Quantum-consistent superstring models in four dimensional space-time

Extra space-time dimensions are predicted by String theory. However, up to date there are not any experimental signals in favor of their existence. It forces to search for consistent string theory formulations in four space-time dimensions. The task can be completed with extending the standard vec...

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Дата:	2009
Автор:	Nurmagambetov, A.J.
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Мова:	English
Опубліковано:	Національний науковий центр «Харківський фізико-технічний інститут» НАН України 2009
Назва видання:	Вопросы атомной науки и техники
Теми:	Ядерная физика и элементарные частицы
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Цитувати:	Quantum-consistent superstring models in four dimensional space-time / A.J. Nurmagambetov // Вопросы атомной науки и техники. — 2009. — № 5. — С. 3-11. — Бібліогр.: 22 назв. — англ.

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spelling	irk-123456789-963962016-03-16T03:02:35Z Quantum-consistent superstring models in four dimensional space-time Nurmagambetov, A.J. Ядерная физика и элементарные частицы Extra space-time dimensions are predicted by String theory. However, up to date there are not any experimental signals in favor of their existence. It forces to search for consistent string theory formulations in four space-time dimensions. The task can be completed with extending the standard vector-type coordinates of four-dimensional space-time with additional tensorial-type bosonic coordinates. The reason of introducing the new set of coordinates is discussed, and calculations of the critical dimension in the Neveu-Schwarz-Ramond tensorial superstring formulation are performed. It is also discussed the role of the new coordinates in the construction of the consistent five-dimensional superstring formulation solely in terms of the tensorial-type coordinates and their world-sheet superpartners. Properties of massless modes casting an open and a closed five-dimensional superstrings spectra are considered in brief. Наявнiсть додаткових просторо-часових вимiрiв передбачено теорiєю струн. Проте, на сьогоднiшнiй день не iснує яких-небудь експериментальних пiдтверджень на користь їх iснування. Ця обставина дає поштовх до пошуку послiдовних формулювань теорiї суперструн у чотиривимiрному просторi-часi. Одним з вирiшень проблеми є розширення стандартного набору чотиривимiрних просторово-часових координат векторного типу додатковими бозонними координатами тензорного типу. У роботi обгово- рюється причина введення саме такого набору додаткових координат, а також приводяться обчислення критичної вимiрностi для формулювання тензорної суперструни типу Невье-Рамона-Шварца. Також обговорюється роль нових координат в побудовi послiдовного формулювання суперструни у просторi- часi вимiрностi п’ять виключно в термiнах координат тензорного типу i їх суперпартнерiв щодо пе- ретворень ‘суперсиметрiї’ на свiтовому листi струни. Стисло розглянутi властивостi безмасових мод в спектрах вiдкритої i замкнутої п’ятивимiрних суперструн. Наличие дополнительных измерений пространства-времени предсказывается теорией струн. Однако, на сегодняшний день не существует каких-либо экспериментальных подтверждений в пользу их суще- ствования. Данное обстоятельство дает толчок к поиску последовательных формулировок теории су- перструн в четырехмерном пространстве-времени. Одним из решений проблемы является расширение стандартного набора четырехмерных пространственно-временных координат векторного типа дополни- тельными бозонными координатами тензорного типа. В работе обсуждается причина введения именно такого набора дополнительных координат, а также приводятся вычисления критической размерности для формулировки тензорной суперструны типа Невье-Рамона-Шварца. Также обсуждается роль но- вых координат в построении последовательной формулировки суперструны в пространстве-времени размерности пять исключительно в терминах координат тензорного типа и их суперпартнеров отно- сительно преобразований ‘суперсимметрии’ на мировом листе струны. Кратко рассмотрены свойства безмассовых мод в спектрах открытой и замкнутой пятимерных суперструн. 2009 Article Quantum-consistent superstring models in four dimensional space-time / A.J. Nurmagambetov // Вопросы атомной науки и техники. — 2009. — № 5. — С. 3-11. — Бібліогр.: 22 назв. — англ. 1562-6016 PACS: 11.10.Kk; 11.25.-w; 11.25.Pm http://dspace.nbuv.gov.ua/handle/123456789/96396 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Ядерная физика и элементарные частицы Ядерная физика и элементарные частицы
spellingShingle	Ядерная физика и элементарные частицы Ядерная физика и элементарные частицы Nurmagambetov, A.J. Quantum-consistent superstring models in four dimensional space-time Вопросы атомной науки и техники
description	Extra space-time dimensions are predicted by String theory. However, up to date there are not any experimental signals in favor of their existence. It forces to search for consistent string theory formulations in four space-time dimensions. The task can be completed with extending the standard vector-type coordinates of four-dimensional space-time with additional tensorial-type bosonic coordinates. The reason of introducing the new set of coordinates is discussed, and calculations of the critical dimension in the Neveu-Schwarz-Ramond tensorial superstring formulation are performed. It is also discussed the role of the new coordinates in the construction of the consistent five-dimensional superstring formulation solely in terms of the tensorial-type coordinates and their world-sheet superpartners. Properties of massless modes casting an open and a closed five-dimensional superstrings spectra are considered in brief.
format	Article
author	Nurmagambetov, A.J.
author_facet	Nurmagambetov, A.J.
author_sort	Nurmagambetov, A.J.
title	Quantum-consistent superstring models in four dimensional space-time
title_short	Quantum-consistent superstring models in four dimensional space-time
title_full	Quantum-consistent superstring models in four dimensional space-time
title_fullStr	Quantum-consistent superstring models in four dimensional space-time
title_full_unstemmed	Quantum-consistent superstring models in four dimensional space-time
title_sort	quantum-consistent superstring models in four dimensional space-time
publisher	Національний науковий центр «Харківський фізико-технічний інститут» НАН України
publishDate	2009
topic_facet	Ядерная физика и элементарные частицы
url	http://dspace.nbuv.gov.ua/handle/123456789/96396
citation_txt	Quantum-consistent superstring models in four dimensional space-time / A.J. Nurmagambetov // Вопросы атомной науки и техники. — 2009. — № 5. — С. 3-11. — Бібліогр.: 22 назв. — англ.
series	Вопросы атомной науки и техники
work_keys_str_mv	AT nurmagambetovaj quantumconsistentsuperstringmodelsinfourdimensionalspacetime
first_indexed	2025-07-07T03:35:33Z
last_indexed	2025-07-07T03:35:33Z
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fulltext	NUCLEAR PHYSICS AND ELEMENTARY PARTICLES QUANTUM-CONSISTENT SUPERSTRING MODELS IN FOUR DIMENSIONAL SPACE-TIME A.J. Nurmagambetov∗ National Science Center ”Kharkov Institute of Physics and Technology”, 61108, Kharkov, Ukraine (Received July 1, 2009) Extra space-time dimensions are predicted by String theory. However, up to date there are not any experimental signals in favor of their existence. It forces to search for consistent string theory formulations in four space-time dimensions. The task can be completed with extending the standard vector-type coordinates of four-dimensional space-time with additional tensorial-type bosonic coordinates. The reason of introducing the new set of coordi- nates is discussed, and calculations of the critical dimension in the Neveu-Schwarz-Ramond tensorial superstring formulation are performed. It is also discussed the role of the new coordinates in the construction of the consistent five-dimensional superstring formulation solely in terms of the tensorial-type coordinates and their world-sheet su- perpartners. Properties of massless modes casting an open and a closed five-dimensional superstrings spectra are considered in brief. PACS: 11.10.Kk; 11.25.-w; 11.25.Pm 1. INTRODUCTION One of the most fascinating predictions of String the- ory is the existence of extra dimensions required for quantum consistency of strings. Extra dimensions, the presence of which was very sceptically conceived at early days of String theory, have become the stan- dard de facto in the construction of various Unifica- tion models of fields and have received a lot of atten- tion on the particle physics community side. Searching for signals from extra dimensions on modern experimental factories is a good way to put String theory on the test. Unfortunately, up to date there are not any experimental observations in favor of extra dimensions. Therefore, a possibility of liv- ing in the World without extra dimensions has not a priori to be ruled out. This fact stimulates search- ing for String based Unification models being already consistent in the observed number of space-time di- mensions. In this notes I discuss models of supersymmet- ric strings which fall into such a criterion. To make the notes self-contained I begin with a brief discus- sion of extra dimensions in frameworks of String the- ory. Then, I discuss benefits and drawbacks of two most popular scenarios of Unification models with ex- tra dimensions, the Kaluza-Klein scenario and the Brane World model. Next, I consider a formulation of String theory which is consistent, i.e. anomaly free, in D = 4, and outline some of features of the construction. My conclusions with summary of the results are collected in the end of the paper. 2. EXTRA DIMENSIONS FROM STRINGS As I have noted in the above, the quantum con- sistency of String theory implies living in extra- dimensional world. This conclusion comes from an infinite-dimensional algebra of quantum operators [1] [L̂m, L̂n] = (m− n)L̂m+n + A(m)δm+n, (1) A(m) = 1 12 D(m3 −m), which corresponds to a classical infinite-dimensional algebra {Lm, Ln} = i(m− n)Lm+n, (2) generating conformal transformations on a two- dimensional strings’ world-sheet. The difference between quantum and classical Vi- rasoro algebras (1), (2) consists in the central ele- ment of the algebra A(m). This term is absent in (2), and appears after the normal ordering of quan- tum oscillators entering the Virasoro generators L̂m. The central extension of (1) encodes the conformal anomaly in the quantized theory, and it manifestly depends on the space-time dimension. In fact, A(m) is only a part of the total anomaly coefficient, since the classical Virasoro operators generate the resid- ual world-sheet symmetry after the conformal gauge fixing. Following the Faddeev-Popov recipe ghosts have to be introduced, and their contribution into the anomaly coefficient is [1] Ab,c(m) = 1 6 (m− 13m3). The total anomaly coefficient Atotal(m) = A(m) + Ab,c(m) + 2aopenm. (3) also contains the contribution from the open string intercept aopen. ∗ajn@kipt.kharkov.ua PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2009, N5. Series: Nuclear Physics Investigations (52), p.3-11. 3 The anomaly absence, i.e. Atotal(m) = 0, implies the space-time dimension D=26 and the string inter- cept aopen = 1. The same result holds for a closed bosonic string [1]. In the case of Neveu-Schwarz-Ramond (open) su- perstring (NSR superstring [2]) the Virasoro algebra is modified with two additional graded commutators, that leads to the set of two total anomaly coefficients. Depending on boundary conditions on world-sheet fermions two sectors of quantum oscillators appears: the Ramond (R) sector with    Atotal(m) = D 8 m3 + 1 6 (m− 13m3)+ + 1 12 (11m3 − 2m) + 2aR openm, Btotal(m) = D 2 m2 − 5m2 + 2aR open , (4) and the Neveu-Schwarz (NS) sector, where    Atotal(m) = D 8 (m3 −m) + 1 6 (m− 13m3)+ + 1 12 (11m3 + m) + 2aNS openm, Btotal(m) = D 2 (m2 − 1 4 ) + ( 1 4 − 5m2) +2aNS open . (5) From A(m) = 0, B(m) = 0 it follows the critical di- mension D=10 and the string intercepts aR open = 0, aNS open = 1/2. Therefore, the presence of extra dimensions is an intrinsic property of the consistently quantized String theory. We get 22 space-like extra dimensions in bosonic string theory and 6 extra spatial dimen- sions for superstrings. The superstring case is more preferable since we have not the tachyonic vacuum state here: aNS open = 1/2 corresponds to the tachyonic vacuum, while aR open = 0 leads to a well-defined zero- energy vacuum state. Though we have aNS open = 1/2 in the Neveu-Schwarz sector of superstring, we get rid off the tachyon state taking the Gliozzi-Scherk-Olive (GSO) projection [3]. Once the GSO projection is applied the spectrum of states in NS+R sectors pos- sesses D=10 space-time supersymmetry. In what follows I will mainly focus on the superstring case, where the number of extra dimensions is 6. 3. LIVING WITH EXTRA DIMENSIONS From the String theory point of view, our World looks as follows (see Fig.1): Macroworld струны Atoms Nuclei элементарные Particles 10 10 cm -13 -32 -1 Strings Fig.1. The Stringy World To make a contact of String theory living in ten- dimensional world to observable physics in D=4 a special procedure of compactifying the extra dimen- sions has to be realized. There are many ways to this end, but the right way, which would reproduce main properties of the Standard Model (or its minimal supersymmetric extension), is up to date missed. Dealing with extra dimensions one may won- der what is their size and what is the nature of extra dimensions? In the Kaluza-Klein picture [4] the extra dimensions have a small size that leads to appearing very massive particles after the com- pactification, with masses M ∼ 1/lcomp. (lcomp. is a characteristic length of a compactified dimension). 3+1 6 3+1 6 Fig.2. The Kaluza-Klein picture Massive modes coming from the Kaluza-Klein com- pactification are too massive to be ever experi- mentally observed, so the best one can do is to consider massless modes, corresponding in part to the gauge bosons of the SM symmetry group SU(3) × SU(2) × U(1), or an extended symmetry group including the SM group as a subgroup (see Fig.2). Common drawbacks of the Kaluza-Klein scheme consist in: • Unsatisfactory spectrum of particles appearing upon the reduction which does not fit well the spectrum of the SM fields. The desired spec- trum of the Kaluza-Klein massless modes has to be realistic. A part of this spectrum should correspond to the gauge bosons of the Standard Model that puts restrictions on the type of the internal six-dimensional manifolds. However, the way of getting masses for the rest of the modes and establishing their correspondence to other Standard Model fields is an open task [5]. • Gauge hierarchy problem still takes place. The Kaluza-Klein scenario does not resolve the hi- erarchy problem, the gravity scale still remains near the Plank scale. • Typically four rather than three generations of quarks and leptons. The exact number of gener- ations coming after the dimensional reduction is strongly depended on geometrical and topolog- ical characteristics of internal six-dimensional manifolds. Roughly speaking, the number of fermion generations is twice less than the main 4 topological number of the internal manifold (the Betti number). It turns out that the min- imal Betti number for phenomenologically rel- evant internal manifolds is equal to 8 (Calabi- Yau manifolds), hence the number of genera- tions is 4. A way to resolve this problem is to consider special manifolds of a Calabi-Yau type with the relevant Betti number [6], or to re- duce on orbifolds [7] which are not manifolds in a common sense. Another perspective direction is to consider branes intersections [8] within the Brane World scenario (see below). • Masses and chiralities of fermions. After the reduction fermions received masses of the compactification scale order, i.e. huge masses. Massless fermions comes from the zero-eigenvalue states of the Dirac operator on a compact internal manifold. In most phe- nomenologically interesting cases such zero- eigenvalue states do not exist. Another prob- lem is to recover chiral fermions after the re- duction. It may not be correctly resolved within the standard Kaluza-Klein scheme (see e.g. [9] and Refs. therein). • Large cosmological constant. This point be- comes important in context of String theory ap- plication to Cosmology and astrophysics, since we have definitely known that the right cosmo- logical constant is small. Problems with Kaluza-Klein motivated searching for other scenarios. One of them became popular last decade is the Brane World scenario [10]. Within the Brane World (BW) scenario it is supposed that fields of the SM do confine on a 4- dimensional brane (3-brane). A 3-brane is embedded in a higher-dimensional World. Gravity takes a spe- cial place in the BW picture since gravity does not confine on a 3-brane and gets trapped in high dimen- sions. From the String theory point of view the BW pic- ture looks like (see Fig.3): 10-dimensional string’s World Our World “Standard Model” 3-brane strings 6-dimensional manifold Fig.3. The Brane-World picture A 3-brane is embedded into ten-dimensional space-time, a connection between 3-brane and extra dimensions is realized through strings. What is im- portant in such a scheme is that extra dimensions are large. It leads to essential decreasing of the effective Plank scale on a Brane, that resolves the hierarchy problem. Substantial progress in the Stringy BW has been achieved, nevertheless several important problems still remain open: • How to break Supersymmetry in a correct way? Indeed, once we are talking about a 3-brane, it naturally appears in type IIB supersymmet- ric String theory. One may wonder why it is so necessary to deal with Superstring theory? The answer is we would like to have a joint coupling constant in high energies that provides by su- persymmetry, and we would like to have a uni- fied theory of gravity and the SM fields that is realized in String theory. However, the SM is not a supersymmetric theory, hence the way of supersymmetry breaking has to be found. • How to set up the right cosmological constant in the end? I recall that Anti-de-Sitter space is actively exploited within the BW. Hence, all the machinery of the AdS/CFT correspondence is applied here. But we have to recover the right, de-Sitter space, cosmological constant in the end, which is the experimentally verified cosmological constant driving the late-time ac- celeration of Universe. • The predicted gravity scale is over TeV, but should we believe in that? The BW scenario is a proposal for the resolving the hierarchy prob- lem. However, we have not any signals on TeV quantum gravity (as well as on extra dimen- sions) up to date that makes the point ques- tionable. 4. LIVING WITHOUT EXTRA DIMENSIONS Living in extra dimensional World makes possi- ble to resolve some of the fundamental problems of the Standard Model. At the same time the major worry on extra-dimensions is the absence of any ex- perimental signals in favor of their existence. Once living in extra-dimensional World will be experimen- tally verified, it will get rid of any doubts on them, and on String theory, which predicts extra dimen- sions, as well. Currently, all possible ways of con- structing Unification models, with or without extra dimensions, are needed to be taken into account on equal footing. 4.1. SUSY algebra and supersymmetric strings in extended superspaces I have noted String theory is good enough to unify gravity with other interactions. But could we find a comprehensive String theory with realistic critical di- mensions? To get an answer let me begin with reviewing an irrelevant at first sight subject. In 1988 Curtright [11] made an analysis of the maximally extended SUSY 5 algebra in D=11 (M-theory algebra [12], [13]). The algebra in particular includes {Q,Q} = γaPa + γabPab + γabcdePabcde. (6) The right hand side of (6) contains different types of ”momenta”. Dynamical charge Pa corresponds to the standard momenta, other ‘momenta’ are topological charges corresponding to ‘electrically’ charged mem- brane and ‘magnetically’ charged 5-brane. Clearly, Pab = −Pba, Pabcde = P[abcde]. Membranes and five-branes appear in eleven- dimensional M-theory, however there is not a room for strings there. What happens if charges on the r.h.s. of (6) will be treated in more democratic way? They are differ- ent, of course, the dynamical momenta have the con- jugated coordinates, whilst topological charges have not. To reach charges democracy Curtright proposed, instead of the standard D=11 superspace (Xa, θα), an ‘extended’ D=11 superspace (Xa, Zab, Zabcde, θα) [11], where Zab, Zabcde are “coordinates” conju- gated to topological charges Pab, Pabcde. Curiously enough, there exists a room for superstrings in such an extended superspace. A general form of the Curtright’s superstring ac- tion with unit tension looks as follows [11] S = ∫ d2ξ √ −det(ωa µωνa + αωab µ ωνab + βωabcde µ ωνabcde) (7) +SWZ . The building blocks of the action consist of the pull-back of D=11 Volkov-Akulov superform ωa µ = ∂µXa + iθ̄γa∂µθ, its extensions to tensorial-type co- ordinates ωab µ = ∂µZab + iθ̄γab∂µθ and ωabcde µ = ∂µZabcde + iθ̄γabcde∂µθ. Two parameters α, β are constants fixed by supersymmetry in the end, and the last terms of the action is the Wess-Zumino term. The term by Wess and Zumino was introduced in (7) to reach the invariance of the action under a local fermionic symmetry, the so-called kappa-symmetry, taking an important place in theory of supersymmet- ric extended objects. Nevertheless, in the original Curtright’s paper the kappa-invariance of the action was rather claimed than exactly proved. Now what about D=4? A similar extension of D=4 superspace was considered by Amorim and Barcelos-Neto [14], and a line of they reasoning was almost the same. The maximally extended N=1 D=4 superalgebra in particular contains [12] {Q,Q} = γaPa + γabPab. (8) Adding new tensor-type coordinates Zab = −Zba, which are conjugated to ”momenta” Pab, we get an extended superspace (Xa, Zab, θα). Pab is commonly treated as a topological charge (due to a D=4 mem- brane), but treating it dynamically it’s possible to construct a Green-Schwarz-type superstring in the extended superspace S = ∫ d2ξ √ −det(ωa µωνa + αωab µ ωνab) + SWZ . (9) The notation in (9) is that of (7). I postpone the discussion of (9) to the end of the paper, currently focussing on the superconformal algebra in the ex- tended superspace and on the superstring critical di- mension. 4.2. Superconformal algebra in tensorial superspace and superstring’s critical dimension To calculate the critical dimension of tensorial su- perstring let us turn back to the bosonic string case. As it has been noted in the above the total conformal anomaly coefficient (eq. (3)) Atotal(m) = 1 12 D(m3−m)+ 1 6 (m−13m3)+2aopenm contains contributions from bosonic fields Xa, confor- mal (anti)ghosts and the string intercept. One could notice that • The critical dimension is calculated from set- ting the terms proportional to m3 to zero. • D bosonic coordinates Xa contribute the rela- tive coefficient D. • The conformal (anti)ghosts contribute the rel- ative coefficient ”−26” independently on the number of space-time dimensions. Hence, we need 26 bosonic coordinates Xa to com- pensate the ghosts contribution, +26− 26 = 0. In the NSR superstring case one of the total su- perconformal anomaly coefficients has the following form Atotal(m) = ( D 12 · 1 + D 12 · 1 2 ) m3 + 1 6 (m− 13m3)+ + 1 12 (11m3 − 2m) + 2aR openm. (10) It’s easy to recognize the contributions of bosonic Xa, fermionic conformal (anti)ghosts, bosonic supercon- formal (anti)ghosts and the contribution of the string intercept. But what about the second term of (10)? This term contains the contribution of the world- sheet fermionic superpartners ψa of the bosonic co- ordinates Xa. Clearly, the fermionic superpartners contribute only 1/2 of the corresponding bosonic co- efficient. Hence, to calculate the critical dimension the fol- lowing mnemonic rule may be used [1]: • D bosons get the coefficient D. • The input of D fermions (boson’s superpart- ners) is D/2. • The (super)conformal ghosts give ‘−26’ for con- formal (fermionic) ghosts, and superconformal (bosonic-type) ghosts contribute ”+11”. 6 The difference ‘−26 + 11 = −15’ has to be compen- sated with contributions of additional bosonic and/or fermionic fields, one of the realizations of which are bosonic coordinates Xa and their world-sheet super- partners ψa. Let us fix the space-time dimension D = 4. Four bosonic coordinates Xm and their four world-sheet superpartners ψm contribute the coefficient 4+4/2 = +6. On account of ghosts contribution it is necessary to compensate the coefficient equal to −15+6 = −9. There are different routes to this end. Say, if one were to use 6 ‘internal’ coordinates yi and their su- perpartners ψi (i = 1, . . . , 6), this choice would be transformed into the standard 4+6 = 10 set of coor- dinates of the NSR superstring in the end. Another productive choice suggested by D = 4 N = 1 superalgebra structure is to consider 6 addi- tional tensorial-type coordinates Zmn = −Znm to- gether with their world-sheet superpartners Ψmn = −Ψnm [15]. This set of coordinates contribute the required coefficient +9, hence we arrive at the con- sistent quantum formulation of superstring in the ob- servable number of space-time dimensions. 5. NEW SET OF COORDINATES AND NEW FIELDS We have established the existence of a NSR-type superstring formulation with realistic critical dimen- sion. The price we paid to this end is the exten- sion of the conventional space-time with additional tensorial-type bosonic coordinates. Let me take an extensive treatment of new coordinates in the so ex- tended space and give more strong evidence for the quantum consistency of the superstring. Note to this end the bosonic subset (Xm, Zmn) could be embedded into the unique set of tensorial coordinates ZMN , but in D = 5, Zm̃5 Ã Xm, Zm̃ñ Ã Zmn, m̃ = 0, . . . , 3 . (11) Such a coordinates embedding of (Xm, Zmn) can be done in any space-time dimension (D− 1) thus lead- ing to D-dimensional space parameterized by ZMN . After that, instead of the standard Nambu-Goto string action functional in the conventional space- time with coordinates Xm(ξ) SNG = T 2 ∫ d2ξ √ −det ∂µXm∂νXm , we have the action in terms of solely tensorial coor- dinates ZMN (ξ) S = T 2 ∫ d2ξ √ − det ∂µZMN∂νZMN . (12) Here, as well as in the action before, T is the string tension. The action (12) possesses the same world-sheet symmetries as that of the Nambu-Goto action, hence after the appropriate gauge fixing the equation of mo- tion of ZMN (ξ) is reduced to ∂µ∂µZMN = 0. (13) In what follows I will consider the closed tensorial string, then the solution to eq. (13) satisfying the closed string boundary conditions is ZMN R = 1 2 zMN + 1 2 l2pMN (τ − σ)+ + l 2 ∑ n6=0 1 n αMN n e−2in(τ−σ) , ZMN L = 1 2 zMN + 1 2 l2pMN (τ + σ)+ + l 2 ∑ n 6=0 1 n α̃MN n e−2in(τ+σ) . (14) Here, as usual, I set l = √ 2α′ = 1/ √ πT , where T is the string tension. ZMN R,L are supposed to be real that leads to αMN −n = (αMN n )†, α̃MN −n = (α̃MN n )†. Now we have to define the Poisson brackets be- tween canonical variables. They are {ŻMN (σ), ZPQ(σ′)}PB = 1 2 T−1× × ( ηMP ηNQ − ηMQηNP ) δ(σ − σ′) , (15) and the overall factor in the r.h.s. of the Poisson brackets has chosen to be one half to get the right canonical Poisson brackets between D − 1 vector co- ordinates and their momenta {Ẋm(σ), Xn(σ′)}PB = T−1ηmnδ(σ − σ′) (16) after identifying √ 2Zm(D) = Xm. Substituting (14) into (15) we derive, by use of ∞∑ n=−∞ ein(σ−σ′) = 2πδ(σ − σ′) , the following non-trivial Poisson brackets {αMN n , αKL m }PB = i 2 nδn+m ( ηMKηNL − ηMLηNK ) , (17) {α̃MN n , α̃KL m }PB = i 2 nδn+m ( ηMKηNL − ηMLηNK ) . (18) Next, we construct the Virasoro generators Lm = T 2 ∫ π 0 e−2imσŻ2 R dσ\|τ=0 = 1 2 ∞∑ −∞ αMN m−kαMN k . (19) Here we have substituted ŻMN R = l ∞∑ −∞ αMN m e−2im(τ−σ) , and have used ∫ π 0 e2imφe−2inφdφ = πδmn . Clearly, αMN 0 = 1 2 lpMN . The left moving Virasoro generators L̃m are identical to (19) with relpacing αMN m → α̃MN m , and α0 = α̃0. 7 Taking into account the Poisson brackets (17), (18) one may calculate the Poisson brackets between the Virasoro generators {Lm, Ln}PB = i(m− n)Lm+n, {L̃m, L̃n}PB = i(m− n)L̃m+n. (20) There is no difference between the Poisson brackets in (20) and those of the standard bosonic string theory. The same happens for the quantum Virasoro algebra, which is [Lm, Ln] = (m− n)Lm+n + A(m)δm+n, [L̃m, L̃n] = (m− n)L̃m+n + A(m)δm+n, (21) and A(m) = c3m 3 + c1m. Two unfixed coefficients are calculated in the standard way [1] that results in A(m) = 1 24 D(D − 1)(m3 −m). (22) The conformal anomaly coefficient possesses the same dependence on m as its standard counterpart entering eq. (1). This is not a surprise since the result is based on properties of a two-dimensional world-sheet of any string. The difference comes from the contribution of the world-sheet coordinates: it is equal to D in the conventional bosonic string the- ory, while for the tensorial string case it becomes D(D − 1)/2 (as it should follow from the mnemonic rule considered in the previous section). The b− c ghosts system contributes Ab,c(m) = 1 3 (m− 13m3), (23) where we have summed over the left and the right modes contributions. Summing up, the total anom- aly, which has to be canceled, is Atotal(m) = [ 1 2 D(D − 1) ] × 1 6 (m3 −m)+ + 1 3 (m− 13m3) + 2acl.m. (24) The expression we got is very similar to that of the total anomaly of a bosonic string with vector-type coordinates (cf. eq. (1)). The difference is just in replacing D in the standard bosonic string case with 1 2D(D − 1). From (24) one may notice that there is not a (integer) critical dimension where anomaly is canceled, and the ordering constant is the same as for a closed bosonic string, acl. = 2. Let me turn to the NSR-type tensorial super- string. The world-sheet superpartners of Xm and Zmn can also be recast into the single world-sheet fermion ΨMN Ψm̃5 Ã ψm, Ψm̃ñ Ã ψmn, m̃ = 0, . . . , 3 . The gauged fixed action of the tensorial superstring (compare to [16], [17]) is in the case S = T 2 ∫ d2ξ ( ∂µZMN∂µZMN − iΨ̄MNρµ∂µΨMN ) . (25) In the NSR-type formulation we are dealing with tensorial-type coordinates ZMN which are scalars w.r.t. the world-sheet diffeomorphisms, and with their superpartners ΨMN which are world-sheet spinors. If we calculate the Virasoro-like superalgebra of the NSR-type string following standard methods of [1], the difference in the total anomaly coefficients, in compare to the standard NSR superstring case, con- sists just in replacing D in (4), (5) with 1 2D(D − 1) [15]. This result is very expected from the previous calculations of the bosonic tensorial string anomaly coefficient. We have    Atotal(m) = 1 12 (D(D−1) 2 + D(D−1) 4 )m3+ + 1 6 (m− 13m3) + 1 12 (11m3 − 2m) + 2aRm, Btotal(m) = D(D−1) 4 m2 − 5m2 + 2aR , (26) in the Ramond sector and    Atotal(m) = 1 12 (D(D−1) 2 + D(D−1) 4 )(m3 −m)+ + 1 6 (m− 13m3) + 1 12 (11m3 + m) + 2aNSm, Btotal(m) = D(D−1) 4 (m2 − 1 4 ) + ( 1 4 − 5m2)+ +2aNS (27) in the Neveu-Schwarz sector. The critical dimension is D = 5 in the case, and we arrive at new formulation of superstring theory living in five-dimensional space-time endowed with coordinates ZMN . This parametrization includes, as a four-dimensional part, the standard set of vector- type coordinates Xm, hence coordinates ZMN seem to be more fundamental than Xm, and the tensorial string theory in D = 5 becomes more fundamental than its four-dimensional analog. All of that reminds of the M-theory–String theory relation, when the more fundamental theory is formulated in a space- time of one spatial dimension higher. Another consequence of introducing the new space-time parametrization may be viewed from the following observation. Recall that calculating the Virasoro-like superalgebra of the NSR-type D = 5 tensorial string theory we have used the canonical re- lation between ZMN and their conjugate generalized momenta PMN {PMN (σ), ZKL(σ′)}PB = 1 2 T−1× × ( ηMKηNL − ηMLηNK ) δ(σ − σ′) . When we pass to operators, the generalized momenta become P̂MN = −i∂MN , ∂MNZMN = 1 2D(D − 1). Hence, in tensorial space there is an exotic one-form of the Yang-Mills-type A = AMNdZMN . Its strength tensor is FMN,KL = ∂MNAKL − ∂KLAMN + i[AMN ,AKL] (28) 8 and the action functional for such a field looks like S = 1 4 Tr ∫ dΩD FMN,KLFMN,KL, where dΩD is the invariant volume form in D- dimensional tensorial space.1 The one-form field A is an analog of a non-abelian Yang-Mills gauge field in the conventional space-time. It corresponds to the massless mode in the spectrum of open tensorial string. Indeed, let me define the vacuum state \|0〉 as α̂MN n \|0〉 = 0 for n > 0, where α̂MN n are the annihila- tion operators. They are quantum analog of classical oscillators αMN n entering (14). The first exited level in the momentum representation is described by AMN (k)α̂MN −1 \|0〉, kMNAMN (k) = 0, which is nothing but the above mentioned one-form gauge field. In the spectrum of the closed tensorial string there are other exotic massless modes GMN \|PQ and BMN \|PQ corresponding to graviton and the Kalb- Ramond antisymmetric tensor fields of the standard superstring spectrum: GMN \|PQ ˆ̃α{MN −1 α̂ PQ} −1 \|0〉, BMN \|PQ ˆ̃α[MN −1 α̂ PQ] −1 \|0〉. The ‘graviton’ mode GMN \|PQ possesses the following properties GMN \|PQ = −GNM \|PQ = −GMN \|QP = GPQ\|MN which are formally the same as that of the curva- ture tensor in the conventional space. For a ”Kalb- Ramond” field we have BMN \|PQ = −BNM \|PQ = −BMN \|QP = −BPQ\|MN . The remaining massless mode in the spectrum is a ‘dilaton’ Φ Φ ˆ̃αMN −1 α̂MN −1\|0〉. Having such exotic modes it is important, from the point of view of various applications, to under- stand the dynamics of these fields. The action func- tional for the ‘dilaton’ and the ‘Kalb-Ramond’ fields is more or less predictable. It is likely S = ∫ dΩ5 ( 1 2 ∂MNΦ∂MNΦ+ + 1 12 HMN, KL\|PQHMN, KL\|PQ), where the ”Kalb-Ramond” field strength is defined by HMN, KL\|PQ = ∂MNBKL\|PQ+ +∂PQBMN \|KL + ∂KLBPQ\|MN . As for the effective action of ‘graviton’ GMN \|PQ, its structure is unclear. It could be recovered from cal- culations of the 3-point tree amplitude of interacting strings in the low-energy approximation (as, for in- stance, in [19]) and I postpone this task for further studies. 6. SUMMARY AND CONCLUSIONS We have discussed a reformulation of superstring theory, the critical dimension of which coincides with the observable space-time dimension. To recover the critical dimension D = 4 an ex- tension of the standard space-time is required. New elements which have to be taken into account are tensorial-type bosonic coordinates. From the point of view of the string world-sheet theory, it does not matter what kind of bosonic coordinates need to be added to compensate the superconformal anomaly. They could be scalars, vectors or tensors under the space-time Poincare. The main point is that they are scalars with respect to the world-sheet diffeomor- phisms. One may wonder, that is a rule for selecting new coordinates then? What kind of the coordinates have to be selected to parameterize the target space? It turns out that the choice of the string’s coordinates describing an immersion of the string world-sheet into a target superspace is governed by the structure of a target space superalgebra. Let me discuss the target-space – world-sheet cor- respondence in more detail. There are two indepen- dent formulations of superstrings: 1. Neveu-Schwarz-Ramond with the world-sheet supersymmetry; 2. Green-Schwarz with the manifest target-space supersymmetry. As I have noted these formulations are equiv- alent in D = 10, since their quantum spec- tra coincide (after truncation of the NSR spec- trum with the GSO projection (see Fig.4)) and their critical dimensions are the same. w.-s. SUSY NSR SS GS SS target-sp. SUSY GSO projection Fig.4. NSR-GS connection The world-sheet SUSY in the NSR formulation just says that there are world-sheet scalars and their superpartners under the world-sheet supersymmetry. However, it doesn’t say anything on properties of these variables under the target-space Poincare trans- formations. In its turn, properties of the string coordinates in the Green-Schwarz formulation are fixed. Indeed, a part of the space-time SUSY algebra is {Q,Q} = γaPa + . . . 1I should recall that the number of coordinates of D-dimensional tensorial space does not coincide with D. 9 and string coordinates are defined as ones conjugated to Pa. They are vector-type coordinates Xa with respect to the target-space Poincare. Precisely this type of the coordinates enter the standard NSR string action. Extending the space-time to superspace re- covers the rest of the coordinates entering the Green- Schwarz superstring action, the space-time fermions θα. They are the target-space superpartners of Xa. Hence, the relation between NSR and GS su- perstrings observes an independent interpretation, in which properties of the space-time SUSY, manifest in the GS formulation, govern the choice of the space- time coordinates to describe the NSR string. Let me now turn to the Green-Schwarz-type ac- tion (9). This action is based on the target space supersymmetry algebra involving the supercharges anticommutator (8). If we give a credit to hav- ing a correspondence between NSR and GS formu- lations in the extended superspace (Xm, Zmn, θα), the NSR-type tensorial superstring variables are (Xm, Zmn) together with their world-sheet super- partners (ψm,Ψmn). Indeed, these variables casting ZMN and ΨMN enter the gauge fixed action (25). As for the NSR-type tensorial superstring the bosonic subset (Xm, Zmn) of the Green-Schwarz-type tensorial superstring coordinates could be embedded into the unique set of tensorial coordinates ZMN , but in D = 5. Zm̃5 Ã Xm, Zm̃ñ Ã Zmn, m̃ = 0, . . . , 3 . (29) The # of fermionic target-space superpartners θα is the same in D = 5 and D = 4. Therefore, it is pos- sible to reformulate the superstring model solely in terms of (ZMN , θα) coordinates that essentially sim- plifies the Green-Schwarz-like tensorial superstring action [18] and proving its kappa-invariance. More- over, the consistency of the Green-Schwarz-type su- perstring model in D = 5 tensorial superspace (ZMN , θα) (kappa-invariance of the action) also re- quires [18] GM [N \|PQ] = 0 . This condition just says that the field GMN \|PQ is in the [2,2] irreducible rep. over the Lorentz in D = 5 tangent space. At the same time I should note that the Green- Schwarz-type formulation of tensorial superstring in D = 4 (or equivalently in D = 5) extended super- space faces with several questionable points. First of all one may encounter an apparent mismatch be- tween bosonic and fermionic degrees of freedom in the case. Hence, it is necessary to understand the root of the problem. A helpful way to this end is to recover the spectrum of open/closed tensorial strings in different formulations and to figure out an analog of the GSO projection to relate spectra of tensorial superstrings. Perhaps, applying the machinery of the twistor-like superembedding approach [20], [21] (and Refs. therein), which ‘closes’ the diagram on Fig.4, may be useful to this end. Another intriguing prob- lem is to construct the effective action of massless modes to check a correspondence of the approach to that of [22] where a new concept of the area metric was introduced. Acknowledgements. I am very indebted to Igor Bandos and Victor Berezovoj for enlightened discus- sions on the subject of this notes. Partial support from the INTAS Grant # 05-08-7928 and the NASU- RFFI project # 38/50-2008 is acknowledged. References 1. M.B. Green, J.H. Schwarz and E. Witten. Super- string theory. Cambridge: “Cambridge Univer- sity Press”, 1987, v.1, 470 p. 2. P. Ramond. Dual Theory for Free Fermions // Phys. Rev. 1971, v. D3, p. 2415-2418; A. Neveu and J.H. Schwarz. Factorizable dual model of pions // Nucl. Phys. 1971, v. B31, p.86- 112. 3. F. Gliozzi, J. Scherk and D. Olive. Supersymme- try, Supergravity Theories and the Dual Spinor Model // Nucl. Phys. 1977, v. B122, p.253-290. 4. M.J. Duff, B.E.W. Nilsson and C.N. Pope. Kaluza-Klein Supergravity // Phys. Rep. 1986, v. 130, p.1-142. 5. V. Braun, Y.-H. He, B.A. Ovrut and T. Pantev. The Exact MSSM Spectrum from String Theory // JHEP. 2006, v. 0605, p.043. 6. P. Candelas and R. Davies. New Calabi- Yau Manifolds with Small Hodge Numbers. arXiv:0809.4681[hep-th] (2008). 7. S. Ramos-Sanchez. Towards Low Energy Physics from the Heterotic String. arXiv:0812.3560[hep- th] (2008). 8. T. Kimura and S. Mizoguchi. Yet Another Al- ternative to Compactification – Heterotic Five- branes Explain Why Three Generations in Na- ture. arXiv:0905.2185[hep-th] (2009). 9. F. Cianfrani and G. Montani. Review on Ex- tended Approaches in the Kaluza-Klein Model. arXiv:0904.0574[gr-qc] (2009). 10. L. Randall and R. Sundrum. A Large Mass Hi- erarchy from a Small Extra Dimension // Phys. Rev. Lett. 1999, v. 83, 3370-3373; An Alternative to Compactification // ibid. 1999, v. 83, p.4690- 4693. 11. T. Curtright. Are there any superstrings in eleven-dimensions? // Phys. Rev. Lett. 1988, v.60, p.393-396. 12. J.W. van Holten and A. van Proeyen. N=1 Supersymmetry Algebras in D=2, D=3, D=4 mod(8) // J. Phys. 1982, v. A15, p.3763-3783. 13. E. Sezgin. The M-Algebra // Phys.Lett. 1997, v.B392, p.323-331. 10 14. R. Amorim and J. Barcelos-Neto. Superstrings with tensor degrees of freedom // Z. Phys. 1994, v. C64, p.345-347. 15. R. Amorim and J. Barcelos-Neto. Extensions of string theories // Z. Phys. 1993, v. C58, p.513- 518. 16. L. Brink, P. Di Vecchia and P. Howe. A locally supersymmetric and reparametrization invariant action for the spinning string // Phys. Lett. 1976, v.B65, p.471-474. 17. S. Deser and B. Zumino. A complete action for the spinning string // Phys. Lett. 1976, v.B65, p.369-373. 18. A.J. Nurmagambetov. Local fermionic symmetry of Green-Schwarz superstring in tensorial super- space // Visnyk of Kharkov Natl. Univ. 2009, v.1(41), p.21-24 (in Russian). 19. R.I. Nepomechie. On The Low-Energy Limit Of Strings // Phys. Rev. 1985, v. D32, p.3201-3207. 20. D.P. Sorokin.Superbranes and Superembeddings // Phys. Rept. 2000, v. 329, p.1-101. 21. I.A. Bandos, J.A. de Azcarraga and C. Miquel- Espanya. Superspace formulations of the (su- per)twistor string // JHEP 2006, v. 0607, p.005. 22. F.P. Schuller and M.N.R. Wohlfarth. Geometry of manifolds with area metric: multi-metric back- grounds // Nucl. Phys. 2006, v. B747, p.398-422. О КВАНТОВО-СОГЛАСОВАННЫХ МОДЕЛЯХ СУПЕРСТРУН В ЧЕТЫРЕХМЕРНОМ ПРОСТРАНСТВЕ-ВРЕМЕНИ А.Ю. Нурмагамбетов Наличие дополнительных измерений пространства-времени предсказывается теорией струн. Однако, на сегодняшний день не существует каких-либо экспериментальных подтверждений в пользу их суще- ствования. Данное обстоятельство дает толчок к поиску последовательных формулировок теории су- перструн в четырехмерном пространстве-времени. Одним из решений проблемы является расширение стандартного набора четырехмерных пространственно-временных координат векторного типа дополни- тельными бозонными координатами тензорного типа. В работе обсуждается причина введения именно такого набора дополнительных координат, а также приводятся вычисления критической размерности для формулировки тензорной суперструны типа Невье-Рамона-Шварца. Также обсуждается роль но- вых координат в построении последовательной формулировки суперструны в пространстве-времени размерности пять исключительно в терминах координат тензорного типа и их суперпартнеров отно- сительно преобразований ‘суперсимметрии’ на мировом листе струны. Кратко рассмотрены свойства безмассовых мод в спектрах открытой и замкнутой пятимерных суперструн. ПРО КВАНТОВО-УЗГОДЖЕНI МОДЕЛI СУПЕРСТРУН У ЧОТИРИВИМIРНОМУ ПРОСТОРI-ЧАСI О.Ю. Нурмагамбетов Наявнiсть додаткових просторо-часових вимiрiв передбачено теорiєю струн. Проте, на сьогоднiшнiй день не iснує яких-небудь експериментальних пiдтверджень на користь їх iснування. Ця обставина дає поштовх до пошуку послiдовних формулювань теорiї суперструн у чотиривимiрному просторi-часi. Одним з вирiшень проблеми є розширення стандартного набору чотиривимiрних просторово-часових координат векторного типу додатковими бозонними координатами тензорного типу. У роботi обгово- рюється причина введення саме такого набору додаткових координат, а також приводяться обчислення критичної вимiрностi для формулювання тензорної суперструни типу Невье-Рамона-Шварца. Також обговорюється роль нових координат в побудовi послiдовного формулювання суперструни у просторi- часi вимiрностi п’ять виключно в термiнах координат тензорного типу i їх суперпартнерiв щодо пе- ретворень ‘суперсиметрiї’ на свiтовому листi струни. Стисло розглянутi властивостi безмасових мод в спектрах вiдкритої i замкнутої п’ятивимiрних суперструн. 11

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