 2020(10), 1-38 (2020). doi:10.1007/JHEP10(2020)004, The decomposition of correlation functions into conformal blocks is an indispensable tool in conformal field theory. For spinning correlators, non-trivial tensor structures are needed to mediate between the conformal blocks, which are functions of cross ratios only, and the correlation functions that depend on insertion points in the d-dimensional Euclidean space. Here we develop an entirely group theoretic approach to tensor structures, based on the Cartan decomposition of the conformal group. It provides us with a new universal formula for tensor structures and thereby a systematic derivation of crossing equations. Our approach applies to a ‘gauge’ in which the conformal blocks are wave functions of Calogero-Sutherland models rather than solutions of the more standard Casimir equations. Through this ab initio construction of tensor structures we complete the Calogero-Sutherland approach to conformal correlators, at least for four-point functions of local operators in non-supersymmetric models. An extension to defects and superconformal symmetry is possible., Published by SISSA, [Trieste]

We use the monodromy method to compute expectation values of an arbitrary number of light operators in finitely excited ("heavy") eigenstates of holographic 2D CFT. For eigenstates with scaling dimensions above the BTZ threshold, these behave thermally up to small corrections, with an effective temperature determined by the heavy state. Below the threshold we find oscillatory and not decaying behavior. As an application of these results we compute the expectation of the out-of-time order arrangement of four light operators in a heavy eigenstate, i.e. a six-point function. Above the threshold we find maximally scrambling behavior with Lyapunov exponent $2\pi T_{\rm eff}$. Below threshold we find that the eigenstate OTOC shows persistent harmonic oscillations., Comment: 25 pages. 2 figures. v2 references added. v3 minor typos fixed

Journal of high energy physics 2020(9), 78 (1-57) (2020). doi:10.1007/JHEP09(2020)078, In this paper we investigate the relation between conformal blocks of Liouville CFT and the topological string partition functions of the rank one trinion theory $T_{2}$. The partition functions exhibit jumps when passing from one chamber in the parameter space to another. Such jumps can be attributed to a change of the integration contour in the free field representation of Liouville conformal blocks. We compare the partition functions of the $T_{2}$ theories representing trifundamental half hypermultiplets in $N$ = 2, $d$ = 4 field theories to the partition functions associated to bifundamental hypermultiplets. We find that both are related to the same Liouville conformal blocks up to inessential factors. In order to establish this picture we combine and compare results obtained using topological vertex techniques, matrix models and topological recursion. We furthermore check that the partition functions obtained by gluing two $T_{2}$ vertices can be represented in terms of a four point Liouville conformal block. Our results indicate that the $T_{2}$ vertex offers a useful starting point for developing an analog of the instanton calculus for SUSY gauge theories with trifundamental hypermultiplets., Published by SISSA, [Trieste]

Journal of high energy physics 2020(1), 159 (2020). doi:10.1007/JHEP01(2020)159, In this work we launch a systematic theory of superconformal blocks for four­point functions of arbitrary supermultiplets. Our results apply to a large class of superconformal field theories including 4-dimensional models with any number $ \mathcal{N} $ of supersymmetries. The central new ingredient is a universal construction of the relevant Casimir differential equations. In order to find these equations, we model superconformal blocks as functions on the supergroup and pick a distinguished set of coordinates. The latter are chosen so that the superconformal Casimir operator can be written as a perturbation of the Casimir operator for spinning bosonic blocks by a fermionic (nilpotent) term. Solu­ tions to the associated eigenvalue problem can be obtained through a quantum mechanical perturbation theory that truncates at some finite order so that all results are exact. We illustrate the general theory at the example of d = 1 dimensional theories with $ \mathcal{N} $ = 2 supersymmetry for which we recover known superblocks. The paper concludes with an outlook to 4-dimensional blocks with $ \mathcal{N} $ = 1 supersymmetry., Published by SISSA, [Trieste]

Journal of high energy physics 06(6), 154 (2021). doi:10.1007/JHEP06(2021)154, We develop the numerical bootstrap technique to study the 2 + 1 dimensional $ \mathcal{N} $ = 1 superconformal field theories (SCFTs). When applied to the minimal $ \mathcal{N} $ = 1 SCFT, it allows us to determine its critical exponents to high precision. This model was argued in [1] to describe a quantum critical point (QCP) at the boundary of a 3 + 1D topological superconductor. More interestingly, this QCP can be reached by tuning a single parameter, where supersymmetry (SUSY) is realized as an emergent symmetry. We show that the emergent SUSY condition also plays an essential role in bootstrapping this SCFT. But performing a ���two-sided��� Pad�� re-summation of the large N expansion series, we calculate the critical exponents for Gross-Neveu-Yukawa models at N =4 and N =8., Published by SISSA, [Trieste]

Conformal blocks are the central ingredient of the conformal bootstrap programme. We elaborate on our recent observation that uncovered a relation with wave functions of an integrable Calogero-Sutherland Hamiltonian in order to develop a systematic theory of conformal blocks. Our main goal here is to review central ingredients of the Heckman-Opdam theory for scattering states of Calogero-Sutherland models with special emphasis to the relation with scalar 4-point blocks. We will also discuss a number of direct consequences for conformal blocks, including a new series expansion for blocks of arbitrary complex spin and a complete analysis of their poles and residues. Applications to the Froissart-Gribov formula for conformal field theory, as well as extensions to spinning blocks and defects are briefly discussed before we conclude with an outlook on forthcoming work concerning algebraic consequences of integrability., 65 pages, 6 figures

In this article which is the first of a series of three, we consider $\mathcal W({\mathfrak{sl}_d})$-symmetric conformal field theory in topological regimes for a generic value of the background charge, where $\mathcal W({\mathfrak{sl}_d})$ is the W-algebra associated to the affine Lie algebra $\widehat{\mathfrak{sl}_d}$, whose vertex operator algebra is included to that of the affine Lie algebra $\widehat{\mathfrak g}_1$ at level 1. In such regimes, the theory admits a free field representation. We show that the generalized Ward identities assumed to be satisfied by chiral conformal blocks with current insertions can be solved perturbatively in topological regimes. This resolution uses a generalization of the topological recursion to non-commutative, or quantum, spectral curves. In turn, special geometry arguments yields a conjecture for the perturbative reconstruction of a particular chiral block., 35 pages, v4: minor changes

We consider renormalization of four-fermion operators in the critical QED and $SU(N_c)$ version of Gross--Neveu--Yukawa model in non-integer dimensions. Since the number of mixing operators is infinite, the diagonalization of an anomalous dimension matrix becomes a nontrivial problem. At leading order, construction of eigen-operators is equivalent to solving certain three-term recurrence relations. We find analytic solutions of these recurrence relations that allows to determine the spectrum of anomalous dimensions and study their properties., Comment: 9 pages; typos are fixed; Journal reference is given

We introduce a series of articles reviewing various aspects of integrable models relevant to the AdS/CFT correspondence. Topics covered in these reviews are: classical integrability, Yangian symmetry, factorized scattering, the Bethe ansatz, the thermodynamic Bethe ansatz, and integrable structures in (conformal) quantum field theory. In the present article we highlight how these concepts have found application in AdS/CFT, and provide a brief overview of the material contained in this series., v2, published version

We develop in this paper the principles of an associative algebraic approach to bulk logarithmic conformal field theories (LCFTs). We concentrate on the closed $gl(1|1)$ spin-chain and its continuum limit - the $c=-2$ symplectic fermions theory - and rely on two technical companion papers, "Continuum limit and symmetries of the periodic gl(1|1) spin chain" [Nucl. Phys. B 871 (2013) 245-288] and "Bimodule structure in the periodic gl(1|1) spin chain" [Nucl. Phys. B 871 (2013) 289-329]. Our main result is that the algebra of local Hamiltonians, the Jones-Temperley-Lieb algebra JTL_N, goes over in the continuum limit to a bigger algebra than the product of the left and right Virasoro algebras. This algebra, S - which we call interchiral, mixes the left and right moving sectors, and is generated, in the symplectic fermions case, by the additional field $S(z,\bar{z})=S_{ab}\psi^a(z)\bar{\psi}^b(\bar{z})$, with a symmetric form $S_{ab}$ and conformal weights (1,1). We discuss in details how the Hilbert space of the LCFT decomposes onto representations of this algebra, and how this decomposition is related with properties of the finite spin-chain. We show that there is a complete correspondence between algebraic properties of finite periodic spin chains and the continuum limit. An important technical aspect of our analysis involves the fundamental new observation that the action of JTL_N in the $gl(1|1)$ spin chain is in fact isomorphic to an enveloping algebra of a certain Lie algebra, itself a non semi-simple version of $sp(N-2)$. The semi-simple part of JTL_N is represented by $Usp(N-2)$, providing a beautiful example of a classical Howe duality, for which we have a non semi-simple version in the full JTL image represented in the spin-chain. On the continuum side, simple modules over the interchiral algebra S are identified with "fundamental" representations of $sp(\infty)$., Comment: 69 pp., 10 figs, v2: the paper has been substantially modified - new proofs, new refs, new App C with inductive limits construction, etc

The low energy limit of a dense 2D adjoint QCD is described by a family of ${\cal N}=(2,2)$ supersymmetric coset conformal field theories. In previous work we constructed chiral primaries for a small number $N < 6$ of colors. Our aim in the present note is to determine the chiral ring in the multicolor limit where $N$ is sent to infinity. We shall find that chiral primaries are labeled by partitions and identify the ring they generate as the ring of Schur polynomials. Our findings impose strong constraints on the possible dual description through string theory in an $AdS_3$ compactification., 25 pages, 4 figures

Journal of high energy physics 06, 015(2015). doi:10.1007/JHEP06(2015)015, Inspired by recent results in the context of AdS/CFT integrability, we reconsider the Thermodynamic Bethe Ansatz equations describing the 1D fermionic Hubbard model at finite temperature. We prove that the infinite set of TBA equations are equivalent to a simple nonlinear Riemann-Hilbert problem for a finite number of unknown functions. The latter can be transformed into a set of three coupled nonlinear integral equations defined over a finite support, which can be easily solved numerically. We discuss the emergence of an exact Bethe Ansatz and the link between the TBA approach and the results by Jüttner, Klümper and Suzuki based on the Quantum Transfer Matrix method. We also comment on the analytic continuation mechanism leading to excited states and on the mirror equations describing the finite-size Hubbard model with twisted boundary conditions., Published by Springer, Berlin

Journal of high energy physics 1505(5), 114(2015). doi:10.1007/JHEP05(2015)114, The periodic sℓ(2|1) alternating spin chain encodes (some of) the properties of hulls of percolation clusters, and is described in the continuum limit by a logarithmic conformal field theory (LCFT) at central charge c = 0. This theory corresponds to the strong coupling regime of a sigma model on the complex projective superspace CP$^{1|1}$ = U(2|1)/(U(1) × U(1|1)), and the spectrum of critical exponents can be obtained exactly. In this paper we push the analysis further, and determine the main representation theoretic (logarithmic) features of this continuum limit by extending to the periodic case the approach of [1] [N. Read and H. Saleur, Nucl. Phys. B 777 (2007) 316]. We first focus on determining the representation theory of the finite size spin chain with respect to the algebra of local energy densities provided by a representation of the affine Temperley-Lieb algebra at fugacity one. We then analyze how these algebraic properties carry over to the continuum limit to deduce the structure of the space of states as a representation over the product of left and right Virasoro algebras. Our main result is the full structure of the vacuum module of the theory, which exhibits Jordan cells of arbitrary rank for the Hamiltonian., Published by Springer, Berlin

The goal of this note is to show that the Riemann-Hilbert problem to find multivalued analytic functions with $SL(2,\mathbb{C})$-valued monodromy on Riemann surfaces of genus zero with $n$ punctures can be solved by taking suitable linear combinations of the conformal blocks of Liouville theory at $c=1$. This implies a similar representation for the isomonodromic tau-function. In the case $n=4$ we thereby get a proof of the relation between tau-functions and conformal blocks discovered in \cite{GIL}. We briefly discuss a possible application of our results to the study of relations between certain $\mathcal{N}=2$ supersymmetric gauge theories and conformal field theory., Comment: 29 pages, 4 figures; v2: minor changes, added refs

This is the 11th article in the collection of reviews 'Exact results in N=2 supersymmetric gauge theories', ed. J. Teschner. It describes an approach to understanding the 4d/2d relations discovered by Alday, Gaiotto and Tachikawa by establishing a triangle of relations between the zero mode quantum mechanics obtained by localisation of class $\cal S$ theories, the quantum theory obtained by quantisation of Hitchin moduli spaces, and conformal field theory.

Physical review / D 045005, 44(2015). doi:10.1103/PhysRevD.91.045005;10.1103/PhysRevD.91.045005, In this second part of our investigation [1] of the analytic structure of the $2\to 5$ scattering amplitude in the planar limit of $\mathcal{N}=4$ super Yang-Mills theory in multi-Regge kinematics we compute, in all kinematic regions, the Regge-cut contributions at leading order. The results are infrared finite and conformally invariant., Published by Soc., [S.l.]