One-loop tests of supersymmetric gauge theories on spheres

Journal of High Energy Physics, Jul 2017

We show that a recently conjectured form for perturbative supersymmetric partition functions on spheres of general dimension d is consistent with the flat space limit of 6-dimensional \( \mathcal{N} \) = 1 super Yang-Mills. We also show that the partition functions for \( \mathcal{N} \) =18-and9-dimensionaltheoriesareconsistentwiththeirknownflatspacelimits.

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One-loop tests of supersymmetric gauge theories on spheres

HJE One-loop tests of supersymmetric gauge theories on spheres Joseph A. Minahan 0 1 3 Usman Naseer 0 1 2 0 Cambridge , MA 02139 , U.S.A 1 Box 516 , SE-751 20 Uppsala , Sweden 2 Center for Theoretical Physics, Massachusetts Institute of Technology , USA 3 Department of Physics and Astronomy, Uppsala University We show that a recently conjectured form for perturbative supersymmetric partition functions on spheres of general dimension d is consistent with the at space limit of 6-dimensional N = 1 super Yang-Mills. We also show that the partition functions for N = 1 8- and 9-dimensional theories are consistent with their known at space limits. Field Theories in Higher Dimensions; Supersymmetric Gauge Theory 1 Introduction 2 3 Review of localization and its analytic continuation One-loop divergences from partition functions 3.1 3.2 3.3 3.4 Eight supersymmetries in 4d Eight supersymmetries in 6d Sixteen supersymmetries in 4d and 6d Sixteen supersymmetries in 8d and 9d 4 Summary and discussions 1 Introduction on Euclidean spaces [ 1 ]. For instance, a single vector multiplet in a 4d N = 1 gauge theory has a Majorana fermion as a superpartner to the gauge eld in Minkowski space. In rotating to Euclidean space it is impossible to maintain the reality condition on the fermion, so the supercharges are naturally complexi ed. As shown by Zumino [ 1 ], requiring the spinors to be real leads to extra supersymmetry with additional elds in the supermultiplet, namely two extra fermion degrees of freedom as well as two scalars, where one scalar has the wrong sign kinetic term. In hindsight we can easily understand these extra elds as arising from a dimensional reduction of a Minkowskian six-dimensional vector multiplet, where one of the reduced directions is the time direction [2, 3]. A similar issue occurs when we consider a vector multiplet in six dimensions. In Minkowski space an N = 1 vector multiplet has a real Weyl spinor. Analytically continuing to Euclidean space complexi es the spinor. In order to have real spinors one must increase the amount of supersymmetry to N = 2 with 16 supersymmetries. Now there will be four scalars where one of them has the wrong sign kinetic term. Note that these issues do not mean that we cannot have minimal supersymmetry on Euclidean spaces, it just means we cannot have it with only real elds. However, the standard localization procedure often starts with ten-dimensional super Yang-Mills and dimensionally reduces to super Yang-Mills on lower dimensional spheres [4{ 7]. As such, the elds are real. Nevertheless, a simple argument shows that it should be possible to put a theory with N vector multiplet on R4, with no chiral multiplets charged under the U(1). This theory is free, hence it is conformal. Therefore one can make a conformal transformation to S4 and preserve the supersymmetry, albeit with complex elds. In any case, one can = 1 supersymmetry on S4. Suppose one has a U(1) { 1 { use the methods of Festuccia and Seiberg, starting with o -shell supergravity to put any N = 1 theory on AdS4 and analytically continue to S4 [8]. However, the full partition function for a generic interacting N = 1 theory has been shown to be scheme dependent [9], questioning whether one could obtain any scheme independent results from a localized partition function. Nonetheless, certain theories might have further symmetries, opening up the possibility of localizing the theory. A prominent example of this type is the N = 1 theory where it was shown that one can obtain unambiguous and scheme independent quantities constructed from the partition function [10]. However, there is no known way to localize an N = 1 theory on S4. The di culty arises in trying to construct a positive de nite localization term. It is possible to construct a positive de nite Q-exact term from a supersymmetry generator, however the known generators that give such terms do not themselves close to a symmetry of the Lagrangian [11].1 This issue is also discussed in [12]. For six-dimensional N = 1 super Yang-Mills the situation is worse. In the four dimensional minimal supersymmetric case one expects to be able to put the theory on S4 because of the existence of an appropriate superalgebra, namely OSp(1j4), which has four supercharges and a bosonic SO(5) subalgebra corresponding to the isometry of S4. For six dimensions we would want a superalgebra with a bosonic SO(7) subalgebra and 8 supercharges transforming in a spinor representation of SO(7), but no such superalgebra exists. The F (4) supergroup has an SO(7) SU(1; 1) bosonic subalgebra and 16 supercharges, hence this is appropriate for N = 2 supersymmetry. We encounter a similar problem when considering super Yang-Mills in eight and nine dimensions. In this case there are sixteen supersymmetries, and again, one would like to be able to put these theories on spheres. But once more there is no corresponding superalgebra with the appropriate bosonic subalgebras, namely SO(9) and SO(10) for S8 and S9 respectively.2 Even if one were able to de ne such theories it might be the case that their partition functions su er from ambiguities just like N = 1 theories in four dimensions. In the next section we will review explicitly the obstacles in putting these supersymmetric theories on the spheres. Nevertheless, there are some indications that there are theories akin to N = 1 six, eight and nine dimensional supersymmetric theories on spheres. In particular, evidence was presented in [14] showing that the perturbative partition functions for super Yang-Mills with 8 supersymmetries on S3, S4 and S5 have a natural analytic continuation, such that one can continue up to six dimensions. Likewise theories with 16 supersymmetries on Sd with d = 3; 4; 5; 6; 7 also have a natural analytic continuation which can then be continued up to d = 8; 9. Although, we do not have an explicit construction of Lagrangians for these theories, it is reasonable to assume that in the decompacti cation limit, they reduce to usual gauge theories in at space. The main objective of this paper is to demonstrate that the partition functions are consistent with this picture. These partition functions include a 1We thank Guido Festuccia for several discussions on this point. 2There are superalgebras with SO(9) and SO(10) bosonic subalgebras, for example, OSp(9j2) and OSp(10j2). However, these algebras have supercharges that transform in the vector representation of SO(9) one-loop determinants produce the well known physics of the at space theories. The paper is organized as follows: in section 2 we brie y review the localization procedure in [5] and the analytic continuation in [14]. In section 3 we compute one-loop divergences from the analytically continued expressions for one-loop determinants and compare them with well known results in the literature. In section 4 we summarize our results and discuss some further issues. 2 Review of localization and its analytic continuation In this section we review the localization procedure for gauge theories with eight supersymmetries on spheres. We use the conventions in [5] which generalizes the procedure on S4 [4], where one starts with N = 1 super Yang-Mills in ten dimensions and dimensionally reduces, including along a time-like direction so that the unreduced dimensions are Euclidean. We then review analytic continuation of the localized result to other dimensions. In ten dimensions the Lagrangian in at space is given by [15] HJEP07(21)4 L = 1 g10 aMb and ~M ab are real and symmetric. The Lagrangian is invariant under the supersymmetry transformations symmetry transformation to where the constants I are given by = 12 MN FMN + I I r ; I 2 FIJ = [ I ; J ] : where is any constant real spinor. We then dimensionally reduce to d dimensions so that the gauge elds are A , = 1; : : : d, while the remaining bosonic elds are scalars with I AI , I = 0; d + 1; : : : 9. The scalar 0 will have a wrong-sign kinetic term in the action because it came from the dimensional reduction of the time direction. This also leads to a noncompact R-symmetry, SO(1; 9 d). The dimensionally reduced eld-strengths become Putting the d-dimensional Euclidean space on the sphere modi es the fermion super(2.1) (2.2) (2.3) (2.4) (2.5) The index I in (2.4) is summed over. The supersymmetry parameters are special cases of the superconformal Killing spinors on Sd that satisfy r = ~ ~; r ~ = 2 : = 21r , which has 32 independent components. We reduce this to 16 components r , ~ = ~ = 1, ; T = for consistency with (2.6). A = 0 ~8 9, showing that this construction works up to d = 7. If we while at the same time modifying the extra terms in the Lagrangian to 5 by imposing the extra condition = + , where = 6789. The fermions then split into the components of a vector multiplet , with = + , and a hypermultiplet with = . The gauge elds A and scalars 0 , d+1 : : : 5 are the bosonic elds of the vector multiplet, while 6; : : : 9 are the bosonic elds for the hypermultiplet. With fewer supersymmetries we can add mass terms to the Lagrangian for the hypermultiplets. This also modi es the I for the hypermultiplet scalars to then the solutions to (2.7) are given by 1 where s is a constant spinor. The supersymmetric Lagrangian is given by { 4 { s ; r2 I = 6 : : : 9 I = 6; 7 I = 8; 9 ; 2(d 4) r and the quadratic terms for the hypermultiplet to where L L 2 d = = 1 1 2 gYM 2 gYM ( imTr ) ; d2 r2I Tr I I d(d 4 ; 2) 1 loop are vector- and hyper-multiplet one-loop determinants. It was observed in [14] that the one-loop determinants of all known examples with eight superd 5, can be written in the more general form Zvec 1 loop ( ) Y h ; i2 = Y n=1 h n2 + h ; i Y 2 (n + d 2)2 + h ; i (n+d 2) 5, but also agrees with the one-loop determinants on S6 and S7 for sixteen supercharges. It now seems quite natural to analytically continue the dimension past d = 5 in (2.16) and (2.17) up to d = 6, even though there is no appropriate and that can accommodate the above construction of vector and hypermultiplets on S6. At this point we will assume that our presumed theory has 8 supersymmetries and reduces to standard N = 1 super Yang-Mills in the at space limit. One possibility is that the SO(7) symmetry is broken to SO(6) with the spinors having opposite chiralities on the north and south poles. Likewise, it seems natural to analytically continue the dimension past d = 7 in (2.18), even though there is no appropriate for the construction. In the d = 8 and d = 9 cases we will again assume that our presumed theory reduces to standard N = 1 SYM in the at space limit. On the spheres we assume that the rotational symmetry is broken to SO(8). { 5 { In this section we will use the analytically continued expressions for one-loop determinants to compute e ective couplings for theories with eight and sixteen supersymmetries in diverse dimensions. The ultraviolet divergences of the gauge coupling at one-loop can then be compared with the counter terms for supersymmetric theories at one-loop. In four dimensions, upon taking the decompacti cation limit one can compute the beta function of the theory. We show that results obtained from the analytically continued one-loop determinants are in agreement with explicit one-loop computations in these theories. 3.1 Eight supersymmetries in 4d Recall that for a gauge theory in four dimensions with Nf Dirac fermions in representation Rf and Ns complex scalars in representation Rs of a semi-simple gauge group, the one loop beta function is given by (g) = 1 C2 (R) is the quadratic Casimir in the representation R of the gauge group. For N = 2 theory with Nh hypermultiplets in the representation R of the gauge group the beta function becomes (g) = g 3 The contribution from the vector multiplet was previously found in [4] by taking the hypermultiplet mass to in nity in the N = 2 theory. We want to reproduce (3.2) by using the analytically continued one-loop determinant for the vector and hyper multiplets given in equations (2.16) and (2.17). To do so we need to determine O To proceed we replace by t in the expressions for the one-loop determinants. The parameter t keeps track of the order of . Focusing only on the vector multiplet, one can 2 terms appearing in the one-loop determinants. easily nd that d log Z1vecloop dt2 where For d = 4 we nd + X 1 >0 t 2 = X >0 F (x; y; z) = 1 X n=0 i 2z (n + x) 1 (n + 1) (x) (n + y)2 + z2 1 y + iz 2F1 (x; y + iz; y + iz + 1; 1) c:c: : , we expand the r.h.s. in powers of t and . Keeping only the leading terms, h ; i2 (F (d 2; 0; t h ; i) + F (d 2; d 2; t h ; i)) ; d log Z1vecloop dt2 = 2 { 6 { (3.2) (3.3) (3.4) (3.5) HJEP07(21)4 2 For a gauge multiplet and Nh hypermultiplets, the contribution to the O the one-loop determinants can be combined with the O as given in equation (2.15) to get 2 term in the xed point action 2 term from From this we can easily obtain where g ( ) is the running coupling constant at the renormalization scale is the bare coupling. From the above equation one can easily obtain the beta function, Since the explicit expression for one loop determinants for eight supersymmetries in 4d are known in terms of in nite products, the above results can be reproduced by regularizing those expressions by introducing a nite cut o parameter r and then taking the decompacti cation limit r ! 1. As explained earlier, it is not known how to localize a six dimensional theory with eight supersymmetries. In this case the expression (2.16) is a genuine ansatz. In this subsection we will perform a non trivial check on that ansatz by computing the e ective coupling. It is well known that the six dimensional theory with eight supersymmetries has a quadratic divergence at one-loop [18, 19]. We will compute the e ective coupling using the one loop determinant (2.16) and show that it has a quadratic divergence in the decompacti cation limit. Since dimensional regularization is only sensitive to logarithmic divergences we will use a hard cuto to isolate the quadratic divergence. At leading order in the divergence this is expected to be consistent with supersymmetry. However, there could be issues with sub-leading divergences, if for example imposing the cuto leaves o the super-partners of modes at or near the cuto . However, assuming that the proposed dimensional regularization respects the supersymmetry we can show that the logarithmic divergences coming from the hard cuto are consistent with the result coming from dimensional regularization, even if the log divergence is sub-leading. { 7 { We use d = 6 in (2.16) and truncate the in nite product at nmax = nd quadratic dependence on the energy cuto . It is straightforward to nd that the divergent contribution to the 2 term from the vector one-loop determinant is By combining this with the xed point action, we nd the e ective coupling given by In the r ! 1 limit only the leading terms in r survive and one obtains 1 g2 = 1 g 2 0 2 96 3 C2 (Adj) : We see that the e ective coupling diverges quadratically with the scale . It is also known that the six dimensional theory can be made nite at one loop by adding a suitable hypermultiplet. This would be the case if the hypermultiplet and the vector multiplet contribute to the quadratic divergence with opposite sign. This is also consistent with the one-loop determinant (2.17). For a hypermultiplet in representation R, the contribution to O 2 term is given by log Z1hy ploop = + log ( r) C2 (R) 2 + : So that the e ective coupling with Nh hypermultiplets in the representation R is given by: 2r2 + 5 r 6 2 1 3 1 g2 = 1 g 2 0 96 3 (C2 (Adj) NhC2 (R)) : In particular, for a single hypermultiplet in the adjoint representation the quadratic divergence vanishes as expected [18]. 3.3 Sixteen supersymmetries in 4d and 6d In four and six dimensions, explicit expressions for the one-loop determinants for sixteen supersymmetries are known. We will compute the e ective coupling at one-loop using both of these expressions and show that it is consistent. For four dimensions, we compute e ective coupling from (2.18) using d = 4. We will truncate the in nite product at nmax = r. By doing so one nds that the contribution to O 2 term vanishes. log Zvec 1 loopZ1hy ploop = 0 + : Hence the coupling is not a ected by one-loop e ects and is independent of the cuto scale . Now we can easily check that this result is consistent with analytically continued (3.12) (3.13) (3.14) (3.15) { 8 { expression, i.e., expanding (2.18) in powers of for d = 4 . Replacing by t one can easily obtain d Which is consistent with the result obtained from the explicit expression. Similarly, in six dimensions the contribution to the O 2 term from (2.18) takes 1 2 + 1 2 + the form given by: log Zvec 1 loopZ1hy ploop = 3 log ( r) xed point action, we obtain the e ective coupling which is 1 We see that the coupling has a logarithmic dependence on the scale . It is easy to see that Logarithmic dependence is produced by using dimensional regularization for d = 6 in the analytically continued expression. Doing so we nd that: log Zvec 1 loopZ1hy ploop = 3 Combining this with the contribution from the xed point action and noting that in 6 dimensions gr22 has mass dimension we get: 1 where a -independent in nite piece is absorbed in g12 . This gives the same logarithmic 0 dependence on the energy scale . Note that in the decompacti cation limit this logarithmic divergence vanishes, consistent with the fact that the six dimensional theory with 16 supersymmetries is nite at one-loop. 3.4 Sixteen supersymmetries in 8d and 9d For d = 8; 9, it is not known how to localize. Here we show that the analytically continued expression for the one-loop determinant is consistent with known results. It is known that for d = 8; 9, none of the terms present in the tree level Lagrangian need a counter term at one-loop [19, 20]. Hence, the e ective coupling determined from the analytically continued expressions for one-loop determinants should not have any divergences. This can be easily demonstrated by using the methods of this section. A short calculation shows that the contribution to O 2 term from the one-loop determinant (2.18) for d = 8 is log Zvec 1 loopZ1hy ploop = 5 6 ( r)2 4 + 3 r 2 ! + 5 log ( r) C2 (Adj) 2 : (3.21) { 9 { (3.18) (3.19) (3.20) This leads to the e ective ccoupling 1 Here we see that in the decompacti cation limit the dependence on the energy scale vanishes. A similar computation for d = 9 yields log Zvec which leads to following expression for e ective coupling: 1 in the decompacti cation limit. The same calculation can be repeated for d = 10 and it can be shown that the one-loop determinants do not contribute any divergences to the gauge coupling in the decompacti cation limit. 4 Summary and discussions In this note we have presented some nontrivial evidence in favor of the analytically continued expressions for one-loop determinants of supersymmetric gauge theories on Sd. We have computed the one-loop contribution to the gauge coupling as predicted from these determinants and showed that it is consistent with the well known one-loop behavior of minimally supersymmetric gauge theories in diverse dimensions. Unfortunately, at this point it is not clear what are the actual Lagrangians for these gauge theories when they are on the sphere. Attempts to construct these theories by starting from ten dimensional SYM have not been successful so far. Another possible way to nd supersymmetric theories on spheres is to consider o shell supergravity coupled to vector multiplets and then take the rigid limit, as in [8, 21]. An o -shell formalism is known for N = 1 supergravity in six dimensions, [22{24]. This possibility was analyzed partially in [25] and it was shown that the o -shell theory with R-symmetry gauging does not admit S6 as a supersymmetric background. It would be interesting to complete this analysis by determining whether or not the o -shell theory without R-symmetry gauging admits S6 as a supersymmetric background, although in this case the supersymmetry might be enhanced to N = 2. For eight and nine dimensions we are unaware of an o -shell supergravity formulation. It is reasonable to believe that the theories, assuming they exist, will have Lagrangians that have explicit terms that break the bosonic symmetries down to SO(6) ' SU(4) in the six dimensional case, leaving an invariance under the SU(4j1) superalgebra. Likewise, we expect the eight and nine dimensional theories to only have an SO(8) symmetry and be invariant under an OSp(8j2) superalgebra. From the supergravity perspective, it would correspond to turning on non trivial background elds in addition to metric. (3.23) (3.24) HJEP07(21)4 Acknowledgments We thank Guido Festuccia and Luigi Tizzano for discussions. 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Joseph A. Minahan, Usman Naseer. One-loop tests of supersymmetric gauge theories on spheres, Journal of High Energy Physics, 2017, 74, DOI: 10.1007/JHEP07(2017)074