Zoltan Bajnok, Chao Wu
We prove the asymptotic large volume expression of diagonal form factors in integrable models by evaluating carefully the diagonal limit of a non-diagonal form factor in which we send the rapidity of the extra particle to infinity.
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We propose an expression for the classical limit of diagonal form factors in which we integrate the corresponding observable over the moduli space of classical solutions. In infinite volume the integral has to be regularized by proper subtractions and we present the one, which corresponds to the classical limit of the connected diagonal form factors. In finite volume the integral is finite and can be expressed in terms of the classical infinite volume diagonal form factors and subvolumes of the moduli space. We analyze carefully the periodicity properties of the finite volume moduli space and …
The boundary conditions with diagonal boundary $S$ matrix and the boundary form factors for the Smirnov--Fateev model on a half line has been considered in the framework of the free field representation. In contrast to the case of the sine-Gordon model, in this case the free field representation is shown to impose severe restrictions on the boundary $S$ matrix, so that a finite number of solutions is only consistent with the free field realization.
We study the heavy-heavy-light (HHL) three-point functions in the planar N = 4 super-Yang- Mills theory using the recently proposed hexagon bootstrap program [arXiv:1505.06745]. We prove the conjecture of Bajnok, Janik and Wereszczynski [arXiv:1404.4556] on the polynomial L-dependence of HHL structure constant up to the leading finite-size corrections, where L is the length of the heavy operators. The proof is presented for a specific set-up but the method can be applied to more general situations.
We perturbatively study form factors in the Landau-Lifshitz model and the generalisation originating in the study of the N=4 super-Yang-Mills dilatation generator. In particular we study diagonal form factors which have previously been related to gauge theory structure constants. For the Landau-Lifshitz model, due to the non-relativistic nature of the theory, we are able to compute all orders in perturbation theory and to resum the series to find quantum form factors for low numbers of external particles. We apply our form factors to the study of deformations of the integrable theory by means …
We study the asymptotic volume dependence of the heavy-heavy-light three-point functions in the $\mathcal{N}=4$ Super-Yang-Mills theory using the hexagon bootstrap approach, where the volume is the length of the heavy operator. We extend the analysis of our previous short letter 1511.06199 to the general case where the heavy operators can be in any rank one sector and the light operator being a generic non-BPS operator. We prove the conjecture of Bajnok, Janik and Wereszczynski 1404.4556 up to leading finite size corrections.
Form factor bootstrap approach is applied for diagonal scattering theories. We consider the ADE theories and determine the functional equations satisfied by the minimal two-particle form factors. We also determine the parameterization of the singularities in two particle form factors. For $A^{(1)}_{2}$ Affine Toda field theory which is the simplest non-self conjugate theory, form factors are derived up to four-body and identification of operator is done. Generalizing this identification to the $A^{(1)}_N$ Affine Toda cases, we fix the two particle form factors. We also determine the additional…
We derive the leading exponential finite volume corrections in two dimensional integrable models for non-diagonal form factors in diagonally scattering theories. These formulas are expressed in terms of the infinite volume form factors and scattering matrices. If the particles are bound states then the leading exponential finite-size corrections ($μ$-terms) are related to virtual processes in which the particles disintegrate into their constituents. For non-bound state particles the leading exponential finite-size corrections (F-terms) come from virtual particles traveling around the finite wo…
This thesis presents Lüscher's $μ$- and $F$-term corrections to volume dependence of non-diagonal finite volume form factors in the scaling Lee-Yang model. An explicit calculation proves the suspected relation that the $μ$-terms known previously from bound state quantization can be obtained from the $F$-term integrals by modifying the contour of integration such that it picks up residues of appropriate poles in the integrand. The fact that these two different approaches for getting the $μ$-terms give the same result underpins the formal derivation of the $F$-term in arXiv:1904.00492 which was …
We derive an exact formula for finite volume excited state mean values of local operators in 1+1 dimensional Integrable QFT with diagonal scattering. Our result is a non-trivial generalization of the LeClair-Mussardo series, which is a form factor expansion for finite size ground state mean values.
Most known four-dimensional cohomogeneity-one Einstein metrics are diagonal in the basis defined by the left-invariant one-forms, though some essentially non-diagonal ones are known. We consider the problem of explicitly seeking non-diagonal Einstein metrics, and we find solutions which in some cases exhaust the possibilities. In particular we construct new examples of neutral signature non--diagonal Bianchi type VIII Einstein metrics with self--dual Weyl tensor.
In this paper we derive from field theory a Lüscher-formula, which gives the leading exponentially small in volume corrections to the 1-particle form-factors in non-diagonally scattering integrable quantum field theories. Our final formula is expressed in terms of appropriate expressions of 1- and 3-particle form-factors, and can be considered as the generalization of previous results obtained for diagonally scattering bosonic integrable quantum field theories. Since our formulas are also valid for fermions and operators with non-zero Lorentz-spin, we demonstrated our results in the Massive Th…
In this paper we consider a special kind of three-point functions of HHL type at weak coupling in N=4 SYM theory and analyze its volume dependence. At strong coupling this kind of three-point functions were studied recently by Bajnok, Janik and Wereszczynski [1]. The authors considered some cases of HHL correlator in the su(2) sector and, relying on their explicit results, formulated a conjecture about the form of the volume dependence of the symmetric HHL structure constant to be valid at any coupling up to wrapping corrections. In order to test this hypothesis we considered the HHL correlato…
Studying perturbatively, for large m, the torus partition function of both (A,A) and (A,D) series of minimal models in the Cappelli, Itzykson, Zuber classification, deformed by the least relevant operator $ϕ_{(1,3)}$, we disentangle the structure of $ϕ_{1,3}$ flows. The results are conjectured on reasonable ground to be valid for all m. They show that (A,A) models always flow to (A,A) and (A,D) ones to (A,D). No hopping between the two series is possible. Also, we give arguments that there exist 3 isolated flows (E,A)-->(A,E) that, together with the two series, should exhaust all the possible …
We give a simple - straightforward and rigorous - derivation that when the eigenvalues of one of the $d=9 (5,3,2)$ matrices in the SU(N) invariant supersymmetric matrix model become large (and well separated from each other) the ground-state wavefunction (resp. asymptotic zero-energy solution of the corresponding differential equation) factorizes, for all $N>1$, into a product of supersymmetric harmonic oscillator wavefunctions (involving the `off-diagonal' degrees of freedom) and a wavefunction $ψ$ that is annihilated by the free supercharge formed out of all `diagonal' (Cartan sub-algebra) d…
We investigate the possible regular solutions of the boundary Yang-Baxter equation for the fundamental $U_q[G_2]$ vertex model. We find four distinct classes of reflection matrices such that half of them are diagonal while the other half are non-diagonal. The latter are parameterized by two continuous parameters but only one solution has all entries non-null. The non-diagonal solutions do not reduce to diagonal ones at any special limit of the free-parameters.
Diagonally gauged anomaly-free 6D supergravities and their vacua
研究对张量数\(T = 1\)且含对角阿贝尔因子\(U(1)_{R +}\)的六维\((1,0)\)无反常模型进行有界搜索，发现斜度规模型图景丰富，还研究其6维真空态，找到超对称6维闵可夫斯基真空态，R - 对称度规对角性必要。
We present a generalization of the coordinate Bethe ansatz that allows us to solve integrable open XXZ and ASEP models with non-diagonal boundary matrices, provided their parameters obey some relations. These relations extend the ones already known in the literature in the context of algebraic or functional Bethe ansatz. The eigenvectors are represented as sums over cosets of the $BC_n$ Weyl group.
On representation matrices of boundary conditions in $SU(n)$ gauge theories compactified on two-dimensional orbifolds
本研究探讨了在二维轨道上紧致化的SU(n)规范理论中，边界条件表示矩阵的对角化问题，证明了在T^2/Z_2和T^2/Z_3上必存在对角代表元，而在T^2/Z_4和T^2/Z_6上可能存在非对角块，且离散Wilson线相位可导致降秩对称性破缺。
We consider the SU_q (N) invariant spin chain with diagonal and non-diagonal integrable boundary terms. The algebraic study of spin chains with different types of boundary terms is used to motivate a set of spectral equivalences between integrable chains with purely diagonal boundary terms and ones with an arbitrary non-diagonal term at one end. For each choice of diagonal boundary terms there is an isospectral one-boundary problem and vice-versa. The quantum group SU_q (N) symmetry is broken by the presence of a non-diagonal boundary term however one can use the spectral equivalence with the …
We derive a Toda-type recurrence relation, in both high and low temperature regimes, for the $λ$ - extended diagonal correlation functions $C(N,N;λ)$ of the two-dimensional Ising model, using an earlier connection between diagonal form factor expansions and tau-functions within Painlevé VI (PVI) theory, originally discovered by Jimbo and Miwa. This greatly simplifies the calculation of the diagonal correlation functions, particularly their $λ$-extended counterparts. We also conjecture a closed form expression for the simplest off-diagonal case $C^{\pm}(0,1;λ)$ where a connection to PVI theory …
Triggered by a controversy surrounding a universal behaviour of the power spectrum in quantum systems exhibiting regular classical dynamics, we focus on a model of random diagonal matrices (RDM), often associated with the Poisson spectral universality class, and examine how the power spectrum and the form factor get affected by two-sided truncations of RDM spectra. Having developed a nonperturbative description of both statistics, we perform their detailed asymptotic analysis to demonstrate explicitly how a traditional assumption (lying at the heart of the controversy) -- that the power spectr…
We study the SYK model with an extra constant source, i.e. a constant matrix or equivalently a diagonal matrix with only one non-zero entry $λ_1$. By using methods from analytic combinatorics, we find exact expressions for the moments of this model. We further prove that the spectrum of this model can have a gap when $λ_1>λ^c_1$, thus exhibiting a phase transition in $λ_1$. In this case, a single isolated eigenvalue splits off from SYK's eigenvalues distribution. We located this single eigenvalue by analyzing the singular behavior of a supercritical functional composition scheme. In certain li…
Using the "Liouville space" (the space of operators) of the massive Dirac theory, we define mixed-state form factors of $U(1)$ twist fields. We consider mixed states with density matrices diagonal in the asymptotic particle basis. This includes the thermal Gibbs state as well as all generalized Gibbs ensembles of the Dirac theory. When the mixed state is specialized to a thermal Gibbs state, using a Riemann-Hilbert problem and low-temperature expansion, we obtain finite-temperature form factors of $U(1)$ twist fields. We then propose the expression for form factors of $U(1)$ twist fields in ge…
The Drinfeld twists or factorizing F-matrices for the open XXZ spin chain with non-diagonal boundary terms are constructed. It is shown that in the F-basis the two sets of pseudo-particle creation operators simultaneously take completely symmetric and polarization free form. The explicit and completely symmetric expressions of the two sets of Bethe states of the model are obtained.
We study the finite-temperature expectation values of exponential fields in the sine-Gordon model. Using finite-volume regularization, we give a low-temperature expansion of such quantities in terms of the connected diagonal matrix elements, for which we provide explicit formulas. For special values of the exponent, computations by other methods are available and used to validate our findings. Our results can also be interpreted as a further support for a previous conjecture about the connection between finite- and infinite-volume form factors valid up to terms exponentially decaying in the vo…
With the help of the F-basis provided by the Drinfeld twist or factorizing F-matrix for the open XXZ spin chain with non-diagonal boundary terms, we obtain the determinant representation of the partition function of the six-vertex model with a non-diagonal reflecting end under domain wall boundary condition.
With the help of the F-basis provided by the Drinfeld twist or factorizing F-matrix for the open XXZ spin chain with non-diagonal boundary terms, we obtain the determinant representations of the scalar products of Bethe states of the model.
With the help of the F-basis provided by the Drinfeld twist or factorizing F-matrix of the eight-vertex solid-on-solid (SOS) model, we obtain the determinant representations of the scalar products of Bethe states for the open XYZ chain with non-diagonal boundary terms. By taking the on shell limit, we obtain the determinant representations (or Gaudin formula) of the norms of the Bethe states.
We use the quantum separation of variable (SOV) method to construct the eigenstates of the open XXZ chain with the most general boundary terms. The eigenstates in the inhomogeneous case are constructed in terms of solutions of a system of quadratic equations. This SOV representation permits us to compute scalar products and can be used to calculate form factors and correlation functions.
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