On the Problems with Background Independence in String Theory
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困难和曲折是垫脚石Life is full of challenges and obstacles. These difficulties and twists and turns may seem daunting at times, but they often serve as stepping stones towards personal growth, resilience, and success. In this essay, we will explore how facing and overcoming difficulties and obstacles can ultimately shape us into stronger and more capable individuals.One of the key aspects of facing difficulties is the opportunity for personal growth that they present. When we encounter challenges, we are forced to push ourselves beyond our comfort zones and develop new skills and coping mechanisms. Whether it is navigating a tough academic course, dealing with a personal setback, or overcoming a fear, each obstacle provides us with a chance to learn more about ourselves and our capabilities. By facing these challenges headon, we can discover hidden strengths and talents that we may not have known we possessed.Moreover, overcoming obstacles can also help us build resilience and mental toughness. Each time we successfully navigate a difficult situation, we become more adept at handling adversity in the future. Just as a muscle grows stronger with each workout, our ability to bounce back from setbacks improves with each challenge we face. This resilience is a valuable traitthat can serve us well in all areas of life, helping us to persevere in the face of adversity and setbacks.In addition to personal growth and resilience, overcoming obstacles can also lead to greater success in the long run. The road to success is rarely smooth, and it is often lined with obstacles and setbacks. However, those who are able to persevere in the face of adversity are more likely to achieve their goals and realize their dreams. By viewing difficulties as stepping stones rather than roadblocks, we can approach challenges with a positive attitude and a determination to succeed.In conclusion, while difficulties and obstacles may be daunting, they are ultimately opportunities for growth, resilience, and success. By facing challenges headon and viewing them as stepping stones rather than barriers, we can learn more about ourselves, build resilience, and ultimately achieve our goals. So, the next time you encounter a difficulty or obstacle, remember that it may be just the stepping stone you need to reach new heights in your personal and professional life.。
人生中遇到的困难及解决方法英语作文In life, we often encounter various difficulties and challenges. These obstacles can come in different forms, such as personal struggles, academic setbacks, or professional hurdles. However, it is important to remember that every problem has a solution. In this article, we will explore some common difficulties that people face in their lives and discuss possible ways to overcome them.One of the most common challenges individuals face is self-doubt and lack of confidence. Many people struggle with low self-esteem and find it difficult to believe in their own abilities. To overcome this, it is important to focus on personal growth and self-improvement. Engaging in activities that promote self-confidence, such as setting achievable goals and celebrating small victories, can help build a positive self-image. Additionally, seeking support from friends, family, or a mentor can provide encouragement and guidance during challenging times.Another difficulty people often encounter is the fear of failure. Fear of failure can be paralyzing and prevent individuals from taking risks or pursuing their goals. However, it is important to understand that failure is a natural part of the learning process. Instead of viewing failure as a setback, it can be seen as an opportunity for growth and learning. Embracing a growth mindset and reframing failures as valuable experiences can help individuals overcome the fear of failure and persevere in the face of challenges.Time management is another common difficulty that individuals face. Balancing multiple responsibilities and commitments can be overwhelming, leading to stress and inefficiency. To effectively manage time, it is important to prioritize tasks and set realistic goals. Breaking down larger tasks into smaller, manageable chunks can make them less daunting and more achievable. Additionally, eliminating distractions and creating a structured schedule can help individuals stay focused and make the most of their time.In relationships, communication issues often arise as a major challenge. Misunderstandings, conflicts, and lack of effective communication can strainrelationships and lead to frustration. To overcome these difficulties, it is crucial to practice active listening and empathy. Taking the time to understand the perspectives and feelings of others can facilitate effective communication and resolve conflicts. Additionally, expressing oneself clearly and respectfully can help avoid misunderstandings and build stronger relationships.Financial difficulties are also a common challenge that people face. Managing finances, dealing with debt, and saving money can be stressful and overwhelming. To overcome financial difficulties, it is important to create a budget and stick to it. Tracking expenses, prioritizing needs over wants, and seeking professional advice can help individuals regain control of their finances. Additionally, developing good financial habits, such as saving regularly and avoiding unnecessary debt, can provide long-term financial security.In conclusion, life is full of challenges and difficulties that can test our patience and resilience. However, it is important to remember that every problem has a solution. By adopting a positive mindset, seeking support from others, and developing effective strategies, we can overcome these obstacles and grow stronger in the process. Remember, difficulties are not roadblocks, but rather opportunities for personal growth and development.。
问题背景英语作文Writing an essay on the background of a problem requires careful analysis and research. Here is an example of an essay discussing the background of a problem: Title: The Background of Climate ChangeIntroduction:Climate change has become one of the most pressing issues of our time. It refers to long-term shifts in temperature and weather patterns, primarily caused by human activities. Understanding the background of this problem is essential to address its impacts effectively.Body:1. Historical Context:The concept of climate change dates back centuries. However, it was not until the late 19th century that scientists began to recognize the potential impact of human activities on the Earth's climate. The Industrial Revolution marked a significant turning point, as the burning of fossil fuels released large amounts of greenhouse gases into the atmosphere.2. Scientific Discoveries:In the mid-20th century, scientists madegroundbreaking discoveries regarding the correlation between greenhouse gas emissions and global warming. The Intergovernmental Panel on Climate Change (IPCC), established in 1988, has since played a crucial role in collecting and analyzing scientific data related to climate change.3. Human Activities:Human activities, particularly the burning of fossil fuels, deforestation, and industrial processes, have contributed significantly to the increase in greenhouse gas emissions. The rise in population, urbanization, and increased consumption patterns have further exacerbated the problem. These activities have led to the accumulation of greenhouse gases in the atmosphere, trapping heat and causing the Earth's temperature to rise.4. Impacts on Ecosystems and Biodiversity:Climate change has disrupted ecosystems worldwide. Rising temperatures have resulted in the melting of polar ice caps, the loss of biodiversity, and an increase in extreme weather events such as hurricanes and droughts. The consequences are felt globally, affecting agriculture, water resources, and human health.5. International Response:Recognizing the severity of the problem, the international community has come together to address climate change. The United Nations Framework Convention on Climate Change (UNFCCC) was adopted in 1992, followed by the historic Paris Agreement in 2015. These agreements aim to reduce greenhouse gas emissions, promote sustainable development, and provide support for vulnerable countries.Conclusion:Understanding the background of climate change is crucial for developing effective strategies to mitigate its impacts. Through scientific research, global cooperation, and individual actions, we can work towards a sustainable future and protect our planet for generations to come. It is a collective responsibility to address this problem and ensure a livable environment for all.。
如何解决生活中的困难英语作文Title: Navigating Challenges in Daily Life.Life is a journey fraught with challenges, and it is how we navigate these obstacles that defines our character and resilience. Facing difficulties head-on, learning from them, and growing stronger is the essence of a fulfilling life. In this essay, we will explore strategies to overcome challenges in various aspects of daily life.The first step in addressing any challenge is recognition. Acknowledging that a problem exists is crucial, as it allows us to move beyond denial and towards solutions. Whether it's a personal struggle, a professional setback,or a relationship issue, acknowledging the difficulty isthe first step towards overcoming it.Once we have recognized the challenge, the next step is to assess it objectively. This involves understanding the nature of the problem, its root causes, and its potentialimpact. By breaking down the issue into smaller, more manageable parts, we can gain a clearer understanding of what needs to be done to resolve it.Once we have a clear picture of the challenge, it's time to develop a plan of action. This plan should be specific, measurable, achievable, relevant, and time-bound (SMART). For example, if the challenge is related to time management, the plan might involve setting clear goals, prioritizing tasks, and allocating appropriate time to each task. If it's a financial issue, the plan might involve creating a budget, cutting unnecessary expenses, and saving towards a goal.Executing the plan is the next crucial step. This requires discipline, perseverance, and the ability to stay focused despite setbacks. It's important to remember that progress is often gradual, and small steps can lead to significant changes over time. It's also essential to stay positive and maintain a growth mindset, believing that with effort and perseverance, the challenge can be overcome.Support is crucial throughout this process. Seeking advice from friends, family, or mentors can providevaluable insights and encouragement. Joining support groups or communities can also help us feel less alone and more motivated to persevere. Additionally, self-care is essential; taking time to recharge, refresh, and care for ourselves is vital for maintaining resilience and apositive mindset.Monitoring progress is another key aspect of overcoming challenges. Regularly checking in on our progress allows us to see how far we have come and what still needs to be done. It also helps us identify any areas where we might need to make adjustments or seek additional support.Finally, when we reach a milestone or overcome a significant challenge, it's important to celebrate our achievements. This not only boosts our confidence and motivation but also helps us appreciate the journey we have taken. However, it's also crucial to remember that life is an ongoing process of growth and learning, and there will always be new challenges to face. By maintaining a positivemindset, staying focused, and seeking support when needed, we can continue to navigate the challenges of life with confidence and resilience.In conclusion, overcoming challenges in daily life requires recognition, planning, execution, support, monitoring, and celebration. By applying these strategies, we can turn obstacles into opportunities for growth and development. Remember, every challenge is an opportunity for personal growth and transformation. With perseverance and a positive mindset, we can overcome any obstacle and achieve our goals.。
遇到困难英语作文初二英文回答:In the face of adversity, it is imperative to maintain a positive and resilient mindset. Embracing a "growth mindset," where setbacks are viewed as opportunities for learning and growth, is crucial. Furthermore, it is important to develop a strong support system of individuals who provide encouragement and support during challenging times. Additionally, focusing on one's strengths and accomplishments can boost self-confidence and resilience. Moreover, maintaining a healthy lifestyle, including regular exercise and a balanced diet, can improve overall well-being and better equip individuals to handle difficulties.When confronted with an obstacle, it is beneficial to adopt a problem-solving approach. Breaking down the challenge into smaller, manageable tasks can make it seem less daunting. Seeking guidance from others, such asmentors or counselors, can provide valuable insights and perspectives. It is also essential to avoid dwelling on setbacks and instead focus on finding solutions. Additionally, maintaining a positive attitude and practicing gratitude can help shift one's perspective and foster resilience.中文回答:遇到困难英语作文初二。
如何克服这些困难英语作文英文,Overcoming difficulties is a part of life, andit's important to develop strategies to deal with them. One way to overcome difficulties is to break them down into smaller, more manageable parts. For example, if I'm struggling with a challenging English essay, I can break it down into smaller tasks such as brainstorming ideas, outlining the essay, and then writing the first draft. By focusing on one task at a time, I can make progress andfeel a sense of accomplishment.Another strategy is to seek support from others. This could be in the form of asking a teacher or tutor for help, or simply talking to a friend or family member about the difficulties I'm facing. Sometimes, just talking about the problem can help me see it from a different perspective and come up with new solutions.Additionally, I find it helpful to stay positive and remind myself of past successes. When I encounter adifficult situation, I can think back to times when I overcame similar challenges and draw strength from those experiences. This positive mindset can help me stay motivated and persistent in the face of difficulties.Lastly, I believe in the power of perseverance. Difficulties are a natural part of life, and it's important to keep pushing forward even when things get tough. By staying determined and not giving up, I can often find a way to overcome the obstacles in my path.中文,克服困难是生活的一部分,发展应对困难的策略是很重要的。
解决身边的困难英语作文Title: Overcoming Challenges Around Us。
Challenges are an inevitable part of life, and they come in various forms, ranging from personal struggles to societal issues. However, it is how we confront and overcome these challenges that define our character and resilience. In this essay, we will explore the importance of addressing difficulties in our lives and provide examples of effective strategies to tackle them.Firstly, it's essential to acknowledge that challenges exist in every aspect of life. Whether it's academic pressure, financial constraints, interpersonal conflicts, or global crises, no one is immune to facing difficulties. However, what sets individuals apart is their ability to confront these challenges head-on rather than succumbing to them.One effective way to address challenges is bycultivating a growth mindset. Instead of viewing obstacles as insurmountable barriers, a growth mindset allows individuals to see them as opportunities for learning and growth. By embracing challenges with a positive attitude and a willingness to learn from setbacks, people can develop resilience and adaptability, enabling them to overcome even the most daunting obstacles.Another crucial aspect of overcoming challenges is seeking support from others. Whether it's friends, family members, teachers, or mentors, having a support system can provide encouragement, guidance, and practical assistance during tough times. By reaching out to others and sharing our burdens, we can lighten the load and gain valuable insights and perspectives that can help us navigate through difficulties more effectively.Furthermore, effective problem-solving skills are instrumental in overcoming challenges. Instead of feeling overwhelmed or defeated by obstacles, individuals can break down complex problems into smaller, more manageable tasks and develop action plans to address them systematically. Byapproaching challenges with a clear strategy and a problem-solving mindset, people can increase their chances of success and overcome hurdles more efficiently.Moreover, maintaining a sense of perspective can be incredibly beneficial when facing challenges. While difficulties may seem overwhelming in the moment, taking a step back and considering the bigger picture can help individuals realize that setbacks are often temporary and can be overcome with time and perseverance. By focusing on long-term goals and staying optimistic about the future, people can maintain a sense of hope and resilience even in the face of adversity.In addition to individual efforts, collective action is often necessary to address larger societal challenges. Whether it's advocating for policy changes, volunteeringfor community initiatives, or participating in grassroots movements, working together with others can amplify our impact and bring about meaningful change. By harnessing the power of collective action, we can address systemic issues and create a more equitable and just society for all.In conclusion, challenges are an inevitable part of life, but they do not have to define us. By cultivating a growth mindset, seeking support from others, developing effective problem-solving skills, maintaining perspective, and engaging in collective action, we can overcome difficulties and emerge stronger and more resilient. Instead of being intimidated by challenges, let us embrace them as opportunities for growth and positive change. Together, we can overcome any obstacle that stands in our way and create a brighter future for ourselves and others.。
面对困境需另辟蹊径的作文英文回答:When faced with difficulties, it is often necessary to think outside the box and find alternative solutions. It is important to approach the problem from different angles and consider unconventional methods. By doing so, we can overcome obstacles and achieve success.One example of this is when I was faced with a challenging project at work. The deadline was approaching, and I was struggling to come up with innovative ideas. Instead of sticking to traditional methods, I decided to brainstorm with my colleagues and gather their input. We held a brainstorming session where everyone was encouraged to share their ideas, no matter how unconventional they seemed. This approach led to the discovery of a unique solution that none of us had thought of before. By thinking outside the box and considering different perspectives, we were able to complete the project successfully.Another example is when I was preparing for a difficult exam. I found myself struggling to understand certain concepts and was worried about my performance. Instead of solely relying on textbooks and lectures, I decided to seek help from my classmates. We formed a study group where we discussed the material and explained it to each other in simpler terms. This collaborative approach not only helped me understand the concepts better, but it also gave me the opportunity to learn from my peers' perspectives. As a result, I was able to overcome my difficulties and perform well in the exam.中文回答:面对困境时,我们常常需要打破常规,寻找另类解决方案。
处理问题和烦恼英语作文Coping with Problems and Troubles.Life is a journey filled with both joy and sorrows, success and failures, happiness and melancholy. Along this path, we encounter various problems and troubles that test our resilience and wisdom. Coping with these challenges effectively is crucial for our growth and well-being.Firstly, it is important to accept that problems and troubles are inevitable. No one can live a life free from difficulties. When faced with a problem, the first step is to recognize it and accept it without fear or denial. This acceptance helps us to approach the issue with a clear mind and an open heart.Secondly, we need to identify the root cause of the problem. Understanding why a particular issue arose is crucial for finding a lasting solution. It might involve self-reflection, analyzing situations, or seeking feedbackfrom others. This process helps us to gain a deeper understanding of the issue and its underlying causes.Thirdly, once we have identified the problem, it's time to formulate a plan of action. This plan should be realistic, practical, and achievable. It should also involve setting clear goals and breaking down the problem into smaller, manageable tasks. This helps us to stay focused and motivated as we work towards resolving the issue.Fourthly, implementing the plan is crucial. We need to take consistent and persistent action to see results. This might involve making changes in our behavior, seeking help from others, or developing new skills and abilities. The key is to stay committed to the plan and not to give up easily.Lastly, we should always remember to learn from our experiences. Each problem and trouble we encounter teaches us something valuable about life and ourselves. Reflecting on these experiences helps us to gain insights and wisdomthat can guide us in the future.In addition to these steps, there are several other strategies that can help us cope with problems and troubles more effectively. One such strategy is positive thinking.By maintaining a positive mindset, we can stay motivatedand optimistic even in the face of difficulties. We should believe in ourselves and our abilities to overcome any challenge that comes our way.Another strategy is seeking support from others. We should never hesitate to ask for help when needed. Talkingto friends, family, or professionals can provide us with valuable advice and support that can help us cope with our problems better.Moreover, practicing relaxation techniques such as yoga, meditation, or deep breathing can help us to manage stress and anxiety. These techniques help us to calm our minds and focus on the present moment instead of worrying about the future.In conclusion, coping with problems and troubles is an essential part of life. By following the steps outlined above and implementing effective strategies, we can turn these challenges into opportunities for growth and development. Remember, every problem has a solution, and with the right approach, we can overcome any obstacle that comes our way.。
如何缓解这一问题英语作文There are various ways to alleviate this problem. One effective method is to engage in physical activities. Exercise can help release stress and tension, allowing you to relax and clear your mind. Whether it's going for a run, practicing yoga, or playing a sport, physical activities can provide a much-needed break from the problem at hand.Another way to find relief is through creative outlets. Engaging in activities such as painting, writing, or playing a musical instrument can help distract your mind and provide a sense of accomplishment. These creative pursuits allow you to express yourself and focus on something other than the problem, giving you a much-needed mental break.Taking a break and spending time in nature can also be beneficial. Going for a walk in a park or spending time at the beach can help you feel refreshed and rejuvenated. The beauty of nature can provide a sense of calm andtranquility, allowing you to temporarily forget about the problem and gain a new perspective.Connecting with others can also provide relief. Talking to a trusted friend or family member about the problem can help you gain new insights and potential solutions. Sometimes, simply sharing your thoughts and feelings with someone who understands can provide a sense of relief and support.Engaging in relaxation techniques can also be helpful. Practices such as deep breathing, meditation, or mindfulness can help you calm your mind and reduce stress. These techniques can be done anywhere, anytime, making them easily accessible and effective in providing relief.Lastly, taking care of yourself is crucial in alleviating this problem. Getting enough sleep, eating a balanced diet, and practicing self-care activities can help improve your overall well-being and resilience. When you prioritize your own well-being, you are better equipped to handle and overcome the challenges that come your way.In conclusion, there are various ways to alleviate this problem. Engaging in physical activities, pursuing creative outlets, spending time in nature, connecting with others, practicing relaxation techniques, and taking care of yourself are all effective methods to find relief. By incorporating these strategies into your daily life, you can better manage the problem and maintain a sense of balance and well-being.。
a rXiv:h ep-th/9311177v13N ov1993IASSNS-HEP-93/66On the Problems with Background Independence in String Theory Samson L.Shatashvili †#School of Natural Sciences Institute for Advanced Study Olden Lane Princeton,NJ 08540Dedicated to my teacher L.D.Faddeev on the occasion of his 60th birthday The problems with background independence are discussed in the example of open string theory.Based on the recent proposal by Witten I calculate the String Field Theoryaction in conformal perturbation theory to second order and demonstrate that the proper treatment of contact terms leads to nontrivial equations of motion.I conjecture the form of the field theory action to all orders.October,93In contrast with difficult problems in realistic4d Field Theory models,where the theory is defined and an explicit analytic solution is not yet known,String Theory isn’t yet even defined.In many cases what we have is just a number of S-matrices for the processes when the background isfixed by our choice of conformalfield theory in thefirst quantized formulation for amplitudes.Satisfactory formulation of String Theory would have been a formulation where we don’t need to refer to any particular classical background and these ”classical backgrounds”are given by solutions of some equations.The latter statement is very vague,because unfortunately it is not even clear(at least to the author)what should be the right terminology to address the question.It is believed,by analogy with the second quantized description of ordinary quantumfield theory,that the understanding of vacuum structure of string theory as well as the nonperturbative character,can be achieved by developing thefield theory language to describe the target space theory.It might well be that the procedure that allows us to construct second quantizedfield theory from Feynman sum over trajectories directly applied to string theory is not the best way to approach the problem and some other new ideas should be introduced.One of the most important ingredients of any construction has to be a background independence.In this paper I will address the question of background independent formulation of string theory in the example of open string theory recently suggested by Witten[1]1.I can’t claim that at present every point is understood for the case of open string;this paper should be considered as an attempt to single out main problems andfind a correct language based on this experience.The calculations and observations presented in section 2,together withfinal result(see below)might serve as a proper guide.This explains the title.I’ll show that the integrals of total derivatives do not decouple inside the correlation functions that defines the String Field Theory action in the formalism of[1]due to contact term contributions.I’ll explain that these contributions have universal character and can’t be removed by change of renormalization scheme(this statement has the same origin as the one for gauge anomalies infield theory).This fact leads to slight modifications of the assumptions made in original paper[1]and also in[3].It was shown in[3]that under the key assumption of decoupling of total derivatives,plus the requirement that BRST operator on the boundary is coupling constant independent,theory has a linear character.I’ll demonstrate here that including the contribution of total derivatives one also has to properly define”BRST”operator on the boundary,which now necessarily should depend on couplings to satisfy the consistency condition.I’ll discuss the ambiguities related to this issue and will make a particular choice.In this setup I’ll calculate thefield theory action in the lines of[3]using the conformal perturbation theory around somefixed point and demonstrate the existence of following relation:∂S=−βi2The fact that the stringfield theory action on the classical equations of motion is given by the world-sheet bosonic partition function was previously suspected in[5][6]form.As a result this formalism doesn’t avoid the usual technical difficulties that are present in any other formulation of stringfield theory,although it is formally background independent and contains all string modes.1.Boundary Problem and Open String Field TheoryIn the beginning of this section wefirst will describe the construction of[1]with the emphasis of places where some assumptions are made.The idea of the construction in [1]is based on BV formalism.Let M be a supermanifold which is equipped with closed, nondegenerate odd simplectic structureωand U(1)symmetry,called ghost number U.This means that in Darboux coordinatesψ,θon M withψfermionic andθbosonicω=dψdθandωhas ghost number1.In analogy with ordinary(bosonic)simplectic manifolds one can define the Poisson Brackets,antibracket,with{A,B}=∂r A∂t j.(1.1)One can show that the following two simple facts take place:i.If V is a vectorfield that generates the symmetries ofω,which means that(di V+ i V d)ω=0,then there exists a function S thatdS=i Vω.(1.2)ii.Vectorfield defined by(1.2)generates a symmetry ofωfor any function S.From the above immediately follows that the Poisson brackets of function S,{S,S}, defined by(1.2)is annihilated by d and thus the function{S,S}is a constantd{S,S}=0.(1.3) The equation{S,S}=0(1.4)is called the BV master equation and S is the action functional if it solves the master equation.Every solution of(1.4)is automatically gauge invariant[1],[7].;the variation of the action under any symplectic transformationδt i={t i,K},(1.5)generated by hamiltonian function K(odd function),is given byδS={S,K}and for {S,K}=0,it vanishes;trivial transformations are given with K={S,Λ}.In the quantization of gauge theories the action S is given on subspace of M with U=0,S0,and we have tofind S.That is in fact what the Faddeev-Popov procedure does in the case of Gauge Theories.In the case of String Field Theory,the idea of[1]was to identify the antibracket and vectorfield in terms of the world-sheet theory and thus identify S as the action of the corresponding target space theory.It was claimed in[1] that this can be done in the case of Open String in a background independent way.The following identifications were proposed:ω:ω(δO,δO)= ∂Σdσ1 ∂Σdσ2<δO(σ1)δO(σ2)>(1.6)V:δV O={Q,O}(1.7) where<...>formally is defined through the world-sheet theory given by path integral corresponding to the2d actionL a= Σd2z(12πb ij D i c j)+ ∂ΣdθV(X,b,c,t)(1.8) Here,thefirst term is the closed string background and the second term describes an arbitrary boundary interaction,parametrised by coupling constants t i(in general there are infinite number of coupling constants),with the condition that the boundary operator V has the formV=b−1O,(1.9) with O being a general operator of ghost number1.Q is a BRST operator defined by BRST current:Q= C dσJ BRST with contour C approaching the boundary∂Σ3and b−1= C b(v),b(v)=v i b ijǫi k dx k with v i being the killing vector that generates the rotation of disc.This world-sheet action is also equipped with an ultraviolet cutoffa.From the above identifications we have the definition of stringfield theory action:1dS=3We will not worry about generality and assume thatΣis just a disc.where d=dt i d/dt i and<...>again denotes un-normalized correlation function.Witten has shown that(1.6)gives a closed form and it is invariant under(1.7).We need for future use to repeat his arguments and stress the points where some assumptions are made.4 The fact thatωdefined by(1.6)is closed follows from the identity:0=<b−1(A1(θ1)...A n(θn))>(1.11)Here we use the definition of b−1and take two limits:first we shrink the contour C to a point and get zero;second we take the limit when the contour approaches the boundary ∂Σand get the right hand side in(1.11).Thus,in the notationδi O=∂ω(δi O,δj O)−∂t k(1.12)−cyclic permutations=<(b−1δk O)δi Oδj O>−cyclic perm.=0and the last step follows from(1.11).BRST invariance of(1.6)is equivalent to exactness of the right hand side in(1.10). This follows from the simple observation that because the transformation law ofωis ω′=ω+ǫ(i V d+di V)ωand we already have shown that dω=0,what we have to show is that di Vω=0.We haved<dO{Q,O}>=<(b−1dO)dO{Q,O}>−.(1.13)−<dO{dQ,O}>−<dO{Q,dO}>If we use the identity(1.11)for thefirst term in(1.13)and the definition(1.9)we get:<(b−1dO)dO{Q,O}>−<dOb−1dO{Q,O}>+(1.14)+<(dO)2[L0,O]>−<(dO)2[Q,V]>=0.We are considering a deformation of the critical string,so we can drop all terms of the type<{Q,...}>0,where subscript0means the expectation value in the unperturbed theory of some number of operators,using the argument of the contour deformation in the definition for Q,and thus in the last term of(1.14)we can integrate by parts in the pathintegral to obtain+<{Q,(dO)2}>;the same is true for the last term in(1.13),which leads to+1<(dO)2[L0,O]>=0(1.15)2and we see thatω,defined by(1.6)is BRST invariant only if the right hand side in the equation(1.15)is identically zero.It was concluded in[1]that(1.6)is BRST invariant because of the following two assumptions:∂QO(θ)dθ=0(1.17)∂θhere,in(1.17),thefirst identity means that L0is a generator of the rotation of circle, and the second identity assumes that total derivatives decouple inside the correlation functions.5Comment:the second correlator in(1.15),the one with total derivative inside the integral,generically is not zero and might receive the contribution from the boundary of moduli space(position of operators on the circle are moduli).Thus,we have to treat such terms and include their contribution.Or,if we want to set up such a scheme during the evaluation of correlation functions in(1.10),when(1.16)is satisfied once inside the correlator,we have to make sure that our regularization scheme leads to decoupling of total derivatives in the second term in(1.15).The latter is a nontrivial statement and in the next section we are going to address this question in detail.The identity in(1.15)should be considered as the requirement for operator Q;so,Q,when the contour approaches the boundary∂Σshould depend on the couplings according to equation(1.15)and this leads to consistency condition on the construction.From the point of view of conformal perturbation theory the above requirement means that we have to use the parallel transport of Q,consistent with(1.15)when we move away from the critical point t CF T.It happens that only the P SL(2,R)subalgebra of Virasoro algebra is relevant,thus what one needs is to deform this subalgebra by including the contributions of boundary term.In thenext section we will evaluate the right hand side in(1.10)and formulate this consistencycondition in more clear terms for the case when the boundary interaction doesn’t mixesghosts and matter.At the end of this section as an illustration I would like to discuss a known example ofperturbation of a conformalfield theory(closed string)by dimension one operator,wherethe decoupling of total derivatives doesn’t takes place and the obstruction is aβ-function[10].Similar calculations will be performed in the next section for open strings.Consider some CFT perturbed by a dimension one operator V i t i.6We will denotethe correlation functions in the perturbed theory by<<...>>and those in the unper-turbed theory by<..>;so,the partition function for can be written as<<1>>,or<exp(i V)>.One can calculate the trace of stress-tensor in the perturbed theory in the following way.We start from the expectation value of the holomorphic part of stresstensor<<T(z,¯z)>>and use the operator expansion algebraT(z,¯z)V i(w)=1z−w∂∂w(1∂w i(1(n−1)!( V)n−1>.(1.19)If we claim that the contribution of the total derivative in the right hand side is zero,we will conclude that the expectation values of stress tensor is zero;but the latter is wrong–we know that the following relation holds:∂∂z <<trT(z,¯z)>>=βi∂6I would like to thank A.Polyakov for pointing out to me the following example.V i (w 1,¯w 1)V j (w 2,¯w 2)=g ij|w 1−w 2|2V k (w 2)+...(1.21)Now we see that because of the poles in (1.19)and (1.21),there is nonzero boundary contribution in the integral over w i in (1.19).For this we substitute (1.21)in (1.19).After integration over w i in (1.19)we are left with the boundary contour integral,with small contour surrounding each point w j ;denoting w i =w j +ρe iθwith small ρ,we get:<<T (z,¯z )>>= ij t i t j < d 2w j dθρe −iθ1ρ4++C kij (n −2)!( V )n −2>.(1.22)If we expand the denominator in (1.22)in the powers of ρand integrate over θ,we see that only second term contributes,with final answer:<<T (z,¯z )>>=C k ij t i t j d 2w 1conformalfield theory with stress tensor T and corresponding BRST operator Q.When the contour C approaches the boundary of the disc the operatorQ= dθc(θ)[T m+12 2πdθ1dθ3<<c(θ1)dV(θ1)c(θ3)∂θ3c(θ3)V(θ3)>>+12 2πdθ1dθ2(2cos(θ1−θ2)−2)<<dV(θ1)V(θ2)>>== dθ1dθ2cos(θ1−θ2)<<dV(θ1)V(θ2)>>−d<< dθV(θ)>>+d<<1>>.(2.4) If we use the notation L n= 2π0dθe inθT m(θ),the second term in(2.2)can be written as a combination of three expressions:1<< dθ1e iθ1dV(θ1)[L−1, dθ2V(θ2)]>>==< dθe iθdV(θ)[L−1,exp(iL int)]>,(2.6) << dθ1dV(θ1)[L−1, dθ2e iθ2V(θ2)]>>==<d(exp(iL int))[L−1, dθe iθV(θ)]>.(2.7) We need two more identities:<d(exp(iL int))[L−1, dθe iθV(θ)]>=d<<[L−1, dθe iθV(θ)]>>−−<<[L−1, dθe iθdV(θ)]>>−<<[dL−1, dθe iθV(θ)]>>,(2.8) andfinally for the second term in(2.8)<<[L−1, dθe iθdV(θ)]>>==−<[L−1,exp(iL int)] dθe iθdV(θ)>+<[L−1,exp(iL int)] dθe iθdV(θ)]>.(2.9) Here we used the notation where L int is the boundary interaction term,and dL−1= dt i∂dS =d i∂θ2sin (θ1−θ2)<<dV (θ1)V (θ2)>>−− dθ1dθ2sin (θ1−θ2)<<dV (θ1)[L 0−∂2(<<[dL −1, dθe iθV (θ)]>>−c.c.).(2.12)Now I would like to require that under proper definition of renormalization scheme and the generators of SL (2,R )subalgebra,the object X is identically zero.One should note that the consistency condition (1.15)requires that X is just an exact form,thus there is an ambiguity in the definition of Q and equivalently the symmetries of (1.6).Moreover it is not guaranteed that in the deformed theory the stress tensor,that enters in the definition of Q ,is deformed accordingly to this requirement.Below I will give arguments that it is indeed the case and that the vanishing of X is a natural choice.They couldn’t be considered as a rigorous proof to all orders in t ;they are just arguments and most likely they can lead to such a proof.Before I turn to this very important question let us evaluate the first term in (2.12),to be sure that the total derivative doesn’t make it zero.We have in the lowest order in t :dθ1dθ2∂sin 1|sin 1the”resonance term”,is convergent in the limit when we remove the cutoff,and others diverge.These divergent terms can be removed by redefinition of couplings or the same, by a subtraction procedure(see below),while the constant term is universal and can’t be removed.Also,there should be a higher order correction in couplings in(2.13).We see that,like in the example for closed string at the end of the previous section, total derivative in X doesn’t decouple and is proportional toβ-function coefficient C k ij; thus(1.17)fails,so does(1.16).Until now we had avoided the question about the transformation properties of bound-ary operator.To proceed further it is necessary to know the action of L−1,L0and L1on V’s.What we need to cancel the contribution of(2.13)in X is:∂∂t i =δj i∆i+4c jikt k+....,(2.18)andβj is theβ-function for operator V j.This deformation is a very natural one.7In fact what we need is(2.17)in thefirst nontrivial order in couplings to compare with(2.13), because the latter we are able to calculate only up to this order.Here are the arguments in support of(2.17):consider the auotomorphizms of disc:z′=αz−z07There are similarities between(2.17)and the boundary problem analog of anomaly equation (1.20),recently derived by Zamolodchikov[12],see also[13].I would like to thank A.Zamolod-chikov for sharing his insight on the problem of boundary deformations with me.with|z0|<1and|α|=1.The requirement,that corresponding P SL(2,R)subalgebra of Virasoro algebra is not broken in perturbed theory,fixes the transformation law(2.17).In the case of closed string,the transformation properties under constant shift is standard, and under dilatation is controlled by the Callan-Symanzik equation,that leads to change of classical dimension to anomalous dimension[8];so these two elements are universal,and the third one isfixed by the requirement,mentioned above.The open string version is given by(2.17)and leads to(2.16).We see from this expression that they are compatible in the lowest order in couplings and guarantee that X vanishes.What is needed to complete the proof is that one has to show the compatibility of(2.17),in particular regularization scheme (note that higher order terms inβ-function are scheme dependent),with the vanishing of X in(2.12)and(2.13)to all orders..Thus,thefirst term in equation(2.11)defines the stringfield theory actioniS=Z(t)+Z(t).(2.22)∂t iIf we remember that structure constants in(2.13)were cutoffdependent and this depen-dence we removed by a subtraction,we might have kept it up to(2.22).The reason isthat as it follows form Poincare-Dulac Theorem8about vectorfields,every vectorfield can be linearized by appropriate redefinition of coordinates up to the resonant terms,and the resonant condition is related to the zero modes of linear partα1,...,αn.The N-th order term can not be removed by this redefinition if and only if there exists the integer relation of the form:αs= N1m iαi(2.23) with(m1,...)integers and m k≥0, m k≥2and(2.23)is called the resonance relation. The linear term forβi is given by(1−∆i)t i,thus the resonance condition in the second order is the one we had written before(2.15):1−∆k=1−∆i+1−∆j.They correspond tofinite terms in(2.13)and can’t be removed by coordinate transformation.This in fact proves that the non-vanishing of total derivative term in S is universal.The expression(2.22)shows that for exactly marginal deformations of base point (which is a particular CFT)action S is the same as partition function and this confirms the statement of[4].Obviously,assuming thatβ=0(∆i=1and c k ij=0in the perturbation theory)from the begining and going through our calculations again we get dS=0.Because any attempt to calculate stringfield theory action in the present approach should use the perturbation theory we can’t make any statement about global properties of action. Also,it is difficult to check above statement about equation of motion directly fromfinal expression(2.22)without going to world-sheet and using the identities described in this Section;it would be very interesting tofind such st important comment related to these questions:we couldn’t compare two actions(note that dependance on t’s enter throughβ)if they are calculated in perturbation theory arround different points in space of t’s.For this we will need the natural paralel transport in the space of theories that we unfortunately don’t have;we only know the deformed relations for SL(2,R)generators in perturbation theory.Thus,the result is not truly background independent even it looks so formally.Acknowledgments:I would like to thank A.Zamolodchikov,E.Verlinde and E.Witten for very useful discussions.References[1] E.Witten,Phys.Rev.,D46(1992)5446,hep-th-9208027.[2] A.Sen,B.Zwiebach,Quantum Background Independence of Closed String Theory,Preprint MIT-CTP-2244,TIFR-TH-93-37,hep-th/9311009.[3]S.Shatashvili,Phys.Lett.B311(1993)83,hep-th-9303143.[4] E.Witten,Phys.Rev.,D47(1993)3405,hep-th-9210065.[5] E.Fradkin and A.Tseytlin,Phys.Lett.B163(1985)123.[6] A.Abouelsaood,C.Callan,C.Nappi and S.Yost,Nucl.Phys.B280(1987)559.[7] A.Sen and B.Zwiebach,A note on gauge transformations in Batalin-Vilkovisky the-ory,Preprint MIT-CTP2240,TIFR-TH-93-38,hep-th/93099027.[8] A.B.Zamolodchikov,Yad.Fiz.46(1987)1819,Sov.J.Nucl.Phys.46(1987)1091.[9]K.K.Li and E.Witten,Phys.Rev.D,48(1993)853,hep-th-9303067.[10] A.Polyakov,Unpublished,A.M.Polyakov,Gauge Fields and Strings,Harwood,1991.[11]K.Ranganathan,H.Sonoda,B.Zwiebach,Connection on the state space over Con-formal Field Theories,Preprint MIT-CTP-2206,hep-th-9304053.[12] A.Zamolodchikov,Unpublished[13]S.Ghoshal and A.Zamolodchikov,Boundary S-matrix and boundary state in two-dimensional integrable quantumfield theory,Rutgers Preprint RU-93-20,hep-th-9306002.[14] A.Zamolodchikov,Adv.St.in Pure Math.,19(1989)641.。