Holomorphic Anomaly Research Articles

On triple-cut of scattering amplitudes

It is analysed the triple-cut of one-loop amplitudes in dimensional regularisation within spinor-helicity representation. The triple-cut is defined as a difference of two double-cuts with the same particle contents, and a same propagator carrying, respectively, causal and anti-causal prescription in each of the two cuts. That turns out into an effective tool for extracting the coefficients of three-point functions (and higher-point ones) from one-loop amplitudes. The phase-space integration is oversimplified by using residues theorem to perform the integration on the spinor variables, via the holomorphic anomaly, and a trivial integration on the Feynman parameter. The results are valid for arbitrary values of dimensions.

Holomorphic anomaly in the quantum Hall system

Experimental and theoretical evidence which is accumulating in favor of the existence of a global sub-modular symmetry in the quantum Hall system is reviewed. The scaling data suggest that the zeros of the beta-function are effectively anti-holomorphic, and it is explained how this leads to a superuniversal scaling function. This motivates the first construction of a candidate beta-function which agrees with all scaling data, as well as the restrictions coming from both the perturbative and dilute instanton gas analysis of the non-linear sigma-model of planar charge transport. The key mathematical concept allowing global analysis of this non-holomorphic function is quasi-holomorphic automorphy, and a firm mathematical foundation is given which exhibits the intimate relationship with the holomorphic anomaly that appears in supersymmetric string- and field-theories in high energy physics. It is therefore natural to conjecture that quasi-holomorphy is simply a consequence of the well-known existence of an effective supersymmetry in the low-energy field theory of a disordered medium.

Lectures on black holes, topological strings and quantum attractors**Lectures delivered at the RTN Winter School on Strings, Supergravity and Gauge theories, (CERN, January 16–20, 2006), the 11th APCTP/KIAS String Winter School (Pohang, Feb 8–15 2005) and the Winter School on the Attractor Mechanism (Frascati, March 20–24, 2006).

In these lecture notes, we review some recent developments on the relation between the macroscopic entropy of four-dimensional BPS black holes and the microscopic counting of states, beyond the thermodynamical, large charge limit. After a brief overview of charged black holes in supergravity and string theory, we give an extensive introduction to special and very special geometry, attractor flows and topological string theory, including holomorphic anomalies. We then expose the Ooguri–Strominger–Vafa (OSV) conjecture which relates microscopic degeneracies to the topological string amplitude and review precision tests of this formula on ‘small’ black holes. Finally, motivated by a holographic interpretation of the OSV conjecture, we discuss the radial quantization of BPS black holes (i.e. quantum attractors) and present a recent conjecture relating exact black hole degeneracies to Fourier coefficients of certain automorphic forms.

Entropy function for heterotic black holes

We use the entropy function formalism to study the effect of the Gauss-Bonnet term on the entropy of spherically symmetric extremal black holes in heterotic string theory in four dimensions. Surprisingly the resulting entropy and the near horizon metric, gauge field strengths and the axion-dilaton field are identical to those obtained by Cardoso et. al. for a supersymmetric version of the theory that contains Weyl tensor squared term instead of the Gauss-Bonnet term. We also study the effect of holomorphic anomaly on the entropy using our formalism. Again the resulting attractor equations for the axion-dilaton field and the black hole entropy agree with the corresponding equations for the supersymmetric version of the theory. These results suggest that there might be a simpler description of supergravity with curvature squared terms in which we supersymmetrize the Gauss-Bonnet term instead of the Weyl tensor squared term.

Kähler quantization ofH3(CY3,R) and the holomorphic anomaly

Studying the quadratic field theory on seven dimensional spacetime constructed by a direct product of Calabi-Yau three-fold by a real time axis, with phase space being the third cohomology of the Calabi-Yau three-fold, the generators of translation along moduli directions of Calabi-Yau three-fold are constructed. The algebra of these generators is derived which take a simple form in canonical coordinates. Applying the Dirac method of quantization of second class constraint systems, we show that the Schr\"{o}dinger equations corresponding to these generators are equivalent to the holomorphic anomaly equations if one defines the action functional of the quadratic field theory with a proper factor one-half.

Open string topological amplitudes and gaugino masses

We discuss the moduli-dependent couplings of the higher derivative F-terms ( Tr W 2 ) h − 1 , where W is the gauge N = 1 chiral superfield. They are determined by the genus zero topological partition function F ( 0 , h ) , on a world-sheet with h boundaries. By string duality, these terms are also related to heterotic topological amplitudes studied in the past, with the topological twist applied only in the left-moving supersymmetric sector of the internal N = ( 2 , 0 ) superconformal field theory. The holomorphic anomaly of these couplings relates them to terms of the form Π n ( Tr W 2 ) h − 2 , where Π's represent chiral projections of non-holomorphic functions of chiral superfields. An important property of these couplings is that they violate R-symmetry for h ⩾ 3 . As a result, once supersymmetry is broken by D-term expectation values, ( Tr W 2 ) 2 generates gaugino masses that can be hierarchically smaller than the scalar masses, behaving as m 1 / 2 ∼ m 0 4 in string units. Similarly, Π Tr W 2 generates Dirac masses for non-chiral brane fermions, of the same order of magnitude. This mechanism can be used, for instance, to obtain fermion masses at the TeV scale for scalar masses as high as m 0 ∼ O ( 10 13 ) GeV . We present explicit examples in toroidal string compactifications with intersecting D-branes.

Black holes, elementary strings and holomorphic anomaly

In a previous paper we had proposed a specific route to relating the entropy of two charge black holes to the degeneracy of elementary string states in N=4 supersymmetric heterotic string theory in four dimensions. For toroidal compactification this proposal works correctly to all orders in a power series expansion in inverse charges provided we take into account the corrections to the black hole entropy formula due to holomorphic anomaly. In this paper we demonstrate that similar agreement holds also for other N=4 supersymmetric heterotic string compactifications.

Computing one-loop amplitudes from the holomorphic anomaly of unitarity cuts

We propose a systematic way to carry out the method introduced in F. Cachazo, hep-th/0410077 for computing certain unitarity cuts of one-loop $\mathcal{N}=4$ amplitudes of gluons. We observe that the class of cuts for which the method works involves all next-to-MHV $n$-gluon one-loop amplitudes of any helicity configurations. As an application of our systematic procedure, we obtain the complete seven-gluon one-loop leading-color amplitude ${A}_{7;1}({1}^{\ensuremath{-}},{2}^{\ensuremath{-}},{3}^{\ensuremath{-}},{4}^{+},{5}^{+},{6}^{+},{7}^{+})$.

Formula omitted] supersymmetric one-loop amplitudes and the holomorphic anomaly of unitarity cuts

Recently, it has been shown that the holomorphic anomaly of unitarity cuts can be used as a tool in determining the one-loop amplitudes in N=4 super-Yang–Mills theory. It is interesting to examine whether this method can be applied to more general cases. We present results for a non-MHV N=1 supersymmetric one-loop amplitude. We show that the holomorphic anomaly of each unitarity cut correctly reproduces the action on the amplitude's imaginary part of the differential operators corresponding to collinearity in twistor space. We find that the use of the holomorphic anomaly to evaluate the amplitude requires the solution of differential rather than algebraic equations.

Gauge Theory Amplitudes In Twistor Space And Holomorphic Anomaly

We show that, in analyzing differential equations obeyed by one-loop gauge theory amplitudes, one must take into account a certain holomorphic anomaly. When this is done, the results are consistent with the simplest twistor-space picture of the available one-loop amplitudes.

Counting higher genus curves in a Calabi-Yau manifold

We explicitly evaluate the low energy coupling F g in a d = 4, N = 2 compactification of the heterotic string. The holomorphic piece of this expression provides the information not encoded in the holomorphic anomaly equations, and we find that it is given by an elementary polylogarithm with index 3 - 2g, thus generalizing in a natural way the known results for g = 0, 1. The heterotic model has a dual Calabi-Yau compactification of the type 11 string. We compare the answer with the general form expected from curve-counting formulae and find good agreement. As a corollary of this comparison we predict some numbers of higher genus curves in a specific Calabi-Yau, and extract some intersection numbers on the moduli space of genus-g Riemann surfaces.

The Holomorphic Anomaly of Topological Strings

We show that the BRS operator of the topological string B model is not holomorphic in the complex structure of the target space. This implies that the so-called holomorphic anomaly of topological strings should not be interpreted as a BRS anomaly.

E-strings and N = 4 topological Yang-Mills theories

We study certain properties of six-dimensional tensionless E-strings (arising from zero size E 8 instantons). In particular we show that n E-strings form a bound string which carries an E 8level- n current algebra as well as a left-over conformal system with c = 12 n−4− (248 n/ n+30), whose characters can be computed. Moreover we show that the characters of the n-string bound state are captured by N = 4 U( n) topological Yang-Mills theory on 1 2 K3 . This relation not only illuminates certain aspects of E-strings but can also be used to shed light on the properties of N = 4 topological Yang-Mills theories on manifolds with b 2 + = 1. In particular the E-string partition functions, which can be computed using local mirror symmetry on a Calabi-Yau threefold, give the Euler characteristics of the Yang-Mills instanton moduli space on 1 2 K3 . Moreover, the partition functions are determined by a gap condition combined with a simple recurrence relation which has its origins in a holomorphic anomaly that has been conjectured to exist for N = 4 topological Yang-Mills on manifolds with b 2 + = 1 and is also related to the holomorphic anomaly for higher genus topological strings on Calabi-Yau threefolds.

R 4 couplings in M- and type II theories on Calabi-Yau spaces

We discuss several implications of R 4 couplings in M-theory when compactified on Calabi-Yau (CY) manifolds. In particular, these couplings can be predicted by supersymmetry from the mixed gauge-gravitational Chem-Simons couplings in five dimensions and are related to the one-loop holomorphic anomaly in four-dimensional N = 2 theories. We find a new contribution to the Einstein term in five dimensions proportional to the Euler number of the internal CY threefold, which corresponds to a one-loop correction of the hypermultiplet geometry. This correction is reproduced by a direct computation in type 11 string theories. Finally, we discuss a universal non-perturbative correction to the type IIB hyper-metric.

Partition functions for BPS states of the non-critical $E_8$ string

We consider the BPS states of the $E_8$ non-critical string wound around one of the circles of a toroidal compactification to four dimensions. These states are indexed by their momenta and winding numbers. We find explicit expressions, $G_n$, for the momentum partition functions for the states with winding number $n$. The $G_n$ are given in terms of modular forms. We give a simple algorithm for generating the $G_n$, and we show that they satisfy a recurrence relation that is reminiscent of the holomorphic anomaly equations of Kodaira-Spencer theory.

Holomorphic anomalies and the nonrenormalization theorem

It has been argued that the superpotential can be renormalized in the presence of massless particles. Possible implications which have been considered include the restoration of supersymmetry at higher loops or a shift to a supersymmetric vacuum state. We argue that even in the presence of massless particles, there are no new contributions to the superpotential at any order in perturbation theory. This confirms the utility of the Wilsonian superpotential for analyzing the moduli space of the low energy theory.

Topological amplitudes in heterotic superstring theory

We show that certain heterotic string amplitudes are given in terms of correlators of the twisted topological (2,0) SCFT, corresponding to the internal sector of the N = 1 space-time supersymmetric background. The genus g topological partition function F g corresponds to a term in the effective action of the form W 2 g , where W is the gauge or gravitational superfield. We study also recursion relations related to holomorphic anomalies, showing that, contrary to the type II case, they involve correlators of anti-chiral superfields. The corresponding terms in the effective action are of the form W 2 g Π n , where Π is a chiral superfield obtained by chiral projection of a general superfield. We observe that the structure of the recursion relations is that of N = 1 spacetime supersymmetry Ward identity. We give also a solution of the tree-level recursion relations and discuss orbifold examples.

The Calabi-Yau moduli and the Toda equations

For a one-parameter family of Calabi-Yau d-fold embedded in C P d+1 we analyze the hermitian metrics on the moduli space of the non-linear sigma model by topological field techniques. The set of equations turns out to be the non-affine A-type Toda equation. It is contrasted with the C P d+1 case where the affine Toda equation appears. We obtain exact solutions of the hermitian metrics and associate their non-perturbative corrections to the holomorphic maps. In particular for the quintic case, the coefficients in the q-expansion are given by the number of rational curves. From these data, we construct the genus one partition function F 1 of the topological sigma model by the method of the holomorphic anomaly. In the asymmetrical limit of the Kähler parameter t → ∞ , the contribution from the A ∗- part is decoupled and we obtain the partition function of A-matter coupled with topological gravity at the stringy one-loop level. The coefficients of the series expansion are related with the integrals of the top Chem class of some vector bundle over the stable maps with definite degrees. We also comment on the behavior of the F 1 near the conifold locus.

N = 2 type II-heterotic duality and higher-derivative F-terms

We test the recently conjectured duality between N = 2 supersymmetric type II and heterotic string models by analysing a class of higher-dimensional interactions in the respective low-energy Lagrangians. These are F-terms of the form F g W 2 g where W is the gravitational superfield. On the type II side these terms are generated at the g-loop level and in fact are given by topological partition functions of the twisted Calabi-Yau sigma model. We show that on the heterotic side these terms arise at the one-loop level. We study in detail a rank-3 example and show that the corresponding couplings F g satisfy the same holomorphic anomaly equations as in the type II case. Moreover we study the leading singularities of F g 's on the heterotic side, near the enhanced symmetry point and show that they are universal poles of order 2 g − 2 with coefficients that are given by the Euler number of the moduli space of genus- g Riemann surfaces. This confirms a recent conjecture that the physics near the conifold singularity is governed by c = 1 string theory at the self-dual point.

On the algebraic structure of the holomorphic anomaly forĉ=3 topological strings

On the algebraic structure of the holomorphic anomaly forĉ=3 topological strings