Anomalies involving the space of couplings and the Zamolodchikov metric

Journal of High Energy Physics, Dec 2017

The anomaly polynomial of a theory can involve not only curvature two-forms of the flavor symmetry background but also two-forms on the space of coupling constants. As an example, we point out that there is a mixed anomaly between the R-symmetry and the topology of the space of exactly marginal couplings of class S theories. Using supersymmetry, we translate this anomaly to the Kähler class of the Zamolodchikov metric. We compare the result against a holographic computation in the large N limit.

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Anomalies involving the space of couplings and the Zamolodchikov metric

HJE Anomalies involving the space of couplings and the Zamolodchikov metric CFT Correspondence 0 1 0 Kashiwa , Chiba 277-8583 , Japan 1 Kavli Institute for the Physics and Mathematics of the Universe, University of Tokyo The anomaly polynomial of a theory can involve not only curvature two-forms of the avor symmetry background but also two-forms on the space of coupling constants. As an example, we point out that there is a mixed anomaly between the R-symmetry and the topology of the space of exactly marginal couplings of class S theories. Using supersymmetry, we translate this anomaly to the Kahler class of the Zamolodchikov metric. We compare the result against a holographic computation in the large N limit. Anomalies in Field and String Theories; Supersymmetry and Duality; AdS- - Zamolodchikov 1 Introduction and summary 2 3 the space of couplings [1{3].1 One example can be constructed as follows. Take an arbitrary target space M and a U(N ) gauge eld A on it. Consider a D-dimensional free theory of a dynamical chiral fermion in the fundamental of U(N ) coupled to a background scalar which is a map to M in the following manner: the fermion is minimally coupled to the pull-back under of the U(N ) eld A. Of course the theory has the anomaly polynomial AD+2 = A^(T X) tr ei (F) (1.1) where X is the worldvolume of the theory and F is the curvature two-form on M. When the scalar is considered dynamical, this is known under the name of the -model anomaly, and makes the theory ill-de ned when not canceled. When is considered as a background eld, this anomaly polynomial involves di erential forms on the space M of couplings, and serves as one of the characteristic properties of the theory, just as the ordinary 't Hooft anomalies do. To readers who found the example above rather arti cial, let us provide a more meaningful case. In [5] Gaiotto introduced a large class of 4d N =2 theories obtained by compactifying a 6d N =(2; 0) theory on a Riemann surface C, possibly decorated with punctures. Here let us consider a simple case where C is of genus g without any puncture. This construction gives rise to a family of 4d N =2 superconformal eld theories (SCFTs), now 1Strictly speaking in these papers the scalars were considered dynamical. But before 't Hooft [ 4 ] the gauge elds in the anomalies were also mostly considered dynamical. In this sense the anomaly involving the space of couplings is known from the early days. { 1 { known as class S theories, whose space of exactly marginal couplings is the moduli space Mg of the genus-g Riemann surfaces. We show below that this theory has a hithertounappreciated term in the anomaly polynomial of the form A6 c2 (R) h ! i 2 (1.2) where c2(R) is the second Chern class of the background SU(2)R gauge eld and [!] is a degree-2 cohomology class on Mg, which can be determined from the known anomaly polynomial of the 6d N =(2; 0) theory [6{8]. Now, a mixed anomaly between the Weyl transformation and the Kahler transformation of the space of exactly marginal couplings of general 4d N =2 SCFTs was described in [9{11].2 When one reads their derivation carefully, one nds that their analysis already implies an anomaly of the form (1.2) above, with an added bonus that [!] is proportional to the cohomology class [!Z] of the Kahler form !Z of the Zamolodchikov metric. Turning the logic around, this means that we can easily x [!Z] in terms of the anomaly polynomial of the 6d N =(2; 0) theory. Finally, we note that the Zamolodchikov metric and therefore !Z is computable in the large N limit by means of the AdS/CFT correspondence [13] using the holographic dual of the class S theories [14, 15]. This is known to be proportional to the standard Weil-Petersson metric on Mg,3 but the precise proportionality coe cient has not been computed to the author's knowledge. We will show below that the class of the Kahler form computed holographically is compatible with the computation from the anomaly as above, using a classic mathematical result by Wolpert [16, 17]. The rest of the note is devoted to implement the computations outlined above: in section 2 we compute A6 of the class S theory from the anomaly of the 6d N =(2; 0) theory and then use it to compute the Kahler class [!Z] of the Zamolodchikov metric. Then in section 3 we compute the Zamolodchikov metric using holography, determine the proportionality coe cient with respect to the standard Weil-Petersson metric, and compare it against the result in section 2. 2 2.1 Field theoretical computations The 4d anomaly from the 6d anomaly We start from the anomaly polynomial [6{8] of the 6d N =(2; 0) theory of type G = An 1; Dn; E6;7;8: h_GdG 24 rG 48 A8 = p2(N Y ) + p2(N Y ) p2(T Y ) + (p1(N Y ) p1(T Y ))2 : (2.1) 1 4 2There they conjectured that the Kahler potential would be globally well de ned but this was answered 3The author does not know who originally noticed this; he forgot from whom he rst learned the fact. This is surely a common knowledge among those who study class S theories using AdS/CFT. { 2 { Here Y is the worldvolume of the theory, T Y is its tangent bundle, N Y is the SO(5) Rsymmetry bundle, and p1, p2 are the Pontryagin classes; h_G, dG and rG are the dual Coxeter number, the dimension and the rank of the Lie algebra of type G. We use the convention that the anomaly polynomial gives the (D + 2)-dimensional phase exp(RYD+2 2 iAD+2). The N =2 class S theory of our interest is obtained by compactifying the 6d theory on a Riemann surface C of genus g without any punctures so that we introduce a nonzero curvature to the subgroup SO(2) SO(5) of the R-symmery which cancels the curvature of C. This means that the Chern roots of N Y is 2 and t where are the Chern roots of the SU(2)R background eld and t is the c1 of the tangent bundle of C. We note that c2(R) = 2 . Using p1 = P i i2 and p2 = Pi<j i2 j2 when the Chern roots are i, we easily get HJEP12(07)4 A6 h_GdG + 6 rG 12 c2 (R) Z C t2: In the last factor, t is considered as the c1 of the relative tangent bundle of C over Mg (i.e. the tangent bundle of the universal bundle U where C , ! U Mg minus the pull back of the tangent bundle of Mg). Then t2 is a 4-form on U , and we obtain a 2-form on Mg by integrating over the ber C. 2.2 Finding the Kahler potential Suppose now that a given 4d N =2 SCFT has a space of exactly marginal couplings parameterized by M with local complex coordinates I marginal operators so that they enter in the deformation of the Lagrangian as . We normalize the corresponding exactly following the convention of [9{11]. We then de ne the Zamolodchikov metric gIZJ by the formula to be accompanied by a shift of the contact terms In [11] the authors identi ed that the Kahler transformation K 7! K + F + F needs S 7! S + 1 where we used the N =2 supergravity super elds as used in [11]. Using the component expansion given in (5.5) of [18], we see that this shift contains the terms of the form that the second Chern class is given by Z hOI (x)OJ (0)i = gIJ : x8 c2(R) = d4x 16 2 R(V )abijR~(V )abij { 3 { (2.2) (2.3) (2.4) (2.5) (2.6) (2.7) we see that K 7! K + F + F is accompanied by The Kahler form of the Zamolodchikov metric is where A = Therefore, K 7! K + F + F does HJEP12(07)4 A 7! A + This means that the shift of the contact term (2.8) is exactly the gauge variation one obtains from the anomaly polynomial 1 2 A6 c2 (R) !Z 24 2 Choosing the type to be AN 1 and taking the large N limit, we nd that our results (2.2) and (2.11) imply as cohomology classes. 3 Holographic computations !Z 2 4N 3 Z C t 2 holds as di erential forms, not just as cohomology classes. !WP 2 2 = Z C t 2 { 4 { Weil-Petersson metric. First let us recall the Weil-Petersson metric on the moduli space Mg of the genus-g Riemann surface. Given Beltrami di erentials I on a Riemann surface C, we de ne the Hermitean structure on them by It is a classic mathematical result by Wolpert [16, 17] that the relation hIWJP = Z C I J dA !WP = gIWJPd I d J = 12 hIWJPd I d J : where dA is the area form of curvature form of the Weil-Petersson metric is given by 1, so that RC dA = 4 (g 1). Then the Kahler Maldacena-Nun~ez solution. The holographic dual of the class S theory of type AN 1 on a genus-g Riemann surface is given in [14, 15]. The 11d metric is of the form where ds2AdS5 is the AdS5 metric of unit radius, ds2H2= is the metric on the Riemann surface C represented as a quotient of the Poincare disk of curvature radius 1 by a discrete group , the coordinates , , and parameterize the internal space which is topologically of the form S4, and W := 1 + cos2 . Our convention is that the 11d action is space of genus-g Riemann surfaces. The space Mg appears as the target space of the massless scalars in ve dimensional supergravity, which in turn can be identi ed with the space of exactly marginal couplings of the dual SCFT. Reduction to ve dimensions. Let us proceed with our computation. Integrating over the four-sphere part, we have where the indices are appropriately placed. This means that upon reduction to 5d one nds S5 4 (g 16 G(N7) { 5 { with the 7d action where which means that We now reduce it further to ve dimensions, including the deformation of the Riemann surface. In general, under a small deformation g 7! g + h, we have Now note that the Beltrami di erential := I zzI deforms the internal metric as jdzj2 7! jdz + dzj2 g 7! g + h; h := 2( + ) (3.5) (3.6) (3.7) (3.8) (3.9) (3.10) (3.11) (3.12) Translation to the dual SCFT. At this point, we can use the formula [19] for the central charge a c to compute which reproduces the standard result. We are more interested in the Zamolodchikov metric on Mg, for which we use the formula for the two-point function under AdS/CFT given in [20]. The formula says that given the action HJEP12(07)4 for a real scalar in ve dimensions, the corresponding operator has the two-point function (3.13) (3.14) (3.15) (3.16) (3.17) (3.18) with the caveat that the deformation is introduced via the coupling without an additional factor of 2 in the denominator as in (2.3). Carefully collecting all the factors, one nds meaning that gIZJ = 16 G(N7) RAdS5 2 2hIWJP( 2)2 = 4N 3 gIWJP 1 3 24 !Z 2 4N 3 !WP 2 2 = 4N 3 Z C t 2 c = ; in the large N limit. This is consistent with the result (2.12) we obtained from the consideration of the anomaly in the last section. Acknowledgments The author thanks D. R. Morrison for discussions, and T. 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Yuji Tachikawa. Anomalies involving the space of couplings and the Zamolodchikov metric, Journal of High Energy Physics, 2017, 140, DOI: 10.1007/JHEP12(2017)140