aa r X i v : . [ h e p - t h ] M a r One-loop Amplitudes as BPS state sums
Ioannis Florakis ∗ † Max-Planck-Institut für PhysikWerner-Heisenberg-Institut, Föhringer Ring 6, 80805 München, GermanyE-mail: [email protected]
We review a novel method for evaluating one-loop BPS-saturated amplitudes in string theory.Contrary to traditional techniques of unfolding the fundamental domain F against the Narainlattice, which are only valid in certain regions of the moduli space and which obscure the T-duality invariance of the result, we will describe how the elliptic genus can be represented asa linear combination of certain absolutely convergent Poincaré series, against which F can beunfolded. The result can be expressed as a sum of one-loop contributions of perturbative BPS-states in a manifestly T-duality invariant fashion, valid at any point of (the perturbative) modulispace. Within this framework, the singularity structure of amplitudes around points of gaugesymmetry enhancement becomes crystal clear and a series of applications is given in order tobetter illustrate the power of this approach. Proceedings of the Corfu Summer Institute 2012 "School and Workshops on Elementary Particle Physicsand Gravity"September 8-27, 2012Corfu, Greece ∗ Speaker. † Based on joint work with C. Angelantonj and B. Pioline, both of whom I would like to warmly thank for a veryenjoyable collaboration. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ ne-loop Amplitudes as BPS state sums
Ioannis Florakis
1. Introduction
The problem of calculating scattering amplitudes in string theory lies at the core of any attemptto make contact with low-energy phenomenology. Indeed, string phenomenology has been markedwith impressive progress during the last two decades and several semi-realistic models have beenconstructed which, at tree-level, provide viable candidates for the description of supersymmetricextensions of the Standard Model, including particle content and interactions. To this end, the needfor incorporating loop corrections to gauge and gravitational couplings in the effective action isinherently linked with any attempt to make further contact with experimental data. Furthermore,and regardless of the possible direct applications to low-energy phenomenology, the developmentof a powerful framework for the study of stringy corrections to effective couplings acquires a theo-retical importance on its own especially since, in the presence of sufficient supersymmetry, severalBPS-saturated couplings are protected against higher perturbative or even non-perturbative correc-tions and provide useful laboratories in which to test string dualities (see e.g.[1] and referencestherein).In closed (oriented) string perturbation theory, one is typically dealing with a topological(Polyakov) expansion over closed, genus- g Riemann surfaces, which we schematically write as: ¥ (cid:229) g = g ( g − ) s Z moduli Z vertex operatorinsertions– zi Z D X D y . . . V i ( z i ) . . . e − S [ X , y , g ab ,... ] , (1.1)where g s is the string coupling and S [ X , y , g ab , . . . ] is the worldsheet action. After appropriatelygauge-fixing the diffeomorphism and Weyl gauge symmetries and performing the path integralover the various worldsheet fields X , y , one is instructed to integrate over the worldsheet positions z i of the various vertex operator insertions V i ( z i ) and, eventually, integrate over the moduli space ofthe Riemann surface with an appropriate measure. We will focus entirely on one-loop amplitudes( g = t ∈ H of the worldsheettorus, of the generic form: Z F d m A ( t , ¯ t ) . (1.2)The Teichmüller parameter t is initially defined over the upper half-plane H , before gaugingthe residual discrete group PSL ( Z ) of large diffeomorphisms, also known as the modular group,restricts integration down to its fundamental domain under the modular group, F = H / PSL ( Z ) .The SL ( R ) -invariant measure will, henceforth, be denoted by d m ≡ d t / t . The function A ( t , ¯ t ) ,obtained after performing the path integral and integration over the positions of the insertions V i ,is, by consistency, modular invariant.The techniques that we will review in this note concern precisely the evaluation of genus-1 modular integrals of the form (1.2), for certain classes of the integrand functions A , whichappear naturally in string theory. Contrary to the traditional ‘orbit method’ used in the literature[2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], the new methods we present [14, 15] manifestly preserve the T-duality symmetries of the theory, provide a natural modular invariant IR regularization and clearlyexhibit the singularity structure of the associated amplitudes, without depending on the region in2 ne-loop Amplitudes as BPS state sums Ioannis Florakis moduli space around which one is unfolding. There are four major cases of interest, of increasingdifficulty, depending on the holomorphicity of A :Case A TypeI F ( t ) holomorphic function of t II G ( d , d ) ( t , ¯ t ) Narain latticeIII G ( d + k , d ) ( t , ¯ t ) F ( t ) lattice with Wilson lines times elliptic genus F IV Z ( t , ¯ t ) manifestly non-holomorphic functionCase I involves a holomorphic modular invariant function of t and can be treated in a straightfor-ward fashion using Stokes’ theorem [16], in addition to the new methods we will review here. CaseII involves the integral of a d -dimensional lattice and can be considered a special case of III, wherean asymmetric lattice is multiplied together with an holomorphic function F ( t ) of modular weight w = − k /
2, known as the (modified) elliptic genus. Case III is the generic form of special classes of N = N = A into a lattice G ( d + k , d ) times theelliptic genus F ( t ) , perturbative corrections to these couplings may terminate at 1-loop. Typicalexamples are the N = cf. [12, 13]) in Heteroticstring theory compactified on K × T : D G = − i Z F d m (cid:20) t h Tr (cid:26) J e i p J q L − c ¯ q ¯ L − ˆ c (cid:18) Q − pt (cid:19)(cid:27) − b G t (cid:21) , D grav = − i Z F d m (cid:20) t h Tr (cid:26) J e i p J q L − c ¯ q ¯ L − ˆ c (cid:27) ˆ E − b grav t (cid:21) , (1.3)where the traces run over the internal ( , ) superconformal (SCFT) theory, with the right-movers set to their Ramond ground state and J is the U ( ) generator of the internal N = T -lattice and Q denotes one of the Cartan generators of the gauge group factor G whose 1-loop correction is beingcomputed. Furthermore, b G , b grav are the coefficients of the associated 1-loop beta functions. Theholomorphicity breaking term 1 / ( pt ) in the group trace in D G arises from a contact term in theKac-Moody gauge current correlator. Taking the difference of thresholds for two different gaugegroups of the same Kac-Moody level (here we consider k G i = D G − D G will again involve the T -lattice times a weak holomorphic modular function( w = G i , it will not contribute to the dif-ference of quadratic Casimirs Tr ( Q − Q ) and the resulting modular function will be holomorphiceverywhere, including the cusp at t = i ¥ . It is then a standard result in the theory of modular formsthat an everywhere holomorphic modular function ( w =
0) is actually constant and the differenceof thresholds will produce an integral of the G ( , ) lattice alone. This falls precisely in the class of The asymmetry k of the lattice can be thought of as parametrizing the presence of non-trivial Wilson lines of aHeterotic compactification. For notational simplicity, we adopt the convention where the supersymmetric side of the Heterotic string is takento be the right-moving, holomorphic side. ne-loop Amplitudes as BPS state sums Ioannis Florakis modular integrals of Case II and is, indeed, encountered e.g. when one is calculating the differencebetween the E and E group thresholds in K × T compactifications of the E × E Heteroticstring.
2. The unfolding method
For simplicity and concreteness, we will henceforth restrict our attention to the modular inte-grals of Case III: Z F d m G ( d + k , d ) ( G , B , Y ) F ( t ) , (2.1)which appear naturally in 1-loop corrections BPS-saturated couplings in the effective action. TheNarain lattice G ( d + k , d ) depends on the compactification moduli G IJ , B IJ , Y aI parametrizing the coset SO ( d + k , d ) SO ( d + k ) × SO ( d ) , with I = , . . . d and a = , . . . k . In the most general case, F ( t ) is a weak almostholomorphic modular form of SL ( Z ) , with modular weight w = − k / . The latter requirement derives from the reparametrization properties of the stringworldsheet. The major difficulty with evaluating (2.1) lies in the non-rectangular shape of thefundamental domain F = { t ∈ H : | t | ≥ , | t | ≤ / } and the general technique for evaluatingsuch integrals is known as the ‘orbit method’ or the ‘unfolding method’, which we now brieflyreview.Start from the generic integral: I = Z F d m f ( t , ¯ t ) , (2.2)where f ( t , ¯ t ) is a modular function. The idea behind the unfolding method is to exploit modulartransformations in order to simplify the integration region, by rewriting (2.2) as an integral over arectangular integration domain. It relies on one’s ability to express f as a sum over modular orbitsor, technically, in finding a Poincaré series representation for f : f ( t , ¯ t ) = (cid:229) g ∈ G ¥ \ SL ( Z ) j ( g · t , g · ¯ t ) , (2.3)where G ¥ is the stabilizer of the cusp: G ¥ = (cid:26)(cid:18) n (cid:19) : n ∈ Z (cid:27) ⊂ SL ( Z ) , (2.4)and the element g = (cid:16) ac bd (cid:17) ∈ G ¥ \ SL ( Z ) acts on t ∈ H by linear fractional transformations g · t = a t + bc t + d . The function j in (2.3) is known as the ‘seed’, since its modular averaging producesthe modular function f , and is assumed to be invariant under rigid translations g ∈ G ¥ . Using the In the case of SL ( Z ) , the only cusp is the point at t = i ¥ . ne-loop Amplitudes as BPS state sums Ioannis Florakis
Poincaré series representation (2.3) into the integral (2.2) and, assuming absolute convergence, sothat we can pull the sum outside the integral, we obtain after a change of variables t ′ = g · t : I = (cid:229) g ∈ G ¥ \ SL ( Z ) Z F d m j ( g · t , g · ¯ t ) = (cid:229) g ∈ G ¥ \ SL ( Z ) Z g F d m j ( t ′ , ¯ t ′ ) = Z G ¥ \ H d m j ( t , ¯ t ) . (2.5)Hence, summing over SL ( Z ) orbits, we see that F has been ‘unfolded’ to the half-infinite strip S ≡ G ¥ \ H = { t ∈ H : − ≤ t < , t > } . As a result, the original modular integral (2.2)has been reduced to a simpler integral over the strip, involving the seed j instead of the originalfunction f . In fact, the new integration domain being rectangular, the last integral can now begiven a ‘field-theoretic’ interpretation, in the sense that we can now consider the t -integral to beimposing level-matching, whereas the t -integral will later turn out to provide a Schwinger-likerepresentation of the amplitude.
3. Traditional unfolding of F against the lattice So far, the traditional approach in the literature has been to use the orbit decomposition of theNarain lattice in order to evaluate modular integrals of the form (2.1). This has both advantagesand disadvantages that we will shortly discuss. We illustrate the unfolding of F against the latticewith a simple example of this type : I = Z F d m G ( , ) ( R ) j ( t ) , (3.1)involving a 1-dimensional lattice, corresponding to a circle S of radius R : G ( , ) ( R ) = R (cid:229) ˜ m , n ∈ Z e − p R t | ˜ m + t n | , (3.2)times the holomorphic Klein j -invariant function. The latter is the unique holomorphic modularfunction with a first-order pole in the q = e p i t -expansion at the cusp and is conventionally definedwith vanishing constant term, i.e. j ( t ) = q + O ( q ) . The partition function of the lattice (3.2) isgiven in its Lagrangian representation, with ˜ m , n being the two winding numbers parametrizingthe wrapping of the string around S , with respect to the two non-trivial cycles of the world-sheettorus. To obtain a Poincaré series representation of G ( , ) , we separate out the ( ˜ m , n ) = ( , ) termand, in the remaining sum, we factor out the greatest common divisor ( g.c.d. ) N = ( ˜ m , n ) of thenon-vanishing windings. We can then express the windings as ˜ m = N p , n = Nq with ( p , q ) = G ( , ) ( R ) = R + R (cid:229) N ≥ (cid:20) (cid:229) ( p , q )= e − p ( NR ) t | p + t q | (cid:21) . (3.3)The quantity inside the square brackets above is precisely a Poincaré series with seed j ( t , ¯ t ) = exp ( − p ( NR ) t ) . This is easy to verify by noting that Im ( g · t ) = t | p + t q | for a matrix g = ( ⋆ q ⋆ p ) ∈ For simplicity, we set everywhere a ′ = ne-loop Amplitudes as BPS state sums Ioannis Florakis G ¥ \ SL ( Z ) . We then use this Poincaré series in order to unfold F , as in (2.5): I = R Z F d m j ( t ) + R (cid:229) N ≥ ¥ Z d t t e − p ( NR ) t Z d t j ( t ) . (3.4)Since j ( t ) is defined with vanishing constant term in its (Fourier) q -expansion, the sum in the r.h.s. vanishes due to the t -integration (level matching) and the integral (3.1) is simply given by the firstintegral in the r.h.s. of the expression above. Being an integral of a holomorphic function (Case Iof the previous section), this integral can be readily evaluated using Stokes’ theorem, or using thetechniques that we will review in the upcoming sections, R F d m j ( t ) = − p and, hence, we arriveat the following result: Z F d m G ( , ) ( R ) j ( t ) = − p R . (3.5)Now we encounter one of the major deficiencies of the traditional unfolding of F against thelattice. Namely, the result is not invariant under T -duality, even though the lattice G ( , ) ( R ) in the l.h.s. is invariant under R → R , as can be seen by Poisson resummation. The discrepancy is due tothe loss of absolute convergence which, while being automatically ensured by the lattice in the UV,is only conditional in the IR ( t → ¥ ) and breaks down at precisely the T-self-dual radius R = . As a result, we are no longer allowed to exchangethe order of integration and summation (2.5) and the unfolding of F is no longer justified. Theresult (3.5) is, in fact, only valid in the particular chamber of the moduli space, where R >
1. Inorder to evaluate the integral for radii R <
1, one should first double Poisson resum the lattice to itsdual radius and then repeat the unfolding.This simple example served to illustrate one of the most obvious deficiencies of the traditionalunfolding approach. Indeed, using the orbit decomposition of the lattice in order to unfold F is only useful for extracting the large volume behaviour of the integral, with the loss of absoluteconvergence around extended symmetry points ( i.e. fixed points under T-duality) obscuring thebehaviour of the amplitude around these regions. This is, in fact, a reflection of a much deeperdrawback of the traditional unfolding method : using a Poincaré series representation of the latticein order to unfold, does not yield the result in a manifestly T-duality invariant representation. Thereason for this is that, unfolding against the lattice, inevitably starts with the lattice in its Lagrangianrepresentation and relies on decomposing the winding sum into SL ( Z ) orbits, with each orbitbeing used separately in order to unfold F . However, the orbit decomposition of the windingsum leads to the loss of manifest T-duality invariance. Even though this may seem marginal in thesimple example (involving a one-dimensional lattice) we presented above, as soon as one considershigher dimensional lattices, this problem becomes much more serious. For example, for the slightlymore complicated integral involving a two-dimensional lattice and the almost holomorphic modular The extra states becoming massless at R = j -function, responsible for the loss of exponential suppression in the IR. ne-loop Amplitudes as BPS state sums Ioannis Florakis function ˆ E E E / D , one obtains: Z F d mG ( , ) ( T , U ) ˆ E E E D ≃ Re " − (cid:229) k > (cid:18)
11 Li ( e p ikT ) − p T U P ( kT ) (cid:19) − (cid:229) ℓ> (cid:18)
11 Li ( e p i ℓ U ) − p T U P ( ℓ U ) (cid:19) + (cid:229) k > ,ℓ> (cid:18) ˜ c ( k ℓ ) Li ( e p i ( kT + ℓ U ) ) − c ( k ℓ ) p T U P ( kT + ℓ U ) (cid:19) + Li ( e p i ( T − U + i | T − U | ) ) − p T U P ( T − U + i | T − U | ) + z ( ) p T U +
22 log (cid:18) p e − g √ T U (cid:19) + (cid:18) p U T − p U − p T (cid:19) Q ( T − U ) + (cid:18) p T U − p T − p U (cid:19) Q ( U − T ) , (3.6)where P ( z ) = Im ( z ) Li ( e p iz ) + p Li ( e p iz ) , c , ˜ c are the Fourier coefficients of E E / D and E E E / D , respectively, and we are only displaying the IR finite part. Here, ˆ E , E , E are theweight 2 (almost holomorphic), weight 4 and weight 6 (holomorphic) Eisenstein series, respec-tively, and D = h is the weight 12 cusp form. As illustrated by this example, even though thetraditional unfolding method can be useful for extracting the asymptotic behaviour of amplitudesin the large volume limit, the results are generally local , in the sense that they depend on the regionin moduli space around which one is unfolding . Even the task of merely checking the T-duality in-variance of result (3.6) becomes a daunting task, whereas the singularity structure of the associatedamplitude is fully obscured in this representation.
4. New methods of unfolding
The discussion above stresses the necessity for new methods of unfolding, able to overcomethe limitations of the traditional unfolding method outlined in the previous section. In particular,one is ideally looking for a global representation of the result, which preserves the manifest T-duality symmetries of the lattice and which is able to capture the behaviour around the T-self-dualpoints. We will now proceed to briefly review two such techniques: the Rankin-Selberg-Zagiermethod and the unfolding against Niebur-Poincaré series, applicable to integrals of Cases II andIII, respectively. Integrals of Case II and, in general, modular integrals of a function of moderate ( i.e. non-exponential) growth at the cusp, can be treated using a technique known in the mathematics litera-ture as the Rankin-Selberg-Zagier (RSZ) method [17, 14]. Here we will only point out the salientfeatures applied to integrals of Case II. Notice first that the modular integral of a d -dimensionallattice G ( d , d ) has an IR divergence, since the integrand grows polynomially as t d / at t → i ¥ andneeds to be regularized. The idea behind the RSZ method is to regularize the integral by truncating This is especially visible in (3.6) from the Heaviside Q -functions in the last line. We use ‘global’ here in order to stress the independence of the result from the region in moduli space around whichone is unfolding. ne-loop Amplitudes as BPS state sums Ioannis Florakis the fundamental domain to some (large) cutoff value T and to deform the integrand by an insertionof the (completed) non-holomorphic Eisenstein series: Z F d m G ( d , d ) ( G , B ; t , ¯ t ) −→ Z F T d m G ( d , d ) ( G , B ; t , ¯ t ) E ⋆ ( s ; t ) . (4.1)Here, F T = { t ∈ F : t < T } is the truncated fundamental domain and the Eisenstein series isdefined as: E ⋆ ( s ; t ) = z ⋆ ( s ) (cid:229) g ∈ G ¥ \ SL ( Z ) (cid:2) Im ( g · t ) (cid:3) s = z ⋆ ( s ) (cid:229) ( c , d )= t s | c t + d | s , (4.2)with z ⋆ ( s ) = p − s G ( s ) z ( s ) being the completed Riemann zeta function. Using the above Poincaréseries representation for E ⋆ we can unfold F T for Re ( s ) > Z F T d m G ( d , d ) ( G , B ; t , ¯ t ) E ⋆ ( s ; t ) = z ⋆ ( s ) Z S T d m t s G ( d , d ) − Z F − F T d m G ( d , d ) ( E ⋆ ( s ; t ) − z ⋆ ( s ) t s ) , (4.3)making sure to perform the appropriate subtraction of an infinite number of disks S a / c (with a , c coprime integers, such that c ≥ a ∈ Z c ), corresponding to the images of the complement F − F T under g = (cid:0) ac ⋆⋆ (cid:1) ∈ SL ( Z ) , which give rise to the second integral in the r.h.s of (4.3).It turns out that the latter becomes part of the definition of the renormalized integral. The nextthing to notice is that E ⋆ ( s ; t ) is a meromorphic function with simple poles at s = , s = E ⋆ ( s ; t ) = and, furthermore, E ⋆ ( s ; t ) = E ⋆ ( − s ; t ) . Extracting the residue of both handsides of (4.3) at s = Z F d m G ( d , d ) = s = " z ⋆ ( s ) ¥ Z d t t s + d / −
22 1 Z d t (cid:229) ( m , n ) =( , ) e − pt M e p i t m T n = s = " z ⋆ ( s ) G ( s + d − ) p s + d − (cid:229) ( m , n ) =( , ) mT n = [ M ] s + d − , (4.4)where the l.h.s. is the renormalized integral and the r.h.s. is an integral over the half-infinite strip S that can be easily performed. The constrained sum ( m T n = m · n = / m , n , respectively, excluding the origin m i = n i = M is the (physical) BPS mass squared. It is related to the constrained Epsteinzeta series constructed in [18]. The result (4.4) is manifestly invariant under T-duality, since theunfolding against the Eisenstein series does not depend on the point in moduli space around whichwe are unfolding. As a check, it is straightforward to use (4.4) and the known Fourier expansionof E ⋆ ( s ; t ) , in order to reproduce the standard results for the integrals of the d = d = d = The definition of the renormalized integral and further details on the connection to other renormalization schemescan be found in [14]. ne-loop Amplitudes as BPS state sums Ioannis Florakis
We now return to the generic integral (2.1), involving a (possibly asymmetric) lattice G ( d + k , d ) times a weak, almost holomorphic modular form F (elliptic genus) of weight w = − k / q -expansion at the cusp. The presence of the pole in the q -expansioncan be thought of as the contribution of the unphysical tachyon of the bosonic side of the Heteroticstring. Due to the latter pole, the integrand grows exponentially at the cusp and the RSZ techniqueoutlined in subsection 4.1 is no longer applicable. A new method [15] is then required to treatintegrals of Case III.The main idea of [15] is to construct a Poincaré series representation for the elliptic genus F itself and use it in order to unfold F . The elliptic genus F can be expanded in the generators { ˆ E , E , E , D − } of the graded ring of weak almost holomorphic modular forms of weight w andwith a simple pole in q at the cusp as: F ( t ) = (cid:229) m + n + r = + wm , n , r ≥ c mnr ˆ E m E n E r D , (4.5)where c mnr are appropriate coefficients. The problem of determining the correct seed constructingthe Poincaré series representation of F is a highly non-trivial one and we will not endeavor topresent here the full details. Rather, it will suffice to mention only the guiding principles behindthe construction. First, one notices that the hyperbolic Laplacian D w = t ¶ ¯ t ( ¶ t − iw t ) acts as aCasimir operator in the space of modular forms F of weight w and the latter can be organizedinto appropriate linear combinations of its eigenmodes. The main idea is to construct a Poincaréseries whose seed is already an eigenmode of D w . For generality, one imposes a pole of order k at the cusp F ∼ q − k + . . . and we are interested in constructing Poincaré series that are absolutelyconvergent for w ≤
0, so as to justify the unfolding. These conditions essentially lead to the choiceof seed: j ( t , ¯ t ) = M s , w ( − kt ) e − p i kt , (4.6)where M s , w ( y ) = | p y | − w / M w sgn ( y ) , s − ( p | y | ) and M is the Whittaker M -function. Summingover its images under G ¥ \ SL ( Z ) , this seed generates a Poincaré series known in the mathematicsliterature as the Niebur-Poincaré (NP) series [19, 20, 21, 22, 23, 24]: F ( s , k , w ) = (cid:229) ( c , d )= ( c t + d ) − w M s , w (cid:18) − kt | c t + d | (cid:19) exp (cid:26) − p i k (cid:18) ac − c t + dc | c t + d | (cid:19)(cid:27) , where the sum is over coprime integers c , d and the integers a , b are some solution of ad − bc = ( s ) > k >
0, it indeed reproduces the desiredpole of order k in the q -expansion at the cusp. By construction, it is an eigenmode of D w witheigenvalue − s ( − s ) − w ( w + ) . Its spectrum can be obtained by studying its Fourier expansion andwith the help of the modular derivatives D w = i p ( ¶ t − iw t ) , ¯ D w = − i pt ¶ ¯ t , acting as raising andlowering operators of the modular weight by units of 2, respectively.The NP series F ( s , k , w ) transforms, by construction, in the same way as an holomorphicmodular form. However, a careful study of its Fourier expansion reveals that it generically also9 ne-loop Amplitudes as BPS state sums Ioannis Florakis contains a non-holomorphic part. What kind of modular objects do NP series represent ? In fact, theweak almost holomorphic modular forms we are interested in are eigenmodes of D w with eigenvalue − w and the NP series has the exact same eigenvalue for s = − w . It turns out, in general, thatNP series F ( − w , k , w ) are not (weak, almost) holomorphic modular forms but, rather, weakharmonic Maass forms. These are objects transforming like (weak, almost) holomorphic modularforms, but which are the sum of an holomorphic ‘Mock modular’ part plus an infinite tower ofnegative frequency modes, explicitly breaking holomorphicity, called the ‘Shadow’. Hence, eventhough the holomorphic (Mock) part has an anomalous behaviour under modular transformations,this modular anomaly is precisely cancelled by the non-holomorphic (Shadow) part, which providesthe modular completion. How then can we hope to obtain Niebur-Poincaré series representationsof weak almost holomorphic modular forms F ? The answer is that, by taking appropriate linearcombinations of NP series F ( − w , k , w ) with definite coefficients matching the principal part inthe q -expansion of the modular form F we wish to represent, the Shadows cancel each other andthe resulting linear combination precisely represents the given weak, holomorphic modular form.Weak almost holomorphic modular forms can then be formed out of similar –uniquely determined–linear combinations of NP series with s = − w + n .Since all weak, almost holomorphic modular forms F can be uniquely expressed as linearcombinations of absolute convergent Niebur-Poincaré series, we can effectively reduce the problemof evaluating the generic integral (2.1) into calculating the integral of the lattice G ( d + k , d ) times F ( s , k , − k ) . The fundamental domain can now be unfolded using the Niebur-Poincaré series andthe result is given in terms of a BPS sum: R . N . Z F d m G ( d + k , d ) F ( s , k , − k ) = lim T → ¥ Z F T d m G ( d + k , d ) F ( s , k , − k ) + f ( s ) T d + k − s s − d − k = ¥ Z d t t d / − M s , − k ( − kt ) (cid:229) BPS e − pt ( P L + P R ) / . (4.7)The first line provides the natural, modular invariant definition of the renormalized integral interms of the appropriate cutoff-dependent subtraction, with f ( s ) being the zero-frequency modein the Fourier expansion of F ( s , k , w ) . The second line yields the result in terms of a strip integral,with the t -integration imposing the BPS constraint P L − P R = k , whereas the t -integral casts theBPS-contribution in its Schwinger representation. The t -integral can be explicitly performed toyield the BPS sum: I = ( pk ) − d G ( s + d + k − ) (cid:229) BPS 2 F (cid:18) s − k , s + d + k −
1; 2 s ; 4 k P L (cid:19) (cid:18) P L k (cid:19) − s − d − k , (4.8)which can be proven to converge absolutely for Re ( s ) > d + k and can be meromorphically contin-ued to the full s -plane with the exception of a simple pole at s = d + k which requires an additional To be precise, the same eigenvalue also appears for NP series with s = w , but this lies outside the range of conver-gence for w <
0, so that we can safely ignore it for the present discussion. This definition of the renormalized integral is valid for Re ( s ) > d + k and can be extended, by meromorphiccontinuation, to the whole s -plane, except for the pole s = d + k , which requires a slightly modified subtraction of alogarithmic divergence (for details, see [15]). ne-loop Amplitudes as BPS state sums Ioannis Florakis subtraction. The result (4.8) is manifestly invariant under T-duality and chamber independent,providing a global representation of the result, valid at any point in moduli space. Therefore, itcan be applied to study the behaviour of amplitudes around points of extended symmetry, wherethe traditional method of unfolding breaks down. In fact, for all cases of interest to string theoryapplications, it is possible to re-express the hypergeometric function in (4.8) in terms of elemen-tary functions. The general expression is given in [15] and renders the singularity structure of theintegral crystal clear. One finds that that for odd-dimensional lattices, the integral (4.7) alwaysdevelops conical singularities, whereas for d ≥ d ≥ d ≤ s + w . Furthermore, in the absence of Wilson lines, one may prove thatthe universal singularity behaviour in d = Z F d m G ( , ) F ( + n , , ) ≃ − ( n + ) ! n ! log | j ( T ) − j ( U ) | . (4.9)Furthermore, the explicit expression for the BPS sum (4.8) in [15] can be used to prove, in achamber independent fashion, the absence of singularities in gauge thresholds involving ellipticgenera where the unphysical tachyon pole cancels out . In the last section, we will illustrate thepower of our new method by applying it to specific examples.
5. Examples
We will first start with integrals of Case III involving a one-dimensional lattice: Z F d m G ( , ) ( R ) F ( + n , , ) = + n √ p G ( n + ) (cid:18) R + n + R + n − (cid:12)(cid:12)(cid:12)(cid:12) R + n − R + n (cid:12)(cid:12)(cid:12)(cid:12) (cid:19) , (5.1)for any integer n ≥
0. For example, for n =
0, one has F ( , , ) = j ( t ) +
24. With the help of theelementary result R F d m G ( , ) ( R ) = p ( R + R ) , which can be derived e.g. using the RSZ method,we immediately derive: Z F d m G ( , ) ( R ) j ( t ) = − p (cid:18) R + R + (cid:12)(cid:12)(cid:12)(cid:12) R − R (cid:12)(cid:12)(cid:12)(cid:12) (cid:19) . (5.2)The result is manifestly invariant under T-duality and holds for any radius. It should be contrastedwith (3.5), which is only valid in the R > S -lattice. On the other hand, the conical singularity appears naturally within our formalism.Another example is the one-dimensional analogue of gravitational thresholds in E × E Heteroticstring theory compactified on K × T : Z F d m G ( , ) ( R ) ˆ E E E D = p ( R + R − − | R − R − | ) − p ( R + R − ) + p | R − R − | , (5.3) Note that, using the traditional unfolding method of Section 3, such a proof is a priori not possible, even if oneasymptotically approaches the boundary separating different chambers of moduli space around an enhancement point. ne-loop Amplitudes as BPS state sums Ioannis Florakis which can be easily obtained using the Niebur-Poincaré series expansion ˆ E E E / D = F ( , , ) − F ( , , ) − N = E × E Heterotic string compacti-fied on K × T , where we realize K T / Z orbifold. The gauge thresholds for the E and E factors are given by the BPS-sums: D E = − Z F d m G ( , ) ( T , U ) ˆ E E E − E D = (cid:229) BPS h + P R (cid:18) P R P L (cid:19)i +
72 log (cid:16) T U | h ( T ) h ( U ) | (cid:17) + cst , D E = − Z F d m G ( , ) ( T , U ) ˆ E E E − E D = (cid:229) BPS h + P R (cid:18) P R P L (cid:19)i −
72 log (cid:16) T U | h ( T ) h ( U ) | (cid:17) + cst . The case of non-trivial Wilson lines can also be easily treated. If we Higgs the E group factordown to its Coulomb branch, the E threshold becomes: D E = − Z F d m G ( , ) ˆ E E − E D = (cid:229) BPS " + P R (cid:18) P R P L (cid:19) − P L − P L − P L − P L , where the left- and right- moving momenta P L , R now also depend on the Wilson lines Y aI and theBPS constraint now involves also the U ( ) charge vectors Q in the Cartan of E , m T n + Q T Q = at any point in moduli space andvalid in any chamber.A further application of our new methods of unfolding concerns the treatment of integralsinvolving insertions of P L , R lattice momenta of the generic form: Z F d m " t − l / (cid:229) P L , P R r ( P L √ t , P R √ t ) q P L ¯ q P R F ( t ) . (5.4)For consistency, the quantity in the brackets must be a modular form of weight l + d + k . Thegeneral conditions on the function r ( x , y ) for this to take place can be found in [25, 15]. Providedthey are satisfied, the integrand is modular invariant and − w = l + d + k . One may then expandthe elliptic genus F in terms of Niebur-Poincaré series and unfold against each of them to obtainthe corresponding BPS sum: Z F d m t − l / (cid:229) P L , P R r ( P L √ t , P R √ t ) q P L ¯ q P R F ( s , k , w )= ( pk ) + l / ¥ Z dt t + d + k − F (cid:18) s − l + d + k s ; t (cid:19) r P L r t pk , P R r t pk ! (cid:229) BPS e − tP L / k . In order to demonstrate the power of our methods, we will present one final example involvingan ‘exotic’ integral that does not even contain moduli dependence: Z F d m (cid:0) √ t h ¯ h (cid:1) ˆ E E − E E D = − √ . (5.5) This is physically expected, since the unphysical tachyon of the Heterotic string is chargeless with respect to E and E . ne-loop Amplitudes as BPS state sums Ioannis Florakis
This integral is not only interesting as a mathematical exercise but, in fact, appears in [26] as athreshold contribution to certain non-compact Heterotic constructions on ALE spaces in the pres-ence of NS5 branes. This is an example where the traditional orbit method cannot even be applied.Expanding the weak almost holomorphic modular form into Niebur-Poincaré series:ˆ E E − E E D = F ( , , ) − F ( , , ) + j + , (5.6)and using (5.1), it is straightforward to arrive at the explicit numerical value − √
6. Conclusions
In this short review we attempted to briefly portray some of the aspects of our novel approachto evaluating one-loop BPS-saturated amplitudes in string theory. We discussed how the traditionalunfolding method, using the orbit decomposition of the lattice, generically fails to preserve themanifest T-duality symmetries of the theory. Our novel proposal was to exploit the fact that anyweak, almost holomorphic modular form (such as the modified elliptic genus) can be uniquelyrepresented as a linear combination of absolutely convergent Niebur-Poincaré series and we canuse these to unfold the fundamental domain F . BPS-saturated one-loop string amplitudes are thennaturally expressed as sums over the perturbative BPS states in a manifestly T-duality invariantfashion. Within this new framework, the singularity structure of the amplitudes becomes crystalclear and the results in this representation are valid at any point in moduli space (chamber in-dependent). The incorporation of non-trivial Wilson lines and lattice momentum insertions is alsoachieved in a simple manner, as illustrated in several examples. Finally, these methods successfullyapply to cases of ‘exotic’ integrals, that may not even involve moduli dependence.We end this short review with a few comments concerning integrals of Case IV, namely, theclass of integrals where the integrand function A is manifestly non-holomorphic. Examples ofthis class in string theory include, e.g. the non-trivial 1-loop corrections to the effective potential(vacuum energy) of Type II and Heterotic vacua with spontaneously broken supersymmetry, or thetechnically similar case of the free energy of string theories at finite temperature. We would liketo note that, traditionally, the behaviour of the 1-loop effective potential around points of extendedsymmetry has been notoriously hard to study. In fact, understanding these properties is highlyrelated to some of the long-standing puzzles plaguing string thermodynamics and string cosmology,such as the resolution of the Hagedorn phase transition and the initial singularity problem. Eventhough, with our present machinery, attacking the fully non-holomorphic integrals of Case IVseems to be out of reach (aside from certain notable exceptions), it is still hoped that future progressin this direction may provide the tools necessary to study such integrals as well. Acknowledgements
It is a pleasure to thank my collaborators C. Angelantonj and B. Pioline for a very enjoy-able collaboration and the organizers of the
Corfu Summer Institute 2012 and the
XVIII EuropeanWorkshop on String Theory for giving me the opportunity to present this work.13 ne-loop Amplitudes as BPS state sums
Ioannis Florakis
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