Boundary Partition Functions Research Articles

Note on the action with the Schwarzian at the stretched horizon

In this paper, we discuss the quantization of an interesting model of Carlip which appeared recently. It shows a way to associate boundary degrees of freedom to the stretched horizon of a stationary non-extremal black hole, as has been done in JT gravity for near-extremal black holes. The path integral now contains an integral over the boundary degrees of freedom, which are time reparametrizations of the stretched horizon keeping its length fixed. These boundary degrees of freedom can be viewed as elements of $Diff(S^1)/S^1$, which is the coadjoint orbit of an ordinary coadjoint vector under the action of the Virasoro group. From the symplectic form on this manifold, we obtain the measure in the boundary path integral. Doing a one-loop computation about the classical solution, we find that the one-loop answer is not finite, signalling that either the classical solution is unstable or there is an indefiniteness problem with this action, similar to the conformal mode problem in quantum gravity. Upon analytically continuing the field, the boundary partition function we get is independent of the inverse temperature and does not contribute to the thermodynamics at least at one-loop. This is in contrast to the study of near-extremal black holes in JT gravity, where the entire contribution to thermodynamics is from boundary degrees of freedom.

Summing over geometries in string theory

We examine the question how string theory achieves a sum over bulk geometries with fixed asymptotic boundary conditions. We discuss this problem with the help of the tensionless string on {mathrm{mathcal{M}}}_3times {mathrm{S}}^3times {mathbbm{T}}^4 (with one unit of NS-NS flux) that was recently understood to be dual to the symmetric orbifold SymN ( {mathbbm{T}}^4 ). We strengthen the analysis of [1] and show that the perturbative string partition function around a fixed bulk background already includes a sum over semi-classical geometries and large stringy corrections can be interpreted as various semi-classical geometries. We argue in particular that the string partition function on a Euclidean wormhole geometry factorizes completely into factors associated to the two boundaries of spacetime. Central to this is the remarkable property of the moduli space integral of string theory to localize on covering spaces of the conformal boundary of ℳ3. We also emphasize the fact that string perturbation theory computes the grand canonical partition function of the family of theories ⊕N SymN ( {mathbbm{T}}^4 ). The boundary partition function is naturally expressed as a sum over winding worldsheets, each of which we interpret as a ‘stringy geometry’. We argue that the semi-classical bulk geometry can be understood as a condensate of such stringy geometries. We also briefly discuss the effect of ensemble averaging over the Narain moduli space of {mathbbm{T}}^4 and of deforming away from the orbifold by the marginal deformation.

Partition functions of the tensionless string

We consider string theory on AdS3× S3× \U0001d54b4 in the tensionless limit, with one unit of NS-NS flux. This theory is conjectured to describe the symmetric product orbifold CFT. We consider the string on different Euclidean backgrounds such as thermal AdS3, the BTZ black hole, conical defects and wormhole geometries. In simple examples we compute the full string partition function. We find it to be independent of the precise bulk geometry, but only dependent on the geometry of the conformal boundary. For example, the string partition function on thermal AdS3 and the conical defect with a torus boundary is shown to agree, thus giving evidence for the equivalence of the tensionless string on these different background geometries. We also find that thermal AdS3 and the BTZ black hole are dual descriptions and the vacuum of the BTZ black hole is mapped to a single long string winding many times asymptotically around thermal AdS3. Thus the system yields a concrete example of the string-black hole transition. Consequently, reproducing the boundary partition function does not require a sum over bulk geometries, but rather agrees with the string partition function on any bulk geometry with the appropriate boundary. We argue that the same mechanism can lead to a resolution of the factorization problem when geometries with disconnected boundaries are considered, since the connected and disconnected geometries give the same contribution and we do not have to include them separately.

Electrical varieties as vertex integrable statistical models

We propose a new approach to studying electrical networks interpreting the Ohm law as the operator which solves certain local Yang–Baxter equation. Using this operator and the medial graph of the electrical network we define a vertex integrable statistical model and its boundary partition function. This gives an equivalent description of electrical networks. We show that, in the important case of an electrical network on the standard graph introduced in [Curtis E B et al 1998 Linear Algebr. Appl. 283 115–50], the response matrix of an electrical network, its most important feature, and the boundary partition function of our statistical model can be recovered from each other. Defining the electrical varieties in the usual way we compare them to the theory of the Lusztig varieties developed in [Berenstein A et al 1996 Adv. Math. 122 49–149]. In our picture the former turns out to be a deformation of the later. Our results should be compared to the earlier work started in [Lam T and Pylyavskyy P 2015 Algebr. Number Theory 9 1401–18] on the connection between the Lusztig varieties and the electrical varieties. There the authors introduced a one-parameter family of Lie groups which are deformations of the unipotent group. For the value of the parameter equal to 1 the group in the family acts on the set of response matrices and is related to the symplectic group. Using the data of electrical networks we construct a representation of the group in this family which corresponds to the value of the parameter −1 in the symplectic group and show that our boundary partition functions belong to it. Remarkably this representation has been studied before in the work on six vertex statistical models and the representations of the Temperley–Lieb algebra.

Izergin–Korepin analysis on the wavefunctions of the $U_q(\mathrm {sl}_2)$ six-vertex model with reflecting end

We extend the recently developed Izergin–Korepin analysis on the wavefunctions of the U_q(\mathrm {sl}_2) six-vertex model to the reflecting boundary conditions. Based on the Izergin–Korepin analysis, we determine the exact forms of the symmetric functions which represent the wavefunctions and its dual. Comparison of the symmetric functions with the coordinate Bethe ansatz wavefunctions for the open XXZ chain by Alcaraz, Barber, Batchelor, Baxter, and Quispel is also made. As an application, we derive algebraic identities for the symmetric functions by combining the results with the determinant formula of the domain wall boundary partition function of the six-vertex model with reflecting end.

AdS7 black-hole entropy and 5D mathcal{N} = 2 Yang-Mills

We generalise the work of 1810.11442 for the case of AdS7/CFT5. Starting from the 2-equivalent charge, 3-equivalent rotation non-extremal black-hole solution in 7D gauged supergravity, we consider the supersymmetric and then the extremal limit and evaluate the associated thermodynamic quantities. Away from extremality, the black-hole solution becomes complex. The entropy is then given by the Legendre transform of the on-shell action with respect to two complex chemical potentials subject to a constraint. At the conformal boundary we derive the dual background and evaluate the corresponding partition function for the AN–1 6D (2,0) theory at large N in a Cardy-like limit. This is carried out via a 5D mathcal{N} = 2 super Yang-Mills calculation on S5. The gravitational on-shell action is found to be exactly reproduced by the boundary partition function at large N. We argue that this agreement puts strong constraints on the form of possible higher-derivative corrections to the 5D gauge theory that is used in the S5 evaluation.

Quantum inverse scattering method and generalizations of symplectic Schur functions and Whittaker functions

We introduce generalizations of type C and B ice models which were recently introduced by Ivanov and Brubaker–Bump–Chinta–Gunnells, and study in detail the partition functions of the models by using the quantum inverse scattering method. We compute the explicit forms of the wavefunctions and their duals by using the Izergin–Korepin technique, which can be applied to both models. For type C ice, we show the wavefunctions are expressed using generalizations of the symplectic Schur functions. This gives a generalization of the correspondence by Ivanov. For type B ice, we prove that the exact expressions of the wavefunctions are given by generalizations of the Whittaker functions introduced by Bump–Friedberg–Hoffstein. The special case is the correspondence conjectured by Brubaker–Bump–Chinta–Gunnells. We also show the factorized forms for the domain wall boundary partition functions for both models. As a consequence of the studies of the partition functions, we obtain dual Cauchy formulas for the generalized symplectic Schur functions and the generalized Whittaker functions.

Dynamical quantum phase transitions in Weyl semimetals

The quench dynamics in type-I inversion symmetric Weyl semimetals (WSM) are explored in this work which, due to the form of the Hamiltonian, may be readily extended to two-dimensional Chern insulators. We analyze the role of equilibrium topological properties characterized by the Chern number of the pre-quench ground state in dictating the non-equilibrium dynamics of the system, specifically, the emergence of dynamical quantum phase transitions (DQPT). By investigating the ground state fidelity, it is found that a change in the signed Chern number constitutes a sufficient but not necessary condition for the occurrence of DQPTs. Depending on the ratio of the transverse and longitudinal hopping parameters, DQPTs may also be observed for quenches lying entirely within the initial Chern phase. Additionally, we analyze the zeros of the boundary partition function discovering that while the zeros generally form two-dimensional structures resulting in one-dimensional critical times, infinitesimal quenches may lead to one-dimensional zeros with zero-dimensional critical times provided the quench distance scales appropriately with the system size. This is strikingly manifested in the nature of non-analyticies of the dynamical free energy, revealing a logarithmic singularity. In addition, following recent experimental advances in observing the dynamical Fisher zeros of the Loschmidt overlap amplitude through azimuthal Bloch phase vortices by Bloch-state tomography, we rigorously investigate the same in WSMs. Finally, we establish the relationship between the dimension of the critical times and the presence of dynamical vortices, demonstrating that only one-dimensional critical times arising from two-dimensional manifolds of zeros of the boundary partition function lead to dynamical vortices.

Elliptic free-fermion model with OS boundary and elliptic Pfaffians

We introduce and study a class of partition functions of an elliptic free-fermionic face model. We study the partition functions with a triangular boundary using the off-diagonal $K$-matrix at the boundary (OS boundary), which was introduced by Kuperberg as a class of variants of the domain wall boundary partition functions. We find explicit forms of the partition functions with OS boundary using elliptic Pfaffians. We find two expressions based on two versions of Korepin's method, and we obtain an identity between two elliptic Pfaffians as a corollary.

Symmetric functions and wavefunctions of XXZ-type six-vertex models and elliptic Felderhof models by Izergin–Korepin analysis

We present a method to analyze the wavefunctions of six-vertex models by extending the Izergin–Korepin analysis originally developed for domain wall boundary partition functions. First, we apply the method to the case of the basic wavefunctions of the XXZ-type six-vertex model. By giving the Izergin–Korepin characterization of the wavefunctions, we show that these wavefunctions can be expressed as multiparameter deformations of the quantum group deformed Grothendieck polynomials. As a second example, we show that the Izergin–Korepin analysis is effective for analysis of the wavefunctions for a triangular boundary and present the explicit forms of the symmetric functions representing these wavefunctions. As a third example, we apply the method to the elliptic Felderhof model which is a face-type version and an elliptic extension of the trigonometric Felderhof model. We show that the wavefunctions can be expressed as one-parameter deformations of an elliptic analog of the Vandermonde determinant and elliptic symmetric functions.

Inner products in integrable Richardson-Gaudin models

We present the inner products of eigenstates in integrable Richardson-Gaudin models from two different perspectives and derive two classes of Gaudin-like determinant expressions for such inner products. The requirement that one of the states is on-shell arises naturally by demanding that a state has a dual representation. By implicitly combining these different representations, inner products can be recast as domain wall boundary partition functions. The structure of all involved matrices in terms of Cauchy matrices is made explicit and used to show how one of the classes returns the Slavnov determinant formula.Furthermore, this framework provides a further connection between two different approaches for integrable models, one in which everything is expressed in terms of rapidities satisfying Bethe equations, and one in which everything is expressed in terms of the eigenvalues of conserved charges, satisfying quadratic equations.

Combinatorial properties of symmetric polynomials from integrable vertex models in finite lattice

We introduce and study several combinatorial properties of a class of symmetric polynomials from the point of view of integrable vertex models in a finite lattice. We introduce the L-operator related to the Uq(sl2) R-matrix and construct the wavefunctions and their duals. We prove the exact correspondence between the wavefunctions and symmetric polynomials which is a quantum group deformation of the Grothendieck polynomials. This is proved by combining the matrix product method and an analysis on the domain wall boundary partition functions. As applications of the correspondence between the wavefunctions and symmetric polynomials, we derive several properties of the symmetric polynomials such as the determinant pairing formulas and the branching formulas by analyzing the domain wall boundary partition functions and the matrix elements of the B-operators.

Some special features of a rotating-moving Dp-brane

Using the boundary string field theory (BSFT) techniques we study the boundary state and partition function for a dynamical (rotating-moving) Dp-brane coupled to the electromagnetic and tachyonic background fields in superstring theory. By making use of the created partition function, the super BSFT action with a tachyonic field and Dirac-Born-Infeld–type action will be constructed. By analyzing the obtained action interesting features will be revealed.

Boundary conditions and partition functions in higher spin AdS3/CFT2

We discuss alternative definitions of the semiclassical partition function in two-dimensional CFTs with higher spin symmetry, in the presence of sources for the higher spin currents. Theories of this type can often be described via Hamiltonian reduction of current algebras, and a holographic description in terms of three-dimensional Chern-Simons theory with generalized AdS boundary conditions becomes available. By studying the CFT Ward identities in the presence of non-trivial sources, we determine the appropriate choice of boundary terms and boundary conditions in Chern-Simons theory for the various types of partition functions considered. In particular, we compare the Chern-Simons description of deformations of the field theory Hamiltonian versus those encoding deformations of the CFT action. Our analysis clarifies various issues and confusions that have permeated the literature on this subject.

Holography beyond conformal invariance and AdS isometry?

We suggest that the principle of holographic duality can be extended beyond conformal invariance and AdS isometry. Such an extension is based on a special relation between functional determinants of the operators acting in the bulk and on its boundary, provided that the boundary operator represents the inverse propagators of the theory induced on the boundary by the Dirichlet boundary value problem from the bulk spacetime. This relation holds for operators of general spin-tensor structure on generic manifolds with boundaries irrespective of their background geometry and conformal invariance, and it apparently underlies numerous $O(N^0)$ tests of AdS/CFT correspondence, based on direct calculation of the bulk and boundary partition functions, Casimir energies and conformal anomalies. The generalized holographic duality is discussed within the concept of the "double-trace" deformation of the boundary theory, which is responsible in the case of large $N$ CFT coupled to the tower of higher spin gauge fields for the renormalization group flow between infrared and ultraviolet fixed points. Potential extension of this method beyond one-loop order is also briefly discussed.

Localization & exact holography

Abstract We consider the AdS 2 /CFT 1 holographic correspondence near the horizon of big four-dimensional black holes preserving four supersymmetries in toroidally compactified Type-II string theory. The boundary partition function of CFT 1 is given by the known quantum degeneracies of these black holes. The bulk partition function is given by a functional integral over string fields in AdS 2. Using recent results on localization we reduce the infinite-dimensional functional integral to a finite number of ordinary integrals over a space of localizing instantons. Under reasonable assumptions about the relevant terms in the effective action, these integrals can be evaluated exactly to obtain a bulk partition function. It precisely reproduces all terms in the exact Rademacher expansion of the boundary partition function as nontrivial functions of charges except for the Kloosterman sum which can in principle follow from an analysis of phases in the background of orbifolded instantons. Our results can be regarded as a step towards proving ‘exact holography’ in that the bulk and boundary partition functions computed independently agree for finite charges. Since the bulk partition function defines the quantum entropy of the black hole, our results enable the evaluation of perturbative as well as nonperturbative quantum corrections to the Bekenstein-Hawking-Wald entropy of these black holes.

Determinant and Weyl anomaly of the Dirac operator: a holographic derivation

We present a holographic formula relating functional determinants: the fermion determinant in the one-loop effective action of bulk spinors in an asymptotically locally AdS background and the determinant of the two-point function of the dual operator at the conformal boundary. The formula originates from AdS/CFT heuristics that map a quantum contribution in the bulk partition function to a subleading large-N contribution in the boundary partition function. We use this holographic picture to address questions in spectral theory and conformal geometry. As an instance, we compute the type-A Weyl anomaly and the determinant of the iterated Dirac operator on round spheres, express the latter in terms of Barnes’ multiple gamma function and gain insight into a conjecture by Bär and Schopka.

General properties of the boundary renormalization group flow for supersymmetric systems in $1 + 1$ dimensions

We consider the general supersymmetric one-dimensional quantum system with boundary, critical in the bulk but not at the boundary. The renormalization group (RG) flow on the space of boundary conditions is generated by the boundary beta functions β a (λ) for the boundary coupling constants λ a . We prove a gradient formula ∂ ln z/∂λ a = −g S abβ b where z(λ) is the boundary partition function at given temperature T =1 /β, and g S (λ) is a certain positive-definite metric on the

Abelian and Non-Abelian Branes in WZW Models and Gerbes

We discuss how gerbes may be used to set up a consistent Lagrangian approach to the WZW models with boundary. The approach permits to study in detail possible boundary conditions that restrict the values of the fields on the worldsheet boundary to brane submanifolds in the target group. Such submanifolds are equipped with an additional geometric structure that is summarized in the notion of a gerbe module and includes a twisted Chan-Paton gauge field. Using the geometric approach, we present a complete classification of the branes that conserve the diagonal current-algebra symmetry in the WZW models with simple, compact but not necessarily simply connected target groups. Such symmetric branes are supported by a discrete series of conjugacy classes in the target group and may carry Abelian or non-Abelian twisted gauge fields. The latter situation occurs for the conjugacy classes with fundamental group ℤ2×ℤ2 in SO(4n)/ℤ2. The branes supported by such conjugacy classes have to be equipped with a projectively flat twisted U(2) gauge field in one of the two possible WZW models differing by discrete torsion. We show how the geometric description of branes leads to explicit formulae for the boundary partition functions and boundary operator product coefficients in the WZW models with non-simply connected target groups.

One-loop conformal anomalies from AdS/CFT in the Schrödinger representation

We compute the conformal anomalies of boundary CFTs for scalar and fermionic fields propagating in AdS spacetime at one-loop. The coefficients are quantized, with values related to the mass-spectra for Kaluza-Klein compactifications of Supergravity on AdS5xS5 and AdS7xS4. Our approach interprets the boundary partition function of fields propagating on AdS spacetime in terms of wave-functionals that satisfy the functional Schrodinger equation.