<p>We propose a numerical method to estimate one-point functions and the free-energy density of conformal field theories at finite temperature by solving the Kubo-Martin-Schwinger condition for the two-point functions of identical scalars. We apply the method for the critical <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="normal">O</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo></math> model for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>N</mi><mo>=</mo><mn>1</mn></math>, 2, 3 in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>3</mn><mo>≤</mo><mi>d</mi><mo>≤</mo><mn>4</mn></math>. We find agreement with known results from Monte Carlo simulations and previous results for the 3D Ising model, and we provide new predictions for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>N</mi><mo>=</mo><mn>2</mn></math>, 3.</p>

Germany

Julien Barrat

United Kingdom

Enrico Marchetto

Alessio Miscioscia

Elli Pomoni

Thermal Bootstrap for the Critical <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi mathvariant="normal">O</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo></mrow></math> Model

We study the global symmetries of the $\textit{ℤ}$ -orbifold of $$ \mathcal{N} $$ = 4 Super-Yang-Mills theory and its marginal deformations. The process of orbifolding to obtain an $$ \mathcal{N} $$ = 2 theory would appear to break the SU(4) R-symmetry down to SU(2) × SU(2) × U(1). We show that the broken generators can be recovered by moving beyond the Lie algebraic setting to that of a Lie algebroid. This remains true when marginally deforming away from the orbifold point by allowing the couplings of the SU($\textit{N}$) × SU($\textit{N}$) gauge groups to vary independently. The information about the marginal deformation is captured by a Drinfeld-type twist of this SU(4) Lie algebroid. The twist is read off from the F- and D- terms, and thus directly from the Lagrangian. Even though at the orbifold point the algebraic structure is associative, it becomes non-associative after the marginal deformation. We explicitly check that the planar Lagrangian of the theory is invariant under this twisted version of the SU(4) algebroid and we discuss implications of this hidden symmetry for the spectrum of the $$ \mathcal{N} $$ = 2 theory.

Hanno Bertle

China

Xinyu Zhang

South Africa

Konstantinos Zoubos

Hidden symmetries of 4D $$ \mathcal{N} $$ = 2 gauge theories

We study CFTs at finite temperature and derive explicit sum rules for one-point functions of operators by imposing the KMS condition and we explicitly estimate one-point functions for light operators. Turning to heavy operators we employ Tauberian theorems and compute the asymptotic OPE density for heavy operators, from which we extract the leading terms of the OPE coefficients associated with heavy operators. Furthermore, we approximate and establish bounds for the two-point functions.

Sum rules &amp; Tauberian theorems at finite temperature

When a (super) conformal field theory is placed on a non-trivial manifold, the (super) conformal symmetry is broken. However, it is still possible to derive broken Ward identities for these broken symmetries, which provide additional constraints on the theory. We derive and apply the broken Ward identities associated with the (super) conformal group on the thermal manifold   <math> <msub> <mi>M</mi> <mi>β</mi> </msub> <mo>=</mo> <msubsup> <mi>S</mi> <mi>β</mi> <mn>1</mn> </msubsup> <mo>×</mo> <msup> <mi>ℝ</mi> <mrow> <mi>d</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </math>  $$ {\mathcal{M}}_{\beta }={S}_{\beta}^1\times {\mathbb{R}}^{d-1} $$  and   <math> <mi>M</mi> <mo>=</mo> <msup> <mi>T</mi> <mn>2</mn> </msup> <mo>×</mo> <msup> <mi>ℝ</mi> <mrow> <mi>d</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> </math>  $$ \mathcal{M}={T}^2\times {\mathbb{R}}^{d-2} $$ . The novel constraints not only systematically reproduce known results, including an implicit formulation of the generalized Cardy formula, but also elegantly relate the thermal energy spectrum with the conformal spectrum.

Broken (super) conformal Ward identities at finite temperature

The duality between type IIA superstring theory and M-theory enables us to lift bound states of D0-branes and n parallel D6-branes to M-theory compactified on an n-centered multi-Taub-NUT space   <math> <mi>TN</mi> </math>  $$ \mathbbm{TN} $$   n . Accordingly, the rank n K-theoretic Donaldson-Thomas invariants of ℂ3 are connected with the index of M-theory on ℂ3 ×   <math> <mi>TN</mi> </math>  $$ \mathbbm{TN} $$   n . In this paper, we extend this connection by considering intersecting D6-branes. In the presence of a suitable Neveu-Schwarz B-field, the system preserves two supercharges. This system is T-dual to the configuration of tetrahedron instantons which we introduced in [1]. We conjecture a closed-form expression for the K-theoretic tetrahedron instanton partition function, which is the generating function of the D0-D6 partition functions. We find that the tetrahedron instanton partition function coincides with the partition function of the magnificent four model for special values of the parameters, leading us to conjecture that our system of intersecting D6-branes can be obtained from the annihilation of D8-branes and anti-D8-branes. Remarkably, the K-theoretic tetrahedron instanton partition function allows an interpretation in terms of the index of M-theory on a noncompact Calabi-Yau fivefold which is related to a superposition of Kaluza-Klein monopoles. The dimensional reduction of the system allows us to express the cohomological tetrahedron instanton partition function in terms of the MacMahon function, generalizing the correspondence between Gromov-Witten invariants and Donaldson-Thomas invariants for Calabi-Yau threefolds.

Wenbin Yan

Probing M-theory with tetrahedron instantons

Operators with integer scaling dimensions in even-dimensional conformal field theories exhibit well-known type-B Weyl anomalies. In general, these anomalies depend nontrivially on exactly marginal couplings. We study the corresponding fully covariantized anomaly functional on conformal manifolds in several examples. We show that a natural consequence of the Wess-Zumino consistency condition is that the anomalies are covariantly constant with respect to the exactly marginal couplings. The argument is general and applies even when the conformal symmetry is spontaneously broken on moduli spaces of vacua.

Enrico Andriolo

Greece

Vasilis Niarchos

Constantinos Papageorgakis

Covariantly constant anomalies on conformal manifolds

This is the first in a series of papers devoted to the study of spin chains capturing the spectral problem of 4d   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 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   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 symmetry algebra down to the   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 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 ℤ2 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.

Randle Rabe

Dynamical spin chains in 4D   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 SCFTs

Even though for generic   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 1 theories it is not possible to separate distinct branches of supersymmetric vacua, in this paper we study a special class of   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 1 SCFTs, these of Class   <math> <msub> <mi>S</mi> <mi>k</mi> </msub> </math>  $$ {\mathcal{S}}_k $$  for which it is possible to define Coulomb and Higgs branches precisely as for the   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 theories of Class   <math> <mi>S</mi> </math>  $$ \mathcal{S} $$  from which they descend. We study the BPS operators that parameterise these branches of vacua using the different limits of the superconformal index as well as the Coulomb and Higgs branch Hilbert Series. Finally, with the tools we have developed, we provide a check that six dimensional (1, 1) Little String theory can be deconstructed from a toroidal quiver in Class   <math> <msub> <mi>S</mi> <mi>k</mi> </msub> </math>  $$ {\mathcal{S}}_k $$ .

Thomas Bourton

Italy

Alessandro Pini

The Coulomb and Higgs branches of   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 1 theories of Class   <math> <msub> <mi>S</mi> <mi>k</mi> </msub> </math>  $$ {\mathcal{S}}_k $$

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.

Netherlands

Ioana Coman

Joerg Teschner

Trinion conformal blocks from topological strings

We study type-B conformal anomalies associated with $$ \frac{1}{2} $$ -BPS Coulomb-branch operators in 4D $$ \mathcal{N} $$ = 2 superconformal field theories. When the vacuum preserves the conformal symmetry these anomalies coincide with the two-point function coefficients in the Coulomb-branch chiral ring. They are non-trivial functions of exactly-marginal couplings that can be determined from the partition function. In this paper, we examine the fate of these anomalies in vacua of the Higgs-branch moduli space, where conformal symmetry is spontaneously broken. We argue non-perturbatively that these anomalies are covariantly constant on conformal manifolds. In some cases, this can be used to show that they match in the broken and unbroken phases. Thus, we uncover a new class of data on the Higgs branch of 4D $$ \mathcal{N} $$ = 2 conformal field theories that are exactly computable. An interesting application of this matching occurs in $$ \mathcal{N} $$ = 2 circular quivers that deconstruct the 6D (2,0) theory on a torus. In that context, we argue that 4D supersymmetric localisation can be used to calculate non-trivial data involving $$ \frac{1}{2} $$ -BPS operators of the 6D theory as exact functions of the complex structure of the torus.