Anomaly polynomial of E-string theories
Kantaro Ohmori
1
Hiroyuki Shimizu
1
Yuji Tachikawa
0
1
Open Access
c The Authors. Article funded by SCOAP
0
Institute for the Physics and Mathematics of the Universe, University of Tokyo
, Kashiwa, Chiba 277-8583,
Japan
1
Department of Physics, Faculty of Science, University of Tokyo
, Bunkyo-ku,
Tokyo 133-0022, Japan
We determine the anomaly polynomial of the E-string theory and its higherrank generalizations, that is, the 6d N = (1, 0) superconformal theories on the worldvolume of one or multiple M5-branes embedded within the end-of-the-world brane with E8 symmetry.
1 Introduction 2 3
Computations
2.1 Chern-Simons terms of M-theory
2.2 Anomalies of M5-branes
2.3 Anomalies of the E8 end-of-the-world brane
2.4 E-string anomalies Checks
3.1 Terms that do not involve the normal bundle
3.2 Coefficient of (Tr F 2)2
3.3 Behavior when Q = 1 4 Conclusions and future directions A Global angular forms
Introduction
In the last few years, there has been a significant progress in our understanding of 6d
N = (2, 0) superconformal theories and their compactifications to lower dimensions,
starting with [1]. The dynamics of 6d N = (1, 0) superconformal theories, however, remains
quite mysterious.
One class of 6d N = (1, 0) theories is obtained by taking the decoupling limit of Q
coincident M5-branes embedded within the E8 end-of-the-world brane of M-theory. When
Q = 1, the theory is commonly known as the E-string theory, as the stringy degrees of
freedom in this theory has E8 flavor symmetry.1 We call the theories for Q > 1 the
Estring theories of general rank. The objective of this paper is to compute their anomaly
polynomials, thereby adding an item to the list of known properties of these mysterious
theories.
Let us quickly recall the symmetry of E-string theories. As we already mentioned, they
have N = (1, 0) superconformal symmetry and E8 flavor symmetry. The transverse space
to Q M5 branes has the form R4 R>0, and therefore they have SO(4) SU(2)L SU(2)R
symmetry. One of the two SU(2) symmetries is the R-symmetry in the superconformal
algebra. The low-energy limit of this brane system consists of a decoupled single free
hypermultiplet, describing the center-of-mass motion of Q M5 branes within the E8
endof-the-world brane, and a genuinely interacting 6d superconformal field theory.
Without further ado, here we will present the final result. The anomaly polynomial of
the total system, including the contribution from the free hypermultiplet is
AE8+free(Q) = Q
Here we used the symbols F for the E8 background field, T for the tangent bundle of
the worldvolume and N for the SO(4) normal bundle; pi are the Pontrjagin classes and
4(N ) is the Euler class.2 Our Tr is the trace in the adjoint representation divided by
the dual Coxeter number. Therefore, the integral of Tr F 2/4 over a four-cycle gives the
instanton number in the standard normalization.
Under the decomposition SO(4) SU(2)R SU(2)L, we have3
p1(N ) = 2 c2(R) + c2(L) ,
where c2(L), c2(R) are the second Chern classes of the rank-2 bundles L, R such that
L R NC. When Q > 0, SU(2)R is the R-symmetry and SU(2)L is a flavor symmetry;
when Q < 0 the assignment is reversed. In the following we assume Q > 0 unless otherwise
specified. The anomaly polynomial of the system without the decoupled center-of-mass
part is obtained by subtracting from (1.1) the contribution of the free hypermultiplet,
which is a half-hypermultiplet in the doublet of SU(2)L. This is given by
Afree =
7p1(T )2 4p2(T ) + c2(L)p1(T ) +
5760 48
The rest of the paper is organized as follows. In section 2, we will compute the
anomaly polynomial by combining the analysis of Horava and Witten [8] of the anomaly
of the E8 end-of-the-world brane and that of Freed, Harvey, Minasian and Moore [9, 10] of
the anomaly of multiple coincident five-branes.4 In section 3, we perform three checks of
the computation. First, we give another derivation of the terms in (1.1) that do not involve
2Our normalization of the anomaly polynomials is such that the contribution of a Weyl fermion in a
gauge representation is A(T ) tr eiF . In particular, we take F to be anti-hermitian and we include a factor
(2)1 in the definition of F .
3In our convention, a positively charged M5-brane has instanton number 1, and preserves the same
supersymmetry as the K3 manifold in the standard orientation. As RK3 p1 = 48 and RK3 4 = 24, we have
c2(L) = 24 and c2(R) = 0. This means that c2(R) corresponds to the R-symmetry of the 6d supersymmetry.
4Our computation is performed at the origin of the moduli space of vacua. On generic points on the
tensor branch, the same anomaly should be given by a related Hopf-Wess-Zumino term, which we discuss
in section 4.
the normal bundle using the compactification of heterotic string theory on K3. Second,
we compare the coefficient of (Tr F 2)2 with that computed in [6, 7]. Third, we show that
when Q = 1 the c2(L) dependence of (1.1) comes solely from the free hypermultiplet. W (...truncated)