Representation Of Partition Function Research Articles

Carrollian Fluids and Spontaneous Breaking of Boost Symmetry.

In the hydrodynamic regime, field theories typically have their boost symmetry spontaneously broken due to the presence of a thermal rest frame although the associated Goldstone field does not acquire independent dynamics. We show that this is not the case for Carrollian field theories where the boost Goldstone field plays a central role. This allows us to give a first-principles derivation of the equilibrium currents and dissipative effects of Carrollian fluids. We also demonstrate that the limit of vanishing speed of light of relativistic fluids is a special case of this class of Carrollian fluids. Our results shine light on the thermodynamic properties and thermal partition functions of Carrollian field theories.

Large N expansion of superconformal index of k = 1 ABJM theory and semiclassical M5 brane partition function

It was shown in arXiv:2309.10786 that the leading non-perturbative contribution to the large N expansion of the superconformal index of the (2,0) 6d theory (which describes low-energy dynamics of N coincident M5 branes) is reproduced by the semiclassical partition function of quantum M2 brane wrapped on S1×S2 in a twisted version of AdS×7S4 background. Here we demonstrate an analogous relation for the leading non-perturbative contribution to the large N expansion of the superconformal index of the N=8 supersymmetric level-one U(N)×U(N) ABJM theory (which describes low-energy dynamics of N coincident M2 branes). The roles of M2 and M5 branes get effectively interchanged. Namely, the large N correction to the ABJM index is found to be given by the semiclassical partition function of quantum M5 brane wrapped on S1×S5 in a twisted version of AdS×4S7 background. This effectively confirms the suggestion for the “M5 brane index” made in arXiv:2007.05213 on the basis of indirect superconformal algebra considerations.

Bailey pairs for the q-hypergeometric integral pentagon identity

In this work, we construct new Bailey pairs for the integral pentagon identity in terms of q-hypergeometric functions. The pentagon identity considered here represents the equality of the partition functions of certain three-dimensional supersymmetric dual theories. It can be also interpreted as the star-triangle relation for the Ising-type integrable lattice model.

Perverse filtrations, Chern filtrations, and refined BPS invariants for local [formula omitted

We explore connections between three structures associated with the cohomology of the moduli of 1-dimensional stable sheaves on P2: perverse filtrations, tautological classes, and refined BPS invariants for local P2. We formulate the P=C conjecture identifying the perverse filtration with the Chern filtration for the free part of the cohomology. This can be viewed as an analog of de Cataldo–Hausel–Migliorini's P=W conjecture for Hitchin systems. Our conjecture is compatible with the enumerative invariants of local P2 calculated by refined Pandharipande–Thomas theory or Nekrasov partition functions. It provides a cohomological lift of a conjectural product formula of the asymptotic refined BPS invariants. We prove the P=C conjecture for degrees ≤4.

Partition functions of non-Lagrangian theories from the holomorphic anomaly

The computation of the partition function in certain quantum field theories, such as those of the Argyres-Douglas or Minahan-Nemeschansky type, is problematic due to the lack of a Lagrangian description. In this paper, we use the holomorphic anomaly equation to derive the gravitational corrections to the prepotential of such theories at rank one by deforming them from the conformal point. In the conformal limit, we find a general formula for the partition function as a sum of hypergeometric functions. We show explicit results for the round sphere and the Nekrasov-Shatashvili phases of the Ω background. The first case is relevant for the derivation of extremal correlators in flat space, whereas the second one has interesting applications for the study of anharmonic oscillators.

DE-type little strings from glued brane webs

We propose brane web configurations for D-type and E-type mathcal{N} =(1,0) little string theories based on a trivalent or quadrivalent gluing of 5-brane web diagrams. Tri-/quadri-valent gluing is a powerful way of computing 5d/6d partition functions for supersymmetric gauge theories based on the topological vertex. We generalize the gluing techniques to little string theories by introducing a new compact direction and compute their supersymmetric partition functions on Omega-deformed ℝ4×T2. As concrete examples, we consider little string theories arising from Type IIB NS5-branes probing D4 or D5 singularity. Their effective gauge theory descriptions as the affine D4 or D5 quiver gauge theory can be realized with quadrivalent or trivalent gluing, respectively. Based on these gluings of 5-brane webs, we compute their refined partition functions and compare them with the known results. We extend the computation of the partition function to little string theory engineered from IIB NS5-branes probing E6 singularity based on a trivalent gluing. We also discuss the generalization to higher rank cases and the symmetries of the partition functions.

Recurrence relation for instanton partition function in SU(N) gauge theory

We derive a residue formula and as a consequence a recurrence relation for the instanton partition function in mathcal{N} = 2 supersymmetric theory on ℂ2 with SU(N) gauge group.The particular cases of SU(2) and SU(3) gauge groups were considered in the literature before. The recurrence relation with SU(2) gauge group is long well known and was found as the Alday-Gaiotto-Tachikawa (AGT) counterpart of the Zamolodchikov relation for the Virasoro conformal blocks. In the SU(3) case a residue formula for the term with the minimal number of instantons was found and basing on it a recurrence relation was conjectured.We give a complete proof of the residue formula in all instanton orders in presence of any number of matter hypermultiplets in the adjoint and fundamental representations. The recurrence relation however describes only theories with not too much matter hypermultiplets so that the behaviour at infinity is moderate. The guideline of the proof is an algebro-geometric interpretation of the mathcal{N} = 2 supersymmetric gauge theory partition function in terms of the framed torsion-free sheaves. Lead by this interpretation we formulate a refined version of the residue formula and prove it by direct algebraic manipulations.

Instantons on Young diagrams with matters

We present the unrefined instanton partition functions of various 5d gauge theories with matter beyond the fundamental representation as sums over Young diagrams. By using these explicit expressions, we verify a range of identities among the instanton partition functions predicted by Higgsing procedures of fivebrane web diagrams and representation theory.

5d AGT correspondence of supergroup gauge theories from quantum toroidal mathfrak{gl} 1

We discuss the 5d AGT correspondence of supergroup gauge theories with A-type supergroups. We introduce two intertwiners called positive and negative intertwiners to compute the instanton partition function. The positive intertwiner is the ordinary Awata-Feigin-Shiraishi intertwiner while the negative intertwiner is an intertwiner obtained by using central charges with negative levels. We show that composition of them gives the basic Nekrasov factors appearing in supergroup partition functions. We explicitly derive the instanton partition functions of supergroup gauge theories with A and D-type quiver structures. Using the intertwiners, we briefly study the Gaiotto state, qq-characters and the relation with quiver W-algebra. Furthermore, we show that the negative intertwiner corresponds to the anti-refined topological vertex recently defined by Kimura and Sugimoto. We also discuss how superquiver theories should appear in our formalism if they exist. The existence of the AGT correspondence of the theories we study in this paper implies that there is a broader 2d/4d (5d/q-algebra) correspondence, or more generally the BPS/CFT correspondence, where new non-unitary theories play important roles.

Hyperbolic and trigonometric hypergeometric solutions to the star-star equation

We construct the hyperbolic and trigonometric solutions to the star-star relation via the gauge/YBE correspondence by using the three-dimensional lens partition function and superconformal index for a certain mathcal N=2 supersymmetric gauge dual theories. This correspondence relates supersymmetric gauge theories to exactly solvable models of statistical mechanics. The equality of partition functions for the three-dimensional supersymmetric dual theories can be written as an integral identity for hyperbolic and basic hypergeometric functions.

Evidence for an algebra of G2 instantons

In this short note, we present some evidence towards the existence of an algebra of BPS G2 instantons. These are instantonic configurations that govern the partition functions of 7d SYM theories on local G2 holonomy manifolds mathcal{X} . To shed light on such structure, we begin investigating the relation with parent 4d mathcal{N} = 1 theories obtained by geometric engineering M-theory on mathcal{X} . The main point of this paper is to substantiate the following dream: the holomorphic sector of such theories on multi-centered Taub-NUT spaces gives rise to an algebra whose characters organise the G2 instanton partition function. As a first step towards this program we argue by string duality that a multitude of geometries mathcal{X} exist that are dual to well-known 4d SCFTs arising from D3 branes probes of CY cones: all these models are amenable to an analysis along the lines suggested by Dijkgraaf, Gukov, Neitzke and Vafa in the context of topological M-theory. Moreover, we discuss an interesting relation to Costello’s twisted M-theory, which arises at local patches, and is a key ingredient in identifying the relevant algebras.

Topological strings and Wilson loops

We propose the refined topological string correspondence to the expectation values of half-BPS Wilson loop operators in 5d mathcal{N} = 1 gauge theory partition function on the Omega-deformed background {mathbb{R}}_{upepsilon_{1,2}}^4 × S1. We provide the refined topological vertex method and the refined holomorphic anomaly equation method in the topological string theory, from which we have exact computations on the 5d Wilson loops partition functions in both A- and B-models. Finally, with the exact results we have in B-model, we recover the quantum periods of local ℙ1 × ℙ1 model and local ℙ2 model in the study of quantum geometry and we further give a refined generalization of A-period.

3d mathcal{N} = 2 brane webs and quiver matrices

We discuss 3d brane webs and effective Chern-Simons levels for 3d mathcal{N} = 2 gauge theories. We find that turning on real masses for chiral multiplets leads to various equivalent brane webs that are related by flipping positions of D5-branes. We interpret flips as ST-transformations for chiral multiplets. ST-transformations could turn abelian theories into dual theories with mixed Chern-Simons levels that are interpreted as quiver matrices Cij encoding DT-invariants. We notice that each brane web corresponds to a quiver matrix. ST-transformations of holomorphic blocks are discussed to verify results. We also discuss the movement of flavor D5-branes, which leads to double-layer brane webs and manifests fiber-base duality. In the second part, we compute refined vortex partition functions of nonabelian theories with the gauge group U(N) and find corresponding quiver matrices. The computation shows that on Higgs branch nonabelian groups are broken to abelian groups.

Symmetry defects and orbifolds of two-dimensional Yang–Mills theory

We describe discrete symmetries of two-dimensional Yang–Mills theory with gauge group G associated with outer automorphisms of G, and their corresponding defects. We show that the gauge theory partition function with defects can be computed as a path integral over the space of twisted G-bundles and calculate it exactly. We argue that its weak-coupling limit computes the symplectic volume of the moduli space of flat twisted G-bundles on a surface. Using the defect network approach to generalised orbifolds, we gauge the discrete symmetry and construct the corresponding orbifold theory, which is again two-dimensional Yang–Mills theory but with gauge group given by an extension of G by outer automorphisms. With the help of the orbifold completion of the topological defect bicategory of two-dimensional Yang–Mills theory, we describe the reverse orbifold using a Wilson line defect for the discrete gauge symmetry. We present our results using two complementary approaches: in the lattice regularisation of the path integral, and in the functorial approach to area-dependent quantum field theories with defects via regularised Frobenius algebras.

$6$-dimensional FJRW theories of the simple–elliptic singularities

We give explicitly in the closed formulae the genus zero primary potentials of the three 6-dimensional FJRW theories of the simple-elliptic singularity $\tilde E_7$ with the non-maximal symmetry groups. For each of these FJRW theories we establish the CY/LG correspondence to the Gromov-Witten theory of the orbifold $[\mathcal{E}/ (\mathbb{Z}/2\mathbb{Z})]$ --- the orbifold quotient of the elliptic curve by the hyperelliptic involution. Namely, we give explicitly the Givental's group elements, whose actions on the partition function of the Gromov--Witten theory of $[\mathcal{E}/ (\mathbb{Z}/2\mathbb{Z})]$ give up to a linear change of the variables the partition functions of the FJRW theories mentioned. We keep track of the linear changes of the variables needed. We show that using only the axioms of Fan--Jarvis--Ruan, the genus zero potential can only be reconstructed up to a scaling.

Decomposition of BPS moduli spaces and asymptotics of supersymmetric partition functions

We present a prototype for Wilsonian analysis of asymptotics of supersymmetric partition functions of non-abelian gauge theories. Localization allows expressing such partition functions as an integral over a BPS moduli space. When the limit of interest introduces a scale hierarchy in the problem, asymptotics of the partition function is obtained in the Wilsonian approach by i) decomposing (in some suitable scheme) the BPS moduli space into various patches according to the set of light fields (lighter than the scheme dependent cut-off Λ) they support, ii) localizing the partition function of the effective field theory on each patch (with cut-offs set by the scheme), and iii) summing up the contributions of all patches to obtain the final asymptotic result (which is scheme-independent and accurate as Λ → ∞). Our prototype concerns the Cardy-like asymptotics of the 4d superconformal index, which has been of interest recently for its application to black hole microstate counting in AdS5/CFT4. As a byproduct of our analysis we obtain the most general asymptotic expression for the index of gauge theories in the Cardy-like limit, encompassing and extending all previous results.

Modularity of supersymmetric partition functions

We discover a modular property of supersymmetric partition functions of supersymmetric theories with R-symmetry in four dimensions. This modular property is, in a sense, the generalization of the modular invariance of the supersymmetric partition function of two-dimensional supersymmetric theories on a torus i.e. of the elliptic genus. The partition functions in question are on manifolds homeomorphic to the ones obtained by gluing solid tori. Such gluing involves the choice of a large diffeomorphism of the boundary torus, along with the choice of a large gauge transformation for the background flavor symmetry connections, if present. Our modular property is a manifestation of the consistency of the gluing procedure. The modular property is used to rederive a supersymmetric Cardy formula for four dimensional gauge theories that has played a key role in computing the entropy of supersymmetric black holes. To be concrete, we work with four-dimensional mathcal{N} = 1 supersymmetric theories but we expect versions of our result to apply more widely to supersymmetric theories in other dimensions.

Comments on wormholes, ensembles, and cosmology

Certain closed-universe big-bang/big-crunch cosmological spacetimes may be obtained by analytic continuation from asymptotically AdS Euclidean wormholes, as emphasized by Maldacena and Maoz. We investigate how these Euclidean wormhole spacetimes and their associated cosmological physics might be described within the context of AdS/CFT. We point out that a holographic model for cosmology proposed recently in arXiv:1810.10601 can be understood as a specific example of this picture. Based on this example, we suggest key features that should be present in more general examples of this approach to cosmology. The basic picture is that we start with two non-interacting copies of a Euclidean holographic CFT associated with the asymptotic regions of the Euclidean wormhole and couple these to auxiliary degrees of freedom such that the original theories interact strongly in the IR but softly in the UV. The partition function for the full theory with the auxiliary degrees of freedom can be viewed as a product of partition functions for the original theories averaged over an ensemble of possible sources. The Lorentzian cosmological spacetime is encoded in a wavefunction of the universe that lives in the Hilbert space of the auxiliary degrees of freedom.

Mathfrak{gl} N Higgsed networks

We generalize the framework of Higgsed networks of intertwiners to the quantum toroidal algebra associated to Lie algebra mathfrak{gl} N. Using our formalism we obtain a systems of screening operators corresponding to W-algebras associated to toric strip geometries and reproduce partition functions of 3d theories on orbifolded backgrounds.

Harmonic analysis of 2d CFT partition functions

We apply the theory of harmonic analysis on the fundamental domain of SL(2, ℤ) to partition functions of two-dimensional conformal field theories. We decompose the partition function of c free bosons on a Narain lattice into eigenfunctions of the Laplacian of worldsheet moduli space ℍ/SL(2, ℤ), and of target space moduli space O(c, c; ℤ)\\O(c, c; ℝ)/O(c)×O(c). This decomposition manifests certain properties of Narain theories and ensemble averages thereof. We extend the application of spectral theory to partition functions of general two-dimensional conformal field theories, and explore its meaning in connection to AdS3 gravity. An implication of harmonic analysis is that the local operator spectrum is fully determined by a certain subset of degeneracies.