Z ( w, β ) = w − 1 24 f ( w ) ∞ ∑ n $^{=}$−∞$^{w}$$^{(}$$^{n}$$^{+}$$^{β/}$$^{4}$$^{π}$$^{)}$$^{2}$ (34) 1 = − iγ 2 ∫ ∞ 0 dτ ∫ d $^{2}$p (2 π ) $_{2}$Z ( e − τ / 4 α $^{′}$, β ( p$_{1}$ )) e − τ p 2 2 (35) It is interesting to view the tadpole condition (32) in the context of open string/closed string (OS/CS) duality. This duality property is encoded in the modular structure of the string loop amplitude and represents the fact that the evolution of the string worldsheet may be viewed in two equivalent ways. What has been described here so far is the open string viewpoint, in which the Z particle is replaced by a short piece of string which propagates in a closed loop with Neumann boundary conditions on the string coordinates X µ interacting with background λ and A$_{µ}$ fields. But we can also view the amplitude as describing the propagation of a closed string between two boundary states. The field theory limit from this viewpoint is the limit of vanishing propagation time between the two closed string boundary states and the vacuum amplitude reduces to the overlap between the boundary states. In this way, one may interpret the two sides of the large N gap equation (32) as the field theoretic limit of the equation which identifies the open and closed string representations of the vacuum amplitude. Defining we can write the string generalization of the large N gap equation (32) as Actually, from the OS/CS viewpoint the ultraviolet divergent ( τ → 0) part of the integrand on the right hand side of (35) should be moved to the left hand side, since it is properly interpreted as coming from infrared behavior of the closed string channel, using the OS/CS duality transform (28). The closed string view provides additional insight into the nature of the excitations in the tachy- onic crystal vacuum. While the open string represents the Z -particle in the field theory limit, the closed string states which arise by a dual transformation of the open string loop are associated with gauge excitations and Wilson loop operators in the field theory. The boundary states in the closed string channel correspond to different configurations of D-branes in the spacetime background. Low lying excitations in this channel describe collective transverse modes of the codimension-one D-branes in the tachyonic crystal and are thus associated with topological charge fluctuations in the field theory. For finite cutoff α ′− $^{1}$, the ultraviolet limit of the loop integrand in (35) is con- trolled by the limit q → 0 ( w → 1), i.e. by the low energy or long range limit of the closed string channel. This is an example of a general feature of string theory, that ultraviolet divergences can IV. OPEN-STRING CLOSED-STRING DUALITY 15