One-loop quantization of Euclidean D3-branes in holographic backgrounds

In this note we analyze the semi-classical quantization of D3 branes in three different holographic backgrounds in type IIB string theory. The first background is Euclidean AdS$_5$ with $S^1\times S^3$ boundary accompanied with a twist to preserve supersymmetry. We work out the spectrum of fluctuations around the classical D3-brane configuration, compute its one-loop partition function, and match to the non-perturbative correction to the superconformal index of ${\cal N}=4$ SYM. We then study Euclidean D3-branes in the Pilch-Warner geometry dual to the IR Leigh-Strassler fixed point of ${\cal N}=1^*$ with the aim to find non-perturbative corrections to its index. Finally we study Euclidean D3-branes in the non-geometric ${\cal N}=2$ J-fold background which is dual to the gauging of the 3D Gaiotto-Witten SCFT.


Introduction
Since its discovery, the AdS/CFT correspondence has provided a wealth of knowledge in the context of both quantum field theory, and quantum gravity.Due to the advent of exact tools in quantum field theory, such as supersymmetric localization and the analytic bootstrap, the correspondence has been verified in a plethora of non-trivial ways.Additionally, it has proven to be a fruitful pathway to study string and M-theory away from their respective two-derivative low-energy supergravity regimes.The review [1] has summarized recent progress and relevant references in this context.
A particular venture that has sparked our interest is the semi-classical quantization of strings and branes in curved holographic backgrounds of type II-and M-theory.A full quantization of strings and branes in curved backgrounds has proven to be an arduous task.However, a combination of modern QFT tools, and the AdS/CFT correspondence provides a systematic approach to achieve this goal in a semi-classical expansion, at the very least when considering supersymmetric systems.Indeed, recent examples where this approach has proven to be rewarding include the quantization of closed strings on disks with a hyperbolic metric, which are dual to expectation values of supersymmetric Wilson loops in the dual QFT, see for example [2,3].In a recent application we have also studied the quantization of strings on two-cycles of genus-0 with the round metric [4].In particular, we have shown in two cases that such a semi-classical quantization provides a precise holographic match with non-perturbative contributions to the dual QFT partition functions.One of the cases that was studied concerns strings in the type IIA holographic background dual to five-dimensional SYM.This was obtained as a twisted dimensional reduction of AdS 7 × S 4 [5], which is dual to the six-dimensional (2, 0) theory.In this background the semi-classical quantization of the multi-wound genus-0 strings reproduced the full tower of non-perturbative contributions to the S 5 partition function of the SYM theory in the large N limit at strong coupling.This tower has a direct connection to the giant graviton expansion in the (2, 0) theory, as it is simply a Cardy-like limit of these giant gravitons in the AdS 7 × S 4 background of M-theory. 1 Giant graviton expansions have received interest recently as they point to the fact that the full quantum gravitational path integral can in certain cases be computed by simply considering a single saddle (the thermal AdS background) and an infinite series of states consisting of branes on top of this saddle [7][8][9][10][11][12][13][14]. 2

Indices and twists in holography
In this paper we study the semi-classical quantization of D3-branes in three distinct type IIB string theory backgrounds.These backgrounds are respectively dual to N = 4 SYM with gauge group SU(N ) on S 3 × S 1 , the Leigh-Strassler CFT on S 3 × S 1 , and finally the N = 4 SYM theory on S 3 × S 1  J , where the circle contains an S-duality wall.These QFTs can be localized in order to compute their supersymmetric partition functions on S 3 × S 1 as a function of N .In the large N limit it is expected that these partition functions can be factorized from the string theory point of view into a supergravity contribution, and towers of brane excitations wrapping Euclidean cycles in the internal geometry, reminiscent of the giant graviton expansions discussed above.For the N = 4 SYM and the Leigh-Strassler theory the partition functions are directly related to the Witten (or superconformal) indices through a multiplicative factor of the Casimir energy: Z S 1 ×S 3 = e −βEc I . (1.1) For the S-fold theory the situation is more involved, and we will discuss this setup separately below.For the initial two cases, however, the object that one would like to compute is thus their Witten indices, which in an N = 1 language schematically can be expressed as where δ is the anti-commutator of two supercharges, which ensures that only states contribute for which δ = 0, J and J form the Cartan of the Lorentz group, r is the superconformal U(1) R charge, and finally and (µ i , p i ) are respectively the chemical potentials and charges of additional possible global symmetries in the theory.For the specific example of the N = 4 SYM theory, with gauge group SU(N ), this index reduces to where R i are the three Cartan generators of the R-symmetry group SO (6).We will be particularly interested in two limits of this index.In the first limit, known as the 1/2-BPS limit, we take all fugacities but q to be trivial.Furthermore choosing x = √ q this index reduces to the following simple expression This index can be written as follows [9] 3 where ∞ arises in the large N limit as the contributions of fluctuating (super)gravitons in AdS 5 [18].The expansion in q N that multiplies the large N limit has been interpreted as the contribution of giant gravitons to the index.The term which scales as q nN can be thought of as the contribution of n D3-branes in the saddle point expansion and its prefactor arises from the quantization of the D3-branes around their classical solution.Recovering this series directly from quantized D3-branes has been the focus of the recent paper [14].We will briefly comment on their computation below.
The limit that we will be interested in, commonly referred to as the Schur limit [19,20], does not take the fugacity y to be trivial, but instead scales it as y = √ q.In this case, simply taking x = 1, the index reduces to I Schur

N
= Tr e 2πi(J+ J ) q H+J+ J . (1.6) In this limit the index was computed analytically as a function of N in [21] and takes the form where I Schur ∞ = PE q(2+q) (1−q) 2 constitutes the contributions coming from the large N limit.We are interested in the leading non-perturbative contribution (in N ) in this expression which is (1.8) Note that, for both the 1/2-BPS and the Schur index, the fugacity of the Hamiltonian equals q, which scales with the periodicity of the circle q = e −β .The holographic dual of the N = 4 SYM theory is the celebrated AdS 5 × S 5 background of type IIB string theory.To study the aforementioned indices from the string theory point of view one has to impose S 3 × S 1 boundary conditions on Euclidean AdS.Importantly, to ensure that the background remains supersymmetric when Euclidean time is periodically identified with period β, one must twist S 1 with an additional circle in the geometry.We will refer to this twisted Euclidean time S 1 β as the thermal circle, and the corresponding AdS geometry is called thermal.The twist is necessary because the Killing spinors on AdS 5 ×S 5 generically depend on all coordinates including Euclidean time.If Euclidean time is periodically identified without further modifications, the Killing spinors are not globally consistent and supersymmetry is broken.A solution to this problem is to perform a local coordinate transformation φ → φ + iτ , where τ is Euclidean time and φ parametrizes a U(1) direction in AdS 5 × S 5 .This local coordinate transformation cannot be undone without breaking supersymmetry because of the global issues with the Killing spinors mentioned above.The specific choice of the U(1) that is twisted is in one-to-one correspondence with the fugacities turned on in the N = 1 index (1.2).For the 1/2-BPS index the twist is done with the U(1) generated by the Cartan R 1 + R 2 + R 3 , which in the geometry corresponds to the great circle of S 5 (we call this the 1/2-BPS twist).For the Schur index, instead, the twist involves the Cartan J + J, which in the geometry corresponds to the great circle of S 3 inside AdS 5 (which we call the Schur twist).A completely general twist may be implemented by forming a linear combination of all available Cartan generators and would be relevant for the study of the general N = 1 superconformal index.In this paper 3 The Pochhammer symbol is defined as (q we will mostly focus on the Schur twist but also briefly mention the 1/2-BPS twist.A twist analogous to the latter was previously studied for the Euclidean supergravity backgrounds dual to the six-dimensional (2,0) theory in [2,5] (see also [17]).
The non-perturbative corrections to the superconformal index in the 1/2-BPS and Schur limit can be reproduced in the bulk from D3-branes wrapping Euclidean compact cycles in the geometry.More specifically, they wrap the thermal circle in AdS and an S 3 ⊂ S 5 , making them explicitly sensitive to the supersymmetry preserving twists discussed above.We will explicitly show how to semi-classically quantize these D3-branes in order to reproduce the nonperturbative contributions to the Witten index from string theory.Our approach can be viewed complementary to the approach of Imamura et.al. who have localized the QFT living on the D3-branes to find the same result (see [22] for the specific case of the Schur index), and functions as a proof of concept for the procedure that we subsequently also apply in the more intricate geometries that are holographically dual to the Leigh-Strassler, and the S-fold theories, both with a reduced amount of global-and supersymmetries.
The Leigh-Strassler theory is the IR fixed point of an N = 1 RG flow originating in N = 4 SYM [23].Apart from N = 1 superconformal symmetry it also preserves an SU(2) flavor symmetry.Its superconformal index can be written down in an equivalent form to (1.2), where one merely turns on a fugacity for the R-symmetry Cartan and the flavor symmetry.Since the index is a protected quantity under continuous RG-flows one can essentially compute it in the weakly coupled UV regime a la Römelsberger [24] (see also [25]).This procedure was explicitly worked out in the case of the Leigh-Strassler CFT in [26] to find that the superconformal index of the Leigh-Strassler theory equals where R F denotes the cartan of the SU(2) flavor symmetry.In the large N limit this index was matched to the full supergravity spectrum, including all KK-modes, in the dual string theory geometry [26].We are interested in a limit of this index similar reminiscent of the N = 4 Schur index, namely where we scale y = √ q and trivialise all other fugacities.To the best of our knowledge an expression similar to that of the Schur index in N = 4 SYM in (1.7) has not been found for the Leigh-Strassler theory, which is something we hope to address in future work.
In this paper we focus on the gravitational description of this theory, and in particular the non-perturbative D3-brane excitations in the type IIB string theory solution.Explicitly, the geometry is topologically of the form AdS 5 × S 5 , where the AdS metric contains a twist along the thermal circle, identical to the Schur twist in the N = 4 SYM theory, and the fivesphere is squashed.Inspired by the N = 4 SYM index we conjecture that the non-perturbative contributions to the superconformal index of the Leigh-Strassler theory, at large N , are described by Euclidean D3-branes wrapping compact four-cycles in the geometry.We find the stable brane configurations which wrap a squashed three-sphere inside the five-sphere, and the thermal circle in AdS 5 .We show that the D3-brane on-shell action is given by S cl = 3βN/2, providing a prediction for the exponential suppression of the leading non-perturbative correction to the superconformal index of the dual QFT to scale as e −3βN/2 . (1.10) It will be interesting to see if a large N analysis in the dual QFT can reproduce this behavior in this setup.We also study the QFT living on the the squashed D3-branes and determine its spectrum.This spectrum is a first step towards a full 1-loop analysis of the Euclidean D3branes, which will in turn provide the prefactor to the exponentially suppressed contribution to the superconformal index.A computation which we plan to report on in the near future.It would also be interesting to compare to the giant graviton expansion in the Klebanov-Witten theory recently discussed in [27].
The final example in which we study the quantization of D3-branes involves a two-parameter family of non-geometric backgrounds in type IIB string theory first described in [28].Their background metrics take the following form where the five-sphere is squashed, and the S 1 contains an S-duality wall determined by an SL(2, Z) element that we denote with J.This family of backgrounds is dual to a conformal manifold of three-dimensional N = 2 strongly coupled CFTs with a U(1) flavor symmetry arising from compactifying N = 4 SYM on a circle with said S-duality wall.This conformal manifold contains two symmetry enhanced points, one where the flavor symmetry is enhanced to SU(2), and one point where the supersymmetry is enhanced to N = 4.In the latter point the S 3 partition function of the CFT was shown to equal [29][30][31]] and (q) N denotes the Pochhammer symbol.Since the superrsymmetric S 3 partition function is independent of exactly marginal couplings, this result holds all over the conformal manifold.Which was explicitly shown to hold in the large N limit through a holographic analysis in [32][33][34].Furthermore, it was recently shown in [28] that the full partition function of these theories can be split into a perturbative supergravity piece, and a tower of non-perturbative D-brane states on top of this leading saddle where we have explicitly matched the classical on-shell action to the exponential behavior of the non-perturbative contributions in this partition function.This is much alike to the superconformal index of N = 4 SYM and thus one can pose the question if it is possible to define and explicitly compute a supersymmetric index where the thermal circle contains an S-duality wall, and if on the string theory side this index can be reproduced by a giant graviton like expansion.In this paper we take a first step towards a semi-classical quantization of D3-branes in this non-geometric backgrounds, similar as we have discussed for the thermal AdS backgrounds mentioned above, by computing the spectrum of fluctuations consisting of scalars, fermions, and a vector on the squashed D3-brane.To the best of our knowledge it is not known how to apply supersymmetric localization on such non-geometric curved backgrounds with S-duality walls, 4and thus it seems that the quantization of D3-branes is a fruitful approach to study such nonperturbative contributions in string theory.We plan to report on the full one-loop quantization of the D3-branes in this non-geometric background in the near future.This paper is structured as follows.In section 2 we study the semi-classical quantization of D3-branes in thermal AdS 5 × S 5 and reproduce the leading non-perturbative correction to the Schur index of N = 4 SYM from string theory at 1-loop.We also comment briefely on the 1/2-BPS index of N = 4 SYM from this persepective.In Section 3 we study Euclidean D3-branes embedded in the thermal AdS backgrounds dual to the Leigh-Strassler CFT.Finally, in Section 4 we study the semi-classical quantization of D3-branes in S-fold backgrounds of type IIB string theory.

Note added:
In the preparation of this manuscript [35] appeared on the ArXiv which has significant overlap with Section 2 of our paper.

D3-branes in thermal AdS
The background we are interested in is thermal AdS 5 ×S5 with supersymmetric S 1 β ×S 3 boundary conditions.To preserve supersymmetry a globally non-trivial twist involving the Euclidean time must be imposed, as mentioned in the previous section.This is because the Killing spinors of AdS 5 × S 5 generally depends on all coordinates and so compactifying e.g.Euclidean time will eliminate the Killing spinors.One way to identify the correct choice of twisting needed for a particular observable is through a comparison to the dual field theory.In this case we are interested in the index given in (1.7) which indicates that the twist needed is between the time-circle and the great circle of S 3 ⊂ AdS 5 .This results in a ten-dimensional metric 5 (2. 2) The only additional non-trivial background field is the four-form gauge potential The length scale L is related to the rank of the gauge group in the dual theory via We are interested in studying non-perturbative corrections to the S 1 β × S 3 partition function arising from D3-brane instantons.The embedding of the D3-brane in question is rather simple, it wraps the Euclidean time and the S 3 ⊂ S 5 .To describe the dynamics on the world-volume of the brane we will work in static gauge throughout, such that its metric equals6 The embedding is fully specialized by minimizing the classical D3-brane action which in this case is equivalent to imposing that the extrinsic curvature is traceless Usually the right-hand-side of this equation receives a contribution from the coupling of the brane to background F 5 , but in the current case this contribution vanishes.This traceless requirement imposes the brane to localize at (ρ, θ) = (0, π/2), where its action and metric evaluate to The fact that the classical action of the D3-brane equals N β is already a good sign as it means that the D3-partition function scales as Our next task is to determine the prefactor which is obtained by performing a one-loop quantization of the fluctuations around the classical background.To this end we start by computing the spectrum of fluctuations in the next two subsection before computing the one-loop partition function itself.

Bosonic spectrum
Before studying the explicit spectrum of the bosonic fluctuations around the classical configuration (2.7), we discuss some generalities of the bosonic D3-brane action.
We will denote the world-volume coordinates with ζ a , and the transverse fluctuations in tangent basis with Φ i .In this respect a = 1, . . ., 4 is a curved index on the brane world volume and i = 1, . . ., 6 is a flat index on the normal bundle. 7The bosonic action of the D3-brane is given by where F = f − B 2 and f denotes the worldvolume field-strength on the brane.This combination of fields also appears in the WZ coupling of the brane to the RR fields which in explicit terms takes the form For the D-brane configurations in this paper the background values of the field strength and indeed the entire world-volume gauge field F vanishes.We will therefore often drop it in our expressions for simplicity.However, when considering the one-loop fluctuations the gauge field does play an important role.
In static gauge the spectrum of the bosonic sector in the worldvolume theory is determined by the transverse fluctuation of the brane, denoted by Φ i , and the worldvolume field strength around their background values.Instead of explicitly expanding the D-brane action to second order we utilize standard geometric methods to split quantities up into to tangent bundle and normal bundle quantities (see [36,37]).In particular, we find a natural gauge connection on the brane 8A where the first term is the pull-back of the ten-dimensional spin-connection and the second term is a result of the coupling of the D3-brane to the background RR-fields.Here two of the indices on five-form are in the normal bundle while the other three are in the tangent bundle on which the world-volume Levi-Civita tensor acts on.The second order scalar Lagrangian arising from the expansion of the D3 action can then be written as9 where R is the ten-dimensional Riemann curvature and the covariant derivative includes, apart from the world-volume connection, also the gauge connection in (2.11) Coming back now to the specific case of D3-branes in thermal AdS 5 , evaluating the terms in this action and diagonalizing the gauge connection we find we can rewrite the full second order scalar Lagrangian simply as a set of six conformally coupled scalars with R = 6 is the scalar curvature of the D3-brane and the gauge covariant derivative is defined as The fact that the system reduces to such a simple action for the scalars is perhaps not surprising since we are just dealing with D3-branes on a conformally flat background.The non-trivial aspect of this theory is entirely contained in the constant imaginary gauge potential and the couplings of the fields to it.Here we have introduced the charges Q of each of the six modes and we find To analyze this further we want to perform a KK reduction along S 1 β .To this end it is convenient to define the following complex scalar degrees of freedom (2.17) Upon integration by parts, the full scalar Lagrangian can be expanded to find where we have conveniently combined the mass 1 with the 3D spherical Laplacian which is appropriate for conformally coupled scalars The dot denotes the τ -derivative which we deal with by imposing a standard KK-ansatz along S 1 β which reduces to the problem to three towers of complex 3D scalar fields, with masses where n β = 2πn/β, and n ∈ Z denotes the KK-level of the scalar.
Recall that the gauge field vanishes on-shell and so its fluctuations are easily evaluated by expanding in f to second order.After rescaling the gauge field to absorb g s , this results in the standard Maxwell term (2.21)

Fermionic spectrum
We start from the fermionic action in [38][39][40]. 10After setting the worldvolume gauge field F to zero and pulling the L 2 out of the worldvolume metric the action takes the form where Γ a = ∂ a X M Γ M are the gamma matrices pulled back to the world volume, γ ab is the metric on the D3, and Θ is a 32 component spinor in ten dimensions written as a pair of chiral fermions Θ = (θ 1 , θ 2 ), for which Γ 11 θ n = θ n .Finally, we have also defined (2.23) 10 We use the same conventions as in [40] except we work in Einstein frame and we have reversed the sign of B2.
where the pauli-matrices act on the pair of chiral spinors and / H a = / H aµν Γ µν /2.We also note that Γ D3 is a projection operator on the brane which for vanishing worldvolume flux takes the simple expression 11Γ D3 = Γ (4) σ 2 , ( where Γ (4) is the chirality operator on the four-dimensional brane world-volume, i.e. the product of all pull-backed Γ-matrices with flat indices.A common way to fix the kappa symmetry gauge in type IIB is to impose θ 1 = θ 2 ≡ θ.This reduces the fermionic action to (2.25) In the next sections we will choose a different kappa symmetry gauge which is more appropriate there.For the current case of interest, however, the standard gauge is applicable as most of the terms drop out.Note that when the dilaton is constant, the fermions should be rescaled by appropriate powers of g s to absorb the explicit factor of e −Φ/4 in the action.
After inserting the background for twisted AdS 5 × S 5 , the first term in the action gives rise to the Dirac operator on S 1 β × S 3 plus a constant gauge connection along the τ -direction.The last term depends on the RR field strength F 5 , which when slashed takes the form This also gives a coupling to the same constant gauge connection A = idτ .Unsurprisingly, this turns out to be the same constant gauge potential that was felt by the scalar fields.The resulting operator is the kinetic operator of four massless fermions on S 1 β × S 3 which are charged with respect to the background gauge field as controlled by the charges of the fermions which are obtained by taking all possible combinations of s 1,2 = ±1.The signs s 1,2 arise when reducing the ten-dimensional fermions to four dimensions, and can be thought of as the eigenvalues of the original spinors with respect to the ten-dimensional gamma matrices iΓ 45 and iΓ 67 .We now continue by imposing a standard KK-ansatz on the thermal circle and find that the fermionic operator takes the form where once again n β denotes the KK-level on the time circle and the sign ± in the mass is a result of the gamma-matrix along the τ -direction in the original four-dimensional theory which originally multiplies the M n -term.To properly write this as a three-dimensional operator, we should reduce the four-dimensional spinor to three-dimensions using Γ τ to split them up.This means that half of the three-dimensional spinors will have mass n β + iQ and the other half will have negative that mass.In the end this sign does not affect the one-loop partition function of the modes and so effectively the degeneracy of the spinors is simply doubled.We will therefore drop this sign in what follows.Before moving on to the evaluation of the one-loop determinants we list the full spectrum in Table 1.
Table 1: A summary of the spectrum of conformally coupled scalars, fermions, and vector.The fermions get doubled when reducing to three dimensions which is indicated by the doubled degeneracy in a bracket.

One-loop determinants
Quantization of the above fluctuations boils down to evaluating the determinants of the operators derived above.To this end we must first recall the spectrum of the Dirac and Laplace operators on S 3 and multiply all eigenvalues together accounting for all modes.These products usually diverge and in order to regularize each individual contributions to the log of the determinants coming from the scalars, vectors and fermions we apply the regularization scheme used in [41] (see also [17]).For all our fields the quadratic one-loop operator takes the form where K 0 is the Laplacian on S 3 for the appropriate field with the conformal coupling taken into account.Let ω 2 k be the spectrum of K 0 with degeneracy d k .The logarithm of the determinant of K can be expressed as a double sum where we have used that the sum over n runs over positive and negative integers and we have defined The double sums in (2.31) are regularized as [41] with E c the Casimir energy.Exponentiation gives the 1-loop contribution to the D3-brane partition function det where PE represents the plethystic exponential, and Ẑ(β) is the single particle index , where q = e −β . (2.35) Computing the single particle index can often be much simpler than computing the full partition function and is a useful intermediate step.It is also useful because it clearly identifies the zero mode contributions that one has to treat separately.
The spectrum of Laplace operators for conformally coupled scalars, fermions and vector on S 3 is well known and can be summarized by 12scalars : where k = 0, 1, . . .Before evaluating the full 1-loop partition function we note that the spectrum contains 10 scalar zero modes, and 8 fermionic zero modes, at KK-level n = 0 2 × scalars : which we will treat separately after computing the single letter index of the fields.Applying the regularization procedure above, utilizing the mass spectrum summarized in Table 1, we find the following Casimir energies and single letter indices for the scalars, fermions, and the vector and (2.39) Adding the contributions together we find the following simple result (2.40) The constant contribution 1 in the single particle index arises due to the mismatch in scalar and fermion zero-modes listed in (2.37).These, as well as the paired zero-modes should be separated off when computing the partition function and we therefore subtract the constant contribution from the single particle index before taking its plethystic exponential.The remaining single particle index has a simply plethystic exponential given by −q such that the full one-loop partition function of the D3-brane is As mentioned, the zero-modes can be split into eight pairs of fermion and boson zero-modes, and two unpaired bosonic zero-modes.In [4] we argued that the paired zero-modes on a quantized string of genus-0 contributed a factor two in the 1-loop partition function arising from the fact that the string can localize at two points in the target space geometry.This was justified by deforming the background geometry, essentially giving rise to a mass for the zero-modes which localized them.The factor two could therefore be a posteriori justified by carefully counting the number of points where U(1) isometries degenerate.Using similar counting arguments in the current situation is slightly problematic since the location of the D3-brane is fixed to a single position at the classical level, in light of this we postulate the contribution of the paired zero-modes to be one.Finally, one has to integrate over the collective coordinates associated to the two remaining scalar zero-modes, which we can identify with the spherical coordinates ξ 2 and ξ 3 to find that Z zero-modes = T D3 2π 2 = N . (2.42) The final answer for the 1-loop partition function of the D3-branes is thus matching with the QFT result.

Comment on the 1/2-BPS index
We close this section by analyzing the 1/2-BPS index using semi-classical D3-branes.As discussed in the introduction, depending on the index under consideration, the twist changes.For the particular case of 1/2-BPS index the twist involves the equator of S 5 as follows where dΩ 2 3 and d Ω2 3 denotes the metrics on two copies of the round S 3 .Due to this 1/2-BPS twist, which locally is just a simple coordinate transformation but cannot be undone due to global restrictions on the Killing spinor, the four-form now takes a slightly altered form (2.45) It is worth pointing out here that the twist itself breaks supersymmetry by 1/2 as only half of the Killing spinors are independent of τ after performing the twist.The other half are projected out as Euclidean time is periodically identified.
If we now consider probe D3-brane in this background, it is straightforward to see that a similar configuration of the D3-brane as above, i.e. wrapping τ and Ω 3 with θ = π/2 and ρ = 0 again solves the equations of motion. 13Comparing to the giant graviton solutions in Lorentzian signature [43] the twist acts similarly to the velocity of the giant graviton.A crucial difference however is that here we have a unique solution for θ whereas for standard giant gravitons the 'size of the graviton' controlled by θ is an unfixed parameter.
For the classical solution described we find an exact cancellation between the DBI and the WZ term in the D-brane action, and hence the on-shell action vanishes This should already sound some warning bells as it indicates that the semi-classical expansion is problematic.If we ignore this problem for a moment we can carry on and compute the spectrum of fluctuations as we did for the Schur case above.The computation is analogous, and we skip all computational details and summarize our results for the spectrum in table 2.
Using this spectrum, we can compute the one-loop partition function in the same way as we did above.We find that the Casimir energy is the same as for the Schur limit Table 2: A summary of the spectrum of conformally coupled scalars, fermions, and vector on the D3-brane in the 1/2-BPS twisted background (2.44).
the single particle partition function is Ẑ1 = 1.This simple result indicates that all non-zero modes exactly cancel leaving only the zero-mode contribution of two scalar fields.This would then indicate that the one-loop contribution of this brane scales as which does not match the leading non-perturbative term in (1.5) which is −q N +1 /(1 − q).Indeed since the on-shell action of the D3 vanishes, it is not clear whether this result is entirely trustworthy.The mismatch with the QFT is perhaps due to this issue.
Recently quantum D3-branes were analyzed in order to reproduce the 1/2-BPS index of N = 4 SYM in [14].There a few differences when compared to our approach.Instead of doing the one-loop quantization of D3-branes as we do here, they employ supersymmetric localization to study the full D3-brane index.To this end they supersymmetrize the scalar zero-mode sector on the D3-branes using the expected supercharges preserved by the D3-brane.It is not clear to us whether this supersymmetrization gives a result that is consistent with the expansion of the fermionic D3-brane action.Upon a naive comparison of their supersymmetric Lagrangian with our result in table 2, we were unable to match precisely the two theories, but since they only retain a subset of modes the comparison may be more involved than naively expected.Determining exactly what lies behind this mismatch is beyond the scope of this work.It is however encouraging to see that in [14], they are able to find a match with the QFT prediction.

D3-branes in the thermal Leigh-Strassler background
The second type IIB string theory setup in which we study Euclidean D3-branes is dual to the Leigh-Strassler CFT [23], the IR fixed point of an N = 1 RG flow originating from the N = 4 theory in the UV by turning on the mass for one chiral multiplet.The IR CFT preserves a U(1) R-and SU(2) flavor global symmetry.The bulk geometry is similar to the one discussed above, with, however, a squashed five-sphere and a reduced amount of preserved supersymmetry, and was first constructed in [44,45].Explicitly, the metric given by where ds 2 AdS 5 is identical to the AdS 5 metric in (2.1), and thus we have imposed the same Schur twist as before to ensure that the background is supersymmetric. 14Furthermore, we defined w = 1 + sin 2 θ, and the metric on the five-sphere is given by (3.2) 14 It will be interesting in the future to study also the 1/2-BPS twist background for the Leigh-Strassler CFT.
This metric reproduces the global symmetries in the dual field theory through isometries on the internal five-sphere.In particular, the SU(2) × U(1) symmetry is realized through a squashed S 3 ⊂ S 5 spanned by the angles ω i .For future reference we also provide the chosen frame fields The axion vanishes and dilaton is constant in this background.To write down the RR and NSNS two-forms in a compact fashion it is useful to define the following complex two-form and finally, the four-form equals The AdS length here is related to the rank of the gauge group and the string length through As mentioned in the introduction, it is expected that when computing the superconformal index of the Leigh-Strassler theory in the large N limit that there are non-perturbative contributions which in the string theory description come from D3-branes wrapping the Euclidean time circle, and a three-cycle in the internal geometry, very much similar to the index of N = 4 SYM.The obvious choice for this three-cycle is the squashed three-sphere spanned by ω i .Extremizing the D3-brane action in this particular embedding shows that it localizes at (ρ, θ) = (0, 0), where the worldvolume metric reduces to As in the previous case of standard AdS 5 × S 5 , the D3-brane has the topology of S 1 β × S 3 .Now however, the S 3 is squashed while preserving an SU(2) × U(1) isometry.The on-shell action of the brane is computed to In the dual field theory this result predicts the leading non-perturbative correction to the superconformal index to scale as Z 1-loop e −3N β/2 , (3.9) where Z 1-loop is to be determined through a one-loop partition function of the fields living on the squashed D3-brane.We will take initial steps in determining this one-loop partition function by first computing the spectrum.We apply the same procedure and formulae as discussed in the previous section.
In particular the scalar fluctuations are determined using eqs.(2.11) and (2.9).The Lagrangian is just a sum of Gaussian fields that are subject to the operator We now find two gauge field, the first of them is a constant which is due to the Schur twist along AdS 5 while the second one is due to the non-trivial fibration in S 5 For each of the six scalar fields we must therefore specify their four-dimensional mass squared as well as the two charges Q and Q.These are listed in table 3 we find once more it to be massless with the standard Maxwell kinetic term as in (2.21).
To determine the fermionic spectrum it turns out to be convenient to choose a different κ-symmetry gauge than discussed in previous section.This is because the three-form fields H and F 3 contribute non-trivially to various terms in the fermionic action.However by choosing an appropriate kappa-symmetry gauge, all of these terms vanish and we are only left with the contribution from F 5 and the spin connection.This leaves us with massless fermions which like the scalar fields are charged with respect to the two gauge fields A and Ã where D is the same differential operator as appears for the scalar fields in (3.10).All fermions are charged with respect to the same gauge fields as the bosons and they are listed in table 3. Using this spectrum, and the eigenvalues of the Laplacian on the SU(2) × U(1) squashed three-sphere (see for example [46]) one can compute the full 1-loop determinant of the Euclidean D3-brane, providing a full prediction of the leading non-perturbative correction to the Leigh-Strassler Witten index, a computation we hope to report on in the near future.
We would like to bring attention to the fact that the squashing of the three-sphere on which the D3-branes are wrapped preserves SU(2) × U(1) invariance in three-dimensions.Reducing our system to three dimensions employing a KK ansatz as we did above means that we should compute a 3D partition function on the squashed sphere.Such partition functions were studied in [47] for superconformal field theories.In particular, it was shown that SU(2) × U(1) invariant squashing is Q-exact with respect to the localizing supercharge in three dimensions, which means that the full partition function only depends on the squashing parameter through a simple rescaling of length scales.It would be interesting to see if the partition function of the squashed D3-brane discussed in this section and the next section depend non-trivially on the squashing parameter or they show the 'Q-exactness' reported in [47].This would point to an underlying 3D supersymmetry of the D3-brane model which is not manifest in our formulation.

D3-branes in S-fold backgrounds
The final example we will study is a non-geometric background of type IIB string theory.It is holographically dual to SU(N ) N = 4 SYM compactified on a circle, with an S-duality twist.By compactifying the four-dimensional theory on the circle a 3D QFT is obtained that, in the IR, flows to an N = 4 CFT (which we refer to as the N = 4 S-fold), which can be understood as a gauging of the well-known Gaiotto-Witten theory in three dimensions [48].The three-sphere partition function of this theory was computed analytically as a function of N in [29,31], and takes a form reminiscent of the Witten index of N = 4.Our goal in this section is to further study this S 1 J × S 3 supersymmetric partition function of N = 4 SYM from a string theory perspective. 15he string theory background dual to the N = 4 S-fold was first constructed in [49] and analyzed further in [29].Subsequently, in [32,33,50] it was shown that the 3D N = 4 SCFT lives on a N = 2 conformal manifold where at a generic point a U(1) R × U(1) F global symmetry is preserved.Apart from the supersymmetry enhanced N = 4 point, there is another symmetry enhanced point on the conformal manifold, where the flavor symmetry becomes SU(2) F .In this section we will focus on this SU(2) F × U(1) R invariant point to study non-perturbative corrections to the three-sphere partition function.The field theory global symmetries are realized as isometries on the internal geometry in the bulk.In particular, the SU(2) × U(1) symmetry group is realized as a squashed S 3 ⊂ S 5 , very much alike to the squashed three-sphere in the Leigh-Strassler background of the previous section.Exactly marginal operators are exact with respect to the supercharged used to localize three-dimensional theories on a sphere.Consequently the partition function is constant along the conformal manifold, and thus the answer in (1.13) also holds in the SU(2) enhanced point we study in this section.In [28] we showed that the nonperturbative corrections to the S 3 partition function scale as the on-shell action of D3-branes wrapping the compactified S-fold circle, and a three cycle inside S 5 , much alike to the giant graviton expansion in the four-dimensional N = 4 SYM theory.In the following we will further analyze these D3-branes.
We start with spelling out the full ten-dimensional background.The metric in Einstein frame takes the following form [34,50]  where, just as in the section before, we have defined w = 1+sin 2 θ and the squashed three-sphere metric equals is the metric on squashed S 3 .This metric is round of unit radius when w = 2.We have also defined the flat frame fields to simplify the expression for the form fields: Our coordinates take the following range Finally the AdS 4 length scale is set by the rank of the gauge group The angle ϕ is assumed to be periodic, with period β.It appears that the full background above does not allow for such periodic identifications since some of the fields have explicit dependence on ϕ.So in order to make the periodic identification we must simultaneously perform an SL(2, R) rotation of the fields in order to "glue" the circle back onto itself.The SL(2, R) transformation that is required takes the form [50] J = e −β 0 0 e β .( In order for this transformation to be a valid transformation in string theory, it must take values in SL(2, Z) which requires a different SL(2, R) gauge for the background above.The end result is that Tr (J) = 2 cosh β = k , (4.7) must be an integer and is mapped to a Chern-Simons level in the dual 3D conformal field theory.
In this paper we will not bother changing to the proper string theory gauge which renders the J-matrix integer valued.This will not affect any of our results and if done, would only serve to complicate intermediate expressions.
Contrary to the pure AdS 5 solutions discussed in the previous two sections above, we do not need to perform any twist in order to preserve supersymmetry.Indeed in [50] it was explicitly verified that the Killing spinor in the above background does not depend on ϕ and so the compactification of that coordinate does not cause any issue for supersymmetry.
A non-trivial embedding of a D3-brane in this geometry was found in [28] which gives rise to non-perturbative corrections to the S-fold sphere partition function.The brane wraps S 1 J as well as the squashed three-sphere d Ω2 3 .It is easy to show that the embedding is stabilized at θ = 0 but the position of the D3-brane in AdS 4 is entirely unfixed.The brane world-volume metric takes the form16  i.e. it is the metric on S 1 J × S 3 where the metric on S 3 is squashed in such a way that an SU(2) × U(1) isometry is preserved, very much akin to the D3-branes wrapping Euclidean cycles in the Leigh-Strassler background.The on-shell action of the configuration is (4.9) Using the formulae in Sections 2.1 and 2.2 we can compute the spectrum of bosonic and fermionic fluctuations.Computing the bosonic spectrum is a simple application of equations (2.11,2.12)and we obtain the scalar operators and we find that some of our fields are charged with respect to a constant gauge potential which in this case takes the form: A = idϕ .(4.11)We find that four of the six scalar fields are massless and uncharged while the remaining two are charged with Q = ±5/2 and have mass M 2 = 9/4.Next consider the vector field.Largely the expansion of the vector field follows the same pattern as for AdS 5 except that now the rescaling of the vector fields to absorb e −Φ/2 gives rise to a 'connection' in the Maxwell Lagrangian.Ultimately this just leads to a shift of the KK modes n β → n β + i when we reduce the action to 3D.
When dealing with the fermions, it is convenient to return back to the action before fixing the kappa symmetry gauge (2.22).Once again the computation is more involved compared to the maximally supersymmetry thermal AdS background discussed in Section 2 due to the presence on non-trivial three-forms field strengths H and F 3 contributing to the fermion action.When combining the terms we find that the total contribution of three-forms to the action is17 This term motivates that we use the operators σ 1 Γ φ and σ 3 Γ θ to break up the ten-dimensional spinor when reducing to four dimensions.We use the two signs s 1,2 to denote the eigenvalues of the two operators respectively.Furthermore we must rescale the fermions by e ϕ/4 to absorb the explicit factor of e −Φ/4 in the fermion action.This rescaling changes operator slightly since the ϕ-derivative acts on this factor.Doing so results in the fermionic operator We note that the dirac operator on S 1 J × S 3 includes the spin connection on the squashed S 3 .The terms that are proportional to Γ ϕ can be interpreted as the coupling of the fermions to the constant gauge field (4.11) whereas the other terms are a result of the squashing of the S 3 .They give rise to mass terms when reduced down to the round two-dimensional sphere.
We summarize the end result of the bosonic and fermionic spectrum in Table 4.We plan to report on the full 1-loop determinant of this effective theory on the squashed D3-brane in the near future.Similarly to the D3-branes in the thermal Leigh-Strassler background we find that the squashing of the three sphere is part of the class SU(2) × U(1) invariant squashed 3D partition functions studied in [47].It is interesting to note that even though the D3-brane partition function should not depend on the position along the N = 2 S-fold conformal manifold [32], this seems like a highly non-trivial result in string theory.Even though the classical action of D3branes has been verified to be independent of the position [28], the D3-brane metric generically only exhibits U(1) × U(1) isometry.Moreover, we expect that the spectrum of fluctuations will depend non-trivially on the position along the conformal manifold.Nevertheless, the QFT prediction clearly indicates that the same answer for the D3-brane partition function must be obtained no matter the position.We may speculate that this invariance is a non-trivial consequence of some Q-exactness of the effective D3-brane theory or its 3D reduction in line with similar results in 3D localization [47].Table 4: A summary of the spectrum of scalars, fermions, and vector on the D3-brane in the J-fold background.

Table 3 :
The spectrum of fluctuations around the classical D3-brane solution (3.7) in the LS geometry.