ap-3-10.dvi Acta Polytechnica Vol. 50 No. 3/2010 Algebraic Solutions in Open String Field Theory – a Lightning Review M. Schnabl Abstract In this short paper we review basic ideas of string field theory with the emphasis on the recent developments. We show how without much technicalities one can look for analytic solutions to Witten’s open string field theory. This is an expanded version of a talk given by the author over the last year at a number of occasions1 and notably at the conference Selected Topics in Mathematical and Particle Physics in honor of Prof. Jǐŕı Niederle’s 70th birthday. 1 What is string field theory? The traditional rulesof firstquantized string theoryal- low one to compute on-shell perturbative amplitudes, but they tell us little about collective phenomena or non-perturbative effects. Two most prominent exam- ples of such are tachyon condensation (a close relative of the Higgs mechanism) and instanton physics. String field theory is an attempt to turn string the- ory into some sort of field theory by writing a field theory action for each of the single string modes and combining them together with very particular inter- actions. Perturbative quantization of this field theory yields all of the perturbative string theory, and one might hope that one day we could get a truly non- perturbative description of the theory. One of the most interesting applications of string field theory to date has been in studies of the classi- cal backgrounds of string theory. Traditionally, string theories aredefined tobe inone-to-onecorrespondence with worldsheet conformal field theories (CFT’s). As such they correspond to the choice of infinitely many couplings in two dimensional worldsheet theory. The condition of vanishing beta functions for all of these couplings is equivalent to Einstein or Maxwell like equations for the classical backgrounds. Given two CFT’s, the two corresponding string theories look in general entirely different. For CFT’s related by ex- actly marginal deformations, the two theories may bear some resemblance, but for theories related by relevant deformations it is very hard to see how one background can be a solution of a theory formulated around another background. This is in stark contrast to general relativity, where theEinstein-Hilbert action does not depend on any particular background, but it allows for solutions describing very different geome- tries. One of the holy grails of string theory research is a manifestly background independent formulation of string theory. String field theory (SFT) goes half-way towards this goal. It gives us a formulation which is background independent in form (not truly in essence) and which posseses a multitude of classical solutions representing different backgrounds. It is defined us- ing the data of a single reference CFT. It is analogous towriting theEinstein-Hilbert action and substituting the metric gμν(x) with g ref μν(x)+ hμν(x). The funda- mental reason for this difficulty is that what are the field theoretic degrees of freedom in string theory de- pends on the background, unlike in general relativity. Following Sen and Zwiebach [1, 2], it is believed that the space of classical solutions of SFT is in one-to-one correspondence (modulo gauge symmetries and per- haps dualities) with worldsheet CFT’s. In this short paper we review the amazingly sim- ple construction of a class of solutions that can be de- termined purely algebraically. These are just the first steps in a long programof constructing and classifying all solutions and relating them to someCFT’s. In sec- tions 5 and 6we add in a little bit of originalmaterial. Borrowing a few theorems from the theory of distribu- tions and the Laplace transform we are able to shed novel light on what the space of allowed string fields should look like. This seemingly academic question is actually important for distinguishing gauge trivial and non-trivial solutions. 2 Précis of OSFT One of the best understood string field theories is Witten’s covariantChern-Simons type string field the- ory [3] for open bosonic string.2 As is well known, quantization of a single classical string is somewhat subtle, due to the reparametriza- tion invariance of the worldsheet action. This gauge 1Parts of this work have been presented at the Kavli Institute for Theoretical Physics, the Aspen Center for Physics, the Simons Center for Geometry and Physics and the Yukawa Institute for Physics. We thank these institutes for their warm hospitality. 2There aremany other string field theories, also for closed strings or superstrings, and some theories have more than one description, often non covariant. Some are also non-polynomial. 102 Acta Polytechnica Vol. 50 No. 3/2010 symmetry canbe fixed in a number ofways. In the co- variantquantizationprocedureonehas togaugefix the worldsheet metric hαβ and introduce the worldsheet Fadeev-Popov ghost fields b and c. The Virasoro con- straints Tαβ = 0 resulting from gauge fixing can then be conveniently imposed using the BRST formalism.3 The space of physical states of the string is then iden- tified with the cohomology of the BRST operator QB acting on theHilbert spaceHBCF T of thematter-plus- ghost boundary conformal field theory (BCFT) deter- minedbythe stringbackground. Interestingly, andnot for trivial reasons, this BCFT is the most convenient starting point for string field theory. The classical degrees of freedomof open string field theory are fields associated to quantum states of the first quantized open string. It is very convenient to workwith the extended space HBCF T , which contains not only physical states of the string but also various other states. Interestingly, these turn into auxiliary and ghost fields of string field theory. All the fields are neatly assembled into a string field |Ψ〉 = ∑ i ∫ dp+1k φi(k)|i, k〉, (2.1) where the index i runs over all states of the first quan- tized string in a sector of momentum k. The dimen- sionality of the momentum is p+1, as appropriate for open strings ending on a D-p-brane. The coefficients φi(k) aremomentum spacewave functions for particle like excitations of the string, and would become field theory operators if we proceeded to second quantiza- tion. The action of string field theory can be written in the form S = − 1 g2o [ 1 2 〈Ψ∗ QBΨ〉+ 1 3 〈Ψ∗Ψ∗Ψ〉 ] , (2.2) where go is the open string coupling constant and ∗ is Witten’s star product. The action has enormous gauge symmetry given by δΨ= QBΛ+Ψ∗Λ−Λ∗Ψ, (2.3) where Λ ∈ HBCF T (Grassman even), provided the start product is associative, QB acts as a graded derivation and 〈 . 〉 has properties of integration. To summarize, the basic ingredients that oneneeds in order to write down Witten’s OSFT in a particular background are HBCF T , ∗, QB, 〈.〉. For a more comprehensive review, the reader is re- ferred to the excellent reviews [4, 5].4 3 Demystifying the star product The star product has alwaysbeenoneof themostdiffi- cult ingredients of the stringfield tounderstandand to workwith. It can be defined very intuitively using the Schrödinger presentation of string wave functionals (Ψ1 � Ψ2)[X(σ)] =∫ [DXoverlap]Ψ1 [ X̂(σ) ] Ψ2 [ X̌(σ) ] , (3.4) where the hat and checkmeans that the left and right halves of these functions respectively coincide with those of the X(σ). It took some years and many re- searchpapers to understand exactlywhether this path integral makes sense. There is however a modern definitionwhichmakes many of the star product properties manifest. Let us describe string field theory states as linear combina- tions of surfaces with vertex operator insertions, such as inFig. 1.5 These represent theworldsheets of a sin- gle string evolving from the infinite past to the infinite future. A conformal transformation can be used to bring the surface to a canonical form, but this would act nontrivially on the in and out states. We will con- sider only shapeswhichhave the future (upper) part in the canonical shapeof a semi-infinite strip. Byputting various vertex operators in the far future and evaluat- ing the path integral over the surface, we can uniquely probe both the shape of the lower part of the surface and what vertex operators are inserted there. To describe the star product we take two states in the canonical form, cut off the probe strips (in light yellow) and glue the lower parts of the strips along the upper boundaries of the hatched regions. One gets again a state in the form of a surface with insertions, but the shape is different from those we started with. Imaginenowfactorizing thepath integralmeasureover theworldsheetfields in thehatchedareaand in the rest of the surface. The path integral over the hatched re- gion is performed first. Then since there are no vertex operators inserted, one can replace its result by an ef- fective term in the worldsheet action, or equivalently as an insertion of somenonlocal line operator. It turns out that this operator can be written as e−K, where K is a line integral of the worldsheet energy momen- tum tensor in some specific coordinates. The integra- tion extends from the boundary to themidpoint of the worldsheet. 3Alternatively, as in the light cone gauge, one could use the residual infinite dimensional conformal symmetry to gauge fix one of the embedding coordinates and solve the Virasoro constraints algebraically. 4Older reviews are [6, 7] and a more recent development appears in [8]. 5In order to match with Witten’s original definition the σ coordinate must run from right to left. 103 Acta Polytechnica Vol. 50 No. 3/2010 Fig. 1: Witten’s star product is defined by gluing the respective worldsheets The upshot is that the star multiplication is iso- morphic to operator multiplication. To see this more explicitly, consider twovertex operators φ1,2. The cor- responding states |φ1,2〉 star multiply as |φ1〉∗ |φ2〉 = |φ1e−K φ2〉. (3.5) Let us now introduce new (non-local) CFT operators φ̂ = eK/2φeK/2 andassociated states |φ̂〉. Then clearly |φ̂1〉 ∗ |φ̂2〉 = |φ̂1φ2〉. (3.6) Therefore φ → |φ̂〉 is the claimed isomorphism. Usually one does not think of the star product and the operator product as being the same thing. In particular, there are well known short distance sin- gularities for local vertex operators in nearby points, whereas the starproduct isusually thought tobemuch more regular. Well, thanks to the presence of the eK/2 operators, we do indeed get the same type of singu- larities when we try to star multiply two φ̂ states as in the vertex operator algebra. The φ̂ states can be represented by surfaces that differ from those in Fig. 1 in that the lower bluish part ismissing and is replaced by the identification of the left and right parts of the base of the upper semi-infinite strip with a local oper- ator inserted at themidpoint. We say that such string states have no security strips. 4 Algebraic solutions to OSFT To solve the classical equation of motion QBΨ+Ψ∗Ψ=0 (4.7) one could try to restrict the huge star algebra to as small subsector as possible. As we first want to study the tachyon condensation, perhaps we should include the vertex operator of the zero momentum tachyon, which is just the c-ghost. For the subalgebra to be nontrivial we should also include the non-local oper- ator K. One can easily (but not necessarily) add an operator B which is defined by the same type of inte- gral as K with the energy momentum tensor replaced by the b-ghost. Together all these elements obey c2 =0, B2 =0, {c, B} =1 (4.8) [K, B] = 0, [K, c] = ∂c. (4.9) The action of the exterior derivative is equally simple QBK =0, QBB = K, QBc = cKc. (4.10) QB is not the only useful derivation. There is also one called L−, which aside of the usual Leibnitz rule also obeys L−c = −c, L−B = B, L−K = K. (4.11) At a given ghost number, the derivative L− counts the number of K’s and is bounded from below. One could therefore use it to solve the equation of motion order by order in L− within the subalgebra generated by K, B, c. The simplest possible solution is Ψ= αc − cK. (4.12) Clearly QBΨ = αcKc − cKCK = −Ψ2. A more general solution has been found by Okawa [9], follow- ing [10] (see also the works by Erler [11, 12]) Ψ= F c KB 1− F2 cF, (4.13) where F = F(K) is an arbitrary function of K. To prove that it obeys the equation of motion requires some straightforward if a bit tedious algebra. The so- lution can be formally written in the form Ψ=(1− F BcF)QB ( 1 1− F BcF ) , (4.14) whichmakes the proof of the equations ofmotion triv- ial. What isnot so trivial is todistinguisha trivialpure gauge solution fromthenontrivial solutions. Note that 104 Acta Polytechnica Vol. 50 No. 3/2010 Bc is a projector, in the sense that it squares to itself, and therefore (1− F BcF)−1 =1+ F 1− F2 BcF. (4.15) For a solution to be nontrivial, the factor F/(1− F2) must be ill defined, whereas the similar looking factor appearing in the string field itself F̃ ≡ K/(1 − F2) must be well defined. Before we go in depth into what ill/well defined means, let us discuss another interesting property of these solutions. Expanding string field theory around these solutions one obtains a similar looking theory with QB replaced by QΨ = QB +{Ψ, ·}∗. (4.16) The second term acts as a graded star-commutator with Ψ. The fluctuations around the vacuum are de- scribed by the cohomology of this operator. Interest- ingly, one could find a homotopy operator A which formally trivializes the cohomology [13] A = 1− F2 K B, (4.17) i.e. it obeys {QΨ, A} = 1. Therefore, formally, any QΨ closed state χ can be written as QΨ exact: χ = QΨ(Aχ). Absence of nontrivial excitations around a given state Ψ is a property expected by Sen’s conjectures [14] around the tachyon vacuum, but definitely not around a generic state. We thus find an analogous condition to the one above: (1−F2)/K should bewell defined for the tachyon vacuum, but ill defined for the perturbative vacuum (F =0). Assuming that F(K) is a well defined string field, and adopting a simplifying assumption that F is ana- lytic around the origin, we find that F(K)= a + bK + . . . (4.18) gives the tachyonvacuum if a =1and b �=0, and gives the trivial vacuum for a �=1. Solutionswith a =1and b =0might correspond to somethingmore exotic such as multiple brane solutions, but this has not yet been shown convincingly. 5 What constitutes a well defined string field? This is still an open question, so we will rather ask a more specific question of when a function F(K) con- stitutes a well defined string field. Even this question might not have a unique answer, as there are several possible definitions of what might constitute ‘good’ or ‘bad’, depending on the context. We define a set of geometric string field functions F(K) to be those ex- pressible as6 F(K)= ∫ ∞ 0 dαf(α)e−αK . (5.19) The name geometricmeans that we consider superpo- sitions of surfaces, recall that e−αK represents a sur- face. For α ∈ N0 it is the α-th power of the SL(2, R) vacuum |0〉, and for generic α ≥ 0 one canfind a frame (a so called cylinder frame), in which the surface is a strip of size α. What space do we want f(α) to belong to? Obvi- ously a space of functions would be too restrictive, as one would have no hope of representing even the vac- uum corresponding to F(K) = e−K. The theory of distributions, developed to a large extent by Laurent Schwartz more than sixty years ago, is exactly what we need. Let us now remind the reader of some of the useful spaces that are introduced in a beautiful treatise [15]. Schwartz introduces the following spaces D ⊂ S ⊂ DLp ⊂ DLq ⊂ Ḃ ⊂ B ⊂ OM ⊂ E ∩ ∩ ∩ ∩ ∩ ∩ ∩ ∩ E′ ⊂ O′c ⊂ D ′ Lp ⊂ D ′ Lq ⊂ Ḃ ′ ⊂ B′ ⊂ S′ ⊂ D′ where in both lines 1 ≤ p < q < ∞. The first line denotes spaces of infinitely differentiable functions (in general on Rn, but here we restrict to R) which in addition satisfy (together with their derivatives): D are of compact support, S is the space of fast decay- ing Schwartz functions, DLp must also belong to Lp, B ≡ DL∞ are bounded, Ḃ are both bounded and pos- sess a finite limit at infinity, OM cannot grow too fast, but finally E have no restrictions on their growth. On the second line we have spaces of distributions that are defined as continuous linear functionals on some function space from the first line (in the case of O′c and B ′ with an additional restriction). For ex- ample, E′ is dual to E and represents the space of distributions with compact support. O′c are rapidly decaying distributions, i.e. those which together with all their derivatives are bounded even after multipli- cation with (1+ x2)k/2 for all k ∈ R. The space D′L1 is dual to Ḃ. D′Lp for p ∈ (1, ∞) are dual to DLp′ for p′ = p/(p − 1). A useful characterization of the D′Lp spaces for all 1 ≤ p ≤ ∞ is that they are fi- nite sums of derivatives of functions from Lp (hence DLp ⊂ Lp ⊂ D′Lp), or, equivalently, that their convo- lution with any a ∈ D belongs to Lp. The space Ḃ′ is dual to DL1 and the distributions are characterized by convergence towards infinity. Whereas D is dense in Ḃ′, it is not dense in B′ ≡ D′L∞, the space of bounded distributions. S′ is the well known Schwartz space of tempereddistributions (thosewithatmostpolynomial growth) dual to S, and finally D′ is the biggest space of all distributions dual to D. 6A much broader class of interesting non-geometric states has been considered recently by Erler [16], inspired by Rastelli [17]. 105 Acta Polytechnica Vol. 50 No. 3/2010 After this little exposé, we are ready to answer the question which space should f(α) in (5.19) belong to. Let us define geometric states as those for which f is a Laplace transformable distribution, i.e. f ∈ D′, but such that there exists ξ, for which e−ξαf ∈ S′. We could perhaps have beenmore generous and have kept only the f ∈ D′ condition, but such states would be even more meaningless from the string field perspec- tive. Whatweneedarenot themost generalgeometric states, but more restricted ones. Let us define a space of L0-safe geometric string field functions F(K) to be those for which f ∈ D′L1. This conditions comes from considering states F(K)|I〉 expanded in the Virasoro basis. The coef- ficients are given by sums of integrals of the form∫ ∞ 0 dα f(α)(α +1)−n for n = 0,2,4, . . . For these integrals to be absolutely convergent, we must have f ∈ D′L1. This definition gives us a very nice sur- prise. Since the star product of string fields F1(K) and F2(K) is just a multiplication, in terms of its in- verse Laplace transforms f1 and f2 it is a convolution f1 ∗ f2(x) = ∫ x 0 dyf1(y)f2(x − y) (defined in a more sophisticated way when fi are both actual distribu- tions). Now it is known that the space D′L1 is closed under convolution. Therefore the space of L0-safe geo- metric string fields is closed under starmultiplication! We now proceed to define a space of L0-safe ge- ometric string field functions F(K) by the condition f ∈ O′c. This condition comes from considering states F(K)|I〉 expanded in the basis of L0 eigenstates (see [10] for definition), or equivalently from expanding F(K) in the L− eigenstates (see [18]). We demand that ∫ ∞ 0 dα f(α)αn for n ∈ N0 be absolutely conver- gent. This forces f ∈ O′c. Again this space is closed under convolution and hence the space of L0-safe ge- ometric string fields is also closed under star multipli- cation! Both definitions of safe string fields can be recast in terms of the properties of the function F(z) =∫ ∞ 0 f(α)e−αz. String field function F(K) is 1. geometric if and only if there exists ξ such that for all z with Rez > ξ, F(z) is holomorphic and |F(z)| is majorized (i.e. bounded) by a polyno- mial in |z|. 2. L0-safe geometric if and only if F(z) is holomor- phic forall z withRez > 0and |F(z)| ismajorized by a polynomial in |z| for all Rez ≥ 0. 3. L0-safe geometric if and only if F(z) is holomor- phic for all z with Rez > 0, |F(z)| is majorized by a polynomial in |z| for all Rez ≥ 0, and F(z) canbe extended to a C∞ function on the complex half-plane Rez ≥ 0. The proof of the first statement can be found in text- books, and the latter two canbe establishedbya slight modification. To end this mathematical discussion, let us now givea fewexamples. The stringfields (1+K)p areboth L0 and L0-safe geometric since the function (1+ z)p is holomorphic for Rez > −1 and obeys all the above conditions. The inverse Laplace transform f ∈ O′c can be easily computed: 1 Γ(−p) α−p−1e−α, p < 0( 1+ d dα )[p]+1 ·[ 1 Γ([p]+1− p) α[p]−pe−α ] , p > 0, p /∈ N (5.20)( 1+ d dα )p δ(α), p ∈ N0. Here [p] denotes the integer part of p, but in fact any integer greater than that can be taken. For p ∈ N0 the inverse Laplace transform actually belongs to the smaller space E′ of distributions with compact sup- port. Note that for p > 0, p /∈ N distribution theory takes care beautifully of the singularities that would be present if one thought of the inverse Laplace trans- form as a function. Had we considered functions (1 + γ−1K)p, with γ ∈ R+ the domain of holomor- phicity would change, the maximal half-plane being Rez > −γ. The inverse Laplace transform for these functions is γf(γα). The closer γ is to zero, the slower falloff of f we get. If γ were taken negative, the inverse Laplace transform would grow exponentially (definitely not what we want in OSFT) which would manifest itself as singularities of F(z) for Rez > 0. Another example is the string field 1/ √ 1+ K2. It is geometric and L0-finite but neither L0 nor L0-safe. The inverse Laplace transform is the Bessel function J0(α). It belongs to the space D′Lp for p > 2. The reason for L0 finiteness is the cancelations due to the oscillatory behavior of theBessel function. Finally, let us consider string field √ 1+ K2. The inverse Laplace transform is δ′(α)+ 1 2 (J0(α)+ J2(α)) and belongs to D′L1 but not to O ′ c. Correspondingly it is L0-safe, but not L0-safe. 6 Examples Let us go through some of the simplest examples of OSFT algebraic solutions of the form Ψ= F c KB 1− F2 cF in more detail, and let us try to see what the gener- alities of the previous section tell us. Let us remind 106 Acta Polytechnica Vol. 50 No. 3/2010 the reader of the definition F̃ ≡ K/(1− F2) in terms of which Ψ= F cBF̃ cF and the homotopy operator is A = F̃ −1B. • F(K)= a, a �=1 The solution can be simplified as Ψ = a2 1− a2 · Q(Bc). For this object both F/(1 − F2) = a/(1− a2) and F̃ = K 1− a2 are regular, the so- lution is therefore a pure gauge and is thus the perturbative vacuum. Thewouldbehomotopyop- erator A = (1− a2)B/K is singular. This is so, because the inverse Laplace transform of 1/z is 1 (when restricted to R+) which does not belong to neither O′c, nor D ′ L1. • F(K)= √ 1− βK, β �=0 The solution is geometric only for β < 0, but formally for all values one obtains Ψ =√ 1− βKβ−1c √ 1− βK. This is nothing but a real formof the solution (4.12)with the identifica- tion β = α−1. For this solutionboth F̃ = β−1 and the homotopy operator A = βB are very simple and belong to our L0 and L0-safe spaces. Thanks to the vanishing cohomology around the vacuum, it is believed to represent the tachyon vacuum (it can alsobe shownto be formally gauge equivalent to it) but we have not yet succeeded in comput- ing its energy. The reason for the difficulty is that the string field is too identity like and gives rise to divergences in the energy correlator. Perhaps rephrasing the problem in terms of distribution theory could solve this issue. • F(K)= e−K/2 The solution in this case is thefirst discovered an- alytic solution for the tachyon vacuum [10]. There is however one subtlety with this solution. Since F̃ = K 1− e−K = (6.21)∫ ∞ 0 dα ( ∞∑ n=0 δ′(α − n) ) e−αK , the inverse Laplace transform of F̃ does not van- ish for large α. Consequently f̃ ∈ B′ and does not belong to either D′L1 or O ′ c. It is therefore an ex- ample of a geometric string field that is neither L0 nor L0-safe, but is nevertheless L0-finite. This is alsomanifested by the fact that F(z) has poles on the imaginary axes. There are interesting conse- quences to this. Truncating the sum ∑ Ke−nK at some finite value of n = N one gets a remnant Ke−(N+1)K 1− e−K which still contributes significantly to certain observables, in particular to the energy. This is the origin of the so called phantom term in the tachyon vacuum solution. The homotopy operator, on the other hand, is very well defined, and is both L0 and L0-safe. • F(K)= 1√ 1+ K This is the tachyon vacuum solution foundbyTed Erler and the author [19] Ψ= 1 √ 1+ K cB(1+ K)c 1 √ 1+ K . (6.22) The homotopyoperator is simply A = B/(1+K), which is perfectly regular. The inverse Laplace transform of F̃ = 1 + K is f̃ = δ(α) + δ′(α), which actually belongs to E′ and we thus see no need for the phantom term. In fact the energy for this solution can be computed very easily E = −S = 1 6 〈Ψ, QΨ〉= (6.23) 1 6 〈 (c + Q(Bc)) 1 1+ K c∂c 1 1+ K 〉 = 1 6 ∫ ∞ 0 dt1 ∫ ∞ 0 dt2e −t1−t2 〈 c e−t1K c∂c e−t2K 〉 = − 1 6π2 ∫ ∞ 0 duu3e−u ∫ 1 0 dv sin2 πv = − 1 2π2 , (6.24) which is the correct value, minus the tension of the D-brane, according to Sen’s conjecture [14]. The last correlator that we had to evaluate is in- deed very simple: two ghost insertions of c and c∂c on a boundary of a semi-infinite cylinder of circumference t1 + t2 separated by the distance of t1. Acknowledgement I would like to thank Ted Erler for collaborating on many of the topics discussed in this review, and for his insightful comments. I would also like to thank the Kavli Institute for Theoretical Physics, the As- penCenter for Physics, the SimonsCenter forGeome- try andPhysics, the Yukawa Institute for Physics and APCTP in Pohang, Korea, for their hospitality while I was working on parts of the present work. 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[16] Erler, T.: to appear [17] Rastelli,L.: CommentsontheOpenStringC*Al- gebra, Talk at Simons Center for Geometry and Physics Workshop on String Field Theory, Stony Brook, March 23–27, 2009. [18] Okawa, Y., Rastelli, L., Zwiebach, B.: Analytic solutions for tachyon condensation with general projectors, arXiv:hep-th/0611110. [19] Erler, T., Schnabl, M.: A Simple Analytic So- lution for Tachyon Condensation, JHEP 0910 (2009) 066 [arXiv:0906.0979 [hep-th]]. Martin Schnabl E-mail: schnabl.martin@gmail.com Institute of Physics AS CR Na Slovance 2, Prague 8, Czech Republic 108