Superstring Partition Function Research Articles

Investigation of zero-modes for a dynamical Dp-brane

In this article, we investigate zero-modes for a dynamical (rotating-moving) Dp-brane, coupled to the electromagnetic and tachyonic background fields. This work is done by the boundary state methods, in three cases of bosonic and fermionic boundary states and superstring partition function. By analyzing the obtained zero-modes in either of the cases, interesting results will be obtained. Our findings demonstrate the importance of the zero-mode and its effects on the background fields and the defined internal properties of the described system.

Two-loop cusp anomaly in ABJM at strong coupling

We compute the null cusp anomalous dimension of ABJM theory at strong coupling up to two-loop order. This is done by evaluating corrections to the corresponding superstring partition function, weighted by the $AdS_4\times \mathbb{CP}^3$ action in AdS light-cone gauge. We compare our result, where we use an anomalous shift in the $AdS_4$ radius, with the cusp anomaly of $\mathcal{N}=4$ SYM, and extract the two-loop contribution to the non-trivial integrable coupling $h(\lambda)$ of ABJM theory. It coincides with the strong coupling expansion of the exact expression for $ h(\lambda)$ recently conjectured by Gromov and Sizov. Our work provides thus a non-trivial perturbative check for the latter, as well as evidence for two-loop UV-finiteness and quantum integrability of the Type IIA $AdS_4\times \mathbb{CP}^3$ superstring in this gauge.

Two-loop corrections to partition function of Pohlmeyer-reduced theory forAdS5×S5superstring

Pohlmeyer reduction of ${\mathrm{AdS}}_{5}\ifmmode\times\else\texttimes\fi{}{S}^{5}$ superstring, involving the solution of Virasoro conditions in terms of coset current variables, leads to a set of equations of motion following from an action containing a bosonic $Sp(2,2)\ifmmode\times\else\texttimes\fi{}Sp(4)/[SU(2){]}^{4}$ gauged Wess-Zumino-Witten (WZW) term, an integrable potential and a fermionic part coupling bosons from the two factors. The original superstring and the reduced model are in direct correspondence at the classical level but their relation at the quantum level remains an open question. As was found earlier, the one-loop partition functions of the two theories computed on the respective classical backgrounds match; here we explore the fate of this relation at the two-loop level. We consider the example of the reduced theory solution corresponding to the long folded spinning string in AdS. The logarithm of the ${\mathrm{AdS}}_{5}\ifmmode\times\else\texttimes\fi{}{S}^{5}$ superstring partition function computed on the spinning string background is known to be proportional to the universal scaling function which depends on the string tension $\frac{\sqrt{\ensuremath{\lambda}}}{2\ensuremath{\pi}}$. Its quantum part is $f(\ensuremath{\lambda})={a}_{1}+\frac{1}{\sqrt{\ensuremath{\lambda}}}{a}_{2}+\dots{}$, where the one-loop term is ${a}_{1}=\ensuremath{-}3\mathrm{ln}2$ and the two-loop term is the negative of the Catalan's constant, ${a}_{2}=\ensuremath{-}\mathrm{K}$. We find that the counterpart of $f(\ensuremath{\lambda})$ in the reduced theory is $f(k)={\mathrm{a}}_{1}+\frac{2}{k}{\mathrm{a}}_{2}+\dots{}$, where $k$ is the coupling of the reduced theory. Here the one-loop coefficient is the same as in string theory, ${\mathrm{a}}_{1}={a}_{1}$, while the two-loop one is ${\mathrm{a}}_{2}={a}_{2}\ensuremath{-}\frac{1}{4}({a}_{1}{)}^{2}$. Remarkably, the first Catalan's constant term here matches the string-theory result if we identify the two couplings as $k=2\sqrt{\ensuremath{\lambda}}$. Nevertheless, the presence of the additional $({a}_{1}{)}^{2}\ensuremath{\sim}(\mathrm{ln}2{)}^{2}$ term suggests that the relation between the two quantum partition functions (if any) is not a simple equality. Similar results are found in the case of the ${\mathrm{AdS}}_{3}\ifmmode\times\else\texttimes\fi{}{S}^{3}$ string theory where ${a}_{1}=\ensuremath{-}2\mathrm{ln}2$ and ${a}_{2}=0$, while in the corresponding reduced theory ${\mathrm{a}}_{1}={a}_{1}$ and ${\mathrm{a}}_{2}={a}_{2}\ensuremath{-}\frac{1}{4}({a}_{1}{)}^{2}$.

Black hole partition function using the hybrid formalism of superstrings

The type IIA superstring partition function Z_IIA on the euclidean attractor geometry AdS_2 x S^2 x CY_3, computes the modified elliptic genus Z_BH of the associated black hole. The hybrid formalism of superstrings defined as a conformally invariant sigma model on the coset supermanifold PSU(1,1|2)/U(1)xU(1), together with Calabi-Yau and chiral boson CFTs, is used to calculate Z_IIA. The sigma model action on AdS_2 x S^2 is explicitly written in U(1)xU(1) invariant variables. The N=2 generators of AdS_2 x S^2 x CY_3 are enlarged and embedded in an N=4 topological algebra. The world sheet superconformal invariance is then used to construct a nilpotent BRST operator, in contrast to the kappa symmetry analysis used by Beasely et. al. in hep-th/0608021. The sigma model action is explicitly shown to be closed under this BRST operator. Localization arguments are then used to deform the world sheet path integral with the addition of a BRST exact term, where, contributions arise only from the center of AdS_2 and, the north and south poles of S^2. This leads to the OSV result Z_BH = Z_IIA = |Z_top|^2, where |Z_top|^2 is the square of the topological string partition function.

Holomorphic Factorization of Superstring Scattering Amplitudes

The holomorphic factorization of the superstring partition function is verified at arbitrary genus. The evaluation of scattering amplitudes and the implications of genus-dependent estimates on the string coupling are given.

Classical theta constants vs. lattice theta series, and super string partition functions

Recently, various possible expressions for the vacuum-to-vacuum superstring amplitudes has been proposed at genus g = 3, 4, 5. To compare the different proposals, here we will present a careful analysis of the comparison between the two main technical tools adopted to realize the proposals: the classical theta constants and the lattice theta series. We compute the relevant Fourier coefficients in order to relate the two spaces. We will prove the equivalence up to genus 4. In genus five we will show that the solutions are equivalent modulo the Schottky form and coincide if we impose the vanishing of the cosmological constant.

Generalized scaling function from light-cone gauge AdS 5 × S 5 superstring

We revisit the computation of the 2-loop correction to the energy of a folded spinning string in AdS_5 with an angular momentum J in S^5 in the scaling limit log S, J >>1 with J / log S fixed. This correction gives the third term in the strong-coupling expansion of the generalized scaling function. The computation, using the AdS light-cone gauge approach developed in our previous paper, is done by expanding the AdS_5 x S^5 superstring partition function near the generalized null cusp world surface associated to the spinning string solution. The result corrects and extends the previous conformal gauge result of arXiv:0712.2479 and is found to be in complete agreement with the corresponding terms in the generalized scaling function as obtained from the asymptotic Bethe ansatz in arXiv:0805.4615 (and also partially from the quantum O(6) model and the Bethe ansatz data in arXiv:0809.4952). This provides a highly nontrivial strong coupling comparison of the Bethe ansatz proposal with the quantum AdS_5 x S^5 superstring theory, which goes beyond the leading semiclassical term effectively controlled by the underlying algebraic curve. The 2-loop computation we perform involves all the structures in the AdS light-cone gauge superstring action of hep-th/0009171 and thus tests its ultraviolet finiteness and, through the agreement with the Bethe ansatz, its quantum integrability. We do most of the computations for a generalized spinning string solution or the corresponding null cusp surface that involves both the orbital momentum and the winding in a large circle of S^5.

Quantum AdS 5 × S 5 superstring in the AdS light-cone gauge

We consider the AdS_5 x S^5 superstring in the light-cone gauge adapted to a massless geodesic in AdS5 in the Poincare patch. The resulting action has a relatively simple structure which makes it a natural starting point for various perturbative quantum computations. We illustrate the utility of this AdS light-cone gauge action by computing the 1-loop and 2-loop corrections to the null cusp anomalous dimension reproducing in a much simpler and efficient way earlier results obtained in conformal gauge. This leads to a further insight into the structure of the superstring partition function in non-trivial background.

Strong-coupling expansion of cusp anomaly from quantum superstring

We consider the world surface in AdS5 that ends on two intersecting null lines at the boundary. The corresponding superstring partition function describes the expectation value of the Wilson line with a null cusp in dual large N maximally supersymmetric gauge theory and thus determines the cusp anomaly function f(l) of the gauge coupling l or the string tension √λ/2π. The first two coefficients in its strong-coupling or string inverse tension expansion were determined in hep-th/0210115 (a0 = 1) and in arXiv:0707.4254 (a1 = − 3 ln 2). Here we find that the 2-loop coefficient is a2 = −K where K is the Catalan's constant. This is in agreement (expected on the general grounds) with the previous results for f(l) as the coefficient of ln S term in the energy of the closed spinning string in AdS5. The string theory value for a2 is in agreement with the numerical result in hep-th/0611135 and the recent analytic result in arXiv:0708.3933 for the coefficients in strong-coupling Ssolution of the BES equation. We explicitly verify the cancellation of all 2-loop logarithmic divergences thus demonstrating the quantum consistency of the AdS5×S5 superstring action at this order. We also discuss the structure of the three and higher string loop corrections to the cusp anomaly function giving a 2d QFT diagrammatic interpretation to the result of arXiv:0708.3933 for the solution of the BES equation following from the Bethe ansatz prescription for the spectrum of the theory.

Superstring partition functions in the doubled formalism

Computation of superstring partition functions for the non-linear sigma model on the product of a two-torus and its dual within the scope of the doubled formalism is presented. We verify that it reproduces the partition functions of the toroidally compactified type-IIA and type-IIB theories for appropriate choices of the GSO projection.

On open superstring partition function in inhomogeneous rolling tachyon background

We consider open superstring partition function Z on the disc in time-dependent tachyon background T= f(x_i) e^{m x_0} where the profile function f depends on spatial coordinates. We compute Z to second order in derivatives of f and compare the result with some previously suggested effective actions depending only on the first derivatives of the tachyon field. We also compute the target-space stress-energy tensor in this background and demonstrate its conservation in the ``on-shell'' case of the linear profile f= f_0 + q_i x_i corresponding to a marginal perturbation. We comment on the role of the rolling tachyon with linear spatial profile in the decay of an unstable D-brane.

Superstrings on NS5 backgrounds, deformed AdS3and holography

We study a non-standard decoupling limit of the D1/D5-brane system, which interpolates between the near-horizon geometry of the D1/D5 background and the near-horizon limit of the pure D5-brane geometry. The S-dual description of this background is actually an exactly solvable two-dimensional (worldsheet) conformal field theory: { null-deformed SL(2,)} × SU(2) × T4 or K3. This model is free of strong-coupling singularities. By a careful treatment of the SL(2,), based on the better-understood SL(2,)/U(1) coset, we obtain the full partition function for superstrings on SL(2,) × SU(2) × T4/2. This allows us to compute the partition functions for the J33 and J22 deformations, as well as the full line of supersymmetric null deformations, which links the SL(2,) conformal field theory with linear-dilaton theory. The holographic interpretation of this setup is a renormalization-group flow between the decoupled NS5-brane world-volume theory in the ultraviolet (little string theory), and the low-energy dynamics of super Yang-Mills string-like instantons in six dimensions.

Modular invariant critical superstrings on four-dimensional Minkowski space times two-dimensional black hole

Extending the seminal work of Bilal and Gervais, we construct a tachyon-free, modular invariant partition function for critical superstrings on four-dimensional Minkowski x two-dimensional black hole. This model may be thought of as an SL(2,R)/U(1) version of Gepner models and corresponds to a conifold point on the moduli space of Calabi-Yau compactifications. We directly deal with N=2, c=9 unitary superconformal characters. Modular invariance is achieved by requiring the string to have a momentum along an extra noncompact direction, in agreement with the picture of singular CFTs advocated by Witten. The four-dimensional massless spectrum coincides with that of the tensionless strings, suggesting a possible dual description of type II strings on a conifold in terms of two intersecting NS5-branes. An interesting relation to D=6, N=4 gauged supergravity is also discussed.

Open superstring partition function in a constant gauge field background at finite temperature

We find the general expression for the open superstring partition function on the annulus in a constant abelian gauge field background and at finite temperature. We use the approach based on the Green-Schwarz string path integral in the light-cone gauge and compare it with the NSR approach. We discuss the super Yang-Mills theory limit of the string free energy and mention some D-brane applications.

Effective superstring spectral action

A supersymmetric theory in two-dimensions has enough data to define a noncommutative space thus making it possible to use all the tools of noncommutative geometry. In particular, we apply this to the N=1 supersymmetric non-linear sigma model and derive an expression for the generalized loop space Dirac operator, in presence of a general background, using canonical quantization. The spectral action principle is used to show that the superstring partition function is also a spectral action valid for the fluctuations of the string modes.

Low-lying states of the six-dimensional fractional superstring.

The $K=4$ fractional superstring Fock space is constructed in terms of $\bZ_4$ parafermions and free bosons. The bosonization of the $\bZ_4$ parafermion theory and the generalized commutation relations satisfied by the modes of various parafermion fields are reviewed. In this preliminary analysis, we describe a Fock space which is simply a tensor product of $\bZ_4$ parafermion and free boson Fock spaces. It is larger than the Lorentz-covariant Fock space indicated by the fractional superstring partition function. We derive the form of the fractional superconformal algebra that may be used as the constraint algebra for the physical states of the FSS. Issues concerning the associativity, modings and braiding properties of the fractional superconformal algebra are also discussed. The use of the constraint algebra to obtain physical state conditions on the spectrum is illustrated by an application to the massless fermions and bosons of the $K=4$ fractional superstring. However, we fail to generalize these considerations to the massive states. This means that the appropriate constraint algebra on the fractional superstring Fock space remains to be found. Some possible ways of doing this are discussed.

Strings on hyperelliptic supersurfaces

Starting from a definition of a hyperelliptic superRiemann surface (HESS), we study its supermoduli space and the “superVirasoro” algebra on it. We map that algebra into a Z 4 p=2 parafermionic algebra on CP 1,1. Using explicitly supersymmetric bosonization of the first-order superconformal systems, we construct explicitly the primary fields of the parafermionic algebra: the branching and the spin operators. We give a simple prescription for the calculation of sdet D γ on HESS through correlation functions of these branching operators. As an example, we calculate the g = 2 superstring partition function and verify that the cosmological constant vanishes, after integration over the moduli space.

Superstring partition functions and the characters of exceptional groups

We study the relation between the one-loop partition function of supersymmetric four-dimensional heterotic string theories and the level-one characters of exceptional groups. This is used to derive identities among θ-functions which show that the one-loop partition functions vanish identically. It is also shown that the exceptional groups provide the Kač-Moody characters of the underlying conformal field theories.

Global structure of the superstring partition function and resolution of the supermoduli measure ambiguity

We investigate the global structure of the fermionic string partition function on the supermoduli space M g sup. In particular we show how the recently discovered moduli total-derivative ambiguity is due to a non-trivial cocycle on M g sup, present if an atlas of coordinates of M g sup can be found, whose transition functions contain even, nilpotent components. We find a correction to the usual Berezin measure on M g sup, given by the so-called Rothstein volume form, that eliminates the above boundary ambiguities of the supermoduli measure at any genus. We discuss also the so-called theorem of holomorphic factorization in this particular case and its relation to the physical requirement of modular invariance.