Elliptic Genera of Symmetric Products and Second Quantized Strings

In this note we prove an identity that equates the elliptic genus partition function of a supersymmetric sigma model on the N -fold symmetric product M/SN of a manifold M to the partition function of a second quantized string theory on the space M × S1. The generating function of these elliptic genera is shown to be (almost) an automorphic form for O(3, 2, Z). In the context of D-brane dynamics, this result gives a precise computation of the free energy of a gas of D-strings inside a higher-dimensional brane. 1. The Identity Let M be a Kähler manifold. In this note we will consider the partition function of the supersymmetric sigma model defined on the N -fold symmetric product SM of M , which is the orbifold space SM = M/SN (1.1) with SN the symmetric group of N elements. The genus one partition function depends on the boundary conditions imposed on the fermionic fields. For definiteness, we will choose the boundary conditions such that the partition function χ(SM ; q, y) coincides with the elliptic genus [1, 2], which is defined as the trace over the Ramond-Ramond sector of the sigma model of the evolution operator q times (−1)F yL . Here q and y are complex numbers and F = FL +FR is the sum of the leftand right-moving fermion number. (See the Appendix for background.) In particular, χ(M ; q, y) = TrH(M )(−1) yLq (1.2) with H = L0 − c 24 . Of the right-moving sector only the R-ground states contribute to the trace. 198 R. Dijkgraaf, G. Moore, E. Verlinde, H. Verlinde We will prove here an identity, conjectured in [3], that expresses the orbifold elliptic genera of the symmetric product manifolds in terms of that of M as follows:1 ∞ ∑ N=0 pχ(SM ; q, y) = ∏ n>0,m≥0,` 1 (1 − pnqmy`) c(nm,`), (1.3) where the coefficients c(m, `) on the right-hand side are defined via the expansion χ(M ; q, y) = ∑ m≥0,` c(m, `)qy. (1.4) The proof of this identity follows quite directly from borrowing standard results about orbifold conformal field theory [6], and generalizes the orbifold Euler number computation of [7] (see also [8]). Before presenting the proof, however, we will comment on the physical interpretation of this identity in terms of second quantized string theory. 1.1. String Theory Interpretation. Each term on the left-hand side with given N can be thought of as the left-moving partition sum of a single (non-critical) supersymmetric string with space-time SM × S1 × R. This string is wound once around the S1 direction, and in the light-cone gauge its transversal fluctuations are described by the supersymmetric sigma-model on SM . The right-hand side, on the other hand, can be recognized as a partition function of a large Fock space, made up from bosonic and fermionic (depending on whether c(nm, `) is positive or negative) creation operators α n,m,` with I = 1, 2, . . . , |c(nm, `)|. This Fock space is identical to the one obtained by second quantization of the left-moving sector of the string theory on the space M × S1. In this correspondence, the oscillators α n,m,` create string states with winding number n and momentum m around the S1. The number of such states is easily read off from the single string partition function (1.4). In the light-cone gauge we have the level matching condition L0 − L0 = mn, (1.5) and since L0 = 0, this condition implies that the left-moving conformal dimension is equal to h = mn. Therefore, according to (1.4) the number of single string states with winding n, momentum m and FL = ` is indeed given by |c(nm, `)|. (Strictly speaking, the elliptic genus counts the number of bosonic minus fermionic states at each oscillator level. Because of the anti-periodic boundary condition in the time direction for the fermions, only the net number contributes in the space-time partition function (1.3). ) The central idea behind the proof of the above identity is that the partition function of a single string on the symmetric product SM decomposes into several distinct topological sectors, corresponding to the various ways in which a once wound string on SM × S1 can be disentangled into separate strings that wind one or more times around M × S1. To visualize this correspondence, it is useful to think of the string on SM × S1 as a map that associates to each point on the S1 a collection of N points in M . By following the path of these N points as we go around the S1, we obtain a collection of strings on M × S1 with total winding number N , that reconnect the N 1 In case we have more than one conserved quantum number such as FL, the index ` becomes a multiindex and the denominator on the RHS of (1.3) becomes a general product formula as appears in the work of Borcherds [4], see also [5]. Elliptic Genera of Symmetric Products and Second Quantized Strings 199 points with themselves. Since all permutations of the N points on M correspond to the same point in the symmetric product space, the strings can reconnect in different ways labeled by conjugacy classes [g] of the permutation group SN . The factorization of [g] into a product of irreducible cyclic permutations (n) determines the decomposition into several strings of winding number n. (See Fig. 1). The combinatorical description of the