Anomaly of strings of 6d \( \mathcal{N}=\left(1,0\right) \) theories

Journal of High Energy Physics, Nov 2016

We obtain the anomaly polynomial of strings of general 6d \( \mathcal{N}=\left(1,0\right) \) theories in terms of anomaly inflow. Our computation sheds some light on the reason why the simplest 6d \( \mathcal{N}=\left(1,0\right) \) theory has E 8 flavor symmetry, and also partially explains a curious numerology in F-theory.

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Anomaly of strings of 6d \( \mathcal{N}=\left(1,0\right) \) theories

Received: November Published for SISSA by Springer Open Access 0 1 c The Authors. 0 1 0 Kashiwa , Chiba 277-8583 , Japan 1 Kavli Institute for the Physics and Mathematics of the Universe, The University of Tokyo We obtain the anomaly polynomial of strings of general 6d N = (1, 0) theories in terms of anomaly inflow. Our computation sheds some light on the reason why the simplest 6d N = (1, 0) theory has E8 flavor symmetry, and also partially explains a curious numerology in F-theory. Anomalies in Field and String Theories; Field Theories in Higher Dimensions - Anomaly of strings of 6d = (1, 0) theories 1 Introduction 2 Derivation and checks 3 Implications 2.1 2.2 3.1 Anomaly formula from the inflow Comparison with gauge theory description ADE classification of 6d N = (2, 0) theories the seminal work [1]. For example, we now have a better understanding of the landscape of such theories [2–6] and have a general formula for their anomaly polynomials [7, 8]. An essential feature of these theories is that they have supersymmetric strings charged under self-dual tensor fields. The properties of these strings have received much attention, but the analysis has so far been mostly restricted to the case when the description in terms of the refined topological string and/or the 2d gauge theory is available [9–17]. In this note, we provide a formula for the anomaly polynomial of the 2d worldsheet theories of these strings. The only input required is the anomaly of the bulk 6d theory, and therefore our formula applies generically. Our formula is given in (2.4), (2.5). When a 2d gauge theory description of the worldsheet theory of the strings is known, we can compare the outcome of our main formula (2.4), (2.5) and the anomaly polynomial computed from the gauge theory spectrum. We will see below that these two computations indeed do match. When such a 2d gauge theory description is not known, our formula should help us look for one. Prime examples where the 2d gauge theories are not known are the following. Take an F-theory compactification on a complex two-dimensional base with an isolated P1 of nontrivial gauge algebra g on the curve. no additional matter on the curve; for a derivation, see e.g. [18].1 This configuration gives 1As an aside, let us point out curious coincidences concerning table 1: this is an exact subset of table 4 theories were studied. The two tables in [19, 20] contain su(2) and g2 in addition, and k4d there equals n ∨ = 3(n − 2). have been made [13, 15] but not much is known in other cases.2 Our formula (2.4), (2.5) gives at least the anomaly polynomials of these strings, and also explains a curious nuh∨ = 3(n − 2) that is evident in table 1, as we will see below. In the rest of the note, we first provide the derivation of the formula (2.4), (2.5) using the anomaly inflow in section 2. We also provide checks by comparing with the anomaly polynomials computed from 2d gauge theories when available. Then we discuss some of the implications of our formula in section 3 and conclude. Note added. In the final stage of the research leading to this note, a paper [26] appeared, in which the authors also presented our main formula (2.4), although their main concern was the elliptic genera of the strings. In [26] a 2d gauge theory description was also given for the case n = 3. Derivation and checks Anomaly formula from the inflow Review of the tensor branch effective action. Let us briefly recall the tensor branch here. This series of groups with an addition of the empty group, namely ∅ , su(2) , su(3) , g2 , so(8) , f4 , e6 , e7 , e8 , is known as Deligne’s exceptional series of groups, from the papers by Deligne [21, 22] in 1996. The relation of the F-theory list of groups and Deligne’s exceptional series was already noted in a paper by Grassi and Morrison [23], see lemma 7.4 there. It is to be noted that the same series of groups has also appeared in a paper by Mathur, Mukhi and Sen [24] from 1988 when 2d RCFTs with only two characters were systematically looked for. Is there a single unifying principle behind these appearance of the same sequence of groups in various corners of string theory? (Deligne’s exceptional series of groups was also independently found by Cvitanovi´c, see section 21.2 of [25] for the history of multiple independent (re)discoveries of this series. The authors thank L. Rastelli for the information.) 2But see Note added below. plus massive dynamical stringy excitations. Integrating out the massive modes, a part of the effective action is given as dBi ∧ ⋆dBj + Bi ∧ Ij . Here Bi is the self-dual 2-form field normalized so that its field strength is quantized in integer values.3 The strings in the 6d theory are charged under those 2-form fields and the symmetric, positive definite and integral. In the following, indices are raised/lowered by Ii is known to have a concrete form given by ηia Tr Fa2 − (2 − ηii)p1(T ) + h∨Gi c2(I) as derived in [7, 28]. Here the sum over the indices j and a is taken, but the index background tensor multiplets.4 p1(T ) is the first Pontrjagin class of the tangent bundle of the 6d spacetime and c2(I) is the second Chern class of the background SU(2)I R-symmetry charge of the string is specified by a vector Qi with integral entries. The worldvolume theory on the string has the global symmetries SU(2)L × SU(2)R × SU(2)I × Qa Ga. Here SU(2)I × Qa Ga comes from the R, gauge and global symmetries of the bulk 6d theory.5 (2, 1, 2)+ + (1, 2, 2)− under SU(2)L × SU(2)R × SU(2)I , with the subscript ± denoting the chirality. We take I4 = ηij QiQj c2(L) − c2(R) + ηij QiIj . 3Because Bi is self-dual, it is imprecise to write the kinetic term as in (2.1), but we will see that it is for dynamical ones. convenient to include it here for the inflow computation. The indices a, b, . . . are for both dynamical and background fields, while the indices i, j, . . . are only Here we should not confuse SU(2)R with the R-symmetry of 6d supersymmetry. If we assume the general validity of the concrete formula (2.2), this can be more explicitly written as I4 = p1(T ) − 2c2(L) − 2c2(R) + h∨Gi c2(I) . In deriving (2.4) and (2.5), we decompose the 6d p1(T ) as 2d p1(T ) + p1(N ) and we use In the following, we will derive the formula (2.4) using the inflow computation. We will then check (2.5) against concrete examples for which 2d gauge theory description is modified to be Inflow argument. Here we perform the anomaly inflow computation for the self-dual string, using a method pioneered by [29]. The anomaly inflow of self-dual strings was In the presence of the string, the Bianchi identity for the 3-form field strength is Hi = Qi 32 which correctly reproduces the formula (2.4). where Qi is the charge of the string. Its solution is given by where e(30) is the global angular form of the S3 bundle of the tubular neighborhood of the To compute the anomaly polynomial of the 2d theory, it is convenient to use dHi ∧ Hj + Hi ∧ Ij , instead of (2.1). Here Y7 is an auxiliary 7d manifold bounding the physical 6d spacetime. We also extend the worldsheet of the string to Y7 and denote it as M3. In the presence of the string, the most singular term in (2.8) is given as Anomaly inflow tells us that integrating out (2.9) we obtain the anomaly on the string which just reduces the integral over Y7 to M3. The result is explicitly write the representations of the fermions in the multiplets. O(Q) is the gauge symmetry while the other symmetries are global. In the computation of anomalies, we should note that all the fermions listed above are real. The computation of the contribution from the kinetic term of (2.1) or equivalently the first term of (2.8), in the form presented above, involves some amount of hand-waving, due to the self-dual nature of the tensor fields. A more careful derivation, following the one presented in [33] in the case of D3-branes coupled to a self-dual 5-form, gives the same Comparison with gauge theory description is known, we can easily compute its anomaly polynomial by counting the number of multiplets.6 In this subsection, we provide further pieces of evidence for the formula (2.5) by checking the agreement. On the one hand, using our formula (2.5), we see the anomaly 4-form of the bound state of Q E-strings is given as IE-string(Q) = c2(L) − Q2 − Q c2(R) − Tr FE28 − On the other hand, the matter content of the gauge theory was determined in [12] and is summarized in table 2. Noting that all the fermions in the table are real and we have to multiply 1/2 in the anomaly computation, we can check that the SU(2)L,R,I anomaly matches with (2.11). The anomaly of the E8 global symmetry also matches under the assumption that the the fundamental representation of SU(2) gives the anomaly − 2 trfund F numerical factor relating Tr F 2 and trfund F 2 for various groups is summarized e.g. in the appendix of [7]. 2 = − 41 Tr F 2 = −c2 SU(2) . The up those contributions, we reproduce the coefficient of p1(T ) in (2.11). A1 N = (2, 0) theory. According to our formula (2.5), the anomaly on the charge Q I4M-string(Q) = Q2 c2(L) − c2(R) + Q c2(I) − c2(F ) . Both SU(2) symmetries are realized on the worldsheet theory on M-string and appear in the anomaly polynomial (2.12). The gauge theory description of the worldsheet theory [9] is listed in table 3. It is straightforward to check that the counting of multiplets reproduces (2.12). n = 4 minimal 6d N = (1, 0) theory. According to (2.5), the anomaly 4-form of the In=4(Q) = (2Q2 − Q)c2(L) − (2Q2 + Q)c2(R) + Q Tr FS2O(8) + 4 The gauge theory on the worldsheet is determined in [13]. The matter content and representations are given as in table 4. We can check that the multiplets in table 4 correctly reproduce the anomaly (2.13). 6d string chains. engineered by an F-theory compactification on a base with a linear chain of P1’s with selfdetermines ranks of gauge groups, while precise matter content and types of gauge groups depend on which 6d theory we consider. For these 2d theories, we can also check the agreement in anomalies computed by the formula (2.5) and the multiplet counting. As a simple example, let us compute the SU(2)I Sp(Q) × SO(8) is in fact a half-hypermultiplet. other hand, the multiplets in the quiver description having a non-trivial SU(2)I anomaly are U(Qi) vectors, U(Qi)-adjoint hypers and U(Qi) × SU(k)i-bifundamental hypers [15]. However, the contributions from vectors and adjoint hypers cancel out, and the bifunda ADE classification of 6d N = (2, 0) theories Let us start by reviewing a nice argument by Henningson [30] for the ADE classification points on its tensor branch. There will be strings charged under these tensor multiplets. Given two strings with charges Q~ and Q~ ′ respectively, let us write the Dirac pairing as The Dirac quantization law demands that we have hQ~ , Q~ ′i ∈ Z. Let us consider a single string with charge Q~ , and let us determine the term proportional to c2(L), c2(R), p1(T ) of the anomaly polynomial. Since the string breaks translational invariance and supertranslational invariance, there are four bosonic zero-modes and eight chiral Majorana fermionic zero-modes, forming a hypermultiplet of the worldsheet under SU(2)L × SU(2)R × SU(2)I , with the subscript ± denoting the chirality. In total, the anomaly polynomial is = c2(L) − c2(R) . The crucial assumption in [30] was that the worldsheet theory is given purely by these Nambu-Goldstone zero modes. Then the anomaly (3.2) needs to be reproduced from the anomaly inflow. At this point we do not know the 6d Green-Schwarz coupling. We just assume a generic one dHi = cip1(T ) where we neglected contributions from the 6d R-symmetry c2(I) since they do not affect the 2d anomaly terms we are considering here. Using our inflow formula (2.4), we obtain c2(L) − c2(R) + hQ~ , ~ci p1(T ) − 2c2(L) − 2c2(R) . Comparing (3.2) and (3.4), we find that we need hQ~ , Q~ i = 2 , and at the same time we conclude ci = 0 in (3.3). generated by vectors whose length squared is two. This condition is known to be equivalent to the fact that the charge lattice is a simply-laced root lattice, and therefore it has an ADE smallest theory somehow has E8 flavor symmetry. Why is that? Let us try to mimic the argument recalled in the previous subsection. Again, we restrict attention to the terms proportional to c2(L), c2(R) and p1(T ) in the anomaly polynomial. theory as small as possible, let us assume that the Dirac pairing is given by and there is no dynamical gauge field on the tensor branch. As for the Green-Schwarz term, we assume the validity of the general formula dH = I = find the inflow = c2(L) + c2(I) − tersection number of the canonical divisor and the genus-0 curve producing the tensor multiplet [28]. A purely field-theoretical derivation of (3.7) is not known yet to the authors’ knowledge, but it should not be impossible to find one. c2(L) − c2(R) − p1(T ) − 2c2(L) − 2c2(R) + c2(I) On the string worldsheet, there are bosonic and fermionic zero modes coming from the breaking of the bulk translational and supertranslational symmetry. They form a Comparing (3.8) and (3.9), we know that there necessarily are some additional degrees of freedom on the worldsheet since there is a mismatch in the gravitational anomaly by Assuming that the partition function of the string worldsheet theory is well-defined up character. This forces us to choose the E8 current algebra of level one. We cannot say that the argument above is a derivation of the E8 flavor symmetry, but it does at least indicate that the E8 symmetry needs to arise automatically. F-theory was recalled in section 1. We will continue to use the same symbols there. Using the 6d anomaly polynomial computed in [7], we see that a charge Q string has the anomaly Qh∨G I4(n, Q) = nQ2 − (n − 2)Q c2(L) − (n − 2)Q nQ2 + (n − 2)Q p1(T ) + Qh∨Gc2(I) by applying our main formula (2.5). We know that an instanton configuration of g vector multiplet is charged under the tensor field, such that the instanton number is identified with the charge Q of the instantoninstanton moduli space of gauge group g of instanton number Q. This has a quaternionic dimension h∨Q. ADHM construction of charge-Q so(8) instanton. This suggests that, even for other n in moduli space of gauge algebra g. Now, moving along the Higgs branch does not break SU(2)I symmetry and the diffeomorphism symmetry of the worldsheet. Therefore the terms proportional to p1(T ) and c2(I) can be obtained straightforwardly at the generic point on the Higgs branch. There, zero modes and the same number of chiral fermionic zero modes. From this, we see that the anomaly polynomial should contain the terms Comparing with (3.10), we see that we need to have h∨ = 3(n − 2) . This explains the curious numerology (1.1) pointed out in section 1. The authors thank Noriaki Watanabe for helpful discussions, and Sunil Mukhi for discussions concerning footnote 1. HS is partially supported by the Programs for Leading Graduate Schools, MEXT, Japan, via the Leading Graduate Course for Frontiers of Mathematical Sciences and Physics. HS is also supported by JSPS Research Fellowship for Young Scientists. YT is supported in part by JSPS Grant-in-Aid for Scientific Research No. 25870159, and in part by WPI Initiative, MEXT, Japan at IPMU, the University of Open Access. This article is distributed under the terms of the Creative Commons 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. [1] J.J. Heckman, D.R. Morrison and C. Vafa, On the classification of 6D SCFTs and generalized ADE orbifolds, JHEP 05 (2014) 028 [Erratum ibid. 06 (2015) 017] [arXiv:1312.5746] [INSPIRE]. JHEP 02 (2015) 054 [arXiv:1407.6359] [INSPIRE]. Fortschr. 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Hiroyuki Shimizu, Yuji Tachikawa. Anomaly of strings of 6d \( \mathcal{N}=\left(1,0\right) \) theories, Journal of High Energy Physics, 2016, 165, DOI: 10.1007/JHEP11(2016)165