Brane webs and O5-planes

Journal of High Energy Physics, Mar 2016

We explore the properties of five-dimensional supersymmetric gauge theories living on 5-brane webs in orientifold 5-plane backgrounds. This allows constructing quiver gauge theories with alternating USp(2N) and SO(N) gauge groups with fundamental matter, and thus leads to the existence of new 5d fixed point theories. The web description can be further used to study non-perturbative phenomena such as enhancement of symmetry and duality. We further suggest that one can use these systems to engineer 5d SO group with spinor matter. We present evidence for this claim.

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Brane webs and O5-planes

HJE Brane webs and O5-planes Gabi Zafrir 0 1 0 Haifa , 32000 , Israel 1 Department of Physics, Technion, Israel Institute of Technology We explore the properties of ve-dimensional supersymmetric gauge theories living on 5-brane webs in orientifold 5-plane backgrounds. This allows constructing quiver gauge theories with alternating USp(2N ) and SO(N ) gauge groups with fundamental matter, and thus leads to the existence of new 5d xed point theories. The web description can be further used to study non-perturbative phenomena such as enhancement of symmetry and duality. We further suggest that one can use these systems to engineer 5d SO group with spinor matter. We present evidence for this claim. Brane Dynamics in Gauge Theories; Field Theories in Higher Dimensions 1 Introduction 3 4 2 ertheless, there is a lot of evidence that in the N = 1 supersymmetic case, corresponding to 8 supercharges, an interacting UV xed point may exist making the theory UV complete [3{5]. The gauge theory can then be realized as the IR limit of such a SCFT under a mass deformation, corresponding to the inverse gauge coupling square which has dimension of mass in 5d. An interesting question then is how can we study these 5d SCFT's. One way is to embed them in string theory. A convenient embedding is given by 5-brane webs in type II B string theory [6, 7]. This realizes the 5d SCFT as an intersection of 5-branes at a point. The moduli and mass parameters of the SCFT are then realized as motions of the internal and external 5-branes respectively. In particular, the 5d SCFT may posses a deformation leading to a low-energy gauge theory. Thus, 5-brane webs can be used to study various properties of these theories. First of all they give support for the existence of xed points for various gauge theories. Not { 1 { every 5d gauge theory ows to a 5d SCFT, and the demand that such a SCFT exists is expected to constrain the matter content of the theory. If a gauge theory can be realized as the IR theory in a brane web then this strongly suggests that it ows to a UV xed point described by the collapsed web. Therefore, brane webs can be used to study the conditions for the existence of xed points. Another useful application of brane webs is to study 5d dualities. A single SCFT may have more than one gauge theory deformation, in which case these di erent IR gauge theories are said to be dual. This is somewhat similar to Seiberg duality in 4d, except that in this case there are several di erent IR gauge theories all going to the same UV SCFT. This is nicely realized in brane webs, where a SCFT can be deformed in di erent ways in the brane web. Thus, brane webs provide a useful way to motivate these kind of dualities. For several examples of this see [8{12]. Brane webs can also be used to study symmetry enhancement in 5d gauge theories [13, 14]. They can be used to calculate the 5d superconformal index of a SCFT using the methods of topological strings [15]. Even when calculating the 5d superconformal index for a gauge theory using localization [16], brane webs are very useful for evaluating the instanton contribution [9, 17, 18]. They can also realize 5d versions of A type class S theories [19], and thus can be used to study them, and there are many other applications. The purpose of this article is to study brane webs in the presence of an orientifold 5plane. First, this allows constructing SO(N ) and USp(2N ) gauge theories with fundamental matter, as rst done in [20]. This can then be used to study these gauge theories. These systems can also be realized using an orientifold 7-planes, as done in [12], and our results agree with their nds. More interestingly we can use this to realize more elaborate theories. First, we can realize a linear quiver of alternating SO and USp groups connected by half-bifundamentals. This then provides evidence that these theories exist as xed points, and allows us to study some of their properties. Second, a subset of these theories are closely related to quivers of SU group in the shape of the Dynkin diagram of type D (henceforward referred to as D shaped quiver), so these methods can be used to study these theories as well. We also argue that these can even be used to engineer SO(N ) gauge theories with spinor matter for N 12. The structure of this article is as follows. Section 2 introduces the general construction in the more simpli ed case realizing a single gauge group. In section 3 we consider the general case giving a linear quiver with alternating SO and USp groups. We also consider the S-dual system leading to a D shaped quiver of SU gauge groups. Section 4 deals with describing SO gauge groups with spinor matter. We end with some conclusions. The appendix provides a short review of index calculations and instanton counting. 2 The general construction The starting point is the ordinary brane webs used to describe N = 1 supersymmetric SU groups and their quivers [6, 7]. The supersymmetry permits adding an O5-plane parallel { 2 { black dashed line. (b) The quantum picture, where bending occurs so that the D5-brane charge is NS5-brane crosses the O5-plane, in a way preserving N = 1 supersymmetry, it partitions the O5-plane into two parts of di ering types: O5+(O~5+) changes to O5 (O~5 ). Likewise a stuck D7-brane also has a similar e ect, now changing an O5+(O5 ) into an O~5+(O~5 ) and vice versa. The change in the type of O5-plane when crossing an NS5-brane has important implications once quantum e ects are taken into account. For example, let's consider a system consisting of an O5+ with a stuck NS5-brane that changes it to an O5 . Because of the type change, there is a jump in the D5-brane charge across the NS5-brane which should cause the NS5-brane to bend. Then taking into account charge conservation, sypersymmetry and invariance under the orbifolding, one concludes that the correct con guration should be a (2; 1)-brane crossing the O5-plane and becoming an (2; 1)-brane. This is exhibited in gure 1. Next we explore the implications for the simplest cases of N D5-branes stretched between two NS5-branes in the presence of one of the O5-plane types. 2.1 O5+ and USp groups The classical picture consists of an O5+-plane with N D5-branes suspended between two stuck NS5-branes resulting in a USp(2N ) gauge theory. Taking into account the bending caused by the D5-branes and the O5-plane results in the web shown in gure 2. It is now straightforward to generalize to cases with fundamental avor by adding (1; 0) 7{ 3 { case while (b) shows the general case. HJEP03(216)9 of USp(4) + 4F while (b) shows the general case of USp(2N ) + (NFL + NFR )F . branes on top of the D5-branes. These can then be pulled through the external 5-branes, accompanied by Hanany-Witten transitions resulting in webs with semi-in nte external D5-branes. Examples of these are shown in gure 3. Note that the gauge symmetry on Nf external D5-branes is SO(2Nf ), since these sit on top of an O5 , which is the correct global symmetry of USp(2N ) + Nf F . These webs can now be used to study a variety of issues in 5d gauge theories, notably, the existence of xed point, identifying decoupled states in index calculations and motivating dualities. Such a thing was done for a di erent realization of this theory using O7-planes in [12]. One can see that the webs in the reduced space in these systems are similar to the ones, in the reduced space, with the O5+-plane. Thus, most of the results found using the construction with the O7-plane are also true in this case, and we will not repeat them. We do wish to discuss the manifestation of the Higgs branch in this web, as this is di erent from the O7-plane construction. In brane webs, the Higgs branch consists of all the possible motions of the 5-branes along the 7-branes. Uniquely for 5d, at the xed point there can be additional Higgs branch directions besides the ones visible in the perturbative gauge theory [22]. Figure 4 (a) illustrates an example of this for the case of the E1 theory: the xed point has a one dimensional Higgs branch corresponding to separating the (1; 1) 5-brane from the (1; 1) 5-brane. This theory can also be constructed using an O5+-plane, as shown in gure 4 (c). This also exhibits the 1-dimensional Higgs branch, now given by pulling the (3; 1) and (3; 1) 5-branes out of the O5-plane. The addition of fundamental avor is done by adding (1; 0) 7-branes, which we pull out, resulting in Nf semi-in nte D5-branes all in the same direction, as shown in gure 5 (a). To enter the Higgs branch we go to the origin of the Coulomb branch and set the masses of the { 4 { at a generic point on the Coulomb branch (on the left), and at the origin of the Coulomb branch and taking the bare coupling to in nity (on the right), describing the xed point. We have also explicitly drawn the 7-branes, shown as black circles. These span the 8 directions coming out of the picture. (b) The brane web for SU (2), also known as the E~1 theory. The web on the left is for a generic coupling and at a generic point on the Coulomb branch, and the the web on the right is at the origin of the Coulomb branch and taking the bare coupling to in nity. One note that there is no Higgs branch in this case. (c) The web of gure 2 (a) at the origin of the Coulomb branch and in nite coupling constant. avors to zero by coalescing all the D5-branes on the O5-plane. Furthermore, we separate the D7-branes along the O5 -plane. When the O5 -plane is crossed by a D7-brane it changes into an O~5 which is an O5 -plane with a stuck D5-brane. We thus conclude that upon each crossing one 5-brane must end on a D7-brane resulting in the picture shown in gure 5 (b). The Higgs branch now consists of breaking the 5-branes on the 7-branes as shown in gure 5 (c). The possible breakings are limited by the S-rule, which necessitates that at most one 5-brane can be stretched between any given NS5-brane and D7-brane. When { 5 { massless avor, after the D7-branes were separated along the O5-plane. (c) The web at a generic point on the Higgs branch. For ease of presentation we have used a di erent view of the web where the vertical straight lines are the D5-branes, the horizontal wide lines are the D7-branes, and the black dot is the ( 1,-1 ) 5-brane. Nf > 2N this becomes more stringent as one can no longer connect a D7-brane to the other NS5-brane. Counting the possible breakings, with these restrictions, we indeed nd the correct dimensions of the Higgs branch expected from the gauge theory. Additional examples of this are shown in gure 6. Finally we take the xed point limit by collapsing the gauge D5-branes. In this limit additional directions become available. First there is the 1 dimension given by detaching the 5-branes from the O5-plane, similarly to the one in the E1 theory. This exists for any number of avors. When Nf > 2N there are further additional directions. It appears that when the NS5-branes touch, a D7-brane can always be connected to the other NS5-brane. This eases the constraints imposed by the S-rule and allows additional directions. As we shall soon show this is necessary in order to recover the correct Higgs branch dimensions of known theories, like the rank 1 E6 theory. When Nf = 2N + 4 the two external NS5branes become parallel and there is an additional direction given by breaking one of them on a (0; 1) 7-brane. Finally, when Nf = 2N + 5 there are intersecting external legs, where resolving the interaction leads to one of these external legs becoming a D5-brane due to passing thorough the monodromy of the other (see [12] for the details). At the xed point, one can then also break this D5-brane on the 7-branes leading to additional directions. We can count the dimension of the Higgs branch for these theories and compare it with the one found using a di erent realization of these theories, for example using ordinary webs, nding complete agreement. As an example, consider SU(2) with ve avors, the rank 1 E6 theory. As shown in gure 6 (a) the perturbative Higgs branch is 7 dimensional, which is indeed the gauge theory result. An important limitation here is that there are only two gauge D5-branes so only two D7-branes can be connected to the other NS5-brane. This makes the constraint imposed by the S-rule more stringent. We have argued that this constraint should be relaxed at the xed point, where the two NS5-branes coalesce. Indeed, without this constraint we would get a 10 dimensional Higgs branch, as can be seen by comparing with the perturbative component of the Higgs branch for a di erent theory with the same number of avors but with 2N > Nf like the USp(6) + 5F theory in gure 6 (b). Thus, the non-perturbative Higgs branch has 10 + 1 = 11 (remember there { 6 { next to the O5-plane represent the number of D5-branes stuck on it. The Higgs branch is given by detaching a D5-branes and its image from the O5-plane. One can see that the Higgs branch is 7 dimensional (quaternionic) in accordance with the gauge theory result. (b) The web at a generic point in the Higgs branch for USp(6) + 5F . The Higgs branch for this theory is 10 dimensional, again in accordance with the gauge theory result. Note that this di ers from the case of (a) only by the number of color branes which is large enough so that every D7-brane can be connected to the other NS5-brane. { 7 { of SO(8) + 2F while (b) shows the general case of SO(2N ) + (NFL + NFR )F . is an additional direction given by taking the web o the O5-plane) which is indeed the Higgs branch dimension of the rank 1 E6 theory. Similarly, one also get from the web the correct dimensions of both the perturbative and non-perturbative Higgs branches for the E7 and E8 theories. 2.2 O5 and SO groups Changing the O5+ to an O5 leads to an SO(2N ) gauge theory, an example of which is shown in gure 7. The major di erence in the web is that the bending caused by the O5-plane is in the opposite direction. This results in di erent bounds for xed points, so for example one cannot draw a web for pure SO(2), while one existed for the O5+ case, pure USp(2). This is in accordance with the expected UV incompleteness of 5d U( 1 ) gauge theories. The generalization by the addition of fundamental (in the vector representation) avors is straightforward, examples shown in gure 8. Now, the avor D5-branes sit on top of an O5+-plane resulting in a USp(2Nf ) global symmetry again in accordance with the gauge theory expectation. An alternative realization of SO(2N ) + Nf F using an O7-plane also exists, and again the reduced space webs of these two constructions are similar. Thus, all the results seen from that construction are also valid in this one and we will not repeat them here. We do wish to describe the manifestation of the Higgs branch in this case. In the pure case, exactly as in the O5+ case, one nds a 1-dimensional Higgs branch that opens at the xed point. This agrees with the results seen also from the O7-plane constructions as well as other constructions when these are available (such as pure SO(6) = SU(4)). { 8 { massless avor, after separating the 7-branes. Since a D5-brane cannot be stuck on an O~5+ we are forced to stretch an extra D5-brane between the two 7-branes. (c) The web at a generic point on the Higgs branch. the (2; 1) 5-brane. This leads to the con guration identical to separating a D7-brane and its image, each with a D5-brane ending on it, across an O5+-plane. Next we discuss the generalization when avors are present, starting with the case of mirror image, since now there is an O~5+ one avor. We can again go to the origin of the Coulomb branch and the limit of zero mass. The major di erence from the USp case is encountered when separating a 7-brane from it's between them. One cannot have a stuck D5-brane on an O~5+, so we conclude that when separating the 7-branes the two D5-branes must end on the same 7-brane.1 The resulting construction is shown in gure 9. The implication of this is that now there is a 1-dimensional Higgs branch where the gauge theory is broken to SO(2N 1), in accordance with the gauge theory expectation. The generalization to more than one avor is now straightforward. There are now 2Nf 7-branes stuck on the O5-plane which come in alternating pairs with one with no 5-branes ending on it and the other with two 5-branes ending on it. Breaking the 5-branes on the 7-branes, while taking due care of the S-rule, correctly reproduces the breaking pattern and the dimension of the Higgs branch as expected from the gauge theory. Finally, we take the xed point limit and consider non-perturbative Higgs branch directions. Like in the USp case, there is always the direction given by separating the 1One can also arrive to the same conclusion by moving the D5-branes past the NS5-brane, and separate them on the O5 as done in the previous section. The separated 7-branes can now be moved back past the NS5-brane, along the O5-plane, resulting in the same outcome (see gure 10). { 9 { 7-branes, depicted as a thin dot-dash line. (b) The web for SO(5) which one gets by going on the Higgs branch of SO(6) + 1F . (c) Pulling the 7-brane to the other side through two HW transitions results in the web of SO(5) using an O~5 -plane. Note that there is an half monodromy stuck on the O~5 -plane which xes the D5 brane charge conservation. When drawing an O~5 plane we also draw the stuck D5-branes, slightly above the O~5 plane, and the half monodromy line, slightly so the directions associated with this case do not arise. When Nf = 2N legs become parallel while for Nf = 2N 3 the external legs becomes intersecting, and there are extra directions similar to the cases of Nf = 2N + 4; 2N + 5 in the USp case. 2.3 O~5 Finally we want to consider the case of an O~5 plane. At rst one encounters a problem with the fractional NS5-brane on it. This leads to a change between an O~5+ and an O~5 , resulting in a jump in the D5-brane charge. However this jump is now fractional, and it does not appear to be possible to reconcile the bending required by charge conservation with the re ection symmetry implied by the orientifold. One approach to realizing these theories, in the case of an O~5 , is to use the construction for SO(2N +2)+1F and go on the Higgs branch leading to SO(2N + 1). An example of this is shown in gure 11. We can now transform to a description with an O~5 by moving the stuck D7-brane though the external 5-branes to the other end of the O5-plane, while taking due care of the monodromy of the 7-brane. Now, the resolution of the above issue is clear: there is also a half monodromy on the O5-plane which corrects the bending so as to be consistent with the orbifolding. One can now use this construction to realize SO(2N + 1) + Nf F gauge theories. Once again the reduced space web matches the one using an O7-plane and the results found from this description also apply to this case. The Higgs branch is realized exactly as in the SO(2N ) case. Finally, we can inquire about the web with an O~5+. A natural guess is that this describes the USp theory with = .2 Indeed, we expect such theories to exist yet there are no discrete choices in the web save for this. Furthermore, such a thing occurs for example in the construction of the maximally supersymmetric 5d USp(2N ) gauge theory using O4-planes [23, 24]. However, the web for USp(2N ) using an O~5+, an example of which is shown in gure 12, appears to be identical to USp(2N ) + 1F and does not describe a new theory. Thus, it appears that there is no di erence between using an O5+ and an O~5+. 2We think O5+ describes the = 0 case based on the Higgs branch. As mentioned in gure 4, the E1 xed point has a 1-dimensional Higgs branch while E~1 does not. the half monodromy line slightly below the O~5+-plane. (b) The same web where we have added the half 7-branes on which the half D5-branes end. (c) The same web after taking the 7-branes past the NS5-brane, merging them to a full 7-brane and pulling it out of the O5 plane. One can see that this describes USp(2) + 1F . HJEP03(216)9 USp(2) SO(6) where a half-bifundamental is understood to exist whenever an is used between an SO and USp groups. As a result, there does not appear to be a way to describe USp (2N ) with an O5-plane. One can set out to get such a web by starting with the E2 web and giving a mass to a avor. Then depending on the mass sign one gets either the E1 or E~1 theories. This indeed works for ordinary webs and ones with an O7-plane, but not for this case. The problem is that in this case the avor can only be integrated in one direction. It is interesting whether there is a fundamental reason why = cannot be accommodated in this construction, or alternatively if it is possible to incorporate it in the web in a more intricate way. 3 SO USp quivers So far we considered a system with just two external NS5-branes. The web can be generalized to an arbitrary number of such branes which leads to a long quiver with alternating SO and USp groups connected by half-hypers in the bifundamental representation. Two examples of this are shown in gures 13 and 14. The quiver can contain both SO(2N ) and SO(2N + 1) gauge groups which can be achieved by adding a stuck D7-brane. This can also lead to an half-hyper in the fundamental for USp(2N ) which appears in the web as a stuck D7-brane or a stuck external D5-brane. Note that due to a global gauge anomaly, the 5d version of [25], a USp(2N ) gauge theory must have an even number of fundamental half-hypers [5]. This is indeed respected by the web. These webs have several interesting implications. First, they point to the existence of xed point theories for these quiver theories. This can also be generalized by adding avors and the web can be used to argue the limit beyond which a xed point does not exist. The web can also be used to study the Higgs branch, both the perturbative component and the non-perturbative component, as in the previous examples. It can also be used to identify decoupled states in index calculations using the SO USp formalism, as done for other systems in [9, 12, 17, 18]. USp(2) + 1HF where HF stands for a halffundamental. Note that the total number of half-fundamentals is even for each USp(2) so there is no gauge anomaly. Yet another application is to study symmetry enhancement in such theories. In brane webs this is manifested by some of the external legs becoming parallel. For example, consider a linear quiver consisting of nG groups of ranks Ni, for i = 1; 2 : : : ; nG, with Fi fundamental hypers under the i'th group. By inspecting the D5-charges of the NS5-charge carrying external branes, one can see that these are neither parallel nor intersecting as long as 2Ni 4 > Fi + Ni 1+Ni+1 where the + sign is for USp groups and the 2 sign is for SO groups, and we take N0 = NnG+1 = 0. If however 2Ni 4 = Fi + Ni 1+Ni+1 for some series of adjacent groups then a group 2 of NS5-charge carrying external branes become parallel, signaling an enhancement of the enhancement of U( 1 )nG ! SU(nG + 1) (see gure 15 (a) for an example). topological symmetries associated with these groups. In particular, if all the groups obey 2Ni 4 = Fi + Ni 1+Ni+1 then all these external branes become parallel, suggesting an 2 If 2Ni 4 < Fi + Ni 1+Ni+1 occures for one of the groups then the NS5-charge carrying 2 external branes associated with that group intersect (see gure 15 (b) for an example). The intersecting branes can be continued past one another accompanied with a Hanany-Witten transition. If this process terminates after a nite number of such transitions then this 5d gauge theory go to a 5d xed point described by the collapsed web (see gure 15 (c) for an example). One can now use this brane web to try and read the global symmetry of the xed point theory. For example, consider the previously considered class of theories, where every group obeys 2Ni 4 = Fi + Ni 1+Ni+1 . Say we now add one more avor to one of the edge groups, 2 say for i = 1. As seen in gure 15 (b), this leads to a con guration with the leftmost NS5charge carrying external brane intersecting the nG other NS5-charge carrying branes. This can be resolved by continuing them past one another, accompanied with a Hanany-Witten transition, leading to a con guration with no intersection (see gure 15 (c) for an example) so we conclude that this class of theories go to a 5d xed point. Inspecting the web one sees that this con guration as F1 + nG D5-branes on the left side and FnG D5-branes on the right side. Thus, we conclude that in this case there should be an enhancement of U( 1 )nG SO(2F1) ! SO(2F1 + 2nG) or U( 1 )nG on whether the i = 1 group is of type USp or SO respectively. USp(2F1) ! USp(2F1 + 2nG) depending In the rest of this section we concentrate on one further application which is motivating 5d dualities. These are manifested by a di erent gauge theory description, of the same web, generically in a di erent SL(2; Z) frame. We can rst start by generalizing the dualities of [12] also to the quiver case. As a simple example consider the web shown in gure 16 UT ( 1 )3 ! SU(4). (b) The web for 5F + USp(4) USp(4) + 4F where each group obeys 2Ni 4 = Fi + Ni 1+Ni+1 . One can see that the web as 4 parallel NS5-branes suggesting an enhancement of 2 SO(8) USp(4) + 4F . One can see that the (1; 1) 5-brane intersects the 3 NS5-branes. Resolving the intersection via a Hanany-Witten transition leads to the web in (c). One can see that due to passing through the monodromy of the (1; 1) 5-brane, the NS5-branes become D5-branes. This suggests that this xed point as an enhanced SO(16) global symmetry. (a) which describes a 3F + SO(10) USp(4) + 3F gauge theory. The web can be deformed through several op transitions to the one shown in gure 16 (b). The web shows an SU(3) + 2F gauge theory, existing on the (1; 1)-branes, gauging a global SU(3) symmetry in the remaining web shown in gure 16 (c), which describes a 3F + SO(8) gauge theory. The web suggests that this theory has an enhanced SU(3) instantonic global symmetry. This leads to the duality shown in gure 17. Since the gauged global symmetry is instantonic, this does not lead to a gauge theory duality. For this we would need to nd a dual description of the theory in gure 16 (c), in which the SU(3) global symmetry is perturbativly realized. We can consider generalizations of this duality, but in all we encounter the same problem, where we do not nd a complete gauge theory description of the dual side. USp(4) + 3F gauge theory. (b) The same web after several op transitions. One can see that it is identical to an SU(3) + 2F gauging the enhanced SU(3) instantonic symmetry of the web in (c). (c) The web describing an 3F + SO(8) gauge theory. a half-hyper in the bifundamental representation. The gauged SU(3) group on the right side is instantonic and so is not perturbatively visible. In order to make further progress one must do an SL(2; Z) transformation on the entire system, including the O5-plane. This requires understanding the behavior of the O5-plane under these transformations, in particular S-duality. There is one case where this is actually known which is an O5 -plane with a full D5-brane. In that case the total D5-brane charge of the system is zero, and its strong coupling behavior is of the perturbative orbifold R4=Z2, the Z2 being a re ection in the four directions combined with ( 1 )FL [21, 26]. Thus, in this case we might be able to say something explicit about the S-dual theory. The simplest thing to start with is an O5 -plane with N parallel D5-branes crossed by 2k NS5-branes, where an even number is necessary so that asymptotically we still have an O5 -plane so that we can apply S-duality. Doing S-duality on this con guration results in the web shown in gure 18 (b). This clearly shows a long SU(2k) linear quiver where at its end there are the branes connecting the NS5-brane with the orbifold xed plane. The gauge theory existing on such a system is known to be an SU(n1) SU(n2) gauge theory with n1 + n2 = 2k. The numbers n1; n2 arise as there are two di erent types of 5-branes ending on the orbifold xed plane di ering by their charge under the twisted eld living at the xed plane. In the present case, one can see an SU(k) SU(k) gauge theory. The matter content supplied by the crossed NS5-brane is a bifundamental hyper between each SU(k) and the adjacent SU(2k) so we conclude the duality shown in gure 19. The Chern-Simons levels are all 0 as the web is invariant under re ection which is the brane web analogue of charge conjugation in the gauge theory. One can now inquire whether we can nd additional evidence for this duality. First, we can do several simple checks such as comparing the dimension of the Coulomb branch and the number of mass parameters. A short counting shows that they are equal. Matching the global symmetries is harder as not all are classically apparent. The theory with the O5 plane classically has an U( 1 )2k 1 SO(2N )2 which, as suggested by the web, is enhanced to SO(2N )2 SU(2k). However, on the dual side, the classical global symmetry is U( 1 )2N SU(2k), which the duality suggests should enhance to SO(2N )2 SU(2k). Indeed this is supported by the instanton analysis of [27, 28]. Both theories also have discrete global symmetries. The D-shaped quiver theory has a Z2 The SO Z2 symmetry given by charge conjugation and exchanging the two SU(k) groups. USp quiver theory as a Z2 re ection symmetry, which the web suggests matches charge conjugation in the dual theory. The duality suggests that there should be another (b) The S-dual of (a). gure 18. The upper gauge theory is the one described by 18 (a) while the lower one is the one described by 18 (b) (all the groups are SU with CS level 0). Z2 that matches the exchange of the two SU(k) groups. There is indeed an extra Z2 on the SO USp quiver theory, exchanging the two spinor representations of all the SO groups. We expect this to match the exchange of the two SU(k) groups. Note that this operation on the Dynkin diagram of type D indeed corresponds to exchanging the two spinor representations. This matches similar results in the linear A type quiver [9]. Another check that can be done on this duality is comparing their superconformal indices (see the appendix for a review of the 5d superconformal index). As a starting point we can take the simplest example illustrated in gure 20. On one side we have the gauge theory SU(2) SU(4) SU(2) with 4 avors for the SU(4). The Chern-Simons levels of both SU(2) groups should be 0 which corresponds to = 0. The web suggests this USp(2) + 3F . (b) The S-dual web describing the gauge theory SU(2) SU(4) SU(2) with 4 avors for the SU(4). (c) The resulting duality. theory as an SU(4)3 global symmetry where one is visible perturbativly and the others are brought by instantonic enhancement. Indeed by an index calculation we veri ed that such an enhancement is present. In the dual theory we also expect an SU(4)3 global symmetry where now an SO(6)2 = SU(4)2 is perturbativly visible while instantons should provide the conserved currents for the third SU(4). The web suggests that this requires the contributions of (2,0,0) + ( 0,1,0 ) + (0,0,2) + ( 2,1,0 )+ ( 0,1,2 ) + ( 2,1,2 ) instantons. Unfortunately, calculating all of these contributions is technically demanding so it is worthwhile to look at a simpler example. Particularly, we can consider integrating out avors so as to reduce the degree of enhancement. This is done by taking an external D5-branes to in nity. Note that for the two edge avors this is identical to pulling an NS5-brane to in nity, and thus to also integrating out a avor in the dual theory. Thus, we arrive at the duality shown in gure 21 which is the one we shall check using the superconformal index. The theory on the right of gure 21 is SU(2) SU(4) SU(2) gauge theory with 2 avors for the SU(4). The classical global symmetry is SU(2) U( 1 )6. The fugacity allocation is shown in gure 22. We work to order x5 which requires the contributions of the ( 1,0,0 ) + ( 0,1,0 ) + ( 0,0,1 ) + ( 1,1,0 ) + ( 0,1,1 ) + ( 1,0,1 ) + (2,0,0) + (0,0,2) + ( 1,1,1 ) instantons. We nd: where again we displayed terms only up to x3. From the x2 terms we see that the ( 0,1,0 ) instantons bring about an enhancement of U(1) ! SU(2) so that the quantum symmetry is SU(2)5 U( 1 )2 in accordance with the dual IndexD quiver = 1+x2 7+d2 + d2 + x3 y + +O(x4) 1 y 1 + q + 8+d2 + d2 1 1 q + q + where we have only displayed terms up to order x3 even though the calculation was done up to order x5. From the x2 terms in the index we see that the ( 1,0,0 ) + ( 0,0,1 ) instantons provide the conserved currents to enhance U( 1 )4 ! SU(2)4 so that the quantum symmetry appears to be SU(2)5 U( 1 )2. The remainder of the index indeed forms characters of the enhanced symmetry as will become apparent when we next compare it with the index of the SO USp quiver. In that theory the classical global symmetry is SU(2)4 U( 1 )3. We use the fugacity spanning shown in gure 23. To the order we are working with we get contributions from the ( 1,0,0 ) + ( 0,1,0 ) + ( 0,0,1 ) + ( 1,0,1 ) + (0,2,0) + ( 1,1,1 ) instantons. We nd: IndexSO=USp = 1 + x2 7 + f 2 + + x3 y + + O(x4) 1 y 1 1 f 2 + g2 + g2 + h2 + h2 + p2 + 1 1 p2 + A + 1 A 8 + f 2 + 1 1 f 2 + g2 + g2 + h2 + h2 + p2 + 1 1 p2 + A + 1 A c2 + c2 1 (3.1) (3.2) representations of SO(6)). it reads: f = zpq, g = pzq, h = c p theory. Further comparing the two we nd that taking: pA = d, QT = b2, QTpA = aptq, t, p = cpt render the two equal. From the matching we see that the enhanced SU(2) global symmetry of the SO USp quiver matches the perturbative SU(2) global symmetry of the D shaped quiver theory and vice versa, as expected from the web. The index also makes manifest the Z2 Z2 global symmetry. It acts on the theory by permutating the 4 SU(2) global symmetry groups, perturbativly realized in the SO USp quiver, similar to its action on the 4 vertices of a rectangle whose symmetry group is also Z2 Z2. The matching of fugacities is consistent with charge conjugation of the D-shaped quiver mapped to re ecting the SO USp quiver, while exchanging the two SU(2) groups is mapped to charge conjugation of the SO USp quiver (which exchanges the two spinor HJEP03(216)9 The index can be written in characters of the SU(2)5 U( 1 )2 global symmetry where Index = 1 + x2(2 + [3;1;1;1;1] + [1;3;1;1;1] + [1;1;3;1;1] + [1;1;1;3;1] (3.3) + [1;1;1;1;3]) + x3 y[2](3 + [3;1;1;1;1] + [1;3;1;1;1] + [1;1;3;1;1] + [1;1;1;3;1] + [1;1;1;1;3]) + x4 y[3](3 + [3;1;1;1;1] + [1;3;1;1;1] + [1;1;3;1;1] + [1;1;1;3;1] + [1;1;1;1;3]) + [5;1;1;1;1] + [1;5;1;1;1] + [1;1;5;1;1] + [1;1;1;5;1] + [1;1;1;1;5] + 3 [3;1;1;1;1] + 2 [1;3;1;1;1] + 2 [1;1;3;1;1] + 2 [1;1;1;3;1] + 2 [1;1;1;1;3] + 8 + [3;3;1;1;1] + [3;1;3;1;1] + [3;1;1;3;1] + [3;1;1;1;3] + [1;3;3;1;1] + [1;3;1;3;1] + [1;3;1;1;3] + [1;1;3;3;1] + [1;1;3;1;3] + [1;1;1;3;3] + [1;2;2;2;2] 1 + apqt + apqt ( [2;1;2;2;1] + [2;2;1;1;2]) + b2 + b12 ( [1;2;2;1;1] + [1;1;1;2;2]) + x5 y[4](3 + [3;1;1;1;1] + [1;3;1;1;1] + [1;1;3;1;1] + [1;1;1;3;1] + [1;1;1;1;3]) + y[2] [5;1;1;1;1] + [1;5;1;1;1] + [1;1;5;1;1] + [1;1;1;5;1] + [1;1;1;1;5] + 7 [3;1;1;1;1] + 6 [1;3;1;1;1] + 6 [1;1;3;1;1] + 6 [1;1;1;3;1] + 6 [1;1;1;1;3] + 12 + 2 [3;3;1;1;1] + 2 [3;1;3;1;1] + 2 [3;1;1;3;1] + 2 [3;1;1;1;3] + 2 [1;3;3;1;1] + 2 [1;3;1;3;1] + 2 [1;3;1;1;3] + 2 [1;1;3;3;1] + 2 [1;1;3;1;3] + 2 [1;1;1;3;3] 1 + [1;2;2;2;2] + apqt + apqt ( [2;1;2;2;1] + [2;2;1;1;2]) + b2 + b12 ( [1;2;2;1;1] + [1;1;1;2;2]) + O(x6) ordered as [pA;f;g;p;h]. where we use y[d] for the d dimensional representation of SUy(2) and [d1;d2;d3;d4;d5] for the representations of the appropriate dimensions under SU(2)5 where the SU(2)0s are and one avor for each group. 4 Spinor matter In this section we discuss the addition of matter in a spinor representation of an SO gauge group. It is well known that there is no perturbative way to add matter in spinor representations through D-brane constructions in string theory. However, we claim that there is a way to do so for webs in the presence of O5-planes, in a non perturbative manner. We rst present our conjecture for how this is done, and present our argument for why this gives spinor matter. We then proceed to give evidence for this conjecture. We claim that the con guration shown in gure 24, in which we add a stuck NS5brane to an SO(2N ) gauge theory, corresponds to adding a single hyper in the spinor representation of that group. Our motivation for this is as follows. First, consider the system in gure 25, describing an 1F +SU(2) SO(2N +2)+1F gauge theory. Starting from this system, we can get to the the one in gure 24 by going on the Higgs branch described in the web by separating a full D5-brane. In the gauge theory this describes giving a vev to the operator qBQ where q is the SU(2) fundamental, B the half bifundamental, and Q the vector of SO(2N + 2). Perturbativly, this completely breaks the SU(2) gauge group. However, the web suggests that in this limit we do remain with additional degrees of freedom. Thus, these can only come from instantons of the SU(2). It is well known that the 1 instanton of SU(2) with Nf avors carries charges in the spinor representation of SO(2Nf ). Therefore, we conjecture that the remaining state can be described by an hypermultiplet in the spinor of SO(2N ) whose origin is non-perturbative in the brane web. Although we have used SO(2N ) in this example, the same reasoning can also be carried out for the SO(2N + 1) case. Before giving support for our claim, we wish to state some further implications of it. First, we can consider what happens when we attach further D5-branes as shown in gure 26. We can answer this question by again starting with the system of gure 25 where adding the D5-brane corresponds to adding a avor for the SU(2). The instanton is again in the spinor of SO(2Nf ) which decomposes to two spinors of opposite chirality under SO(2Nf 2). We thus conclude that we now get two hypermultiplets both spinors of SO(2N ), but of opposite chirality. Finally, we wish to consider what happens if we preceding argument. similarly add a (2; 1) 5-brane in the other direction. Particularly, we still expect a spinor hyper, but we inquire whether it has the same chirality or not as it opposing friend. We can answer this by considering the appropriate equivalence of the system in 25, where the spinor should appear as the instanton of the SU(2) gauge group. The chirality of this spinor is determined by the SU(2) angle, and as this is identical in the two constructions, we conclude that the two spinors have the same chirality. To change the chirality between the two spinors, one would have to switch the angles of one of the SU(2)'s. As previously mentioned, we do not know if this can be done. We can now proceed to give evidence for our conjecture. First, we look at the Higgs branch. In gures 27 and 28 we show the webs for a variety of SO groups with two spinors of the same chirality. These theories then have a Higgs branch breaking them to an SU group. We show that the web correctly reproduces this branch, giving the expected theory. Furthermore, this branch can only be accessed when the spinor is e ectively massless which in the web corresponds to the point where the would be SU(2) instanton is massless as expected from our interpretation. Note that we are essentially limited by the requirement that the would be SU(2) gauge group sees less than 8 avors. Naively, this would imply the we cannot get spinors of SO(N ) for N > 10. However, we can by a slight generalization get one also for N = 11; 12. In these theories, the spinors are pseudo-real, so a half-hyper is possible. Figure 29 shows the webs we conjecture for SO(11) and SO(12) with one half or full hyper in the spinor representation. One can see that the Higgs branch of these webs agrees with what expected from the gauge theory. One issue with the webs for the full spinor cases is that they have only one mass deformation, in contrary to the two expected from the gauge theory. This is reminiscent of the web for USp(2N ) + AS which also has just one mass deformation. In that case the web is for a massless antisymmetric. Likewise this web appears to have a massless spinor. Using these webs we can look at the existence of xed points for SO groups with both vector and spinor matter. First, holding the spinor matter xed, we can use the web to determine what is the maximal number of vectors one can add while still having a 5d xed point. We generally nd agreement with the expectations from [29]. We can also ask what this implies about the limit of spinor matter. As mentioned, we are limited by the requirement that the would be SU(2) gauge group sees less than 8 avors. Yet, we argue that this does not represent a limitation on spinor matter for SO gauge theories, rather a breakdown of the interpretation of these webs. A limitation on 3; 4; 5. Starting from the initial web on the right, we take the massless spinor limit, corresponding to taking the distance between the two pairs of NS5-branes to zero. Then a Higgs branch opens up given in the web by detaching the web from the orientifold. This is shown on the right where for ease of presentation we have shown only half the web. (a) The case of N = 3. We know from the gauge theory that there is a Higgs branch breaking the theory to SU(3). (b) The case of N = 4. We know from the gauge theory that there is a Higgs branch breaking the theory to SU(4). (c) The case of N = 5. We know from the gauge theory that there is a Higgs branch breaking the theory to SU(5) + 2F . In all 3 cases the Higgs branch is correctly reproduced in the web. the matter content due to a lack of 5d xed point manifests as a lack of a brane web when taking the Yang-Mills coupling to in nity, generally due to intersecting legs that cannot be resolved by a nite number of HW transitions. This is not the case here, rather the intersection arises when we take the massless spinor limit indicating that there are in fact additional states in this case. Therefore, we do not think this gives a limit on spinor matter for SO gauge theories, rather being a limitation of the method. 4.1 Dualities As our nal piece of evidence, we examine dualities between systems involving SO groups with spinor matter. The idea is to use the webs to motivate dualities and then test them using the superconformal index. This then provides independent evidence for the duality and thus also for the original identi cation leading to it. There is one limitation in this test as instanton counting for SO groups with spinor matter is currently unknown. Thus, we are limited to comparing the perturbative parts. for N = 2; 3; 4. Starting from the initial web on the right, we take the massless spinor limit, corresponding to taking the distance between the two pairs of NS5-branes to zero. Then a Higgs branch opens up given in the web by detaching the web from the orientifold. This is shown on the right where for ease of presentation we have shown only half the web. (a) The case of N = 2. We know from the gauge theory that there is a Higgs branch breaking the theory to SU(2). (b) The case of N = 3. We know from the gauge theory that there is a Higgs branch breaking the theory to SU(3). (c) The case of N = 4. We know from the gauge theory that there is a Higgs branch breaking the theory to SU(4) + 2F . In all 3 cases the Higgs branch is correctly reproduced in the web. SO(11)+1S. One can see that the Higgs branch correctly agrees with the gauge theory expectation. The cases with a half-hyper, (a)+(c), do not have a Higgs branch. For (b) the gauge theory has a Higgs branch leading to SU(6) which is correctly reproduced in the web. For (d) the gauge theory has a Higgs branch leading to SU(5) which is again correctly reproduced in the web. SU0(2) gauge theory. The USp (4) group can be seen by pulling the (1; 1) and (1; 1) 7-branes through the 5-branes, and merging them to an O7 plane (see [12]). As our rst example, consider the theory shown in gure 30 (a). From gure 26, we claim that this describes an SO(8)+2S +2C gauge theory where we use S and C for the two Weyl spinor representations of SO(8). We can take the S-dual description leading, as shown in gure 30 (b), to the quiver theory SU(2) USp(4) SU(2). Thus we conjecture that these two theories are dual. We want to also check this using the superconformal index. We start with the SO(8) theory. The classical global symmetry is U( 1 ) USp(4)2 coming from the topological and avor symmetries. There is also a Z2 discrete symmetry coming from exchanging the two spinor representations. The analysis of [29] suggests that there is no enhanced symmetry so it seems to also be the quantum symmetry. We preform the calculation up to order x5 nding: IndexSO(8) = 1 + x2(1 + [10; 1] + [1; 10]) + x3 y[2](2 + [10; 1] + [1; 10]) (4.1) + x4 + x5 y[3](2 + [10; 1] + [1; 10]) + [35(4;0); 1] + [1; 35(4;0)] + [14; 1] + [1; 14] + [10; 10] + [5; 5] + [10; 1] + [1; 10] + [5; 1] + [1; 5] + 3) y[4](2 + [10; 1] + [1; 10]) + y[2]( [35(4;0); 1] + [1; 35(4;0)] + [35(2;1); 1] + [1; 35(2;1)] + [14; 1] + [1; 14] + 2 [10; 10] + [5; 5] + 4 [10; 1] + 4 [1; 10] + [5; 1] + [1; 5] + 4)) + O(x6) where we use [d1; d2] for the representation of dimension d1 (d2) under the rst (second) USp(4) symmetry. Since USp(4) has two 35 dimensional representations, both appearing in the index, we have also written their Cartan weight to distinguish between them. Note that the index is symmetric under the exchange of the two global USp(4)'s which is the manifestation of the discrete Z2 symmetry. Next we move to the SU0(2) USp (4) SU0(2) theory. The classical global symmetry is U( 1 )3 SU(2)2, but as we will show this is enhanced at least to U( 1 ) USp(4)2 by the SU(2)'s 1-instanton. There is also a discrete Z2 symmetry of exchanging the two SU(2) groups that matches the corresponding one in the SO(8) theory. Next, we evaluate the index of this theory to order x5. To that order we have contributions from the ( 1,0,0 ) + ( 0,0,1 ) + ( 1,0,1 ) + (2,0,0) + (0,0,2) - instantons. Using the fugacity spanning shown in t + c2 + 1 + 1 t 1 1 c2 Although we evaluated the index to order x5, we have written it only up to x3 to avoid over-clutter. From the x2 terms one can see the conserved currents of the classical global symmetry as well as ones provided by the ( 1,0,0 ) + ( 0,0,1 ) - instantons results in the enhancement of U( 1 )2 SU(2)2 ! USp(4)2. This matches the global symmetries of the two theories and one can see that also the indices match to order x3. We have also con rmed that the matching persists up to order x5. 4.1.2 Our next examples involves SO(N ) groups with N odd. As the webs for these theories plane with a stuck monodromy, performing S-duality is di cult. However, we can still overcome this by simply considering the guage theory on the NS5-branes. For example, consider the web of gure 33 (a) describing SO(7)+1V +2S. We can mass deform it as shown in gure 33 (b). From the point of view of the NS5-branes, this describes an SU0(2) gauging of the SCFT described by USp(4) + 1AS + 2F (see gure 33 (c)). The gauging is done into the topological symmetry of the USp(4) gauge theory, but as this theory has an enhancement of symmetry to SU(3) [5], we can rotate it so as to sit in the avor sector. So we conclude that the dual is AS + USp(4) SU0(2) where all that's left is to determine the angle of USp(4). Comparing global symmetries, we see that they match: the SO(7) theory having a UT ( 1 ) USpS(4) global symmetry while the quiver having a UT ( 1 ) SUAS(2) USp(4) one (here we have used the enhancement of U( 1 ) SUBF (2) ! USp(4) coming from the 1-instanton of the gauge SU(2) seen in the previous this web. One can see that it describes an SU(2) gauging of the web in (c). (c) The web for USp(4) + 1AS + 2F , where we have used that SO(5) = USp(4) under which the fundamental and antisymmetric representations of USp(4) are the spinor and vector representations of SO(5) respectively. example). We thus see that the dual must be AS + USp (4) SU0(2), as otherwise there would be an additional enhancement not expected in the SO(7) theory.3 We want to further test this duality by comparing the superconformal index. There are two problems with this calculation. One, due to the presence of the spinor matter, we cannot calculate the SO(7) instanton contribution so we can only calculate the perturbative part. Two, there is a problem calculating di-group instantons in the AS +USp (4) SU0(2) theory. Calculating instantons in USp + AS requires removing decoupled states where the full instanton partition function, Z, contains extraneous contributions that must be removed by hand. This case is well understood, and the form of the decoupled states was worked out in [30]. De ning Zc for the full instanton partition function with the extraneous contributions removed, we nd: 2 c (4.3) where c stands for the antisymmetric SU(2) fugacity, a for the USp(4) instanton fugacity, and SU(2)[2] is the character for the fundamental of the gauge SU(2) (which is a global symmetry from the USp(4) point of view). We also use P E[x] for the plethystic exponent of x, and this term in (4.3) gives the contribution of the decoupled states. One notes that they carry gauge charges under the SU(2) gauge group. These are responsible for the lack of enhancement, but also imply non-trivial interaction between these decoupled states and the SU(2) gauge group degrees of freedom. Therefore, while Zc should properly capture USp(4) or SU(2) instantons, we expect additional extraneous contributions for di-group instantons making these calculations unreliable. Bearing this in mind, we next state our result. We start with the SO(7) + 1V + 2S theory. The classical global symmetry consists of a topological U( 1 ), an SU(2) associated 3Again, the results of [29] suggests no enhancement in this case. with the vector and a USp(4) associated with the 2 spinors. We calculate the perturbative index to order x5 nding: IndexSO(7) = 1+x2(1+ [3; 1]+ [1; 10])+x3 ( y[2](2+ [3; 1]+ [1; 10]) + [1; 35(4;0)]+ [1; 14]+ [3; 10]+ [1; 10]+ [1; 5]+ [5; 1]+ [3; 1]+4 + y[2]( [1; 35(4;0)]+ [1; 35(2;1)]+ [1; 14]+2 [3; 10]+4 [1; 10]+ [1; 5] + [2; 1])+O(x6) (4.4) (4.5) (4.6) Now we wish to compare this to the index of AS + USp (4) SU0(2). We continue to use z; q for the fugacities of the gauge SU(2) bifundamental and topological symmetries while the rest of the fugacities are as in (4.3). To order x5 we get contributions from the ( 0,1 ) + (0,2) + ( 1,0 ) + (2,0) instantons, where only the (2,0) instantons contribute states charged under the USp(4) topological U( 1 ) (the ( 1,0 ) instantons are gauge-charged and only contribute through an invariant with the anti-instanton). We rst separate them out, since we expect these states to match the instantons of SO(7). For the others we nd: IndexAS+USp(4) SU(2) = 1 + x2 4 + z2 + 1 1 q + x3 One can see that the two indices match, and, indeed, we have checked that they match up to order x5. we get: Finally, we can consider the contributions of states charged under the USp(4) topological U( 1 ). To order x5, the only contributions we nd come from the (2,0) instanton where Index(A0S;2+) USp(4) SU(2) = x 5 a2 + 1 a2 1 c c + + O x 6 We expect this to match against instanton contribution of SO(7) + 1V + 2S, but unfortunately we cannot verify it by direct calculation. 4.1.3 As our nal example, we consider the gauge theory SO(9) + 1V + 2S. By arguments similar to the previous ones, we conjecture that the dual should be USp (4) USp0(4) + AS. The brane web for this theory is shown in gure 34 (a). We can mass deform it to the web of 34 (b). Looking from the NS5-branes point of view one can see that it describes a USp (4) gauging of the SCFT described by USp(4) + 1AS + 4F . Since the gauged symmetry is realized on the NS5-branes, it is instantonic from the D5-branes point of view. However which describes a USp(4) gauging of the web describing USp(4) + 1AS + 4F , shown in (c). That the gauging is done by a USp(4) group can again be seen by pulling the (1; 1) and (1; 1) 7-branes through the two D5-branes. The resulting pair of 7-branes can be interpreted as the S-dual of a resolved O7 -plane. dual is USp (4) symmetries match. the USp(4) + 1AS + 4F theory as an enhancement of UT ( 1 ) SOF (8) ! SO(10) [5] which we can use to rotate the gauging to the avor symmetry. Thus, we conclude that the USp0(4) + AS where the last angle was chosen so that the global Particularly, the SO(9) theory as a classical symmetry given by UT ( 1 ) USpS(4) while the USp quiver as a classical UT (1)2 SUBF (2) SUAS(2) global symmetry. However we nd that when = 0, for the USp group with the antisymmetric, there is an additional enhancement of UT ( 1 ) SUBF (2) ! USp(4). Thus, with this choice of angle, the global symmetries of the two theories agree again up to additional enhancements on either side.4 We next test this by calculating and comparing the superconformal index. There are two major limitations in this calculation. First we cannot calculate the SO(9) instanton contribution due to the presence of spinor matter. Second, we cannot reliably calculate di-group instantons for the USp2 theory. The reasons are the same as before: instanton counting for USp(4) + AS requires removing the contributions of decoupled states. The precise form of these decoupled states was worked out in [30], and in our case the removal is done by: c (4.7) where c is the antisymmetric fugacity, z the bifundamental, q the topological one for the group with the antisymmetric and USp(4)[5] is the character for the antisymmetric of the gauge USp (4) (which is a global symmetry from the USp(4) + AS point of view). Again The gauge charges imply additional corrections for di-group instantons are expected so (4.7) is insu cient for the evaluation of di-group instanton contributions. 4Like in the previous cases the analysis in [29] suggests no enhancement for the SO(9) theory. where we are working to order x4. ( 0,1 )+(0,2) instantons nding: IndexUSp(4)2+AS = 1 + x2 4 + z2 + Finally, we state our results. For the perturbative index of SO(9) + 1V + 2S we nd: IndexSO(9) = 1 + x2(1 + [3; 1] + [1; 10]) + x3 ( y[2](2 + [3; 1] + [1; 10]) (4.8) + [2; 10]) + x4 y[3](2 + [3; 1] + [1; 10]) + y[2] [2; 10] + [1; 35(4;0)] + 2 [1; 14] + [3; 10] + [1; 10] + 2 [1; 5] + [5; 1] + [3; 1] + 4) + O(x5) To this order, in the USp (4) USp0(4) + AS theory, we get contributions of the + x3 + O x 4 1 2 + z2 + q + (4.9) HJEP03(216)9 where the fugacities are the ones used in (4.7). One can see that the indices match to the order shown, and we have further checked that the x4 order also matches. 5 In this article we studied brane webs in the presence of an O5-plane. This supports the existence of a wide class of new xed points, and can be used to further study various aspects of these theories, such as dualities and enhancement of symmetry. The gauge theories that can be constructed in this way include alternating linear quivers of SO and USp groups as well as D shaped quivers of SU groups. We have also argued that one can engineer SO(N ) groups with spinor matter, where the spinor matter is thought to arise non-perturbatively. We would like to see if further evidence can be found for this. It will be interesting to further study the gauge theory leaving on the D1-brane associated with the SO(N ) instanton. These gauge theories are known to play an important role in instanton counting, and so may lead to a better understanding of instanton counting for SO groups with spinor matter, which is currently unknown. Finally, when su cient avors are introduced a 5d gauge theory may go to a 6d N = (1; 0) SCFT, instead of a 5d SCFT. A well known example is SU(2) + 8F which has the 6d rank 1 E-string theory as its UV xed point [31]. These sort of relations have been studied extensively recently for theories with ordinary brane web representations [32{35]. This phenomenon appears to also occur for some of the theories considered in this article, as seen for example by the apparent presence of a ne global symmetries [29]. It will be interesting if this can be better understood, and if the 6d N = (1; 0) SCFT's that these theories go to can be uncovered. Acknowledgments I would like to thank Oren Bergman, Soek Kim, and Hee-Cheol Kim for useful comments and discussions. G.Z. is supported in part by the Israel Science Foundation under grant no. 352/13, and by the German-Israeli Foundation for Scienti c Research and Development under grant no. 1156-124.7/2011. A Index computation This appendix provides a brief review of the 5d superconformal index, and it's calculation using localization. The superconformal index is a sum of the BPS operators of a theory where if two or more operators can merge to form a non-BPS multiplet, they sum to zero. This is a useful quantity as it is invariant under continuous deformations since the spectrum of BPS operators can only change via this merging. Speci cally for the case of 5d N = 1 SCFT, the representations of the superconformal group are labeled by the highest weight of its SOL(5) SUR(2) subgroup. We will call the two weights of SOL(5) as j1; j2 and those of SUR(2) as R. Then following [16] the index is: (A.1) (A.2) I = Tr ( 1 )F x2 (j1+R) y2 j2 qQ where x; y are the fugacities associated with the superconformal group, while the fugacities collectively denoted by q correspond to other commuting charges Q, generally avor and topological symmetries. For a 5d gauge theory the index can be evaluated by localization where it is given by [16]: I = Z d ZpertjZNekj2 where the integral is over the Cartan subalgebra of the gauge group. The terms Zpert and ZNek are the contributions of the perturbative and instanton sectors respectively. The perturbative contribution, Zpert, can be easily evaluated using the results of [16]. The instanton contribution, also known as the 5d Nekrasov partition function [36], is harder to evaluate. In general Zpert is expanded in a power series in the instanton fugacity, each term providing the contribution of the associated instantons. These terms can in turn be evaluated by a contour integral where the integrand receiving contributions from the various matter and gauge content of the theory. The expressions for most of these contributions that we need have appeared elsewhere, notably [9, 11, 12, 16, 30], and we won't repeat them here. The only exception being the SO USp bifundamental and half-bifundamental whose expressions we provide below. In addition one also has to supplement this with a pole prescription detailing which poles are inside the contour. A good review of these is given in [30]. Finally, the evaluation of the Nekrasov partition function is sometimes plagued with the contributions of extraneous degrees of freedom that must be removed by hand. These can materialize in the partition function as a breakdown of x ! x1 invariance, which must be obeyed as it is part of the superconformal algebra. Another way these can appear in the partition function is as an in nite tower with representations of increasing dimension under a avor symmetry. Examples and ways of dealing with the former can be found in [9, 17, 18], and those for the latter in [16, 30]. A.1 SO USp In this subsection we state the contributions of the matter content to the Nekrasov partition function in the SO USp formalism. The gauge contributions for both SO and USp groups were already written elsewhere so we will not restate them. We concentrate on the contributions of bifundamentals and half-bifundamentals. In 4d these were considered in [37]. The 5d results in the O+ sector can be derived by lifting the 4d ones, but for the O sector one has to derive these directly using the methods in [38]. We start with the contribution of a full SO(M ) USp(2N ) bifundamental hyper to the integrand for the (k; K) instanton. The dual gauge group in this case is USp(2k) O(K). We shall employ the notation M = 2n1 + 1 and K = 2n2 + 2 where = 0; 1. The O(K) group has two disconnected parts, denoted as the O+ and O , which must both be taken into account. We also separate the O case to two distinct cases depending on whether k is even or odd. Throughout this subsection we use the fugacities: z for the bifundamental U( 1 ), a for the SO(M ) gauge symmetry, b for the USp(2N ) gauge symmetry, u for the USp(2k) dual gauge group, and v for the O(K) dual gauge group. The complete HJEP03(216)9 2 n1 Y i=1 1 Y z+ z1 umy umy 1 ai ai m=1 z+ z1 umx umx 1 z+ z1 uym z+ z1 um x x y 3 2 5 2 n2 4 Y j=1 N;k Y n;m=1 1 Y k;n2 z+ z1 umvjy umvjy 1 m;j=1 z+ z1 umvjx umvjx 1 1 3 1 1 1 vj vj 5 p z p 1 bnum bnum 1 2 nY1;n2 i;j=1 z um z+ z1 umvj y z+ z1 umvj x bn x umvj umvj for the O+ part. ZBSOF UOSp = Y pz+ p1 2 4 N;k Y n;m=1 n1 i=1 n2 Y z j=1 bnum bnum Y k;n2 z+ z1 umvjy umvjy 1 m;j=1 z+ z1 umvjx umvjx 1 1 Y z+ z1 +umy+ um1 y z+ z1 + uym + uym z+ z1 + uxm + uxm 1 vj vj 5 3 1 n1;n2 Y i;j=1 z um z+ z1 umvj y z+ z1 umvj x bn y x umvj umvj aivj aivj 1 z vj ai z+ z1 vujmy vjy um vjx z+ z1 vujmx um vj z+ z1 yum vj z+ z1 xum yum vj xum vj 1 z aivj aivj 1 1 ai vj z vj ai z+ z1 vujmy vjy um vjx z+ z1 vujmx um vj z+ z1 yum vj z+ z1 xum yum vj xum vj (A.3) (A.4) ZBSOF UESp = Y 2 4 z N;k Y n;m=1 1 n1 i=1 n2 1 Y j=1 1 z + bnum Y k;n2 1 z + z1 umvjy m;j=1 z + z1 umvjx part and even K. 2 n1 Y i=1 n1;n2 Y i;j=1 k;n2 Y n1 i=1 n1;n2 Y i;j=1 k;n2 Y pai ai + 1 pai 1 ai 1 um + u1m m;j=1 um + u1m pai + pai 1 ai + vj um + u1m m;j=1 um + u1m z + a 2 i 1 1 bnum umvjy 1 1 umvjx 1 a 2 i Y z2 + z12 u2my2 m=1 z2 + z12 u2mx2 3 1 n1;n2 1 5 z + 1 Y i;j=1 z + z1 umvj y z + z1 umvj x z + bn 1 x umvj umvj aivj 1 1 u2my2 1 aivj z2 + z12 z2 + z12 z + 2 um y2 u2m x2 u2m x2 u2m 1 z vj ai vj z + z1 vujmy vjy vjx z + z1 vujmx um vj z + z1 yum vj z + z1 xum yum xum ZHSOBFU+Sp = 4 ZHSOBFUSOp = Y is to assume x; p; d at the end of the calculation. The contributions of this bifundamental also add additional poles to the integrand. The prescription for dealing with them follows directly from the work of [30]. Doing the following rede nitions in the above expressions: p = z1x and d = xz , the correct prescription 1 taking all the poles within the circles and reset p = z1x ; d = xz only The generalization to half-bifundamentals follows straightforwardly, similarly to the 4d case done in [37]. To avoid the need to add an half-fundamental, we specialize to the case 1 = 0. When taking the limit of a massless half-bifundamental, that is z ! 1, the expressions (A.4){(A.6) become a total square, and the expression squared is identi ed with the contribution of a half-bifundamental. Explicitly these are given by: for the O+ part. um + u1m m=1 um + u1m N;k Y n;m=1 1 vjy 1 vjx um + um + u1m um + u1m N;k Y n;m=1 1 vjy 1 vjx um + Y 1 vj y vj x Y 1 vj vj y vj x m=1 um + u1m + x + x1 x x 1 3 2 1 5 1 vj vj x vj y vj x 1 um bn vj x vj y vj x vj 1 bn 1 bn (A.5) (A.6) (A.7) n1 i=1 n1;n2 1 Y i;j=1 1 ai + 1 Y k;n2 1 um + u1m Y u2m + u12m m=1 u2m + u12m vj y vj x x2 1 y2 1 x2 um + 1 N;k Y n;m=1 1 vjy 1 vjx vj vj y x for the O part and even K. In some cases the contributions add additional poles to the integrand, and the prescription then follows from the previous case where one de nes p = x1 in the denominators of (A.7){(A.8). The prescription is then to assume x; p of the calculation. 1, and set p = x1 only at the end Open Access. Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. Quantum 5D super-Yang-Mills, JHEP 01 (2011) 083 [arXiv:1012.2882] [INSPIRE]. [3] N. 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Gabi Zafrir. Brane webs and O5-planes, Journal of High Energy Physics, 2016, 109, DOI: 10.1007/JHEP03(2016)109