Twisted Index on Hyperbolic Four-Manifolds

We introduce the topologically twisted index for four-dimensional $\mathcal N=1$ gauge theories quantized on ${\rm AdS}_2 \times S^1$. We compute the index by applying supersymmetric localization to partition functions of vector and chiral multiplets on ${\rm AdS}_2 \times T^2$, with and without a boundary: in both instances we classify normalizability and boundary conditions for gauge, matter and ghost fields. The index is twisted as the dynamical fields are coupled to a R-symmetry background 1-form with non-trivial exterior derivative and proportional to the spin connection. After regularization the index is written in terms of elliptic gamma functions, reminiscent of four-dimensional holomorphic blocks, and crucially depends on the R-charge.


Introduction and Summary
Supersymmetric field theories in both flat and curved spaces have been extensively studied over the years, serving as a crucial arena for advancing our theoretical understanding of quantum field theory (QFT), especially in the regime of strong interactions [1][2][3].While the complete information of a QFT is contained in its generating functional of correlation functions, exact computations of this functional in interacting theories remain challenging.Nevertheless, the technique of supersymmetric localization [4] has proven to be an extremely ductile tool, enabling exact non-perturbative computations of specific generating functionals and other observables in a large class of supersymmetric field theories defined on curved manifolds.In particular, localization techniques have been employed to study supersymmetric field theories on compact Riemannian manifolds, where a class of BPS observables can be precisely evaluated by reducing functional integrals to Gaussian integrals around a supersymmetric locus.Several such computations have been performed in various dimension and for diverse topologies, leading to valuable insights [5][6][7][8][9][10][11][12][13][14][15][16][17], see also [18] and references therein.
Building on this success, this paper shifts attention towards studying supersymmetric gauge theories on non-compact hyperbolic manifolds, focussing on AdS 2 × T 2 , where by AdS d we indicate d-dimensional Anti-de Sitter space with Euclidean signature.Gauge theories in AdS have been investigated in connection with monodromy defects [19], black-hole entropy [20,21], chiral algebras [22] and holomorphic blocks [23,24].Moreover, the isometry group of AdS d being the (global) conformal group in (d − 1)-dimensions, QFT in AdS can be studied via conformal bootstrap [25,26].
Applying supersymmetric localization to QFTs on non-compact manifolds is also interesting from a technical viewpoint as it requires the study of the behaviour at infinity of the degrees of freedom contributing to the path integral.This is necessary in order to make sure that not only the final result for the partition function is convergent and well-defined, but also supersymmetry is preserved.
In this paper we present a detailed calculation of partition functions for N = 1 supersymmetric gauge theories defined on AdS 2 × T 2 .The construction of supersymmetric theories in a fixed background geometry involves taking a suitable limit of new minimal supergravity, leading to background fields coupled to a supersymmetric gauge theory with an R-symmetry, incorporating ordinary vector and chiral multiplets [35][36][37].We turn on a non-trivial R-symmetry background field equal to half the spin connection, which is the usual setup corresponding to the topological twist, preserving on AdS 2 × T 2 two Killing spinors of opposite chirality and R-charge.The metric we use on the fourmanifold is refined by parameters that in the partition function combine into two complex moduli: one is a fugacity p = e 2πiτ for momentum on T 2 , with τ being the torus modular parameter, while the other is a fugacity q probing angular momentum on AdS 2 .Such fugacities allow for linking partition functions of supersymmetric gauge theories on AdS 2 × T 2 to flavoured Witten indices for theories quantized on AdS 2 × S1 , namely where H is the Hilbert space of BPS states on AdS 2 ×S 1 whereas the operator H is the Hamiltonian, F is the fermionic number, J is the angular momentum on AdS 2 , P is the translation operator along S 1 while ϕ i and Q i are the chemical potential and the charge operator for the i-th flavour symmetry 1 , respectively.Technically, supersymmetric localization provides the plethystic exponential of the single-letter index (1.1).For instance, let T be the gauge theory represented by a dynamical vector multiplet in the adjoint representation of the gauge group G coupled to a chiral multiplet of Rcharge r < 1 in a representation R G of G.The topologically twisted index of T , defined on the four-dimensional hyperbolic manifold AdS 2 × T 2 in absence of boundaries, is given by the following integral over the Cartan torus of G involving a ratio of elliptic Gamma functions: e 2πi Ψ Γ e e 2πi(2ϕR) e 2πiα(u) ; q, p Γ e e 2πi(2−r)ϕR e −2πiρ(u) ; q, p , where ϕ R and u respectively are gauge and R-symmetry fugacities.We refrain from explicitly writing down the phase factor Ψ as its form is not particularly illuminating; anyhow, Ψ can be read off from a suitable combination of the Ψ CM D,R and Ψ VM D,R reported in the main text.We specified that (1.2) is the expression valid for a chiral multiplet of R-charge lesser than one because the index involving the same multiplets in the case r ≥ 1 has a different form: the reason is that on non-compact AdS 2 × T 2 the normalizability of the fields contributing to the partition function dramatically depends on the R-charge.This phenomenon was already observed in lower dimension in relation to the topologically twisted index computed via localization on AdS 2 × S 1 in absence of boundaries [24].
In presence of boundaries two dual sets of boundary conditions does not break supersymmetry: either Dirichlet conditions, requiring the vanishing of fields at the boundary; or Robin conditions, requiring the vanishing of derivatives of fields at the boundary.In fact, derivatives are linear combinations of partial derivatives in directions that can be parallel and normal to the boundary, hence Robin conditions effectively are generalized Neumann conditions.A boundary then allows for constructing many different theories by just combining multiplets satisfying a priori different sets of supersymmetric boundary conditions.For example, (1.2) can be interpreted as the twisted index of a gauge theory where Robin boundary conditions were imposed on the vector multiplet and Dirichlet boundary conditions were imposed upon the chiral multiplet.An intriguing feature peculiar to the presence of boundaries is how boundary degrees of freedom induce a flip of boundary conditions [29,38]: for instance, suitably coupling a lower-dimensional matter multiplet to a four-dimensional chiral multiplet fulfilling Dirichlet boundary conditions effectively yields a chiral multiplet with Robin boundary conditions.At the level of the partition function such a flip of boundary conditions is realized thanks to the multiplication properties of the special function appearing in the 1-loop determinants whose integral defines the index: where θ 0 (z, q) = (z, q) ∞ (q/z; q) ∞ , with (z; q) ∞ = j≥0 1 − zq j being the indefinite q-Pochhammer symbol, Z CM 1-L | D,R are the four-dimensional chiral-multiplet 1-loop determinants of fluctuations satisfying Dirichlet and Robin boundary conditions, respectively, while Z CM 1-L | ∂ is the 1-loop determinant of the three-dimensional boundary multiplet.
In summary, generalizing localization techniques to the case of non-compact manifolds naturally opens up new avenues for exploration.Compelling future directions include a careful study of the phase factors Ψ CM D,R and Ψ VM D,R appearing after regularization of the 1-loop determinants for chiral and vector multiplets, as such phases encode important scheme-independent informations about anomalies, vacuum energy and central charges of the corresponding gauge theory in hyperbolic spacetime [13,[39][40][41][42]. Furthermore, it would be very interesting to analyze the large-N limit of N = 4 supersymmetric Yang-Mills theory with gauge group SU (N ) on AdS 2 × T 2 as such limit should unveil BPS configurations similar to the black-strings in AdS 5 detected in the topologically twisted index on the compact manifold S 2 × T 2 [43][44][45][46].In particular, we predict that the constraint which we derive in the main text, should also appear in the dual gravity theory with γ R , ω being related to the electrostatic potential and the angular velocity of the supergravity solution, respectively.
Besides, it would be very intriguing to investigate non-perturbative dualities for gauge theories on AdS 2 × T 2 ; especially in relation of boundary degrees of freedom, which are known to be affected by such transformations in a non-trivial way [47].Eventually, possible generalizations of this paper comprehend the addiction of BPS defects, vortices or orbifold structures [48][49][50][51] on AdS 2 × T 2 , as all these objects yield further refinements of the index [52].
Outline.In Section 2 we set up the background geometry by introducing the chosen metric and frame on AdS 2 × T 2 .We then find its rigid supersymmetric completion by solving the conformal Killing spinor equation on AdS 2 × T 2 endowed with a background field for the U (1) R R-symmetry that is proportional to the spin connection and encodes the topological twist.Thus, we show that such conformal Killing spinors also solve the Killing spinor equation with a suitable choice of background fields descending from new minimal supergravity.Hence, we study periodicities and global smoothness of Killing spinors on topologically twisted AdS 2 × T 2 .In Section 3 we write down the supersymmetric multiplets involved in our analysis, their supersymmetry variations in component fields and Lagrangians.Then, we rewrite the supersymmetry transformations in cohomological form by introducing a new set of fields that makes manifest the fundamental degrees of freedom contributing to the partition function.Moreover, we show how supersymmetric boundary conditions emerge from either supersymmetry-exact deformations of the Lagrangian or the equations of motion.In Section 4 we calculate the path integral of topologically twisted gauge theories on AdS 2 × T 2 by means of supersymmetric localization.We first solve the BPS equations for vector and chiral multiplets, thus obtaining the supersymmetric locus over which dynamical fields fluctuate.
Then, we calculate the contribution to the partition function of such fluctuations, giving rise to a non-trivial 1-loop determinant expressed as an infinite product that can be regularized in terms of special functions.We explicitly display the various possibilities corresponding to different choices of either boundary or normalizability conditions imposed on supersymmetric fluctuations.

Supersymmetric Background
We choose the following line element on AdS 2 × T 2 : where the four-dimensional metric g µν can be read off from the usual relation ds 2 = g µν dx µ dx ν , with x µ = (θ, ϕ, x, y).In particular, θ ∈ [0, +∞) and ϕ ∈ [0, 2π) are coordinates on AdS 2 , while x, y ∈ [0, 2π) are coordinates on T 2 .The parameter L has dimension of length and encodes the radius of AdS 2 appearing e.g. in the Ricci scalar R AdS 2 = −2/L 2 .The dimensionless parameters Ω 1 , Ω 2 ∈ R introduce in the partition function of the theory a fugacity for the angular momentum on AdS 2 , as in [14,43]; whereas τ 1 , τ 2 ∈ R respectively are real and imaginary part of the modular parameter τ = τ 1 + iτ 2 of the torus T 2 .Finally, the dimensionless parameter β ∈ R parametrizes the scale of T 2 with respect to the radius of AdS 2 .We shall also consider a boundary at θ = θ 0 > 0 to explore the interplay between bounday conditions and boundary degrees of freedom.
We adopt the orthonormal frame e 1 = Ldθ , e 2 = L sinh θ(dϕ + Ω 3 dx + Ω 4 dy) , satisfying e.g.g µν = δ ab e a µ e b ν and δ ab = g µν e a µ e b ν .In the frame (2.2) the non-trivial components of the spin connection read (2.3) On AdS 2 × T 2 the conformal Killing-spinor equations, is solved by where k 0 ∈ C is a normalization constant and α 2,3,4 ∈ R parametrize non-trivial phases of ζ, ζ along the three circles inside AdS 2 × T 2 , while A C is the background field Moreover, the spinors (2.5) fulfil the Killing-spinor equations with background fields where κ is an arbitrary constant and the 1-forms A C and A R are smooth on AdS 2 if α ϕ = 1.Thus, the ζ and ζ reported in (2.5) are Killing spinors of R-charge ±1, respectively.As the field strength ) of the R-symmetry field is non-trivial and satisfies F (R) = dA = (1/2)dω 12 , the Killing spinors ζ and ζ describe a supersymmetric AdS 2 × T 2 background with a topological twist on AdS 2 , analogous to those investigated in the case of compact manifolds e.g. in four [43] and three [14] dimensions.
On a compact two-dimensional manifold M 2 the direct link between F (R) and dω 12 characterizing the topological twist implies that the R-symmetry flux equals the Euler characteristic of M 2 , up to a sign.On a two-dimensional manifold with boundary B 2 the R-symmetry flux f R is proportional to the line integral of A along the one-dimensional boundary ∂B 2 .Indeed, applying Stokes' theorem to the smooth R-symmetry field where where f is the field strength of the gauge field a with components The constant q G is the gauge charge appearing in the covariant derivative where • RG represents the action upon a field Φ in the representation R G of the gauge group G.The bosonic fields a µ and D of the vector multiplet satisfy the reality conditions whereas there is no need to impose reality conditions upon the vector-multiplet fermionic fields λ, λ.
The supersymmetry transformations (3.1) can be rewritten in cohomological form as follows: where we introduced the Grassmann-even 0-forms Φ G , ∆ as well as the Grassmann-odd 0-form Ψ and 1-form Λ µ given by with being the Killing-spinor bilinears defined in [43].The norms |ζ| 2 and | ζ| 2 descend from the complex conjugates of the Killing spinors, which are providing in turn the reality conditions on Killing-spinor bilinears: In particular, the Killing-spinor equations (2.7) imply that K µ , which in our setup reads is a Killing vector: In (3.4) the supersymmetry variation δ manifests itself as an equivariant differential fulfilling where the Lie derivative L K generates a spacetime isometry of the manifold while represent the action of gauge transformations upon fields.In the case of weakly gauged theories with background vector multiplets, (3.12) is interpreted as the action of the flavour group G ≡ G F .
The vector-multiplet Lagrangian, is δ-exact up to boundary terms, with deformation term In absence of boundaries the total derivative in (3.14) is irrelevant and the corresponding action S CM is δ-exact and then manifestly supersymmetric.In presence of boundaries the total-derivative terms in (3.14) drop out if the following dual sets of supersymmetric boundary conditions are imposed: Robin : together with the vanishing of the corresponding supersymmetry variations.Especially, Dirichlet conditions only affect the components of the gauge field a µ that are parallel to the boundary, whereas Robin conditions mix with each other components that are either parallel or orthogonal to the boundary.After including Faddeev-Popov ghosts c, c and their supersymmetric completion, as e.g. in [4,53], the BRST-improved supersymmetry variation (δ (3.17)

Chiral multiplet
N = 1 chiral multiplets in a representation R G of the gauge group G contain a 0-form φ, a lefthanded spinor ψ and a 0-form F , where the latter is a non-dynamical field that, in analogy with D, allows the closure of the supersymmetry algebra on the chiral multiplet without using the equations of motion.The fields (φ, ψ, F ), whose R-charges are (r, r − 1, r − 2), are related to each other by the following supersymmetry variations: The bosonic fields φ and F of the chiral multiplet fulfil the reality conditions where φ and F , together with the right-handed spinor ψ, form an anti-chiral multiplet2 in the conjugate representation R G of the gauge group G. Their supersymmetry transformation reads If we define cohomological fields correspoding to the Grassmann-odd 0-forms B, C and the Grassmann- where is the covariant Lie derivative along the vector v, the relations (3.18) can be written in cohomological form: with i = 1, 2, where X 1 = φ, X 2 = B, X ′ 1 = C and X ′ 2 = Ξ.The supersymmetry variation of the auxiliary field F in cohomological form is The structure of (3.23) implies that the supersymmetry variation δ behaves as an equivariant differential also on chiral-multiplet fields, where Φ R is the R-symmetry counterpart of the 0-form Φ G , More generally, δ 2 acts upon a field X of R-charge q R , flavour charge q F and gauge charge q G as where where the bulk terms vanish on the solution of the equations of motion.The boundary terms descending from the equations of motion cancel out if the following dual sets of supersymmetric boundary conditions are imposed:

Supersymmetric Localization
We now compute the partition function of gauge theories coupled to matter via supersymmetric localization [4].We focus on Abelian gauge theories as the generalization to the non-Abelian case is straightforward.We start by deriving the supersymmetric locus solving the BPS equations; then, we will compute the 1-loop determinant of the fluctuations over the BPS locus.

BPS Locus
The vector-multiplet BPS equations are which in cohomological form read We employ the following ansatz: where we set to zero the pure-gauge component a x (θ), while a ϕ , a x , a y are complex functions and the flat connections b ϕ , b x , b y are complex constants, a priori.The gauge field above is smooth on AdS 2 if b ϕ = −a ϕ (0).By plugging the ansatz (4.3) into the BPS equations (4.2) we obtain the complex BPS locus for the vector multiplet: implying that the BPS value of the gauge fugacity Φ G is manifestly constant, The BPS equations for the chiral multiplet read which in cohomological form are with φ = φ(θ, ϕ, x, y) and φ = φ(θ, ϕ, x, y) being periodic in ϕ, x, y.For generic values of Φ R , Φ G the trivial locus is the only solution to (4.9, regardless of the presence of a boundary at θ = θ 0 .In particular, the value of fields reported in (4.10) trivializes the classical contribution to the partition function given e.g. by superpotential terms.

One-Loop Determinant
The 1-loop determinant of supersymmetric fluctuations over the BPS locus for a chiral multiplet is implying that φ nϕ,nx,ny is non singular at θ = 0 if (−n ϕ ) = ℓ ϕ ∈ N. The modes φ nϕ,nx,ny are eigenfunctions of the operator δ 2 with eigenvalue which contributes to the denominator of Z CM 1-L .In both λ B and λ φ the R-charges q φ R = r, q B R = (r − 2) as well as the gauge charge q G respectively multiply the same quantities γ R and γ G , where Especially, the first line in (4.18) can be interpreted as a constraint on the chemical potentials γ R , ω, τ , as in the case of gauge theories on S 3 × S 1 dual to AdS 5 black holes [54].
In presence of boundaries we have two possible 1-loop determinants: on the one hand, if we impose Dirichlet conditions, the modes φ nϕ,nx,ny have to satisfy a first-order homogeneous differential equation with boundary condition φ nϕ,nx,ny | ∂ = 0, implying φ nϕ,nx,ny = 0. Therefore, Dirichlet conditions kill the modes φ nϕ,nx,ny contributing to the denominator of Z CM 1-L , leaving the modes B mϕ,mx,my unaffected.The result is which can be regularized by means of (A.4), yielding /Γ e e 2πi(2γR−qiγi) ; q, p , with q i γ i = (rγ R + q G γ G ). On the other hand, if we impose Robin conditions, the modes B mϕ,mx,my have to satisfy a first-order homogeneous differential equation with boundary condition B mϕ,mx,my | ∂ = 0, which sets B mϕ,mx,my = 0 everywhere.As a consequence, Robin conditions trivialize the modes B mϕ,mx,my contributing to the numerator of the chiral-multiplet 1-loop determinant and leave the modes φ nϕ,nx,ny untouched because L Y φ nϕ,nx,ny = 0 on the whole four-manifold by definition.Thus, whose regularized form provided by (A.4) reads R Γ e e 2πiqiγi ; q, p , As observed e.g. in [29,38], dual 1-loop determinants are mapped to each other by multiplication of 1-loop determinants corresponding ot boundary multiplets: [1 + 6q i γ i (q j γ j − 1 − ω) + ω(3 Eventually, in absence of boundaries, we require that both φ nϕ,nx,ny and B mϕ,mx,my are square integrable on AdS 2 × T 2 according to the integral measure In summary, and Z CM 1-L | r=1 = 1 as there are no normalizable modes for r = 1.Similarly to what happens in three dimensions [53], the 1-loop determinant for a non-Abelian vector multiplet enjoying N = 1 supersymmetry is where Ψ, c and c contribute as modes B mϕ,mx,my , in the adjoint representation of the gauge group G, with R-charge r = 2, while ι Y a, ι K a and ι Y a contribute as modes φ nϕ,nx,ny in the adjoint of G with R-charges (2, 0, −2), respectively.Nonetheless, if Dirichlet conditions upon vector multiplet modes are imposed, only ι Y a, ι Y a survive and after simplifications we find where the product over roots of the adjoint representation of G is understood.On the other hand, Robin conditions kill ι Y a, ι Y a, leaving the other modes invariant; therefore, after a few other simplifications, Instead, in absence of boundaries, all modes appearing in (4.32) do contribute, a priori.In fact, various contributions drop out, giving at the end of the day where the last equality holds if we restrict to normalizable modes only, as in (4.31).

) with B 2
being AdS 2 with a boundary at θ = θ 0 and S 1 0 = ∂B 2 being the circle in AdS 2 at θ = θ 0 .As observed in [20], on AdS 2 fluxes are not quantized, as opposed to what happens e.g. on the two-sphere, where the single-valuedness of transition functions between different patches requires all fluxes to take integer values.Imposing either periodicity or anti-periodicity of the Killing spinors ζ, ζ along the torus circles parametrized by x and y yields α x , α y ∈ Z .(2.10) Furthermore, smoothness of the R-symmetry field A requires α ϕ = 1, implying the anti-periodicity of the Killing spinors along the shrinking circle in AdS 2 parametrized by ϕ.The Killing spinors reported in (2.5) are manifestly smooth in every point of the four-manifold apart from the origin as ζ, ζ are written in the frame (2.2), which is singular at θ = 0 due to ϕ being undefined at the origin.Smoothness at θ = 0 can be examined by first rotating (2.2) into a frame that is non-singular at the origin via a local Lorentz transformation ℓ a b , induces the following rotation upon ζ, ζ:

. 13 )
encode the action of local Lorentz transformations upon left-and right-handed spinors, respectively.The spinors ζ ′ , ζ ′ are independent of the coordinate ϕ if and only if α ϕ = 1, which is then the value making the Killing spinors smooth on the whole four-manifold.3Supersymmetry and Cohomology3.1 Vector MultipletA vector multiplet enjoying N = 1 supersymmetry consists of a 1-form a µ encoding the gauge field, two complex spinors λ, λ of opposite chirality parametrizing the gauginos and a 0-form D corresponding to an auxiliary field ensuring off-shell closure of the supersymmetry algebra.The fields a µ , λ, λ, D have R-charges (0, +1, −1, 0), transform in the adjoint representation of the gauge group G.The vector-multiplet supersymmetry variations with respect to ζ, ζ read